[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control Conference - Chicago, Illinois ()] AIAA Guidance, Navigation, and Control Conference - Linear

American Institute of Aeronautics and Astronautics

1

Linear Dynamics and PID Flight Control of a Powered Paraglider

Yoshimasa OchiTPF

1FPT, Hiroyuki KondoTPF

2FPT, and Masahito WatanabeTPF

3FPT

National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan

This paper describes analytical derivation of a linear dynamic model of a powered paraglider (PPG) from a nonlinear dynamic model, which was derived by the authors. The linear model indicates that the longitudinal dynamics are decoupled from the lateral-directional one. The linear model obtained on the canopy coordinates is transformed into the one on the payload coordinates by linear state transformation, since measurement units are mounted on the payload. The transformation matrix is obtained by linearizing the relations of the velocity and the angular velocity between the canopy and the payload, and also by linearizing the relation among the coordinate transformation matrices. The authors are developing a new design method of a proportional-integral-derivative (PID) control system, which is based on plant model reduction with ν-gap metric and the integral-type optimal servomechanism. This method is applied to flight controller design for three single-input-single-output systems of the PPG. A design example and computer simulation results show linear dynamic properties of the PPG and illustrate good control performance and desirable stability margins of the PID controllers.

I. Introduction powered paraglider (PPG), shown in Fig. 1, can be a useful unmanned aerial vehicle (UAV) for land observation, surveillance, space vehicle retrieval, etc. Although it is subject to wind, its light and foldable wing

or canopy makes it portable equipment for both civil and military uses. Actually PPG-type UAVs have been developed and sold on a commercial basis. Such a UAV needs an autonomous/automatic flight control system, and in order to design a control system we need a dynamic model of a PPG, unless we take an empirical design approach, which is often employed for a black-box plant. However, a PPG is not a totally unknown plant. In fact, many papers have been reported on modeling of PPG or paraglider (PG) nonlinear/linear dynamics. P

1-10P The authors also proposed

a nonlinear dynamic model with eight degrees of freedom, where all the internal forces between the canopy and the suspended payload are analytically eliminated to obtain a model in the form of state equation.P

11P Its validity was

demonstrated through numerical simulation and comparison of the results with flight experiment data of a manned paraglider.

In this paper, first we derive a linear model from the nonlinear model by analytical first-order approximation of Taylor series expansion as well as numerical partial differential. Since the original nonlinear model is described with the state variables of the canopy, the linear model also explicitly expresses motion of the canopy. However, measurement sensors such as accelerometers and gyros are mounted on the payload; hence, it would be more convenient in dynamics analysis and/or control system design, if the model is expressed by the state variables of the payload. This motivated us to derive a linear transformation matrix between the canopy states and the payload states. With the state transformation, we can obtain a payload-state linear model from a canopy one, and vice versa. Then, we consider designing a flight control system for the PPG using the linear model. Although a number of studies on PPG or PG flight control have been reported, many of them employ complicated control methods such as model predictive controlP

12P or inversely consider just open-loop control.P

13, 14P In our study, we employed PID (proportional-

integral-derivative) control, which is most commonly used in industry. The authors are proposing a new design method of a PID controllerP

15P based on integral-type optimal servomechanism (IOS), which is a derivative of the

linear quadratic regulator or LQR. In the design method, first we reduce a given linear plant model to a second-order

TP

1PT Professor, Department of Aerospace Engineering, 1-10-20 Hashirimizu, Senior Member AIAA.

TP

2PT Graduate student, Department of Aerospace Engineering, 1-10-20 Hashirimizu.

TP

3PT Graduate student, Department of Aerospace Engineering, 1-10-20 Hashirimizu.

A

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6318

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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system so that ν-gapP

16P between the original plant and the reduced plant becomes as small as possible. The plant

parameters are determined by parameter-space search, since the number of parameters is just two or three. Of course, the three-term controller cannot always stabilize any plant; however, when stabilizable, it usually provides a set of control gains with good control performance and substantial stability margins, which would be attributed to the properties of the LQR. Moreover, the control gains can be adjusted through selection of weighting matrices for trade-off between control performance and robust stability. In application to a PPG, three PID controllers are designed to control the altitude by thrust, the forward speed by collective brake deflection of the canopy, and the heading angle by differential brake deflection. Time responses to appropriate reference outputs are computed as well as stability margins for each control loop, where other loops are closed.

