Research ArticleDesigning a Robust Nonlinear Dynamic Inversion Controller forSpacecraft Formation Flying

Inseok Yang1 Dongik Lee2 and Dong Seog Han2

1 Center for ICT and Automobile Convergence Kyungpook National University Daegu 702-701 Republic of Korea2 School of Electronics Engineering Kyungpook National University Daegu 702-701 Republic of Korea

Correspondence should be addressed to Dong Seog Han dshanknuackr

Received 18 April 2014 Accepted 1 July 2014 Published 17 July 2014

Academic Editor Vivian Martins Gomes

Copyright copy 2014 Inseok Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The robust nonlinear dynamic inversion (RNDI) control technique is proposed to keep the relative position of spacecrafts whileformation flying The proposed RNDI control method is based on nonlinear dynamic inversion (NDI) NDI is nonlinear controlmethod that replaces the original dynamics into the user-selected desired dynamics Because NDI removes nonlinearities in themodel by inverting the original dynamics directly it also eliminates the need of designing suitable controllers for each equilibriumpoint that is NDI works as self-scheduled controller Removing the original model also provides advantages of ease to satisfythe specific requirements by simply handling desired dynamics Therefore NDI is simple and has many similarities to classicalcontrol In real applications however it is difficult to achieve perfect cancellation of the original dynamics due to uncertaintiesthat lead to performance degradation and even make the system unstable This paper proposes robustness assurance method forNDI The proposed RNDI is designed by combining NDI and sliding mode control (SMC) SMC is inherently robust using high-speed switching inputs This paper verifies similarities of NDI and SMC firstly And then RNDI control method is proposed Theperformance of the proposed method is evaluated by simulations applied to spacecraft formation flying problem

1 Introduction

Spacecraft formation flying (SFF) problem is a cooperativecontrol problem that distributes the task of a single spacecraftinto a group of spacecrafts to improve the robustness of aspace mission by decreasing the possibility of a single failurethat can lead to total mission loss [1ndash3] For this reason SFFhas attracted considerable interest owing to its advantagesof not only increased mission success probability but alsoincreased feasibility flexibility and so forth According to thedistribution strategies the control methods for SFF can becategorized into two ways centralized and decentralized [1]In decentralized control each spacecraft in a formation groupcan communicate with each other In particular by transmit-ting the condition of each vehicle they can avoid the fault-sensitive problem that can lead to serious problems suchas collision The main drawbacks of decentralized formationcontrol are the difficulties in analyzing the global SFF stabilityand the increase of system complexity significantly if forma-tion member is big In contrast centralized formation flying

such as the leader-follower approach is easy to implementbecause formation can be achieved by controlling the relativeposition or velocity The objective of SFF problem in thispaper is to evaluate the formation keeping performance usingthe proposed controller Therefore the leader-follower for-mation problem of two spacecrafts is considered This paperconsiders the full nonlinear dynamics describing the relativeposition to design a suitable nonlinear controller for SFF

In this paper spacecraft formation flying using the robustnonlinear dynamic inversion (RNDI) control method isproposed The proposed RNDI control method is basedon well-known nonlinear dynamic inversion (NDI) NDIis a nonlinear control synthesis technique that steers thesystem states to track a user-designed desired trajectory [4ndash6] By inverting the original dynamics to remove the systemnonlinearities directly NDI does not require linearizingand designing gain-scheduled controllers for each equilib-rium point Therefore NDI provides a solution for thedifficulties in ensuring stabilities and performances betweenvarious operational points in gain-scheduled controllersThe

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 471352 12 pageshttpdxdoiorg1011552014471352

2 Mathematical Problems in Engineering

Dynamic inversion

PI-typedesired

dynamics

Dynamicinversion Plant

(a) PI-type of desired dynamics based method

Dynamic inversion

Robustlinear

controller

Dynamicinversion Plant

Desireddynamics

(b) Robust linear controller based method

Figure 1 Block diagrams that have been proposed to achieve robustness

advantages of NDI control are that it is conceptually simpleand has many similarities to classical control methods Inaddition it naturally handles nonlinear systemswithout gain-scheduling because it works as a self-scheduled controllerFor this reason NDI has become a popular flight controllerand been applied to various high-performance aircrafts suchas X-38 [4 5] F-18 HARV [7 8] and F-16 [8 9] On theother hand accurate knowledge of the nonlinear systemdynamics is needed to obtain a perfect cancellation of theoriginal dynamics Actually in real applications such anassumption of taking all accurate information about systemnonlinearities is rarely met due to uncertainties such asmodel mismatches disturbances and measurement noises[4 9ndash13] As these uncertainties lead to the closed-loopsystem control performance degradation and make the sys-tem unstable robustness issues against uncertainties must beconsidered when designing NDI controller

