1 Adaptive, Optimal and Reconfigurable Nonlinear Control Design for Futuristic Flight Vehicles Radhakant Padhi Assistant Professor Dept. of Aerospace Engineering

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Adaptive, Optimal and Reconfigurable Nonlinear Control

Design for Futuristic Flight Vehicles

Radhakant Padhi

Assistant Professor

Dept. of Aerospace EngineeringIndian Institute of Science, Bangalore, India

Abha TripathiProject Assistant

2

Project Plan

Date of Commence: 1st October 2006 Project duration : 2.5 Years Staff members:

Shree Krishnamoorthy, Project Assistant, Oct-Dec 2006.

Kaushik Das, Ph.D. student, January-July, 2007.

Abha Tripathi, Project Assistant, Aug.2007…continuing.

Apurva chunodhkar, a B. Tech. student from IIT-Bombay and Siddharth Goyal, a B.E. student from Punjab Engineering College have worked in sporadic engagements

Jagannath Rajshekharan, Project Assistant, has also worked in sporadic engagements

3

Summary Two parallel directions have been explored in this project.

Firstly, a new dynamic inversion approach has been developed and is experimented on a low-fidelity model of a high performance aircraft (F-16). Comparatively, it leads to some potential benefits:

Elimination of non-minimum phase behavior of the closed loop response

Less oscillatory behavior

Lesser magnitude of control Robustness study was carried out for the above approach with

uncertainties in aerodynamic force and moment coefficients and inertia parameters

4

Summary Secondly, a structured neuro – adaptive control design

idea has been developed which treats the kinematics and dynamics of the problem separately.

Modeling and parameter inaccuracies are considered by using neural network which dynamically capture the unknown functions that are used to design a model-following adaptive controller.

Sigma correction was done in the weight update rule. This idea is found to be successful on a satellite attitude

problem.

5

Command Tracking in High Performance Aircrafts: A New Dynamic Inversion

Design

6

Airplane Dynamics(F-16): Six Degree-of-Freedom

A1 2 3 4 A

2 25 6 7

8 2 4 9

PQ L = c +c +c +c

= c

= c

A

A A

P QR N

Q PR c P R c M

R PQ c QR c L c N

1 sin

1 sin cos

1 cos cos

X X

Y

Z

A T

A

A

U VR WQ g F Fm

V WP VR g Fm

W UQ VP g Fm

= sin tan cos tan

= cos sin

= sin cos sec

P Q R

Q R

Q R

= sin cos sin cos cosh U V W

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Definitions and Goal Total Velocity:

Roll Rate (about x-axis):

Roll Rate (about velocity vector):

Normal Acceleration:

Lateral Acceleration:

Goal:

P

where are pilot commands

/ 1/zz z An F m m F

/ 1/yy y An F m m F

P*, Pw*, nz*, ny*, VT*

* * * * *, , ,w w z z y y T TP P or P P n n n n V V

WP

TV

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Control Synthesis Procedure

Define new variables:

Key observation:

Known:

* *,y y y ya n V a n V

* *,z z z za n W a n W

* * TT

z y z yn n n n * * TT

z y z ya a a a

z z

y y

z n n c

y n n c

n f g U

n f g U

T T

P P c

T V V c

P f g U

V f g U

0V W

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Control Synthesis Procedure Longitudinal Maneuver

Pilot commands:

• Roll Rate (bank angle rate):

• Normal Acceleration:

• Lateral Acceleration:

• Total Velocity:

Lateral Maneuver Pilot commands:

• Roll Rate (bank angle rate):



• Total Velocity:

*

*

0

0

z

y

T

n

n

V

* 0y

T

or P

h

n

V

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Control Synthesis Procedure

Combined Longitudinal and Lateral Maneuver

Pilot commands:

• Roll Rate (about velocity vector):



• Total Velocity:

*

* 0

w

z

y

T

P

n

n

V

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Control Synthesis Procedure Design a controller such that

After some algebra, Finally:

ˆ ˆ 0,TT V TV K V ˆ ˆ 0T TX KX

1[ ]TT

c U UU A b T *

z y z y

T TT T T T T T

U P a a n nPA g g g K g g g

*

* *

z y z y

TT

U P a a n z n yPb f f f K P f f n f n

1 1 1, ,

z yP n n

K diag

1T

T

VV

K diag

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Results: Longitudinal

Tracked Variables Control Variables

0 20 40 60 80-1

-0.5

0

0.5

1

Time (Sec)