This paper is organized as follows. In the next section, we describe an outline of the linear model and transformation between the canopy-state model and the payload-state model. Then we present the design method of a PID controller based on the IOS and the ν-gap metric. In Sections IV and V, numerical analysis and simulation results are presented, followed by summary and conclusions in Section VI.

Figure 1. Configuration of PPG Figure 2. Coordinate systems

II. Linear Dynamic Model and State Transformation

A. Coordinate Systems and Degrees of Freedom of Motion Figure 1 shows coordinate systems of the PPG. The inertial coordinate system Σ BI B is defined as (XBI B, YBI B, ZBI B). The

XBI BY BI B-plane is horizontal and the positive direction of the ZBI B- axis is taken vertically downward. The location of the origin and the positive direction of the XBI B-axis are appropriately chosen. Definition of the canopy coordinate system Σ Bc B = (XBc B, YBc B, ZBc B) and the payload coordinate system Σ BpB = (XBp B, YBpB, ZBpB) is also shown in Fig. 2. The origin OBc B of the canopy-fixed coordinate system is chosen at the center of mass (CM) of the canopy and the ZBc B-axis is chosen in the direction from OBc B to OBmB. The XBc B-axis is perpendicular to the ZBc B-axis in the symmetry plane of the canopy and the positive direction of the X Bc B-axis is taken forward. The YBc B-axis is defined so that the coordinates form a right-hand coordinate system. The payload is assumed to be symmetric, and the origin is taken at the CM. The XBpB-axis is taken forward along the thrust direction, the ZBpB-axis is downward and perpendicular to the XBp B-axis in the symmetry plane, and then the YBpB-axis is defined to form a right-hand system. It is assumed that the XBc BZBc B-plane corresponds to the XBp BZBp B-plane in a trimmed straight flight.

The payload is connected with the suspension lines of the canopy at two points, OBmRB and OBmLB. O BmB is the middle point between OBmRB and OBmLB. It is assumed that the motion of the canopy has six degrees of freedom (DOF) and that the suspension lines are deformed only about the ZBc B-axis. The payload then has two DOF, i.e., the relative yawing about the ZBc B-axis and the relative pitching about the line OBmRBO BmLB. Thus, the PPG is modeled as a system with eight DOF.

Canopy

Payload

Control line

Suspension lines

Air-Intake

Propelling unit


3

B. Nonlinear and Linear State Equations Let the velocity of the canopy at OBc B represented in ΣBc B be VBc B = [uBc B vBc B wBc B]P

TP, and the angular velocity of the canopy

also in ΣBc B be ωBc B = [pBc B qBc B r Bc B]P

TP. Let the relative yaw angle and rate between the canopy and the payload be ψBpc B and

r Bpc B=dψBpc B/dt, respectively, and similarly let the relative pitch angle and rate be θBpc B and qBpc B=dθBpc B/dt, respectively. In addition, let the pitch and roll angles of the canopy be θBc B and φBc B, respectively. We then define the state vector xBc B = [uBc B vBc B wBc B p Bc B qBc B r Bc B qBpc B r Bpc B θ Bpc B ψ Bpc B φ Bc B θ Bc B]P

TP = [VBc PB

TP ω Bc PB

TP ω Bpc PB

TP θ Bpc B ψ Bpc B φ Bc B θ Bc B]P

TP, where ωBpc B = [qBpc B r Bpc B]P

TP. The actual control inputs are

thrust FBpthx B, the right and left brake angles, δBR B and δBLB, which are deflection angles of the right and left rear-portions of the canopy. The right (left) brake angle is defined as dBR B/cBc B (dBLB/cBc B,), where dBR B (dBLB) is the pull-length of the right (left) control lines and cBc B is the mean aerodynamic-chord length of the canopy. For convenience and for derivation of linear state equation, we define collective deflection angle δBe B = δBR B + δBLB, and differential one δBrB = δBR B − δBLB. We then define the control input vector as u = [FBpthx B δBe B δ BrB]P

TP. With the state and control vectors, the 8-DOF motion of the PPG

is described by the nonlinear state equation,P

11P

uxgxfx )()( ccc +=& (1) where f(xBc B) ∈ℜP

12×1P and g(xBc B) ∈ℜP

12×3P are nonlinear functions of xBc B.