A variety of methods have been proposed to achieverobustness inNDI controllerMany studies have reported thatthe system controlled by NDI with a proportional-integral-(PI-) type of desired dynamics can improve the robustness[10 13 14] As shown in Figure 1(a) the PI controller isdesigned as the desired dynamics satisfying the specificrequirements that the states track after removing the originaldynamics by NDI The PI controller takes advantages of sim-ple design and similaritieswith classical controlHowever thedesired dynamics types are selected by considering the sys-tem requirements so there are some systems where the PI-type of desired dynamics is unsuitable One of the mostwidely used methods for solving the robustness issue is toemploy an additional linear controller such as the structuredsingular value (120583-analysis) and 119867

infinsynthesis [4 5 8 9 11]

In these methods NDI works as an inner-loop controllerwhile an additional linear robust controller is employed asan outer-loop controller as shown in Figure 1(b) Thereforethe linear controller attempts improve the robustness of theoverall control system These methods however are basedon linearized system equations and lead to an increase of theorder of the control system For example the controller orderincreases to 14while designing the119867

infincontroller for the X-38

[5] Recently Yang et al proposed a robust dynamic inversion(RDI) controlmethod using slidingmode control (SMC) [15]SMC is a robust nonlinear control method that takes distinctvalues as control inputs to steer the system states into a user-designed sliding surface and to maintain the states on it [1617] By combining dynamic inversion (DI) and SMC the RDIcontroller guarantees stability against uncertainties withoutusing any additional outer-loop controller and moreoverit takes the advantages of both controllers (Figure 2) [15]However the proposed RDI is designed as a linear controllerso it cannot cover the advantages of self-scheduling of NDIcontrollerThis paper extends the results of [15] for nonlinearsystems and verifies the feasibility of the proposed RNDIcontroller with application to the SFF problem

2 Mathematical Model ofSpacecraft Formation Flying

This section presents the derivation of a nonlinear SFFmodelFormation flying considered in this paper is assumed tobe composed of two spacecrafts leader and follower Theleader spacecraft provides the reference trajectory assumedto be a circular orbit with constant velocity 120596

0while the

follower spacecraft navigates the neighborhood of the leader

Mathematical Problems in Engineering 3

Robust dynamic inversion

Desireddynamics

Robustdynamicinversion

Plant

Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion

p

Leader

Follower

Ez

Ex

Ey

eh

rf

er

e120579

rl

Figure 3 Reference coordinate frame [1]

according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E

119909E119910E119911 whose origin is located in the center of the

Earth and the leader orbit frame e119903 e120579 eℎ whose origin is

attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e

119903 e120579 eℎ frame can be represented as follows [1 3]

r119897= minus

120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

(1)

r119891= minus

120583

1199033

119891

r119891+F119889119891

119898119891

+u119891

119898119891

(2)

where u119897isin R3 and u

119891isin R3 are control input vectors of the

leader and follower spacecrafts respectively And F119889119897isin R3

and F119889119891

isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898

119897 119898119891 so 119866(119872 + 119898

119897) asymp 119866119872

and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the

follower spacecraft p = r119891minus r119897 the second derivative of the

relative position can be represented as follows

p = r119891minus r119897= (minus

120583

1199033

119891

(r119897+ p) +

F119889119891

119898119891

+u119891

119898119891

)

minus (minus120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

)

= minus119872119866(r119897+ p

1003817100381710038171003817r119897 + p10038171003817100381710038173minus

r119897

1199033

119897

) minusF119889

119898119891

+u119905

119898119891

(3)

where F119889= F119889119891minus (119898119891119898119897)F119889119897and u

119905= u119891minus (119898119891119898119897)u119897

In this paper it is assumed that the leader spacecraft is freeflying that is u

119897= 0 then it satisfies that u

119905= u119891

If the relative position p in the leader orbit frame isgiven as p = 119909e

119903+ 119910e120579+ 119911eℎfor the unit vectors e

119903 e120579

and eℎalong with the E

119909E119910E119911-frame then the relative

acceleration vector p can be obtained as follows

p = ( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ (4)

In the moving leader orbit frame the position vector r119897is

constant that is r119897= 119903119897e119903 By substituting (4) into (3) the

nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as

( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ

= minus119872119866((119903119897+ 119909) e

119903+ 119910e120579+ 119911eℎ

1003817100381710038171003817r119897 + p10038171003817100381710038173

minus119903119897e119903

1199033

119897

)

minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ

119898119891

+119906119909e119903+ 119906119910e120579+ 119906119911eℎ

119898119891

997904rArr [

[

119910

]

]

= minus[

[

0 minus212059600

21205960

0 0

0 0 0

]

]

[

[

119910

]

]

+

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

[

[

119909

119910

119911

]

]

4 Mathematical Problems in Engineering

minus119872119866[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

minus1

119898119891

[

[

119865119889119909

119865119889119910

119865119889119911

]

]

+1

119898119891

[

[

119906119909

119906119910

119906119911

]

]

(5)

Denote the relative position and velocity vectors as p119897=

(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit

frame respectively Then (5) can be represented as follows

p119897= k119897

k119897= minus C (120596

0) k119897minusD (p

119897 r119897 1205960) p119897minus N (p

119897 r119897) minus

F119889

119898119891

+u119905

119898119891

(6)

where

C (1205960) = [

[

0 minus212059600

21205960

0 0

0 0 0

]

