OBSERVABILITY AND CONTROLLABILITYmocha-java.uccs.edu/ECE5520/ECE5520-CH05.pdfEXAMPLE: Controllability canonical form: x.t/P D 2 6 4 00"a 3 10"a 2 01"a 1 3 7 5x.t/C 2 6 4 1 0 0 3 7

ECE4520/5520: Multivariable Control Systems I. 5–1

OBSERVABILITY AND CONTROLLABILITY

5.1: Continuous-time observability: Where am I?

■ We describe dual ideas called observability and controllability.

■ Both have precise (binary) mathematical descriptions, but we need to

be careful in interpreting the result.

■ We develop some other techniques to help quantify the concepts.

Continuous-time observability

■ If a system is observable, we can determine the initial condition of the

state vector x.0/ via processing the input to the system u.t/ and the

output of the system y.t/.

■ Since we can simulate the system if we know x.0/ and u.t/ this also

implies that we can determine x.t/ for t ! 0. e.g., for LTI,

x.t/ D eAtx.0/C

Z t

0

eA.t"!/Bu.!/ d!:

■ Consider the LTI SISO system LCCODE

«y.t/C a1 Ry.t/C a2 Py.t/C a3y.t/ D b0«u.t/C b1 Ru.t/C b2 Pu.t/C b3u.t/:

■ If we have a realization of this system in state-space form

Px.t/ D Ax.t/C Bu.t/

y.t/ D Cx.t/CDu.t/;

Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett

ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–2

and we have initial conditions y.0/, Py.0/, Ry.0/, how do we find x.0/?

y.0/ D Cx.0/CDu.0/

Py.0/ D C.Ax.0/C Bu.0/„ ƒ‚ …

Px.0/

/CD Pu.0/

D CAx.0/C CBu.0/CD Pu.0/

Ry.0/ D CA2x.0/C CABu.0/C CB Pu.0/CD Ru.0/:

■ In general,

y.k/.0/ D CAkx.0/C CAk"1Bu.0/C $ $ $C CBu.k"1/.0/CDu.k/.0/;

or,2

64

y.0/

Py.0/

Ry.0/

3

75 D

2

64

C

CA

CA2

3

75

„ ƒ‚ …

O.C;A/

x.0/C

2

64

D 0 0

CB D 0

CAB CB D

3

75

„ ƒ‚ …

T

2

64

u.0/

Pu.0/

Ru.0/

3

75 ;

where T is a (block) “Toeplitz matrix”.

■ Thus, if O.C; A/ is invertible, then

x.0/ D O"1

8

ˆ<

ˆ:

2

64

y.0/

Py.0/

Ry.0/

3

75 " T

2

64

u.0/

Pu.0/

Ru.0/

3

75

9

>=

>;

:

■ We say that fC; Ag is an observable pair if O is nonsingular.

CONCLUSION: For a SISO system, if O is nonsingular, then we can

determine/estimate the initial state of the system x.0/ using only u.t/

and y.t/ (and therefore, we can estimate x.t/ for all t ! 0).

EXTENSION: For a MIMO system, if O is full rank, then we can

determine/estimate the initial state of the system x.0/ using only u.t/

and y.t/ (and therefore, we can estimate x.t/ for all t ! 0).



EXAMPLE: Observability canonical form:

Px.t/ D

2

64

0 1 0

0 0 1

"a3 "a2 "a1

3

75 x.t/C

2

64

ˇ1

ˇ2

ˇ3

3

75u.t/

y.t/ Dh

1 0 0i

x.t/:

■ Then

O D

2

64

C

CA

CA2

3

75 D

2

64

1 0 0

0 : : : 0

0 0 1

3

75 D In:

■ This is why it is called observability form!

EXAMPLE: Two unobservable networks

(Redrawn)

1"

1"1"1"

1"

1"1F1F

1H1H

2"2"

x

x1x1

x2x2y yyuu

■ In the first, if u.t/ D 0 then y.t/ D 0 8 t . Cannot determine x.0/.

■ In the second, if u.t/ D 0, x1.0/ ¤ 0 and x2.0/ D 0, then y.t/ D 0 and

we cannot determine x1.0/. [circuit redrawn for u.t/ D 0].

Observers

■ An observer is a device that has as inputs u.t/ and y.t/—the input

and output of a linear system. The output of the observer is the

(estimated) state of the linear system.