Let the equilibrium state and input vectors be xP

*P and uP

*P, i.e.,

f(xBc PB

*P) + g(xBc PB

*P)uP

*P = 0. (2)

Under the assumption of symmetry of the PPG with respect to the XZ-plane, the non-trivial equations in Eq. (2) are four equations corresponding to forward and downward force trim and pitching moment trim of the canopy and the payload. Defining small perturbations of the state and control variables around the equilibrium point (xBc PB

*P, uP

*P) as ΔxBcB

= xBc B −xBc PB

*P and Δu = u −uP

*P, and applying the first-order approximation of Taylor series expansion, we obtain the

linearized state equation, uBxAx Δ+Δ=Δ cccc& , (3)

where ABc B∈ℜP

12×12P is a constant matrix given by partial differentiation of f(xBc B) + g(xBc B)u with respect to xBc B at the trim

point (xBc PB

*P, u P

*P), and BBc B∈ℜP

12×3P is given by g(xBc PB

*P). We have analytically derived the linear state equation, although we

have no space to show details here. Since the velocity of the PPG is small and its angle of attack is relatively large in trim flight, we approximate small perturbation of the angle of attack by

( )cccc

cc uw

uΔ⋅−Δ=Δ *

*

2*

tan)(cosα

αα , (4)

where αBc B = tanP

−1P(wBc B/u Bc B). This approximation provides a better correspondence of the system matrix ABc B between the

analytical linearization and the numerical one. Appropriately rearranging the elements of the state vector makes the matrices ABc B and BBc B block-diagonal, so that Eq. (3) is separated into two state equations, i.e., for the longitudinal motion we have

longclongclongclongclong uBxAx Δ+Δ=Δ& (5) and for the lateral-directional motion

latclatclatclatclat uΔ+Δ=Δ BxAx& , (6) where ΔxBclongB = [ΔuBc B ΔwBc B ΔqBc B Δθ Bc B ΔqBpc B Δθ Bpc B]P

TP, Δu BlongB = [ΔFBbthB Δδ Be B]P

TP, ΔxBclatB = [ΔvBc B ΔpBc B Δr Bc B Δφ Bc B Δr Bpc B ΔψBpc B Δψ Bc B]P

TP and

ΔuBlatB = ΔδBrB.

C. State Transformation Since the canopy generates most of the aerodynamic forces and moments, using the canopy states makes the derivation of the dynamic equation easier and straightforward. However, sensors such as accelerometers, gyros, camera, etc. are mounted on the payload; hence, expressing the state equation in terms of the payload states, which are defined below, would be more convenient for analysis and synthesis of a flight control system. Although we could derive a nonlinear payload-state equation, we choose to derive a linear payload-state equation, considering application of a linear control method. The following relations hold between the canopy states and the payload states. First, the velocity of the payload expressed in ΣBc B is given by

VBpc B = VBc B + KBpc1 BωBc B + KBpc2 Bω Bpc B. (7) In Eq. (7), when we define lBc B (lBpB) to be the distance between OBmB and OBc B (OBp B), KBpc1 B and KBpc2 B are defined, respectively, as


4

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−

−+=

0cossinsinsincossin0)cos(sinsincos0

1

pcpcppcpcp

pcpcpcpcp

pcpcpcpcp

pc

llllllll

ψθψθψθθψθθ

K (8)

and

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

0sincossinsincos

sinsincoscos

2

pc

pcpcpcpc

cpcpcpc

ppc lθ

ψθψθψθψθ

K . (9)

Defining the velocity of the payload expressed in ΣBpB as VBpB = [uBp B vBpB wBp B]P

TP, we can also write VBpc B as

VBpc B = TBpc BVBpB (10) where TBcpB is a coordinate transformation matrix from Σ BpB to Σ Bc B and defined as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

pcpc

pcpcpcpcpc

pcpcpcpcpc

cp

θθψθψψθψθψψθ

cos0sinsinsincossincoscossinsincoscos

T . (11)

Eliminating VBpc B from Eqs. (7) and (10), we have the first relation: VBc B = TBcpBVBp B − KBpc1 Bω Bc B − KBpc2 Bω Bpc B. (12)