]

N (p119897 r119897) = 119872119866

[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

F119889= [

[

119865119889119909

119865119889119910

119865119889119911

]

]

D (p119897 r119897 1205960)

=

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

(7)

3 The Proposed Robust NonlinearDynamic Inversion

In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed

31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains

Consider the following system

x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)

Sliding surface

Sliding phase

Reaching phase

120590(x) = 0

x2

x1

x(t1)

x(t0)

Figure 4The behavior of the system states in sliding mode control

where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ

119894be the regular (119899 minus 1)

dimensional submanifold in R119899 defined in [16] such as

Σ119894≜ x isin R119899 | 120590

119894(x) = 0 119894 = 1 2 119898 (9)

where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function

Define120590119894(x) = 0 as the individual sliding surface or individual

switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]

Σ ≜

119898

⋂

119894=1

Σ119894=

119898

⋂

119894=1

x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0

(10)

where 120590(x) = [1205901(x) 1205902(x) 120590

119898(x)]119879 If the states x(119905) of

the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906

119894(119905) (119894 = 1 2 119898) of the sliding mode

control law can be obtained by taking one of the followingvalues [16]

119906119894(119905) =

119906+

119894 if 120590

119894(x (119905)) gt 0

119906minus

119894 if 120590

119894(x (119905)) lt 0

(11)

where 119906+119894= 119906minus

119894

As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is

Mathematical Problems in Engineering 5

considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905

0and a sliding mode exists for 119905 ge 119905

0 then

(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the

equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]

(x) = 120597120590120597x(f (x) + g (x)ueq) = 0

997904rArr ueq = minus(120597120590

120597xg(x))

minus1

(120597120590

120597xf (x))

(12)

Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905

0is governed by

x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x

g(x))minus1

(120597120590

120597x)] f (x)

(13)

The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface

One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]

119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)

where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example

(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0

(ii) relay with state dependent gains 119906sw119894(x) =

120572119894(x) sgn(120590

119894(x)) with 120572

119894(sdot) lt 0

(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0

32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific

dynamicsDynamicinversion

x x

xPlant

xcmd asymp1

s

Figure 5 Structure of dynamic inversion [4]

performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth

For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]

uNDI (119905) = [g (x (119905))]minus1

[xdes (119905) minus f (x (119905))] (15)

where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI

control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]

By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields

xdes = x (16)

Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler

To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Dynamic inversion

PI-typedesired

dynamics

Dynamicinversion Plant

(a) PI-type of desired dynamics based method

Dynamic inversion

Robustlinear

controller

Dynamicinversion Plant

Desireddynamics

(b) Robust linear controller based method

Figure 1 Block diagrams that have been proposed to achieve robustness

advantages of NDI control are that it is conceptually simpleand has many similarities to classical control methods Inaddition it naturally handles nonlinear systemswithout gain-scheduling because it works as a self-scheduled controllerFor this reason NDI has become a popular flight controllerand been applied to various high-performance aircrafts suchas X-38 [4 5] F-18 HARV [7 8] and F-16 [8 9] On theother hand accurate knowledge of the nonlinear systemdynamics is needed to obtain a perfect cancellation of theoriginal dynamics Actually in real applications such anassumption of taking all accurate information about systemnonlinearities is rarely met due to uncertainties such asmodel mismatches disturbances and measurement noises[4 9ndash13] As these uncertainties lead to the closed-loopsystem control performance degradation and make the sys-tem unstable robustness issues against uncertainties must beconsidered when designing NDI controller

A variety of methods have been proposed to achieverobustness inNDI controllerMany studies have reported thatthe system controlled by NDI with a proportional-integral-(PI-) type of desired dynamics can improve the robustness[10 13 14] As shown in Figure 1(a) the PI controller isdesigned as the desired dynamics satisfying the specificrequirements that the states track after removing the originaldynamics by NDI The PI controller takes advantages of sim-ple design and similaritieswith classical controlHowever thedesired dynamics types are selected by considering the sys-tem requirements so there are some systems where the PI-type of desired dynamics is unsuitable One of the mostwidely used methods for solving the robustness issue is toemploy an additional linear controller such as the structuredsingular value (120583-analysis) and 119867

infinsynthesis [4 5 8 9 11]

In these methods NDI works as an inner-loop controllerwhile an additional linear robust controller is employed asan outer-loop controller as shown in Figure 1(b) Thereforethe linear controller attempts improve the robustness of theoverall control system These methods however are basedon linearized system equations and lead to an increase of theorder of the control system For example the controller orderincreases to 14while designing the119867

infincontroller for the X-38

[5] Recently Yang et al proposed a robust dynamic inversion(RDI) controlmethod using slidingmode control (SMC) [15]SMC is a robust nonlinear control method that takes distinctvalues as control inputs to steer the system states into a user-designed sliding surface and to maintain the states on it [1617] By combining dynamic inversion (DI) and SMC the RDIcontroller guarantees stability against uncertainties withoutusing any additional outer-loop controller and moreoverit takes the advantages of both controllers (Figure 2) [15]However the proposed RDI is designed as a linear controllerso it cannot cover the advantages of self-scheduling of NDIcontrollerThis paper extends the results of [15] for nonlinearsystems and verifies the feasibility of the proposed RNDIcontroller with application to the SFF problem

