1

Chapter 11Design of State Variable Feedback Systems

This chapter deals with the design of controllers utilizing state feedback. We will consider three major subjects: Controllability and

observability and then the procedure for determining an optimal control system. Ackermann’s formula can be used to determine the state variable feedback gain matrix to place the system poles at the

desired locations. The closed-loop system pole locations can be arbitrarily placed if and only if the system is controllable.

When the full state is not available for feedback, we utilize anobserver. The observer design process is described and the applicability of Ackermann’s formula is established. The state variable compensator is obtained by connecting the full-state

feedback law to the observer.We consider optimal control system design and then describe the use of internal model design to achieve prescribed steady-state

response to selected input commands.

2

Pole Placement Using State Feedback

• The state-space design method is based on the pole-placement method and the quadratic optimal regulator method. The pole placement method is similar to the root-locus method. In that we place closed-loop poles at desired locations. The basic difference is that in the root-locus design we place only the dominant closed loop poles at the desired locations, while in the pole-placement method we place all closed-loop poles at desired locations.

• The state variable feedback may be used to achieve the desired pole locations of the closed-loop transfer function T(s).

• The approach is based on the feedback of all the state variables, and therefore u = Kx.

• When using this state variable feedback, the roots of the characteristic equation are placed where the transient performance meets the desired response.

3

State Variable Compensator Employing Full-State Feedback in Series with a Full State Observer

uBAxx +=.

yu ˆˆˆ.

LBxAx ++=

C

-K

C

System Model

Observer

xu

+

-

Control Law

y

x̂

Compensator

4

Controllability and Observability• The concept of controllability and observability were

introduced by Kalman in 1960.

• They play an important role in the design of control systems in state space.

• The conditions of controllability and observability may govern the existence of a complete solution to the control system design problem.

• The solution of the problem may not exist if the system is not controllable.

5

Controllability• A system described by the matrices (A, B) can be said to be controllable

if there exists an unconstrained control u that can transfer any initial state x(0) to any other desired location x(t). That means that over time, some or all of the scalar time functions in u can be arbitrarily large in magnitude.

• Another method of determining whether a system is controllable is to draw the state variable flow diagram and determine whether the control signal, u, has a path to each state variable. If a path to each state exists, the system may be controllable.

[ ]lecontrollab is system thenonzero, is P If

P)matrix bility (Controlla B B...AA AB BP

BuAxx

c

1-2c

.

n=

+=

6

Example of a Controllable system!

( )

( )nonzero is P oft Determinan

- 1

- 1 01 0 0

P

- 1

BA ,-10

AB ,100

B

100

x - - -

1 0 00 1 0

x

1)()()(

c

1222

2c

122

22

2

210

.

12

23

−

=

−

=

=

=

+

=

+++==

aaa

a

aa

aa

uaaa

asasassT

sUsY

o

7

Continue..

• Uncontrollable system has a subsystem that is physically disconnected from the input.

• For a partially controllable system, if the uncontrollable modes are stable and the unstable modes are controllable, the system is said to be stabilized. For example such system

• Is not controllable. The stable mode that corresponds to the eigenvalue of -1 is not controllable. The unstable mode that corresponds to the eigenvalue of 1 is controllable. Such a system can be made stable by the use of a suitable feedback. Therefore the system is stabilizable.

uxx

x

x

01

1- 00 1

2

1

2.

1.

+

=

8

Observability• All the roots of the characteristic equation can be placed where desired

in the s-plane if, and only if, a system is observable and controllable.• Observability refers to the ability to estimate a state variable. Thus we

say a system may be observable if the output has a component due to each state variable.

• A system is observable if, and only if, there exists a finite time T such that the initial state x(t) can be determined from the observation history y(t) given the control u(t). Consider the single-input, single-output system

=

=+=

1

.

CA

..CAC

Q

wherenonzero, isQ oft determinan when theobservable is system This

Cxy andBu Axx

n-

9

Example of Observable System!

[ ]

[ ] [ ]

!observable is system theand 1, QDet , 1 0 0

0 1 00 0 1

Q

1 0 0CA and 0 1 0 CA

0 0 1C and - - -1 0 0 0 1 0

A

2

210

=

=

==

=

=

aaa

10

Is this System Controllable and Observable?