Second, the relation between the angular velocities is given by ωBc B = TBcpBωBpB − KBpc3 Bω Bpc B, (13)

where ωBpB = [pBpB qBpB r BpB]P

TP is the angular velocity vector of the payload in ΣBpB, and KBpc3 B is defined by

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

100cos0sin

3 pc

pc

pc ψψ

K . (14)

Third, from the definition of Euler angles and relative attitude angles between the canopy and the payload, we have the identical equation

TBcI B(φBc B, θ Bc B, ψBc B) =TBcpB(θBpc B, ψBpc B)TBpI B(φ BpB, θBpB, ψBpB), (15) where ψBc B is the yaw angle of the canopy, and φBpB, θBpB, and ψBpB are the roll, pitch, and yaw angles of the payload, respectively. TBpI B (TBcI B) is the coordinate transformation matrix from the inertial coordinates to ΣBpB (ΣBc B). Thus, we have obtained three relations, which are given by Eqs. (12), (13), and (15). Applying the first-order approximation of Taylor series expansion to the equations yields a linear relation between the canopy state vector and the payload one as follows. First, from Eq. (12) we have

ΔVBc B + KBpc1 PB

*Pω Bc B + KBpc2 PB

*Pω Bpc B = TBpc PB

*PΔVBpB +KBpc4 PB

*P[Δθ Bpc B ΔψBpc B]P

TP, (16)

where the superscript ‘*’ means that the functions and the variables are evaluated at the trim states, and KBpc4 PB

*P is

defined as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

0)cos()cos(0

0)sin(

**

**

**

**4

pcp

pcp

pcp

ppc Vθα

θαθα

K , (17)

where αBp PB

*P = tanP

−1P(wBpPB

*P/u BpPB

*P). Next, Eq. (13) is linearized as

ΔωBc B = TBpc PB

*PΔωBp B − KBpc3 PB

*PΔω Bpc B. (18)

Finally, from Eq. (15) the following linear equations are obtained: Δθ BpB = ΔθBc B+Δθ Bpc B (19)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ΔΔ

c

pc

c

ppcpc

cc

pp

p

ψψφ

θθθθθ

θψφ

***

**

* coscossin0sincos

cos1 . (20)

Adding ψBc B to the state vector xBc B, defining the payload state vector as xBpB = [VBpPB

TP ω Bp PB

TP ωBpc PB

TP θ Bpc B ψ Bpc B φ BpB θ BpB ψBpB]P

TP, and

combining Eqs. (16), (18), (19), and (20), we obtain the relation ΔxBpB = TΔxBc B, (21)

where ΔxBpB =xBpB −xBpPB

*P and T ∈ℜP

13×13P is a constant transformation matrix.

With Eq. (21), Eq. (3) is transformed into the equation expressed by the payload state as


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uBxAx Δ+Δ=Δ pppp& , (22)

where ABpB = TABc BTP

−1P and BBpB = TBBc B. As Eqs. (5) and (6) resulted from Eq. (3), Eq. (22) is rewritten into the

longitudinal state equation longplongplongplongplong uBxAx Δ+Δ=Δ& (23)

and the lateral-directional state equation latplatplatplatplat uBxAx Δ+Δ=Δ& , (24)

where ΔxBplongB = [ΔuBpB ΔwBpB ΔqBpB ΔθBpB ΔqBpc B ΔθBpc B]P

TP and ΔxBplatB = [ΔvBp B ΔpBpB Δr Bp B ΔφBp B ΔrBpc B ΔψBpc B ΔψBpB]P

TP. This means that the

state transformation matrix itself becomes block-diagonal by rearranging the state variables.

III. PID Control based on Integral-Type Optimal Servomechanism

A. Plant Representation Consider a second-order delay system described by a transfer function GBCB(s), i.e.

y(s) = GBCB(s)u(s), (25) where y(s) is an output and u(s) is an input, and GBCB(s) is defined as

012

0)(dsds

nsGC ++= . (26)

The plant is obviously controllable and observable. To give a state-space representation of the system, state vector is defined as x= [xB1B xB2B]P

TPB B= [ ]Tyy & . The system is then expressed as

uBAxx +=& (27) y = Cx, (28)

where

⎥⎦

⎤⎢⎣

⎡−−

=10

10dd

A , ⎥⎦

⎤⎢⎣

⎡=

0

0n

B , [ ]01=C . (29)