2 Mathematical Model ofSpacecraft Formation Flying

This section presents the derivation of a nonlinear SFFmodelFormation flying considered in this paper is assumed tobe composed of two spacecrafts leader and follower Theleader spacecraft provides the reference trajectory assumedto be a circular orbit with constant velocity 120596

0while the

follower spacecraft navigates the neighborhood of the leader

Mathematical Problems in Engineering 3

Robust dynamic inversion

Desireddynamics

Robustdynamicinversion

Plant

Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion

p

Leader

Follower

Ez

Ex

Ey

eh

rf

er

e120579

rl

Figure 3 Reference coordinate frame [1]

according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E

119909E119910E119911 whose origin is located in the center of the

Earth and the leader orbit frame e119903 e120579 eℎ whose origin is

attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e

119903 e120579 eℎ frame can be represented as follows [1 3]

r119897= minus

120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

(1)

r119891= minus

120583

1199033

119891

r119891+F119889119891

119898119891

+u119891

119898119891

(2)

where u119897isin R3 and u

119891isin R3 are control input vectors of the

leader and follower spacecrafts respectively And F119889119897isin R3

and F119889119891

isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898

119897 119898119891 so 119866(119872 + 119898

119897) asymp 119866119872

and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the

follower spacecraft p = r119891minus r119897 the second derivative of the

relative position can be represented as follows

p = r119891minus r119897= (minus

120583

1199033

119891

(r119897+ p) +

F119889119891

119898119891

+u119891

119898119891

)

minus (minus120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

)

= minus119872119866(r119897+ p

1003817100381710038171003817r119897 + p10038171003817100381710038173minus

r119897

1199033

119897

) minusF119889

119898119891

+u119905

119898119891

(3)

where F119889= F119889119891minus (119898119891119898119897)F119889119897and u

119905= u119891minus (119898119891119898119897)u119897

In this paper it is assumed that the leader spacecraft is freeflying that is u

119897= 0 then it satisfies that u

119905= u119891

If the relative position p in the leader orbit frame isgiven as p = 119909e

119903+ 119910e120579+ 119911eℎfor the unit vectors e

119903 e120579

and eℎalong with the E

119909E119910E119911-frame then the relative

acceleration vector p can be obtained as follows

p = ( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ (4)

In the moving leader orbit frame the position vector r119897is

constant that is r119897= 119903119897e119903 By substituting (4) into (3) the

nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as

( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ

= minus119872119866((119903119897+ 119909) e

119903+ 119910e120579+ 119911eℎ

1003817100381710038171003817r119897 + p10038171003817100381710038173

minus119903119897e119903

1199033

119897

)

minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ

119898119891

+119906119909e119903+ 119906119910e120579+ 119906119911eℎ

119898119891

997904rArr [

[

119910

]

]

= minus[

[

0 minus212059600

21205960

0 0

0 0 0

]

]

[

[

119910

]

]

+

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

[

[

119909

119910

119911

]

]

4 Mathematical Problems in Engineering

minus119872119866[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

minus1

119898119891

[

[

119865119889119909

119865119889119910

119865119889119911

]

]

+1

119898119891

[

[

119906119909

119906119910

119906119911

]

]

(5)

Denote the relative position and velocity vectors as p119897=

(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit

frame respectively Then (5) can be represented as follows

p119897= k119897

k119897= minus C (120596

0) k119897minusD (p

119897 r119897 1205960) p119897minus N (p

119897 r119897) minus

F119889

119898119891

+u119905

119898119891

(6)

where

C (1205960) = [

[

0 minus212059600

21205960

0 0

0 0 0

]

]

N (p119897 r119897) = 119872119866

[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

F119889= [

[

119865119889119909

119865119889119910

119865119889119911

]

]

D (p119897 r119897 1205960)

=

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

(7)

3 The Proposed Robust NonlinearDynamic Inversion

In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed

31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains

Consider the following system

x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)

Sliding surface

Sliding phase

Reaching phase

120590(x) = 0

x2

x1

x(t1)

x(t0)

Figure 4The behavior of the system states in sliding mode control

where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ

119894be the regular (119899 minus 1)

dimensional submanifold in R119899 defined in [16] such as

Σ119894≜ x isin R119899 | 120590

119894(x) = 0 119894 = 1 2 119898 (9)

where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function

Define120590119894(x) = 0 as the individual sliding surface or individual

switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]

