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Chapter 3. Controllability and Observability

Modern Control Theory (Course Code: 10213403)

Professor Jun WANG

( )

Department of Control Science & Engineering

School of Electronic & Information Engineering

Tongji University

Spring semester, 2012


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Outline

1 3.1 Introduction

2 3.2 Analysis of controllability

3 3.3 Analysis of observability

4 3.4 Principle of duality

5 3.5 Obtaining controllable and observable canonical forms

6 3.6 Canonical decomposition

7 3.7 Simulations with MATLAB


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Outline

1 3.1 Introduction








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3.1 Introduction

Who introduced the concepts?

Rudolf E. Kalman(Hungary, 1821- )

NationalityHungarian-born American

FieldsElec Engn; Mathematics; Applied

Engn Systems TheoryInstitutionsStanford Univ; Univ of Florida;

Swiss Federal Inst of Tech

Alma materMIT; Columbia Univ

Notable awardsIEEE Medal of Honor;

National Medal of Science; Charles Stark

Draper Prize ; Kyoto Prize

Obama awards National Medals of

Science, Techonology and Innovation

U.S. President Barack Obama (R) presents a 2008 National Medal

of Science to Rudolf Kalman (L) of Swiss Federal Institute of

Technology in Zurich during an East Room ceeremony October

7, 2009 at the White House in Washington, DC.

Prof J Wang (Tongji Uni) Chap 3. Controllability and observability Spring 2012 4/52

d


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3.1 Introduction

What are controllability and observability?

Controllability determines whether the state of a state equationcan be controlled by the input.

Observability determines whether the initial state can be observed

from the output.

The conditions of controllability and observability govern the

existence of a solution to the control system design problem.


3 1 I t d ti


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3.1 Introduction

How to understand the concepts?

Lets first look at some examples

Example (Simple circuits)

V(t)

R

R

R y

R1F

x

V(t)

C x1

R

R

C x2I(t)

R1R2

L x1

C x2



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Example (A numerical example)

Suppose a system can be described by the following state-space model

x1x2

= 4 00 5

x1x2

+ 12

u

y= 0 6

x1

x2

=

x1 =4x1+u

x2 = 5x2+2u

y= 6x2

What can we observe from these equations?

Both the statex1andx2can be moved from the their initial values

to zero by a proper control inputu;

The outputyis associated with the statex2, and is completely

isolated from the statex1.

Therefore, the system is completely controllable, but NOT completely

observable.

3 1 Introduction


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3.1 Introduction

Definitions of controllability and observability?

Definition (Controllability)A system is completely controllable if there exists anunconstrained

controlu(t)that can transferanyinitial state x(t0)toanyother desired

locationx(t)in afinitetime,t0 t T.

Definition (Observability)

A system is completely observable if, given the control u(t),everystate

x(t0)can be determined from the observation ofy(t)over afinitetime

interval,t0 t T.



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Outline

1 3.1 Introduction








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How to judge the controllability of an LTI system?

Consider the continuous-time single-input system

x=Ax + bu

where the state x(t) Rn1, control inputu(t) R, and the matrices

A Rnn andb Rn1.

The solution of the equation is

x(t) =eAtx(0) +

t0

eA(t)bu() d

Applying the definition of complete state controllability, we have

x(t1) =0 =eAt1 x(0) +

t10

eA(t1)bu() d

or

x(0) = t1

0

eAbu() d

A


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As the matrix exponential functioneA can be written as

eA =n1

k=0

k()Ak

we have

x(0) = t1

0

n1

k=0

k()Akbu() d

=

n1k=0

t10k()A

kbu() d

=

n1k=0

Akb

t10k()u() d

=

n1

k=0

Akbk

, where k=

t10k()u() d

i iti t t th lt b th f ll i t


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we are now in a position to get the result by the following steps

x(0) = n1

k=0 Akbk

= b1 Ab2 An1bn1

=

b Ab An

1b

1

2

...

n1

If the system is completely state controllable, the above equation must

be satisfied foranyinitial state x(0), which requires that

rank

b Ab An1b

=n

It can be extended to a general case whereuis a vector.