(

deg)

0 20 40 60 80-1

-0.5

0

0.5

1

Time (Sec)

n y(g)

0 20 40 60 800

1

2

3

Time (Sec)

n z(g)

0 20 40 60 80500

600

700

800

Time (Sec)

VT (

ft/s

)

0 20 40 60 800

50

100

Time (Sec)

Thr

ust

(%)

0 20 40 60 80-1

-0.5

0

0.5

1

Time (Sec)

Aile

ron

defle

ctio

n (d

eg)

0 20 40 60 80-4

-2

0

2

Time (Sec)

Ele

vato

r de

flect

ion

(deg

)

0 20 40 60 80-1

-0.5

0

0.5

1

Time (Sec)

Rud

der

defle

ctio

n (d

eg)

New Method

Existing Method

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Results: Lateral Mode

0 20 40 60-20

-10

0

10

20

Time (Sec)

P (

deg/

sec)

0 20 40 600.99

0.995

1

1.005

1.01x 10

4

Time (Sec)

Alt

(ft)

0 20 40 60-0.1

-0.05

0

0.05

0.1

Time (Sec)

n y(g)

0 20 40 60570

575

580

585

590

Time (Sec)

VT (

ft/s

)

New Method

Existing Method

0 20 40 600

20

40

60

80

Time (Sec)

Thr

ust

(%)

0 20 40 60-4

-2

0

2

Time (Sec)

Aile

ron

defle

ctio

n (d

eg)

0 20 40 60-5

0

5

10

Time (Sec)

Ele

vato

r de

flect

ion

(deg

)

0 20 40 60-4

-2

0

2

4

Time (Sec)

Rud

der

defle

ctio

n (d

eg)


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Results: Combined Longitudinal and Lateral


0 20 40 60

-10

0

10

Time (Sec)

Pw

(deg/s

ec)

0 20 40 60-0.1

-0.05

0

0.05

0.1

Time (Sec)

n y (g)

0 20 40 600

0.5

1

1.5

2

2.5

Time (Sec)

n z (g)

0 20 40 60570

575

580

585

590

Time (Sec)

VT (

ft/s

)

0 20 40 600

20

40

60

Time (Sec)

Thr

ust

(%)

0 20 40 60-1

-0.5

0

0.5

1

Time (Sec)

Aile

ron

(deg

)

0 20 40 60-2.2

-2

-1.8

-1.6

-1.4

Time (Sec)

Ele

vato

r (d

eg)

0 20 40 60-0.6

-0.4

-0.2

0

0.2

Time (Sec)

Rud

der

(deg

)

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Summary

Existing Method:

Assumption:

Need of integral control

More number of design parameters (10-12)

Works

New Method:

Assumption: No such need (No wind-up)

Less number of design parameters (5-7)

Works better...!• Lesser control magnitude

• Smoother transient response

• Better turn co-ordination

* * *

0

0

V W

0V W

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Robustness Study

Nominal Controller given to the actual system having uncertainties

Perturbation assumed in the inertia parameters and aerodynamic force and moment coefficients

Normal distribution used for introducing randomness in the parameters with mean value as the nominal value of the parameters and standard deviation as 1/3 of maximum allowed perturbation in that parameter.

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Robustness Study

Inertia parameters varied from 5 to 10% Aerodynamic coefficients varied from 1%

to 10%. Simulation were carried out for 50 cases

in each mode. In each simulation study, the aim was to

declare it as a success or failure

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Longitudinal Mode

0

y

z

Pilot Command Given:

0; ; 0;

1sec; 0.9965 ;

15sec; 2 ;

Limits Imposed on the steady state error:

: 3 ; n : 0.05 ; 585 575 / sec;

n : 20%;

T T y

z

z

T

V V n

for t n g

for t n g

g V to ft

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Longitudinal Mode

Aerody-namic

Coefficients

1% 1% 2% 2% 5% 5% 10% 10%

Inertia

Parameters

5% 10% 5% 10% 5% 10% 5% 10%

Percentage

Success

100% 100% 96% 92% 76% 70% 48% 40%

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Lateral Mode

0 0

y


40 ; ; ; 0;


h: 1%; n : 0.05 ; 585 575 / sec

10%;