Σ ≜

119898

⋂

119894=1

Σ119894=

119898

⋂

119894=1

x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0

(10)

where 120590(x) = [1205901(x) 1205902(x) 120590

119898(x)]119879 If the states x(119905) of

the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906

119894(119905) (119894 = 1 2 119898) of the sliding mode

control law can be obtained by taking one of the followingvalues [16]

119906119894(119905) =

119906+

119894 if 120590

119894(x (119905)) gt 0

119906minus

119894 if 120590

119894(x (119905)) lt 0

(11)

where 119906+119894= 119906minus

119894

As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is

Mathematical Problems in Engineering 5

considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905

0and a sliding mode exists for 119905 ge 119905

0 then

(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the

equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]

(x) = 120597120590120597x(f (x) + g (x)ueq) = 0

997904rArr ueq = minus(120597120590

120597xg(x))

minus1

(120597120590

120597xf (x))

(12)

Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905

0is governed by

x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x

g(x))minus1

(120597120590

120597x)] f (x)

(13)

The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface

One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]

119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)

where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example

(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0

(ii) relay with state dependent gains 119906sw119894(x) =

120572119894(x) sgn(120590

119894(x)) with 120572

119894(sdot) lt 0

(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0

32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific

dynamicsDynamicinversion

x x

xPlant

xcmd asymp1

s

Figure 5 Structure of dynamic inversion [4]

performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth

For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]

uNDI (119905) = [g (x (119905))]minus1

[xdes (119905) minus f (x (119905))] (15)

where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI

control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]

By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields

xdes = x (16)

Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler

To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Robust dynamic inversion

Desireddynamics

Robustdynamicinversion

Plant

Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion

p

Leader

Follower

Ez

Ex

Ey

eh

rf

er

e120579

rl

Figure 3 Reference coordinate frame [1]

according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E

119909E119910E119911 whose origin is located in the center of the

Earth and the leader orbit frame e119903 e120579 eℎ whose origin is

attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e

119903 e120579 eℎ frame can be represented as follows [1 3]

r119897= minus

120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

(1)

r119891= minus

120583

1199033

119891

r119891+F119889119891

119898119891

+u119891

119898119891

(2)

where u119897isin R3 and u

119891isin R3 are control input vectors of the

leader and follower spacecrafts respectively And F119889119897isin R3

and F119889119891

isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898

119897 119898119891 so 119866(119872 + 119898

119897) asymp 119866119872

and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the

follower spacecraft p = r119891minus r119897 the second derivative of the

relative position can be represented as follows

p = r119891minus r119897= (minus

120583

1199033

119891

(r119897+ p) +

F119889119891

119898119891

+u119891

119898119891

)

minus (minus120583

1199033

119897

r119897+F119889119897

119898119897

+u119897

119898119897

)

= minus119872119866(r119897+ p

1003817100381710038171003817r119897 + p10038171003817100381710038173minus

r119897

1199033

119897

) minusF119889

119898119891

+u119905

119898119891

(3)

where F119889= F119889119891minus (119898119891119898119897)F119889119897and u

119905= u119891minus (119898119891119898119897)u119897

In this paper it is assumed that the leader spacecraft is freeflying that is u

119897= 0 then it satisfies that u

119905= u119891

If the relative position p in the leader orbit frame isgiven as p = 119909e

119903+ 119910e120579+ 119911eℎfor the unit vectors e

119903 e120579

and eℎalong with the E

119909E119910E119911-frame then the relative

acceleration vector p can be obtained as follows

p = ( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ (4)

In the moving leader orbit frame the position vector r119897is

constant that is r119897= 119903119897e119903 By substituting (4) into (3) the

nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as

( minus 21205960119910 minus 1205962

0119909) e119903+ ( 119910 + 2120596

0 minus 1205962

0119910) e120579+ eℎ

= minus119872119866((119903119897+ 119909) e

119903+ 119910e120579+ 119911eℎ

1003817100381710038171003817r119897 + p10038171003817100381710038173

minus119903119897e119903

1199033

119897

)

minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ

119898119891

+119906119909e119903+ 119906119910e120579+ 119906119911eℎ

119898119891

997904rArr [

[

119910

]

]

= minus[

[

0 minus212059600

21205960

0 0

0 0 0

]

]

[

[

119910

]

]

+

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

[

[

119909

119910

119911

]

]

4 Mathematical Problems in Engineering

minus119872119866[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

minus1

119898119891

[

[

119865119889119909

119865119889119910

119865119889119911

]

]

+1

119898119891

[

[

119906119909

119906119910

119906119911

]

]

(5)

Denote the relative position and velocity vectors as p119897=

(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit

frame respectively Then (5) can be represented as follows

p119897= k119897

k119897= minus C (120596

0) k119897minusD (p

119897 r119897 1205960) p119897minus N (p

119897 r119897) minus

F119889

119898119891

+u119905

119898119891

(6)

where

C (1205960) = [

[

0 minus212059600

21205960

0 0

0 0 0

]

]

N (p119897 r119897) = 119872119866

[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

F119889= [

[

119865119889119909

119865119889119910

119865119889119911

]

]

D (p119897 r119897 1205960)