B. Integral-type Optimal Servo Controller The first step is to design an integral-type optimal servo controller for the plant given by Eqs. (27) and (28). In

this study, we employ a design method of an IOS controller by Smith and Davison.P

17P With the definition of control

error e = r − y, where r is a constant reference output, we defined an augmented system as

ueedt

d&

&&⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡00

0 BxC

Ax . (30)

For a quadratic cost function defined as

∫∞

++=0

22 )( dtuRQeJ xT &&& xQx , (31)

the optimal control law that minimizes J can be derived as eKu ex += xK && (32)

by using the LQR theory. Integrating Eq. (32) yields

∫+= τedKu ex xK . (33)

Defining KBx B = [KBP B KBD B]P

TP and KBI B = KBe B, we can rewrite the control law as

∫++= τedKyKyKu IDP & . (34)

This is exactly the I-PD (proportional and derivative preceded integral) control law, which means that the PID gains are determined at a time as optimal state feedback gains. Once the I-PD control law is obtained, it can be arbitrarily converted to the PID, PI-D or generic 2-DOF form. P

18P Note that any of the controller forms has the same closed-loop

property such as stability. In practical use, the pure derivative of y is replaced with an approximate one s/(TBD Bs+1), where TBD B is an appropriate constant.

C. Plant Reduction based on ν-Gap Metric As stated above, we can design an optimal PID controller, if the plant dynamics are given by Eq. (25). However,

the order of a plant GBP B(s) is generally higher than the second. For a higher-order plant, we need to reduce its transfer function to Eq. (26) and to achieve this we employ the model reduction technique based on the ν-gap metric.


6

Specifically, we search plant parameters of GBCB(s) that minimize the ν-gap between GBP B(s) and GBCB(s). It is known that if the ν-gap is small, closed-loop properties for the two plants are close to each other.P

19, 20P Reference 21 also uses this

method in conjunction with LMI to find a reduced-order plant for controller design. In our study, parameter-space search is adopted, since the number of parameters is small.

Before giving the definition of ν-gap, let us define the following function

)()(1)()(1)()(

))(),((sGsGsGsG

sGsGsGsG

CCPP

CPCP

−+−+

−=Ψ (35)

and the conditions 1+GBCB(−jω)GBP B(jω) ≠ 0, ∀ω (36) wno (1+GBCB(−jω)GBP B(jω))+η(GBP B(s)) −η(GBCB(s)) −ηB0B(GBCB(s)) = 0, (37)

where η(GBP B(s)) denotes the number of poles of GBP B(s) in the open right-half complex plane and ηB0 B(GBCB(s)) the number of imaginary axis poles of GBCB(s). ‘wno’ denotes the winding number valuated on the standard Nyquist contour indented around any imaginary axis poles of GBP B(s) and GBCB(s). With the definitions, the ν-gap is defined as

⎩⎨⎧ Ψ

= ∞

1),(

),( CPCP

GGGGνδ (38)

From our design experience, we have found that dB0B of GBCB(s) can be set to zero for GBP B(s) having integral element 1/s. This also reduces the number of parameters to be searched. Note that depending on GBP B(s), there exists no GBCB(s) for which the ν-gap is small. In that case, we cannot design a PID controller that possesses good closed-loop properties or even stability. However, this is natural, since the three-term controller cannot stabilize all plants. Thus, the design procedure is summarized as follows. 1) Approximately cancel closely located poles and zeros of GBP B(s), if necessary. 2) Determine the parameters of GBCB(s) based on the ν-gap metric via parameter-space search. 3) Design an integral-type optimal servo using GBCB(s). Features of the design method are as follows. 1) PID gains are obtained at a time. 2) Trade-off between control performance and control effort can be performed through weight selection. 3) Trade-off among P, D, and I actions is also possible through weight selection of the diagonal elements of QBx B and

Q, respectively. 4) The controller has desirable properties as an LQR such as good time response and desirable stability margins,

although some degradation may occur due to plant model reduction.

IV. Linear Dynamics Analysis We use data of a manned paragliderP

22P for numerical analysis and simulation. The span and the chord length of

the canopy are 9.95 m and 3.05 m, respectively, and the weights of the canopy and the payload are 6.4 kg and 93.0 kg, respectively. The trim condition without control inputs is steady-state gliding, where the airspeed is 9.54 m/s, the angle of attack is 15.1 deg, the pitch angle of the canopy is 3.45 deg, and the relative pitch angle is −4.83 deg. The system and control matrices of the linear dynamic model were computed by analytical linearization as well as by numerical partial differentiation. The two methods gave almost the same results. The matrices of the state equations and state transformation matrices are given in Appendix.