[ ]

[ ]

observable is system the

2, is 1 01 1

AC] [C ofrank theexamine condition,ity observabil test theTo

lecontrollab statefully is system the

2, is 1- 11 0

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0 1

10

1- 2-1 1

2

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x.

.

11

Full-State Feedback Control Systemto achieve the desired pole locations of the closed loop system

First we should assume that all the states are available for feedbackThe system input u(t) is given by

u = -KxDetermining the gain matrix K is the objective of the full-state

feedback design procedure

-K

uBAxx +=.

xy

System Model

Control Law

12

The closed-loop system has no input. The objective is to maintain zero output. Because of disturbances the output will deviate from zero. The nonzero output will be returned to zero reference input because of the

state feedback scheme.

regularK

KBAx

BKAλIxBKABKxAxBAxx

KBAxx

BKA

is problemdesign control the allfor 0)(When input. reference theis )( where)()x(-)(

as considered becan input reference a ofaddition Thele.controllab completely is system theif

plane halfleft in the poles loop closed system theallplace to determinecan we), ,(pair Given the

as 0)(ex(t)

stable. is system loop closed then theplane-halfleft in the lie

equation sticcharacteri theof roots theall If0))(det(

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.

.

tt trtrtNrttu

tt

u

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t

⟩=+=

∞→→=

=−−−=−=+=

−=+=

−

13

Design of a Third Order System

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overshoot no has response step The8.170;1.79;4.9 require Then we

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thens, 1 toequal timesettling a want weIf overshoot. minimalfor 0.8 Chose;2 :equation sticcharacteri desired theChose

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)-(-5 )-(-3 )-(-21 0 00 1 0

BK-A

ismatrix feedback state The x;BK-ABKx-Axx

-Kx and K :isK matrix feedback variablestate theIf

BAx 100

x5- 3- 2-1 0 00 1 0

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/;/; :as variablesstate Select the

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===+++=+++

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dtydxdtdyxyx

uydtdy

dtyd

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nnn

s

nnn

ωωξω

ξξωωξω

14

Ackermann’s FormulaFor a single-input, single-output system, Ackermann’s formula is useful

for determining the state variable feedback matrix

IAAA

AP

KxK

nnnn

c

nn-n

n

Aq

q

q

u ....kkk

ααα

αλαλλ

+++=

=

+++=

==

−−

11

1

1-

11

2 1

....)(

)(1] ... 0 [0K

ismatrix gain feedback state The....)(

equation sticcharacteri desired Given the-

] [

15

Observer DesignIf the system is completely observable with a given set of outputs, then

it is possible to determine or estimate the states that are not directly measured

x̂ xCˆ~ −= yy

C

+

-

yu

yu ~ˆˆ.

LBxAx ++=

ObserverEstimate of the matrix

L is the observer gain matrix and to be determined

16

plane. halfleft in the roots all its has 0))(-Det( as as 0)(

)()()(

)ˆ(ˆ

)ˆ(ˆ

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equation previous theoferror estimation theof derivative time theTake.

as 0)( produce shoulddesign observer The ).(ˆ-)()(iserror estimationobserver The observer. the to)(ˆ estimate

intialan provide should weprecisely; )( knownot do We. as ˆ

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.

.

.

...

0

0

=∞→→−=

−−−−+=

−++=

=

∞→→=

∞→→

LC-AIeeLCAe

xCLBxABAxe

xCLBAxx

xxe

exxex

xxx

x

λtttt

yuu

yu

ttttt

tt

t

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E11.3: A system is described by the matrix equation. Determine whether the system is controllable and observable.

[ ]

le!unobservab is system the thereforezero, toequal is QDet Since

6- 02 0

CAC

Q

ismatrix ity observabil Thele.controllab is system thezero, toequalnot is P Since

3- 11 0

AB B P

2 ; 10

x 3- 01 0

x

c

c

2

.

=

=

==

=

+

= xyu

18

E11.4: A system is described by the matrix equation. Determine whether the system is controllable and observable.

[ ]

le.unobservab is system the0, QDet Since0 4-0 1

CAC

Q

ismatrix ity observabil Theable.uncontroll is system the0, PDet Since

1- 10 0

ABA P

matrix ility controllab thefindFirst

and ; 10

x 1- 00 4-

x

c

c

1

=

=

=

=

==

=

+

= xyu