=

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

(7)

3 The Proposed Robust NonlinearDynamic Inversion

In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed

31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains

Consider the following system

x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)

Sliding surface

Sliding phase

Reaching phase

120590(x) = 0

x2

x1

x(t1)

x(t0)

Figure 4The behavior of the system states in sliding mode control

where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ

119894be the regular (119899 minus 1)

dimensional submanifold in R119899 defined in [16] such as

Σ119894≜ x isin R119899 | 120590

119894(x) = 0 119894 = 1 2 119898 (9)

where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function

Define120590119894(x) = 0 as the individual sliding surface or individual

switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]

Σ ≜

119898

⋂

119894=1

Σ119894=

119898

⋂

119894=1

x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0

(10)

where 120590(x) = [1205901(x) 1205902(x) 120590

119898(x)]119879 If the states x(119905) of

the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906

119894(119905) (119894 = 1 2 119898) of the sliding mode

control law can be obtained by taking one of the followingvalues [16]

119906119894(119905) =

119906+

119894 if 120590

119894(x (119905)) gt 0

119906minus

119894 if 120590

119894(x (119905)) lt 0

(11)

where 119906+119894= 119906minus

119894

As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is

Mathematical Problems in Engineering 5

considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905

0and a sliding mode exists for 119905 ge 119905

0 then

(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the

equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]

(x) = 120597120590120597x(f (x) + g (x)ueq) = 0

997904rArr ueq = minus(120597120590

120597xg(x))

minus1

(120597120590

120597xf (x))

(12)

Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905

0is governed by

x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x

g(x))minus1

(120597120590

120597x)] f (x)

(13)

The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface

One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]

119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)

where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example

(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0

(ii) relay with state dependent gains 119906sw119894(x) =

120572119894(x) sgn(120590

119894(x)) with 120572

119894(sdot) lt 0

(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0

32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific

dynamicsDynamicinversion

x x

xPlant

xcmd asymp1

s

Figure 5 Structure of dynamic inversion [4]

performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth

For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]

uNDI (119905) = [g (x (119905))]minus1

[xdes (119905) minus f (x (119905))] (15)

where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI

control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]

By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields

xdes = x (16)

Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler

To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

minus119872119866[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

minus1

119898119891

[

[

119865119889119909

119865119889119910

119865119889119911

]

]

+1

119898119891

[

[

119906119909

119906119910

119906119911

]

]

(5)

Denote the relative position and velocity vectors as p119897=

(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit

frame respectively Then (5) can be represented as follows

p119897= k119897

k119897= minus C (120596

0) k119897minusD (p

119897 r119897 1205960) p119897minus N (p

119897 r119897) minus

F119889

119898119891

+u119905

119898119891

(6)

where

C (1205960) = [

[

0 minus212059600

21205960

0 0

0 0 0

]

]

N (p119897 r119897) = 119872119866

[[[

[

119903119897

1003817100381710038171003817r119897 + p10038171003817100381710038173minus1

1199032119897

0

0

]]]

]

F119889= [

[

119865119889119909

119865119889119910

119865119889119911

]

]

D (p119897 r119897 1205960)

=

[[[[[[[[

[

1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0 0

0 1205962

0minus

119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

0

0 0 minus119872119866

1003817100381710038171003817r119897 + p10038171003817100381710038173

]]]]]]]]

]

(7)

3 The Proposed Robust NonlinearDynamic Inversion

In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed

31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains

Consider the following system

x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)

Sliding surface

Sliding phase

Reaching phase

120590(x) = 0

x2

x1

x(t1)

x(t0)

Figure 4The behavior of the system states in sliding mode control

where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ

119894be the regular (119899 minus 1)

dimensional submanifold in R119899 defined in [16] such as

Σ119894≜ x isin R119899 | 120590

119894(x) = 0 119894 = 1 2 119898 (9)

where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function

Define120590119894(x) = 0 as the individual sliding surface or individual

switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]

Σ ≜

119898

⋂

119894=1

Σ119894=

119898

⋂

119894=1

x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0

(10)

where 120590(x) = [1205901(x) 1205902(x) 120590

119898(x)]119879 If the states x(119905) of

the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906

119894(119905) (119894 = 1 2 119898) of the sliding mode

control law can be obtained by taking one of the followingvalues [16]

119906119894(119905) =

119906+

119894 if 120590

119894(x (119905)) gt 0

119906minus

119894 if 120590

119894(x (119905)) lt 0

(11)

where 119906+119894= 119906minus

119894

As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is

Mathematical Problems in Engineering 5

considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905

0and a sliding mode exists for 119905 ge 119905

0 then

(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the

equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]

(x) = 120597120590120597x(f (x) + g (x)ueq) = 0

997904rArr ueq = minus(120597120590

120597xg(x))

minus1

(120597120590

120597xf (x))

(12)

Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905

0is governed by

x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x

g(x))minus1

(120597120590

120597x)] f (x)

(13)

The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface

One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]