To illustrate the difference between the canopy-state system and the payload-state system, we computed time responses for the initial relative pitch angle Δθ Bpc B(0) = 0.1 rad, where other initial states were zero. Figures 3 shows time responses of the pitch angle of the canopy, ΔθBc B. Although the same initial states are given, the time responses of θ Bc B are very different. The reason for this is that the initial pitch angle ΔθBc B(0) for the payload-state system is −0.1 rad, whereas it is exactly zero for the canopy-state system, where ΔθBbB(0) = 0.1 rad. Computing time responses of the lateral-directional motion for the initial relative yaw angle ΔψBpc B(0) = 0.1 rad, we observe the similar difference between the canopy-state and payload-state systems. Thus, we should consider physical meaning of the states, when we deal with the systems. Table 1 summarizes the eigenvalues of the linear models of the PPG and motion modes corresponding to the complex conjugate pairs. Figure 4 shows time responses of the pitch rate of the canopy computed using the payload-state model for Δθ Bpc B(0) = 0.1 rad. We can see from the figure that the relative pitch rate has a faster, good-damping mode, and that the canopy pitch rate has slower, more oscillatory mode. Hence, the former mode corresponds to the

if Eqs. (36) and (37) are satisfied. otherwise.


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relative pitching motion and the latter to the pitching motion of the canopy. Figure 5 shows time responses of the yaw rate of the canopy computed using the payload-state model for ΔψBpc B(0) = 0.1 rad. We can also see from the figure that the relative yaw rate has a faster, oscillatory mode, and that the canopy yaw rate has a slower, less oscillatory mode. The former mode corresponds to the relative yawing motion and the latter to the Dutch-roll motion of the canopy.

0 10 20-6

-4

-2

0

2

time, s

Cano

py pi

tch an

gle, d

eg

canopy-state system payload-state system

Figure 3. Time responses of the canopy pitch angle for ΔθBpc B(0) = 0.1

Table 1 Eigenvalues of the linear systems (longitudinal and lateral-directional) motion mode eigenvalues natural frequency, rad/s damping ratio

first-order modes −1.96, −0.557 ⎯ ⎯ relative pitching −3.36±1.97i 3.89 0.863 canopy pitching −0.285 ±0.941i 0.983 0.290 first-order modes −121, −1.35, 0 ⎯ ⎯ relative yawing −0.150±5.03i 5.03 0.0297 Dutch-roll −0.224±0.721i 0.755 0.297

0 10 20-4

-2

0

2

4

time, s

Pitch

rate,

deg/s Δqc

Δqpc

Figure 4. Time responses of the canopy pitch rate for ΔθBpc B= 0.1 rad

0 10 20-30-20-10

0102030

time, s

Yaw

rate,

deg/s Δrc

Δrpc

Figure 5. Time responses of the canopy yaw rate for ΔψBpc B= 0.1 rad


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V. Flight Control System Design and Simulation

A. Flight Control System Design The design method of a PID controller has been applied to the linear model of the PPG. Since the PPG has three control inputs, we design three PID controllers for respective SISO systems as follows. The transfer function from thrust, ΔFBpthx B(s), to the altitude variation, Δh(s), is given by

)96647.056976.0)(170.157208.6)(5567.0)(9578.1()678.142294.6)(4408.0)(0410.2)(28.180(101214.7)( 22

25

1 +++++++++++×

=−

sssssssssssssGP

(39)

We have then obtained a second-order plant with a ν-gap of 0.0177 as

sssGC += 21

01061.0)( . (40)

Choosing the weighting matrices as QBx B = 0B2×2B, Q =100, and R =1 and the time constant as TBD B =0.025 s, the resultant PID control gains are KBP B = −52.33, KBI B = 10.00, and KBD B = −42.67. The transfer function from collective brake deflection angle, ΔδBe B(s), to the forward speed, ΔuBpB(s), is given by

)96647.056976.0)(170.157208.6)(5567.0)(9578.1()0135.45840.2)(5482.1)(736.12)(954.69(19986.0)( 22