3.2 Analysis of controllability


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y y

Algebraic controllability criterion

TheoremThe system(A, B)is completely controllable if and only if the rank of the

controllability matrix

Qc =

B AB A2

B An1

B

is n, i.e.rank(Qc) =n




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Example

Consider the system described by

x1x2x3

=

2 2 0

0 0 10 3 4

x1x2x3

+

1 00 11 1

u1u2

Solutions

For this case,

Qc =

B AB A2B

=

1 0 2 2 2 2

0 1 1 1 4 7

1 1 4 7 13 25

Since rank(Qc) =3, the system is completely state controllable.




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Example

Prove that an SISO system must be completely controllable if it can be

described by a state-space model of the controllable canonical form.

Solutions

Since the model is in controllable canonical form,

A=

0 1...

. . .

0 1

a0 a1 an1

, b=

0...

0

1



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PBH controllability criterion

Theorem

The system(A, B)is completely state controllable if and only if all the

eigenvalues iof the state matrixAsatisfy

rank [iI A, B]=n, i=1, 2, , n




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Corollary (Diagonal canonical criterion)

If the eigenvalues ofAare distinct and the corresponding diagonal canonical

form after similarity transformation is

x(t) =P1APx(t) + P1Bu(t)

where

P1AP= A=

1

2. . .

n

The original system(A, B)is completely state controllable if and only if there

is no zero row in B=P1B.




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Corollary (Jordan canonical criterion)

Suppose the system matrixAhas repeated eigenvalues, and the corresponding

Jordan canonical form after similarity transformation is


where

P1AP= A=

J1J2

. . .Jl

, Ji =

i 1i 1

. . . . . .i 1

i

and each distinct eigenvalue is associated with only one Jordan block.Then the original system(A, B)is completely state controllable if and only if

the row ofB= P1Bcorresponding to the last row of each Jordan block is not

zero row.




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Example


x1x2x3x4x5x6

=

1 1

12 1

2 12

3

x1x2x3x4x5x6

+

b21 b22

b51 b52b61 b62

u1u2

where denotes any real constant value.

Solutions

The system is completely state controllable if and only if

1 b21andb22are not all zero;





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Outline

1 3.1 Introduction







3.3 Analysis of observability


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Algebraic observability criterion

TheoremThe system(A, C)is completely observable if and only if the observability

matrix

Qo =

C

CACA2

...

CAn1

or its transpose has rank n, i.e. rank(Qo) =rank(QTo) =n




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Example


x1x2

=

1 12 1

x1x2

+

01

u

y=[1 0] x1

x2

Is the system controllable and observable?

Solutions

Calculating the controllability and observability matrices yields

Qc =[ b Ab]= 0 1

1 1

Qo =

ccA

=

1 01 1

Since rank(Qc) =rank(Qo) =2, the system is completely state

controllable and observable.




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PBH observability criterion

TheoremThe system(A, C)is completely state observable if and only if all the

eigenvalues iof the state matrixAsatisfy

rankiI A

C

=n, i=1, 2, , n




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Corollary (Diagonal canonical criterion)

If the eigenvalues ofAare distinct and the corresponding diagonal canonical

form after similarity transformation is


y(t) =CPx(t) + Du(t)

where

P1AP= A=

1

2.

. .n

The original system(A, C)is completely state observable if and only if there is

no zero column in C=CP.




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Corollary (Jordan canonical criterion)

If the system matrixA has repeated eigenvalues, and the corresponding

Jordan canonical form after similarity transformation is


y(t) =CPx(t) + Du(t)

where

P1AP= A=

J1J2

. . .Jl

, Ji =

i 1i 1

. . . . . .i 1

i

and each distinct eigenvalue is associated with only one Jordan block, then the

original system(A, C)is completely state observable if and only if the column

ofC=CPcorresponding to thefirst columnof each Jordan block is not zero

column.Prof J Wang (Tongji Uni) Chap 3. Controllability and observability Spring 2012 26/52

E l


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Example

Are the following systems(A, B, C)state controllable and observable?

