lecture 34 of iit

Lecture 31

Gain Scheduling and Dynamic Inversion

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Topicsz A Brief Overview of Gain Scheduling

Philosophy Steps Issues

z Dynamic Inversion (DI) Design Philosophy Steps Advantages

Gain Scheduling An Accepted Practice in Industry




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Philosophy of Gain Scheduling

z Design local linear controllers, based on linearization of the nonlinear system at several expected operating points (based on the operating values of the scheduling variable(s)).

z Obtain a nonlinear controller for the nonlinear system by interpolating (scheduling) the linear gains, based on the actual value of the scheduling variable(s).


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Practice in Industryz The scheduling variable may or may not be one/more

state variable(s). Usual practice is to choose a slowly varying meaningful output variable as the scheduling variable. In flight control applications, usually dynamic pressure and Mach number are chosen as the scheduling variables

z The robustness, performance, stability etc. are usually justified from extensive simulation studies.

z The design results in a semi-global controller, depending on the selection of the operating points and validity of linear gains.


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Steps of Gain Schedulingz Linearize the dynamics about an

operating point

z Design the controller gain based on it

z Repeat this procedure about several operating points (within the domain of interest) and store the gains

z Interpolate the gains online and obtain a nonlinear controller

X AX BU= +

U K X= K


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Issuesz Is the linearized system good enough?z How many operating points to consider?z Does the interpolated gain lead to stability?

There are examples of loss of stability because of scheduling!

z How easy it is to select a set of schedulingvariables?

z What if the parameters of the model are updated? z Higher dimensional scheduling vector leads to

complexity in interpolations.z Loss of Minimum-phase property: A serious issue!


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Reference

J. S. Shamma and M. Athans: Gain Scheduling:

Potential Hazards and Possible Remedies, IEEE

Control System Magazine, June 1992, pp.101-107.

Dynamic Inversion: A Promising Substitute for Gain Scheduling




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Philosophy ofDynamic Inversion

x

yxy

?


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Philosophy of Dynamic Inversionz Carryout a co-ordinate transformation such

that the problem appears to be linear in the transformed co-ordinates.

z Design the control for the linear-looking system using the linear control design techniques.

z Obtain the controller for the original system using an inverse transformation.

z Intuitively, the control design is carried out by enforcing a stable linear error dynamics.


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Problem

z System Dynamics:

z Goal (Tracking):

z Special Class:(control affine & square)

( )( )

,

, ,n m p

X f X U

Y h X

X U Y

==

R R R

( )( )

*

*

,

Assumption: is smooth

Y Y t t

Y t

( ) ( )( ), non-singular Y

X f X g X U

p m g X t

= + =


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Dynamic Inversion Design

z Derive the output dynamics:

z Define Error of Tracking:

( ) ( ){ }( ) ( )Y Y

hY XXh f X g X UX

f X g X U

= = +

= +

( ) ( ) ( )*E t Y t Y t

( ) ( )( )

X f X g X U

Y h X

= + =Known:

1 1

1

1

n

p p

n

h hx x

hX

h hx x

"

# % #

"


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Dynamic inversion

z Select a fixed gain such that:

z Carry out the algebra:

z Solve for the controller:

0K >

( ) ( ) ( ){ }1 * *Y YU g X Y K Y Y f X=

( ) ( )( ) ( ) ( )

* *

* *

0

Y Y

Y Y K Y Y

f X g X U Y K Y Y

+ =+ =

00 0,KtE K E E e E t+ = =

( )Usually 1/ , 0i iK diag = >


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A Demonstrative Toy Problemz System Dynamics:z Objective:z Solution:

Desired output: Desired error dynamics:

Control solution:

( ) ( )2 21x x x x u= + + +( ) 0 as y x t=

* *0 , 0y t y t= = ( ) ( ) ( )( ) ( )

( ) ( ){ }* *

2 2

0, 1/ 0

0 0 0

1 0

y y k y y k

x k x

x x x u k x

+ = = > + =

+ + + + =

( )2211u x x k xx = + +


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Is a first-order error dynamics always enforced?

Answer: No!The order of the error dynamics is dictated by the relative degree of the problem, which is defined as the number of times the output needs to be differentiated so that the control variable appears explicitly.If a second-order error dynamics needs to be enforced, then the corresponding equation is

0V PE K E K E+ + = ( ) ( ) ( )2Usually 2 , , , 0i i iV i n P n i nK diag K diag = = >


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Example

z System Dynamics:

z Objective:

z Output:

( ) ( )2 2 21 1 2 1 1 22 1

2 3 2 5 1

3

x x x x x x u

x x

= + + + +=

[ ]1 2 0,Tx x as t ( )

( ) ( )2

2 1

2 2 21 1 2 1 1 2

selecting a proper is crucial!3

3 3 2 3 2 5 1

y x yy x x

y x x x x x x u

== =

= = + + + +


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Examplecontd.z Solution:

Desired output: Desired error dynamics:

Control solution:

* * *0 0y t y y t= = = ( )*e y y2

22 2 2

2 0

2 0n n

n n

e e ex x x

+ + =

+ + =

0 1, 0n < < >

( ) ( )2 21 2 1 1 22 21 21 3 2 3 2 6

15 1 n nu x x x x x

x x = + + + +


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When Does the DynamicInversion Fail?

:

1. Differentiate repeatedly until the input appears. 2. Design to cancel the nonlinearity.

Q : Is it always possible

Fundamental Principle of Dynamic Inversion

y uu

to design this way?

Ans: Not necessarily ! It is possible only if the relative degree is "well-defined".

u


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Undefined Relative degree

0

Undefined relative degree : It may so happen that upon sucessive

differentiation of , appears . However , the coefficient of mayvanish at , whereas it is non-zero at points arbitrarly close to

y u uX

( )

( )2

0

0

1

2 1 221

1 1 1 2

1 2 1 2 1 1

.In such cases, the relative degree is undefined at .

02Ex : 1,

, : Some nonlinear functionThen 2 2

2 2 2 ,

XX

xxu

x x x

y xy x x x xy x x x x x x x

= +

== == + = +

[ ] 222u x+


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Undefined Relative degree( )

( ) ( )N

( )2 2

21 1 2 2 1

1

1

1

1 2

1

2 , 2 + 2

As 0 , ( ) 0.

Hence, at 0 the relative degree is .

: If one choses then ,and the coeffic

,

y x x x x x ug Xf X yy

x g xyx

y xy x xy x x x u

= +

= ==

== == = +

Note

not defined

ient of 1 0 globally.In this case the relative degree is .globally

u = well defined


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Advantages of DI designz Simple design: No need of tedious gain scheduling

(hence, sometimes dynamic inversion is known as a universal gain scheduling design).

z Easy online implementation: It leads to a closed form solution for the controller.

z Asymptotic (rather exponential) stability is guaranteed for the Error dynamics (subjected to the control availability, this is true globally).

z No problem if parameters are updated (the updated values can simply be used in the formula).


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lecture 34 of iit

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