119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)

where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example

(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0

(ii) relay with state dependent gains 119906sw119894(x) =

120572119894(x) sgn(120590

119894(x)) with 120572

119894(sdot) lt 0

(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0

32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific

dynamicsDynamicinversion

x x

xPlant

xcmd asymp1

s

Figure 5 Structure of dynamic inversion [4]

performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth

For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]

uNDI (119905) = [g (x (119905))]minus1

[xdes (119905) minus f (x (119905))] (15)

where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI

control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]

By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields

xdes = x (16)

Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler

To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905

0and a sliding mode exists for 119905 ge 119905

0 then

(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the

equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]

(x) = 120597120590120597x(f (x) + g (x)ueq) = 0

997904rArr ueq = minus(120597120590

120597xg(x))

minus1

(120597120590

120597xf (x))

(12)

Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905

0is governed by

x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x

g(x))minus1

(120597120590

120597x)] f (x)

(13)

The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface

One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]

119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)

where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example

(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0

(ii) relay with state dependent gains 119906sw119894(x) =

120572119894(x) sgn(120590

119894(x)) with 120572

119894(sdot) lt 0

(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0

32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific

dynamicsDynamicinversion

x x

xPlant

xcmd asymp1

s

Figure 5 Structure of dynamic inversion [4]

performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth

For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]

uNDI (119905) = [g (x (119905))]minus1

[xdes (119905) minus f (x (119905))] (15)

where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI

control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]

By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields

xdes = x (16)

Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler

To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)

1205902(xlowast) 120590

119898(xlowast)]119879 = 0 as a sliding surface If the system

states intercept the sliding surface at 119905 = 1199050and the sliding

mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0

for all 119905 ge 1199050 Then the equivalent input ueq that the state

trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]

(xlowast) = 120597120590

120597xlowastxlowast = 120597120590

120597xlowast(xdes minus x) = 0

997904rArr120597120590

120597xlowastxdes minus

120597120590

120597xlowast(f (x) + g (x) ueq) = 0

997904rArr ueq = (120597120590

120597xlowastg (x))

minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))

(17)

If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a

form of NDI input shown in (15)

ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)

Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput

Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]

119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix

It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows

x = f (x) + g (x)ueq

= [f (x) + g (x) ( 120597120590120597xlowast

g (x))minus1

(120597120590

120597xlowastxdes minus

120597120590

120597xlowastf (x))]

= xdes(19)

In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem

33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906

119894(119905) (119894 = 1 2

119898) of the RNDI control law can be obtained by introducingthe following values

119906119894(119905) =

119906+

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0

119906minus

119894 if 120590

119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0

(20)

where 119906+119894= 119906minus

119894satisfies the following for 119891

119894(x) the 119894th entry

of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)

lim119909119894rarr119909

+

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus

119894lt des119894

lim119909119894rarr119909

minus

des119894

119891119894(119909) +

119899

sum

119895=1

119895 = 119894

119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+

119894gt des119894

(21)

where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically

Theorem 2 Consider the following input

u = u119873119863119868

+ gminus1 (x) u119904119908 (22)

where u119873119863119868

is the conventional NDI input and u119904119908

isthe switching input that satisfies the following for u

119904119908=

[1199061199041199081 1199061199041199082 119906

119904119908119899]119879

119906119904119908119894

gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894gt 0

119906119904119908119894

lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909

119889119890119904119894minus 119909119894lt 0

(23)

Then the closed-loop system controlled by (22) is globally stable

Proof Select a Lyapunov candidate as

119881 =1

2120590(xdes minus x)119879120590 (xdes minus x) (24)

From (22) and (23)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 (xdes minus x)

= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]

= 120590(xdes minus x)119879

times [xdes minus f (x) + g (x) uNDI minus usw]

= minus120590(xdes minus x)119879usw

= minus

119899

sum

119894=1

120590119894(xdes minus x) 119906sw119894

(25)

From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0

Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence

the closed-loop system controlled by (22) is globally stable

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

NDI surface

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)u+i

fi(x) +n

sumj=1jnei

gij(x)uj + gii(x)uminusi

120590i( minus ) gt 0x xdes

120590i( minus x) lt 0xdes

120590i( minus ) = 0xdes x

Figure 6 State trajectory on the NDI surface

Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6

Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081

1199061199041199082 119906

119904119908119899]119879 is designed with 119906

119904119908119894= 119896119894sgn(120590

119894(x119889119890119904minus x))

for a positive 119896119894 then the control system is globally stable

Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely

if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to

Theorem 2 the control system is globally stable

Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896

119894) (119894 = 1 2 119899) with 119896

119894gt 0

the RNDI control law yields

uRNDI = uNDI + usw

= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)

It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows

(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896

119894gt 0

(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus

x) sgn(120590119894(xdes minus x)) with 119896

119894(sdot) gt 0

(iii) univector nonlinearity with scale factor usw =

119896119894120590(xdes minus x)120590(xdes minus x) with 119896

119894gt 0

It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the

controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]

