11 Gain Scheduling

7/27/2019 11 Gain Scheduling

1/8

1 Gain SchedulingThe basic limitation of the design via linearization approach is the fact that the controller

is guaranteed to work only in some neighborhood of a single operating point. Gain

Scheduling is a technique to increase the region of attraction to a range of possibleoperating points.

In many situations, it is known how dynamics of a system change with its operatingpoints. It might even be possible to model the system in such a way that the operating

points are parameterized by one or more variables, which we call scheduling variables. In

such situations we may linearize the system at several equilibrium points, design afeedback controller at each point, and implement the resulting family of linear controller

as a single controller whose parameters are changed by monitoring the scheduling

variables. Such controller is called gain-scheduled controller.

Idea:

Linearize the system at a family of operating points, parameterized by thescheduling variables

Design a linear controller for each equilibrium point to achieve the specifiedperformance

Design rules to switch from one controller to another Check non local performance via analytical tools and simulation

Remark 1.5: The region of attraction of one equilibrium point contains the neighboringequilibrium point

Consider the system


2/8


3/8

f

Ax

=

,

fB

u

=

,

hC

x

=

, and m

hC

x

=

with all the Jacobian matrices evaluated at ( ) ( ), , , ,SS SS vx u v x u = . The new element hereis allowing the controller gain to depend on .

cA is the linearization of

( )

( )

( )

1 2 3

1 2

, ,

,

,

,f x Lz M M h M e v wx

h x w r

z Fz G G h x w

+ + +

=

+ +

&

&

&

In the state feedback case, we can drop and its state equation and takez my x= , 0L = ,

1 2M K= , 2 1M K= and 3 0M = , where [ ]1 2K K K= is designed such that

1 2

0

A BK BK

C

is Hurwitz for every .( ), ww D D

Remark 1.6: The fact that is stable for every frozencA = does not guaranteestability of the closed loop system when ( )t =

Example 1.5: Consider the second order system

1 1

2 1

2

tan 2x x x

x x u

y x

= +

= +

=

&

&

where is the measured signal; that is,y my y= . We want to track a reference signal

. We use as a scheduling variable. When

y

r r r const = = , the equilibrium equationshave the unique solution

( )1tan

SSx

=

, ( ) 1tanSSu

=


4/8

We use the observer base integral controller

( ) ( )( )

( ) ( )1 2

e y r

x A x Bu H y Cx

u K x K

= =

= + +

=

&

&

where

( )21 1

1 0A

+=

,

0

1B

=

, [ ]0 1C =

( ) ( )( )2 2 21 21

1 3 3 3

1

K

= + + + + +

+

, 2 21

1

K =

+

( )( )( )

( )

2 2

2

10 4 1

4H

+ + + = +

and A , B , C are evaluated at 11 tanSSx = , 2SSx = and

1tanSSu = .

The next two plots illustrate the behavior of a such controller. The first show response of

the closed-loop system to some step sequence when a fixed-gain controller evaluated at

0 = is used.


5/8

For small reference inputs, the response is as good as the one with the gain-scheduled

controller, but as the reference signal increases, the performance deteriorates and the

system goes unstable.

The second show the closed-loop system under an unmodified controller to the same

sequence of steps as before.

While stability and zero steady-state tracking error are achieved, the transient response

deteriorates rapidly as the reference signal increases. Such bad transient behavior could

lead to instability as it could take the state of the system out of the finite region of

attraction, although instability was not observed in this example.

If is available or can be reasonably well estimated, modifications can be made that

increase the system stability. For, then, we can represent the controller

y&

( ) ( ) ( )( ) ( ) ( ) ( )

1 2

1 2 3

m

m

e y r

z F z G G y

u L z M M y M e

= =

= + +

= + + +

&

&

as

( ) ( )( ) ( ) ( )3

z F z G

u L z M M e

=

= +

= + +

&

&

where


6/8

m

e

y

=

&, [ ]1 2G G G= and [ ]1 2M M M=

he controller is equivalent to

measurement is not available, we can use the gain-scheduled controller

here

T

( ) ( )F G

If y&

wm

y& is replaced by its estimate , provided by the filter

herew is a sufficiently sm and the filter is always initialized at

r some . Since

all positive constant

( )0 such that

fo 0k >m

y is measured, we can always meet this initial condition.

e, wFurthermor henever the system is initiated from an equilibrium point, the upper

condition is always satisfied, since, at equilibrium, my = . The filter acts as derivative

approximation when is sufficiently small, as it can be seen from its transfer function

hich approximate the differentia unction for frequencies much smaller w tor transfer f m

sy

than 1 .

( ) ( )( )3

L M

u M e

= +

= +

&

= +&

( ) ( ) ( )

( ) ( ) ( )

( )

1 1

1 2

3

L M e M

u M e

F G e G

= + +

= +

&

= + +&

( )1

m

m

y

y

= +

= +

&

( ) ( )0 0my k

1m

ys

s

= +


7/8

Example 1.6: (Continuation of example 5)

ince is not available, we implement the gain scheduled controller with

y& 0.01 =S . The

hange the reference slowly from one set point to another. The next

next figure shows the response of the closed loop system to a sequence of step changes inthe reference signal. If the initial state is in the region of attraction of the new equilibrium

point, the system reaches steady state at that point. Since our controller is based onlinearization, it guarantees only local stabilization. Therefore, in general, step changes inthe reference signal have to be small. Reaching a large value of the reference signal can

be done by a sequence of step changes, as in the figure, allowing enough time for the

system to settle down after each step change.

Another method is to c

figure shows the response of the closed loop system to a slow ramp that take the set pointfrom zero to one over a period of 100 seconds.


8/8

This response is consistent with our conclusions about the behavior of gain scheduledcontrollers under slowly varying scheduling variables. As the slope ramp increases,

tracking performance decreases. If we keep increasing the slope of the ramp, the system

will eventually become unstable.

So far, our analysis of the closed loop system under gain-scheduled controller has focused

on the local behavior in the neighborhood of a constant operating point. Can we say more

about the behavior of the nonlinear system? In applications of gain scheduling, thepractice has been that you can schedule on time-varying variables as long as they are slow

enough relative to the dynamics of the systems. This practice is justified by next theorem.

heorem 1.1: Consider the closed loop system under the stated assumptions. SupposeT

( )t is continuously differentiable, ( )t S ), and ( )t & (a compact subset of Dfor all 0t . Then, there exist positive 1k , 2k , k, and T such that if 1k < and

( ) ( )( ) 20 0 ,SS w k < , then ( )t will be uniformly bounded for all 0t and

( )e t k , t T

Furthermore, if ( ) SSt and ( ) 0t & as t , then

( ) 0 as t e t

er words, the theorem shows thIn oth at if ( )t is slowly varying and the state not too farfrom the initial equilibrium point, then the tracking error will remain bounded and tend to

( )t&zero when tend to zero.

Documents

11 Gain Scheduling