■ The observer “observes”

the internal state x

(estimated as Ox) from

external signals u and y.

u.t/ y.t/A; B; C; D

Observer

x

Ox

■ Note that our equations yield an observer:

u.t/ y.t/A; B; C; D

u

Pu

Ru

y

Py

Ry

11

ss

s2s2

"T

O"1

x

Ox

■ Later, we’ll design more practical observers that don’t use

differentiators.



5.2: Continuous-time controllability: Can I get there from here?

■ Can we generate an input u.t/ to set an initial condition quickly?


y.t/ D Cx.t/CDu.t/:

■ If u.t/ D ı.t/ and x.0"/ D 0, then

X.s/ D .sI " A/"1BU.s/ D .sI " A/"1B:

■ So, via the Laplace initial-value theorem,

x.0C/ D lims!1

sX.s/

D lims!1

s.sI " A/"1B

D lims!1

!

I "A

s

""1

B

D B:

■ Thus, an impulse input brings the state to B from 0.

■ What if u.t/ D ı.k/.t/?

■ Then

X.s/ D .sI " A/"1Bsk D1

s

!

I "A

s

""1

Bsk

D1

s

!

I CA

sC

A2

s2C : : :

"

„ ƒ‚ …

holds for large s

Bsk

D Bsk"1 C ABsk"2 C A2Bsk"3 C $ $ $CAkB

sC

AkC1B

s2C $ $ $

■ The first terms are impulsive: they have zero value for t > 0.



■ Thus,

x.0C/ D lims!1

s

!

AkB

sC

AkC1B

s2C $ $ $

"

D AkB:

So, if u.t/ D ı.k/.t/ then x.0C/ D AkB.

■ Now, consider the input

u.t/ D g1ı.t/C g2Pı.t/C $ $ $gnı.n/.t/:

Since x.0"/ D 0, x.0C/ D g1B C g2AB C $ $ $C gnAn"1B, or

x.0C/ D#

B AB $ $ $ An"1B$

„ ƒ‚ …

C

2

64

g1

g2

g3

3

75

where C is called the “controllability matrix.”

CONCLUSION: For a SISO system, if C is nonsingular, then there is an

impulsive input u such that x.0C/ is any desired vector if x.0"/ D 0.

EXTENSION: For a MIMO system, if C is full rank, then there is an

impulsive input u such that x.0C/ is any desired vector if x.0"/ D 0.

■ In fact, we may use

u.t/ Dn"1X

iD0

giı.i/.t/

where 2

64

g1

g2

g3

3

75 D

#

B AB $ $ $ An"1B$"1

xd

where xd is the desired x.0C/ vector.



■ If C is nonsingular, we say fA; Bg is a controllable pair and the system

is controllable.

EXAMPLE: Controllability canonical form:

Px.t/ D

2

64

0 0 "a3

1 0 "a2

0 1 "a1

3

75x.t/C

2

64

1

0

0

3

75u.t/

y.t/ Dh

ˇ1 ˇ2 ˇ3

i

x.t/:

■ Then

C D ŒB AB $ $ $ An"1B#

D

2

64

1 0 0

0 : : : 0

0 0 1

3

75 D In:

■ This is why it is called controllability form!

■ If a system is controllable, we can instantaneously move the state

from any known state to any other state, using impulse-like inputs.

■ Later, we’ll see that smooth inputs can effect the state transfer (not

instantaneously, though!).

DUALITY: fA; B; C; Dg controllable”fAT ; C T ; BT ; DT g observable.

EXAMPLE: Two uncontrollable networks.

1" 1"

1"

1"

1"

1"

1F 1Fx

x1 x2y

u u



■ In the first one, if x.0/ D 0 then x.t/ D 0 8 t . Cannot influence state!

■ In the second one, if x1.0/ D x2.0/ then x1.t/ D x2.t/ 8 t . Cannot

independently alter state.

Diagonal systems, controllability and observability

■ Recall the diagonal form

Px.t/ D

2

66664

$1 0

$2

: : :

0 $n

3

77775

x.t/C

2

66664

%1

%2:::

%n

3

77775

u.t/

y.t/ Dh

ı1 ı2 $ $ $ ın

i

x.t/Ch

0i

u.t/:

u.t/ y.t/

x1.t/

xn.t/

%1

%n

ı1

ın

$1

$n

R

R

:::

■ When controllable? When observable?