(1) 1 :A = 1 00 4 ,B = 01 ,C =[1 2]

(2) 2 :A =

4 1 0 0 0

0 4 0 0 00 0 2 1 00 0 0 2 10 0 0 0 2

,B =

010

21

,C = 1 0 1 0 10 0 1 1 0

(3) 3 :A = 2 1 0

0 2 11 0 2

,B = 0

11

,C =[1 0 1]

Solutions

(1)1is uncontrollable, but observable;

(2)2is controllable and observable;(3) How about 3? Can we get the answer directly?

rank [ B AB A2B]=rank

0 1 51 3 71 1 2

=3, rank

CCA

CA2

=rank

1 0 13 1 17 5 2

=3

3is controllable and observable.

Outline


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Outline

1 3.1 Introduction







3.4 Principle of duality


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Principle of duality

Definition (Dual systems)

1 :

x(t) =Ax(t) + Bu(t)y(t) =Cx(t)

x Rn,u Rl,y Rm.

2 :

(t) =A

T(t) + CT(t)

(t) =BT(t)

Rn, Rm, Rl.

The LTI systems 1(A, B, C)and 2(AT, CT, BT)are dual of each other

and the dimensions of the input and the output are exchanged

between dual systems.

u B

C y

A

x+

+

CT

BT

AT

+

+


Theorem (Principle of duality)


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Theorem (Principle of duality)

The system1(A, B, C)is completely controllable (observable) if and only if

its dual system2(AT, CT, BT)are completely observable (controllable).

For system 1(A, B, C)

Controllability criterion:

rank

B AB

A

n1

B

=n

Observability criterion:

rank

C

CA...

CAn1

=n

For system 2(AT, CT, BT)

Controllability criterion:

rank

CT

AT

CT

A

Tn1C

T=n

Observability criterion:

rank

BT

BTAT

...

BTAT

n1

=n

Outline


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Outline

1 3.1 Introduction







3.5 Obtaining controllable and observable canonical forms


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Controllable canonical form for SISO systems

Given a completely controllable SISO system(A, b, c), find a similaritytransformation matrixPc, which can transform it into the controllable

canonical form(Ac, bc, cc), i.e.

Ac=P1

c AP

c, b

c=P1

c b, c

c=cP

c

Ac =

0 1..

.

. .

.0 1

a0 a1 an1

, bc =

0..

.0

1


From the definition, we have


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Ac =P1c APc P

1c A=AcP

1c

Let pcidenotes theith row vector ofP1c . Hence

pc1

pc2...

pcn

A=

0 1...

. . .

0 1

a0 a1 an1

pc1

pc2...

pcn

Expanding the above equation and comparing the two sizes, we have

pc1A=pc2

pc2A=pc3...

pc(n1)A=pcn

pcnA= a0pc1a1pc2 an1pcn

Ifpc1is known,

then we can work out

pc2,pc3, ,pcn

successively.

Next, we will use equation bc =P1c bto obtainpc1.


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P

1

c b=bc

pc1

pc2

...

pcn

b=

0...

0

1

Then we have

pc1b=0

pc2b= pc1Ab=0

...

pc(n1)b=pc1An2b=0

pcnb=pc1An1b=1

pc1 b Ab A2b An1b= 0 0 0 1



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Therefore, we have

pc1 = 0 0 0 1 b Ab A2b An1b

1

=

0 0 0 1

Q1c

pc1A=

pc2pc2A=pc3

...

pc(n1)A=pcn

= P1c =

pc1

pc1A

pc1A2

...

pc1An1




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Similarity transformation matrix for controllable canonical form

Given a completely controllable SISO system:(A, b, c), the following

similarity transformation matrixPccan transform into thecontrollable canonical form c :(Ac, bc, cc).