2

2 +++++++++++−

=ssssss

ssssssGP (41)

We have then obtained a second-order plant with a ν-gap of 0.298 as

20624.69)( 22 −−

−=

sssGC

. (42)

Choosing the weighting matrices as QBx B = 0B2×2B, Q =1, and R =1 and the time constant as TBD B =0.025 s, we obtained PID control gains of KBP B = 0.9671, KBI B = −1.000, and KBD B = 0.2749. The transfer function from differential brake deflection angle, ΔδBrB(s), to the heading angle, ΔψBp B(s) +ΔβBpB(s), is given by

)5698.04488.0)(35.252990.0)(347.1)(0.121()5392.88946.1)(670.371269.2)(30072.0(434.35)( 22

22

3 +++++++++++

=sssssssssssssGP

(43)

Before designing a PID controller, we first design a roll damper, whose control law is ΔδBrB = −0.5ΔpBp B. Figure 6 shows time responses of the canopy roll rate, ΔpBpB, to the step input ΔδBrB = 0.1 rad with and without the roll damper. It is obvious that the damping of the roll rate is improved.

0 10 20

-5

0

5

10

time, s

Paylo

ad ro

ll rate

, deg

/s

without roll damper with roll damper

Figure 6. Time responses of the roll rate to step input, ΔδBrB = 0.1 rad

The transfer function of the closed-loop system then becomes

)26189.047395.0)(537.2879272.0)(6908.2)(09.119()5392.88946.1)(670.371269.2)(30072.0(434.35)( 22

22

3 +++++++++++

=sssssss

ssssssG dmpP (44)

For GBP3dmp B(s), we have then obtained a second-order plant with a ν-gap of 0.310 as

sssGC 9.0

288.1)( 23 += . (45)

We designed a PID controller for the reduced-order model of Eq. (45), choosing the weighting matrices as QBx B = 10IB2B, Q =10, and R =1 and the time constant as TBD B =0.025 s. The obtained gains are KBP B = −6.193, KBI B = 3.162, and KBD B = −3.785.


9

Stability margins are summarized in Table 2. The margins are computed by opening each closed-loop, while other loops are closed. The controllers have large gain and phase margins. The margin of gain increase is ∞ for the speed controller.

Table 2 Stability margins controller gain margin phase margin crossover frequency altitude 28.63 dB 89.09 deg 0.5150 rad/s speed −18.35 dB 42.44 deg 9.172 rad/s heading ∞ 85.05 deg 2.597 rad/s

In this design example, we use the PID control law

)()( ses

KsKKsu IDP ⎟

⎠⎞

⎜⎝⎛ +−−= . (46)

where e = yBrefB −y and yBrefB is given by

rss

synrefnrefref

nrefref 22

2

2)(

ωωζω

++= . (47)

We have chosen ζBrefB = 1 and ωBnrefB = 0.5. A problem with flight control of a PPG is input saturation. Since the control lines can only be pulled and cannot be pushed, the right and left brakes, δBR B and δBLB, can take a positive angle only. Hence, the control inputs tend to become saturated. Particularly, when a controller has an integral action, the control system may suffer integral-windup due to input saturation, which can significantly degrade control performance. To alleviate the degradation, we have employed an anti-reset-windup schemeP

23P and modified the PID control law as

)(ˆ)()(2

suks

kseks

KsKsKsu IPD

++

++−−

= (48)

where u(s) is a controller output and )(ˆ su is a plant input; namely, if u(s) violates input range limits, )(ˆ su takes the maximum or minimum value, and otherwise, u(s) = )(ˆ su . k is a constant design parameter, and we have chosen k=10.

B. Simulation We conducted computer simulation, applying the controllers to the nonlinear model of the PPG. The reference outputs are 0 m for the altitude variation, 8 m/s for the forward speed, and 90 deg for the heading angle. The initial condition is the steady-state gliding, where the forward speed is 9.22 m/s and the heading angle is 0 deg. As seen from the initial and the target outputs, this simulation considers the case where the PPG changes its flight from the steady-state gliding to level flight with thrust, while turning to the right by 90 deg. Figures 7 through 11 show time histories of the outputs. The altitude undergoes undershoot about 2.4 m, since the PPG is going down at the initial time; however, it recovers to zero by increasing thrust to achieve level flight. Also, as is expected, the forward speed is controlled to 8 m/s, and the heading angle is changed by 90 deg. Time histories of the control inputs are shown in Figs. 12 and 13. Note that the brake deflection angles are saturated, since the neutral deflection angle is zero and the inputs cannot take negative angle. In this design example, we have added the anti-reset windup compensator and it really works. In fact, without it control performance considerably degrades as Fig. 14 shows. The thrust converges to about 200 N, which is required to maintain level flight.