O D

2

66664

C

CA:::

CAn"1

3

77775

D

2

66664

ı1 ı2 $ $ $ ın

$1ı1 $2ı2 $ $ $ $nın

: : :

$n"11 ı1 $n"1

2 ı2 $ $ $ $n"1n ın

3

77775



D

2

66664

1 1 $ $ $ 1

$1 $2 $ $ $ $n

: : :

$n"11 $n"1

2 $ $ $ $n"1n

3

77775

„ ƒ‚ …

Vandermonde matrixV

2

66664

ı1 0

ı2

: : :

0 ın

3

77775

:

■ Singular?

detfOg D .ı1 $ $ $ ın/ detfVg D .ı1 $ $ $ ın/Y

i<j

.$j " $i /:

CONCLUSION: Observable” $i ¤ $j , i ¤ j and ıi ¤ 0 i D 1; $ $ $ ; n.

u.t/ u.t/ y.t/y.t/

x1.t/x1.t/

x2.t/x2.t/

1

s C 1

1

s C 1

1

s C 1

1

s C 2

■ If $1 D $2 then not observable. Can only “observe” the sum x1 C x2.

■ If ık D 0 then cannot observe mode k.

■ What about controllability? Use duality and switch ıs and %s.

CONCLUSION: Controllable” $i ¤ $j , i ¤ j and %i ¤ 0 i D 1; $ $ $ ; n.

u.t/u.t/ y.t/ y.t/

x1.t/x1.t/

x2.t/ x2.t/

1

s C 1

1

s C 1

1

s C 1

1

s C 2

■ If $1 D $2 then not controllable. Can only “control” the sum x1 C x2.

■ If %k D 0 then cannot control mode k.



5.3: Discrete-time controllability and observability

Discrete-time controllability

■ Similar concept for discrete-time.

■ Consider the problem of driving a system to some arbitrary state xŒn#

xŒk C 1# D AxŒk#C BuŒk#

xŒ1# D AxŒ0#C BuŒ0#

xŒ2# D A ŒAxŒ0#C BuŒ0##C BuŒ1#

xŒ3# D A#

A2xŒ0#C ABuŒ0#C BuŒ1#$

C BuŒ2#

:::

xŒn# D AnxŒ0#Ch

B AB A2B $ $ $ An"1Bi

„ ƒ‚ …

C

2

64

uŒn " 1#:::

uŒ0#

3

75 :

■ Which leads to2

64

uŒn " 1#:::

uŒ0#

3

75 D C"1 ŒxŒn# " AnxŒ0## :

If C has no inverse (det.C/ D 0; C is not full-rank) then these control

signals don’t exist. In that case, the input is only partially effective in

influencing the state.

■ If C is full-rank, then the input can move the system to any arbitrary

state for any xŒ0#.

NOTE I: State transition is not instantaneous. Takes n time steps.



NOTE II: In continuous-time, we used input u.t/ D g0ı.t/C g1Pı.t/C $ $ $ , a

signal we could only approximate in practice. Here, the input is a

perfectly good input signal.

Discrete-time reachability

■ In the literature, there are three different controllability definitions:

1. Transfer any state to any other state.

2. Transfer any state to zero, called controllability to the origin.

3. Transfer the zero state to any state, called controllability from the

origin, or reachability.

■ In continuous time, because eAt is nonsingular, the three definitions

are equivalent.

■ In discrete time, if A is nonsingular, the three definitions are also

equivalent.

■ However, if A is singular, (1) and (3) are equivalent but not (2) and (3).

EXAMPLE:

xŒk C 1# D

2

64

0 1 0

0 0 1

0 0 0

3

75xŒk#C

2

64

0

0

0

3

75uŒk#:

■ Its controllability matrix has rank 0 and the equation is not controllable

in (1) or (3).

■ However, Ak D 0 for k ! 3 so xŒ3# D A3xŒ0# D 0 for any initial state

xŒ0# and any input uŒk#.

■ Thus, the system is controllable to the origin but not controllable from

the origin or reachable.



■ Definition (1) encompasses the other two definitions, so is used as

our definition of controllable.

Discrete-time observability

■ Can we reconstruct the state xŒ0# from the output yŒk# and input uŒk#?

yŒk# D CxŒk#CDuŒk#

yŒ0# D CxŒ0#CDuŒ0#

yŒ1# D C ŒAxŒ0#C BuŒ0##CDuŒ1#

yŒ2# D C#

A2xŒ0#CABuŒ0#C BuŒ1#$

CDuŒ2#

:::

yŒn" 1# D C#

An"1xŒ0#CAn"2BuŒ0#C $ $ $C BuŒn " 1#$

CDuŒn" 1#:

■ In vector form, we can write2

66664

yŒ0#

yŒ1#:::

yŒn" 1#

3

77775

D

2

66664

C

CA:::