T T y

T

V V h h n

g V to ft

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Lateral Mode

Aerody-namic

Coefficient

1% 1% 2% 2% 5% 5% 10% 10%

Inertia

Parameter

5% 10% 5% 10% 5% 10% 5% 10%

%

Success

100 100 100 100 94 88 86 80

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Lateral Mode

0 0

y


10 / sec; ; ; 0;


h: 1%; n : 0.05 ; 585 575 / sec

10%;

T T y

T

P V V h h n

g V to ft

P

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Lateral Mode

Aerody-namic

Coefficient

1% 1% 2% 2% 5% 5% 10% 10%

Inertia

Parameter

5% 10% 5% 10% 5% 10% 5% 10%

Percentage

Success

100% 100% 100% 100% 98% 94% 76% 76%

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Lateral Mode

0 0

y


7 / sec; ; ; 0;


h: 1%; n : 0.05 ; 585 575 / sec

10%;

w T T y

T

w

P V V h h n

g V to ft

P

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Combined Mode

Aerody-namic

Coefficient

1% 1% 2% 2% 5% 5% 10% 10%

Inertia

Parameter

5% 10% 5% 10% 5% 10% 5% 10%

Percentage

Success

100% 100% 96% 94% 54% 42% 28% 24%

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Conclusion

When aerodynamic coefficients are perturbed by 5% and the inertia parameters by 10%, the controller is robust

Increase in inertia parameters does not affect the percentage success

Aerodynamic coefficients are more sensitive than inertia parameters

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Enhancement of Robustness

Augment Dynamic inversion with Neuro -Adaptive Design

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Adaptive Approach(Lateral case)

Nominal Outputs:

Actual Outputs:

Approximate Outputs:

),( ddy

Td

yd

d

d

UXf

V

n

Q

P

)(),( XdUXf

V

n

Q

P

y

T

)()(ˆ),( aaaay

Ta

ya

a

a

XXKXdUXf

V

n

Q

P

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Adaptive Approach

Goal: Strategy:

Steps for assuring :

Solve for adaptive controller

tasYY d

tasYYY da

da YY

0

d a d

d d d

E Y Y

E K E

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Adaptive Approach

Steps for assuring Error

Error Dynamics

aYY

( ),a a ai i aiE Y Y e y y

aiaiii

aiiai

iT

ii

iiT

ii

aiaiiyiai

iyii

eKXdXd

yye

XWXd

XWXd

eKXdUXfy

XdUXfy

)(ˆ)(

)(ˆ)(ˆ

)()(

)(ˆ),(

)(),(

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Adaptive Approach

Error Dynamics

NN Training

Lyapunov Function Candidate

aiaiiT

iiiT

iai eKXWXWe )(ˆ})({

2

2

)ˆ1)((

~

ˆ~

)~~

(2

1)(

2

1

aiiaiiaiiii

iaiiT

ii

iii

iT

iaiii

epkepWXepWL

WWWwhere

WWepL

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Adaptive Approach

Weight Update Rule:

Condition For stability:

)(ˆ XepW iaiiii

0

( / )i

ai i ai

L if

e k

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A STRUCTURED Approach for

Attitude Maneuver of Spacecrafts

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Neuro-adaptive Control: Generic Theory

Actual plant

Total tracking error

Tracking error dynamics

( )1 1 2X = h X X

D( , ) ( , )( ) 2 1 2 1 2X = f X X g X X U+

n1X R

n2X R nU R

1 1Z X X

( , ) ( , ) ( ) ( , )X 1 2 X 1 2 1 1 2Z = F X X + G X X U + h X d X X

Assumption

Unknown function

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Neuro-adaptive Control: Generic Theory

Objective of adaptive controller:

Approximate System:

Model-following strategy:

d da a aˆ( , ) ( , ) ( ) ( , ) ( )X 1 2 X 1 2 1 1 2Z = F X X +G X X U+h X d X X K Z Z

a (0) (0)Z ZNN Approximation

d asZ Z t

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Universal approximation property:

Error : Error dynamics for the individual i th error

channel:

Step I: Assuring

a a1 2 1 2 1 2 i( ) ( )Ti i id X ,X = W X ,X ,X ,X +

aZ Z

a aE Z Z

Weight vectorBasis function vector

a a i i

Tj ij 1 j 1 2 1 2 i a a

1

( ) ( )

n

aj

e W h X X ,X ,X ,X + k e

i ij 1 j1

ˆwhere and ( )

n

i i ij

W W - W h X

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Neural Network Training by Lyapunov Analysis