P1c =

pc1

pc1Apc1A

2

...

pc1An1

where

pc1 =

0 0 0 1

Q1c




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Example

Consider a system described by

x1x2x3

=

1 0 0

1 2 00 5 0

x1x2x3

+

100

u

Find the similar transformation matrixPcwhich can transform the

system into the controllable canonical form.

Solutions

(1) Check the controllability of the system

Qc =[ b Ab A2b]= 1 1 1

0 1 30 0 5

rank(Qc) =3

The system is controllable and can be transformed into the

controllable canonical form.Prof J Wang (Tongji Uni) Chap 3. Controllability and observability Spring 2012 37/52


(2) C


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(2) Compute pc1.

Q1c = 15

5 5 20 5 30 0 1

=

1 1 250 1 35

0 0 1

5

pc1 =[0 0 1] Q

1c =[0 0

15]

(3) Compute P1c andPc.

P1c =

pc1pc1A

pc1A2

=

0 0 150 1 01 2 0

, Pc =

0 2 10 1 05 0 0

(4) Compute the controllable canonical form.

Ac =P1c APc=

0 1 00 0 10 2 3

, bc =P

1c b=

001

.



Ob bl i l f f SISO t


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Observable canonical form for SISO systems

Similarity transformation matrix for observable canonical form

Given a completely observable SISO system :(A, b, c), the followingsimilarity transformation matrixPocan transform into the

observable canonical form o :(Ao, bo, co).

Po =

po1 Apo1 A2

po1 An1

po1

where

po1 =Q1o

0...

0

1

=

c

cA...

cn1A

1

0...

0

1


Outline


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1 3.1 Introduction







3.6 Canonical decomposition

Decomposition according to controllability


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Decomposition according to controllability

Suppose annth-order SISO system:(A

,b

,c)

is not completely statecontrollable, say

rank Qc =rank

b Ab A2b An1b

=n1


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By using the similar matrixPdefined above, the system can be

transformed into:(A,b, c).

xcxnc

=Ac A12

0 Anc

xcxnc

+bc

0

u

y= cc cnc

xc

xnc

where Ac Rn1 n1 , bc R

n11, cc R1n1 .

Note that, then1dimensional subsystemc

xc = Acxc+ bcu

y=ccxc

is completely state controllable.Prof J Wang (Tongji Uni) Chap 3. Controllability and observability Spring 2012 42/52

Features of controllability decomposition


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y p

System:

xcxnc

=

Ac A12

0 Anc

xcxnc

+

bc0

u

y=[cc cnc]

xcxnc

Systemc:

xc = Acxc+ bcu

y=ccxc

u bc

cc

Ac

A12 cnc

Anc

y

+

+y1+

+y2

+

+xc

xnc

G(s) =G(s) =Gc (s)

=cc(sI Ac)1bc

3.6 Canonical decomposition

Example


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Example

Consider the following state-space model

x=

1 1 00 1 00 1 1

x +

010

u

y=[1 1 1] x

Determine whether the system is controllable. If not, make adecomposition according to controllability.

Solutions

Since rank(Qc

) =2> A = [0 1 0;0 0 1;-6 -11 -6];

>> B = [0;0;1];

> > C = [ 5 6 1 ] ;

>> D = [0];

>> CONT = ctrb(A,B)

CONT =

0 0 1

0 1 - 6

1 - 6 2 5

>> rank(CONT)

ans =

3

>> OBSER = obsv(A,C)

OBSER =

5 6 1

- 6 - 6 0

0 - 6 - 6

>> rank(OBSER)

ans =

2

Chapter 3 Controllability and Observability


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Chapter 3. Controllability and Observability

Modern Control Theory (Course Code: 10213403)

Professor Jun WANG

(

)

Department of Control Science & Engineering

School of Electronic & Information Engineering

Tongji University

Spring semester, 2012

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