CAn"1

3

77775

„ ƒ‚ …

O

xŒ0#C

2

66664

D 0 $ $ $ 0

CB D $ $ $ 0

CAB CB $ $ $ 0::: ::: : : : D

3

77775

„ ƒ‚ …

T

2

66664

uŒ0#

uŒ1#:::

uŒn " 1#

3

77775

:

■ So,

xŒ0# D O"1

2

64

2

64

yŒ0#:::

yŒn " 1#

3

75 " T

2

64

uŒ0#:::

uŒn " 1#

3

75

3

75 :

■ If O is full-rank or nonsingular, xŒ0# may be reconstructed with any

yŒk#; uŒk#. We say that fC; Ag form an “observable pair.”



■ Do more measurements of yŒn#; yŒnC 1#; : : : help in reconstructing

xŒ0#? No! (Caley–Hamilton theorem: next section).

■ So, if the original state is not “observable” with n measurements, then

it will not be observable with more than n measurements either.

■ Since we know uŒk# and the dynamics of the system, if the system is

observable we can determine the entire state sequence xŒk#, k ! 0

once we determine xŒ0#

xŒn# D AnxŒ0#Cn"1X

iD0

An"1"iBuŒk#

D AnO"1

2

64

2

64

yŒ0#:::

yŒn" 1#

3

75 " T

2

64

uŒ0#:::

uŒn" 1#

3

75

3

75C C

2

64

uŒn" 1#:::

uŒ0#

3

75 :

■ A perfectly good observer (no differentiators...)



5.4: Cayley–Hamilton Theorem

■ A square matrix A satisfies its own characteristic equation. That is, if

&.$/ D det.$I " A/ D 0

then

&.A/ D 0:

■ We can easily show this if A is diagonalizable. Let

A D V "1ƒV:

■ Then

A2 D V "1ƒV V "1ƒV

D V "1ƒ2V

Ak D V "1ƒkV:

■ The characteristic polynomial is:

&.$/ D $n C an"1$n"1 C $ $ $C a1

so if we replace $ with A we get

&.A/ D An C an"1An"1 C $ $ $C a1I

D V "1#

ƒn C an"1ƒn"1C $ $ $C a1I

$

V:

■ To “prove” the Cayley–Hamilton theorem, we just need to show that

the quantity inside the brackets is zero.

■ It is a diagonal matrix, and each diagonal element has the form

$ni C an"1$

n"1i C $ $ $C a1 D 0

because $i is an eigenvalue of A.



■ So, each diagonal element is zero, and we have shown the proof.

■ If A is not diagonalizable, the same proof may be repeated using the

Jordan form and Jordan blocks:

A D T "1JT:

■ Consider a sketch of the proof for a Jordan block of size 2 and

&.$i / D $3i C a2$

2i C a1$i C a0 D 0:

■ Then

Ji D

"

$i 1

0 $i

#

&.Ji/ D

"

$3i 3$2

i

0 $3i

#

C a2

"

$2i 2$i

0 $2i

#

C a1

"

$i 1

0 $i

#

C a0

"

1 0

0 1

#

:

■ We can easily see that the diagonal and lower-diagonal components

are zero so

&.Ji/ D

"

0 ˛

0 0

#

where

˛ D 3$2i C 2a2$i C a1

but ˛ Dd

d$&.$/ D 0 which completes the sketch.

SIGNIFICANCE: The Cayley–Hamilton theorem shows us that An is a

function of matrix powers An"1 down to A0. Therefore, to compute any

polynomial of A it suffices to compute only powers of A up to An"1

and appropriately weight their sum. A lot of proofs use the

Cayley–Hamilton theorem.



■ As we just saw with the section on discrete-time observability, the

Cayley–Hamilton theorem implies that if we cannot observe the state

with n measurements, we cannot observe it with more measurements

either.

EXAMPLE: With A D

"

1 2

3 4

#

we have &.$/ D det.$I " A/, so

&.$/ D det

"

$ 0

0 $

#

" A

!

&.$/ D $2 " 5$ " 2

&.A/ D A2 " 5A " 2I

D

"

7 10

15 22

#

" 5

"

1 2

3 4

#

" 2

"

1 0

0 1

#

D

"

0 0

0 0

#

:

Disclaimer: Does observability/controllability matter, practically?

■ The singularity of C has only one “bit” of information: Is the realization

mathematically controllable or not? This may not tell the whole story.

A D

"

1 0

0 1C '

#

; B D

"

1

1

#

; C D

"

1 1

1 1C '

#

■ fA; Bg are a controllable pair, but barely.