119879 inorder to design a continuous switching input as follows for apositive 120575

119894

sat119894(119909des119894 minus 119909119894

120575119894

) =

1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894

120575119894

if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575

minus1 if (119909des119894 minus 119909119894) lt minus120575(27)

Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat

1((119909des1 minus 1199091)1205751)

sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]

119879 and K =

diag(119896119894) (119894 = 1 2 119899) with 119896

119894gt 0

x = f (x) + g (x)u

= f (x) + g (x)

times [(g (x))minus1 (xdes minus f (x))

+(g (x))minus1Ksat((xdes minus x)

120575)]

= xdes + Ksat((xdes minus x)

120575)

= xdes

+ [1198961(119909des1 minus 1199091)

1205751

1198962(119909des2 minus 1199092)

1205752

sdot sdot sdot

119896119899(119909des119899 minus 119909119899)

120575119899

]

119879

(28)

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

uf

RefTime

Angular

Position of Referencetrajectorygenerator

the leader spacecraft

Reffcn

PositionControl

Dynamics of the Robust nonlinear dynamic inversion follower spacecraft

PositionScope

DisturbancesFd

++

t

ww

plplinputs

velocity

Figure 7 Block diagram using MATLAB Simulink

34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899

x (119905) = [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)

It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))

2lt 120582min(K) where 120582min(K) is the

minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows

119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)

Then the derivative of 119881(xlowast(119905)) yields

(xlowast (119905))

= xlowast(119905)119879xlowast (119905)

= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]

+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]

= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))

minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]

= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]

le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le 120585 (119905 x (119905) u (119905))1

1003817100381710038171003817119909lowast

(119905)10038171003817100381710038171 minus 120582min (K)

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]

1003817100381710038171003817xlowast

(119905)10038171003817100381710038171

(31)

By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative

of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches

Table 1 Parameters for spacecraft formation flying [1 3]

Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2

119872 5974 times 1024 kg119898119891

410 kg119898119897

1550 kgr119897

[4224 times 1024 0 0]119879 m1205960

7272 times 10minus5 rads

4 Simulation Results

In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller

41 Simulation Description The aim of SFF is to design thefeasible control input u

119905(119905) such that p

119897rarr p119889as 119905 rarr infin

for a given reference relative position trajectory p119889isin R3

of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p

119889=

[100 sin(41205960119905) 100 cos(4120596

0119905) 0]119879 that is the follower space-

craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e

119903and e120579

with an angular velocity 41205960 The initial relative position and

velocity are assumed to be respectively as follows

p119897(0) = [10 90 minus20]

119879

k119897(0) = [0 0 0]

119879

(32)

Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F

119889= [290532 31775 minus112298]

119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e

119903and e

ℎaxes are also considered in this

simulation

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

e r-d

irect

ion

(m)

(a) Position tracking error in e119903-direction

0 10 20 30 40 50 60 70 80 90 100

0

2

Time (s)

No disturbanceDisturbance

minus2

minus4

minus6

minus8

minus10

e 120579-d

irect

ion

(m)

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

No disturbanceDisturbance

minus5

minus10

minus15

minus20

e h-d

irect

ion

(m)

(c) Position tracking error in eℎ-direction

Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft

42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e

119903 e120579 and e

ℎaxes respectively

Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described

in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-

jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory

5 Conclusion

In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (m)

NDIRNDI

e r-d

irect

ion

(m)

minus5

minus15

minus10

(a) Position tracking error in e119903-direction

0

2

0 10 20 30 40 50 60 70 80 90 100Time (s)

NDIRNDI

e 120579-d

irect

ion

(m)

minus6

minus2

minus4

minus8

minus10

(b) Position tracking error in e120579-direction

0 10 20 30 40 50 60 70 80 90 100

0

5

10

Time (s)

NDIRNDI

e h-d

irect

ion

(m)

minus20

minus15

minus10

minus5

(c) Position tracking error in eℎ-direction

Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(a) Control inputs generated by NDI

20 30 40 50 60 70 80 90 100

0

100

200

300

400

Time (s)

Inpu

t for

ces (

N)

minus200

minus100

ux along with er-axis

uz along with eh-axis

uy along with e120579-axis

(b) Control inputs generated by RNDI

Figure 10 Control inputs generated by NDI and RNDI

due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the

conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it

(i) is easy to design and implement(ii) eliminates the need of gain-scheduling

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 50 100 150050100150

0

10

e120579 axis er axis

e hax

is

minus50minus50minus100minus100 minus150minus150

minus10

minus20

Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller

(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and

(iv) is inherently robust

However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function

Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying

Nomenclature

G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft

119898119897 Mass of the leader spacecraft

r119891 Distance from the center of the Earth to

the center of the follower spacecraftr119897 Distance from the center of the Earth to

the center of the leader spacecraftxdes Desired dynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)

support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)

References

[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011

[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999

[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000

[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003

[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002

[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999

[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994

[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994

[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000

[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001

[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012

[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997

[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012

[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of