10

0 10 20 30 40 50-5

0

5

10

time, sSp

eed,

m/s up

vp wp

Figure 7. Time histories of the payload speeds

0 10 20 30 40 50-20-10

0102030

time, s

Angu

lar ra

te, de

g/s pp qp rp

Figure 8. Time histories of the payload angular rates

0 10 20 30 40 50-40

-20

0

20

40

time, sRelat

ive an

gular

rate,

deg/s relative pitch rate, qpc

relative yaw rate, rpc

Figure 9. Time histories of the relative angular rates

0 10 20 30 40 500

20406080

100

time, s

Euler

angle

, deg roll angle, φp

pitch angle, θp yaw angle, ψp reference for ψp

Figure 10. Time histories of the Euler angles


11

0 10 20 30 40 50-5

0

5

time, sAl

titude

varia

tion,

m

altitude variation reference output

Figure 11. Time histories of the altitude variation

0 10 20 30 40 50-5

0

5

10

time, sBrak

e defl

ectio

n ang

le, de

g

left deflection angle, δL right deflection angle, δR

Figure 12. Time histories of the brake deflection angles

0 10 20 30 40 500

100

200

300

time, s

Thru

st, N

Figure 13. Time histories of the thrust

0 10 20 30 40 500

50

100

time, s

Euler

angle

, deg

roll angle, φp pitch angle, θp yaw angle, ψp reference for ψp

Figure 14. Time histories of the Euler angles without anti-reset windup compensation


12

VI. Conclusions We have obtained a linear dynamic model of a PPG, which is analytically derived or numerically computed from

the nonlinear dynamic model. We have also derived the state transformation matrix between the canopy states and the payload states. With the matrix we can obtain a linear model expressed by the payload states from the linear model by the canopy states, and the payload-state linear model facilitates dynamical analysis and controller design, since the measurement sensors and cameras are mounted on the payload. The design example of PID controllers and the simulation results illustrate that the design method based on the integral-type optimal servomechanism and the ν-gap metric actually works as an efficient design tool of a PID controller with good control performance and adequate stability margin. A problem with PPG flight control is saturation of the brake deflection angles. We may need to add an anti-reset windup compensator, in addition to carefully giving reference outputs.

Appendix The coefficient matrices in the linear state equations, Eqs. (5) and (6), are as follows:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−−−

−−−

=

010000844.157271.60502.18436.24250.129554.0

00010066983.038030.00546.198936.019233.002926.038697.012228.086154.01585.15516.123140.061035.02382.08799.24840.2071012.053697.0

clongA ,

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−×−

×−×−×

=

−

−

−

−

008906.4102710.100

2030.5104230.1646.18102210.8647.32102966.1

3

3

4

3

clongB

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

××−−×−−−

−×−−−

=

−−

−

−

00000018.10000100000106.2631324.048119.007.12359.152339.200000060345.010064332.0109074.7105409.248.12259.152311.200078414.0107908.91901.177299.0062424.0019645.00091265.0108945.51958.12798.32418.219269.0

34

4

4

clatA,

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

=

00

04.5290

09.5290059881.0024676.0

clatB

Rearranging the state variables appropriately, we can also make the state transformation matrix diagonal; namely, by the same transformation from ABc B to block-diag(ABlongB, ABlatB), we can transform T in Eq. (21) to a block-diagonal matrix. Thus, we obtain the following state transformation matrices for the longitudinal states and the lateral-directional states, respectively,

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−−

−

=

100000010000101000010100

3928.90054898.099645.0084212.06954.133200.008278.6084212.099645.0

longT and

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−−−−

=

199674.00084236.0000010000000100000060252.0099847.00000099645.0099645.0084212.0000084212.00084212.099645.0002167.9027958.00027958.08498.61

latT,

which constitute a block-diagonal matrix, block-diag(TBlongB, TBlatB).


13

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P

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