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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
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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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.
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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.
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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.
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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
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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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.
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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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3.2 Analysis of controllability
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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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3.2 Analysis of controllability
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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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3.2 Analysis of controllability
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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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3.2 Analysis of controllability
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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
x(t) =P1APx(t) + P1Bu(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, 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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3.2 Analysis of controllability
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Example
Consider the system described by
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;
2 b51andb52are not all zero;
3 b61andb62are not all zero;
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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
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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3.3 Analysis of observability
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Example
Consider the system described by
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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3.3 Analysis of observability
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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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3.3 Analysis of observability
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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)
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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3.3 Analysis of observability
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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
x(t) =P1APx(t) + P1Bu(t)
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
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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
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
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
+
+
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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
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
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
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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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3.5 Obtaining controllable and observable canonical forms
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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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3.5 Obtaining controllable and observable canonical forms
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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
3.5 Obtaining controllable and observable canonical forms
(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
.
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3.5 Obtaining controllable and observable canonical forms
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
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Outline
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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
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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