Lyapunov function candidate:

i i

T 1a i a i i i

1

1 1p

2 2

n

i

L e e W Γ W

i a a

i i i

T T 1a i j ij 1 j 1 2 1 2 i i i

1 1

a i i a a i1

ˆ( ) ( )

n n

i j

n

i

L e p W h X X ,X ,X ,X W Γ W

+ e p k e

38

Neural Network Training with Stability

Weight Update Rule:

Sufficient condition:

where

ji i i 1 2 a j ji 1 i i i1

ˆ ˆ( ) ( ) W

n

j

W = Γ X ,X e p h X Γ

n2

aii 1

i i

0 if ,max

( )(i 1,n)

iL < ep

2 2 2 2i i i i i i i

i ai

1 ˆ( p ( W W W ))2

1k

2

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SATELLITE Attitude Dynamics

Attitude kinematics Angular rate dynamics

Nominal Dynamics Actual Dynamics

Objective of Control Design:

,

( )d d d d d d = B I + U

( ) =

I = I + U x

( ) =

D( ) = B I + U + x

T TT T *T 0 0(0)

( )d d d = x

40

Nominal Control : Problem Specific Formulation

Tracking error for nominal system:

Tracking error dynamics:

Solving for nominal control

d d ds

d d d d d d d, , ds f g U

d d1 d d

dd d

d d d d d d d d d

dU B

B I K

x

41

Neuro-adaptive Control : Problem Specific Formulation

Tracking error for actual plant:

Expanding the following terms as:

Tracking error dynamics:

Basis

function

selection:

s

d d, , ( ) s f g U d

1 1 12 2 21 1 1

2 2 22 2 22 2 2

3 3 32 2 23 3 3

2 2 2i a a

2 2 2

2 2 2 T

( , , , ) [

]

2 2 2

2 2 2

2 2 2

( x- ) ( x- ) ( x- )

( x- ) ( x- ) ( x- )

( x- ) ( x- ) ( x- )

e e e

e e e

e e e

d d B = B + B I = I + I

42

Simulation Results:Nominal vs. Adaptive Control for actual system

MRPs Angular rates

(I) Constant disturbances & parameter uncertainties

43

Simulation Results:Nominal vs. Adaptive Control for actual system

Control Unknown function capture

(II) Constant disturbances & parameter uncertainties

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Publications Conference Publications

Radhakant Padhi, Narayan P. Rao, Siddharth Goyal and S.N. Balakrishnan, “Command Tracking in High Performance Aircrafts: A new Dynamic Inversion Design”, 17th IFAC Symposium on Automatic control in Aerospace, Touolose, France.

Apurva Chunodkar and Radhakant Padhi, ”Precision attitude Manoeuvers of Spacecrafts in Presence of Parameter Uncertainities and disturbances: A SMART Approach”, 17th IFAC Symposium on Automatic Control in Aerospace, Touolose, France.

Radhakant Padhi and Apurva Chunodkar, “Model-Following Neuro - adaptive Control Design for attitude maneuvers for rigid bodies in Presence of Parametric Uncertainties and disturbances", International Conference on advances in Control and Optimization of Dynamical Systems, Bangalore, India, 2007.

Abha Tripathi and Radhakant Padhi ,”Robustness Study of A Dynamic Inversion Control Law For A High Performance Aircraft”, International Conference on Aerospace Science And Technology, to be held on 26 – 28 June 2008, Bangalore, India.

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Publications Journal Publications

Radhakant Padhi, Siddharth Goyal, Narayan P. Rao and S.N. Balakrishnan, “A Direct Approach for Nonlinear Flight Control Design of High Performance Aircrafts”, Submitted to Control Engineering Practice.

Jagannath Rajsekaran, Apurva Chunodkar and Radhakant Padhi, ” Precision Attitude Maneuver of Spacecrafts Using Structured Model-Following Neuro -Adaptive Control”, Submitted to Control Engineering Practice.

Radhakant Padhi and Apurva Chunodkar, “Precision Attitude Maneuver of Spacecrafts Using Model - Following Neuro – Adaptive Control”, To appear in Journal of Systems Science & Engineering.

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Questions And comments

Documents

1 Adaptive, Optimal and Reconfigurable Nonlinear Control Design for Futuristic Flight Vehicles Radhakant Padhi Assistant Professor Dept. of Aerospace Engineering