EXAMPLE: Controlling an airplane. (Ideas only, no details). System state

x4Dh

( P( ) P)iT

; ( D Pitch; ) D Roll:

■ Control with elevator?



Px D

"

F( '

0 F)

#

x C

2

66664

0

1

0

0

3

77775

ıe

where ıe is the elevator angle. C is singular ➠ can’t influence roll with

elevators.

■ Control with ailerons?

Px D

"

F( '

0 F)

#

x C

2

66664

0

0

0

1

3

77775

ıa

where ıa is the aileron angle. C is nonsingular! So, we can control

both pitch AND roll with ailerons.

■ THIS IS NONSENSE ! Physically think of the system! Do you want to

roll plane over every time you need to pitch down?

■ Physical intuition can be better than finding C. Other tools can help. . .



5.5: Continuous-time Gramians

Continuous-time controllability Gramian

■ If a continuous-time system is controllable, then

Wc.t/ D

Z t

0

eA!BBT eAT ! d!

is nonsingular for t > 0.

SIGNIFICANCE: Consider

x.t1/ D eAt1x.0/C

Z t1

0

eA.t1"!/Bu.!/ d!:

■ We claim that for any x.0/ D x0 and any x.t1/ D x1 the input

u.t/ D "BT eAT .t1"t /W "1c .t1/

#

eAt1x0 " x1

$

will transfer x0 to x1 at time t1.

PROOF: Substitute the expression for u.t/ into the convolution

expression:

x.t1/ D eAt1x.0/ "

Z t1

0

eA.t1"!/BBT eAT .t1"!/ d! W "1c .t1/

#

eAt1x0 " x1

$

D eAt1x.0/ "

Z t1

0

eAˇBBT eAT ˇ dˇ W "1c .t1/

#

eAt1x0 " x1

$

D eAt1x.0/ "Wc.t1/W"1

c .t1/#

eAt1x0 " x1

$

D eAt1x.0/ " eAt1x0 C x1 D x1:

■ Therefore, we can compute the input u.t/ required to transfer the

state of the system from one state to another over an arbitrary interval

of time. The solution is also the minimum-energy solution.



EXAMPLE: Consider the system in diagonal form

Px.t/ D

"

"0:5 0

0 "1

#

x.t/C

"

0:5

1

#

u.t/:

■ The controllability matrix is:

C D

"

0:5 "0:25

1 "1

#

which has rank 2, so the system is controllable.

■ Consider the input required to move the system state from

x.0/ D Œ10 " 1#T to zero in two seconds.

Wc.2/ D

Z 2

0

"

e"0:5! 0

0 e"!

#"

0:5

1

#h

0:5 1i"

e"0:5! 0

0 e"!

#!

d!

D

"

0:2162 0:3167

0:3167 0:4908

#

;

and

u.t/ D "h

0:5 1i"

e"0:5.2"t / 0

0 e".2"t /

#

Wc.2/"1

"

e"1 0

0 e"2

#"

10

"1

#

D "58:82e0:5t C 27:96et :

■ If a continuous-time system is controllable, and if it is also stable, then

Wc D

Z 1

0

eA!BBT eAT ! d!

can be found by solving for the unique (positive-definite) solution to

the (Lyapunov) equation

AWc CWcAT D "BBT :

Wc is called the controllability Gramian.



PROOF: We proved this identity when considering Lyapunov stability.

■ Wc measures the minimum energy required to reach a desired point

x1 starting at x.0/ D 0 (with no limit on t)

min

(Z t

0

ku.!/k2 d!

ˇˇˇˇˇ

x.0/ D 0; x.t/ D x1

)

D xT1 W "1

c x1:

■ In fact, for any specific “t”, the minimum energy is xT1 W "1

c .t/x1.

■ If A is stable, W "1c > 0 which implies “we can’t get anywhere for free”.

■ If A is unstable, then W "1c can have a nonzero nullspace W "1

c ´ D 0 for

some ´ ¤ 0 which means that we can get to ´ using u’s with energy

as small as you like! (u just gives a little kick to the state; the

instability carries it out to ´ efficiently).

■ Wc may be a better indicator of controllability than C.

Continuous-time observability Gramian

■ If a system is observable, Wo.t/ is nonsingular for t > 0 where

Wo.t/ D

Z t

0

eAT !C T CeA! d!:

SIGNIFICANCE: We can prove that

x.0/ D W "1o .t1/

Z t1

0

eAT tC T Ny.t/ dt

where

Ny.t/ D y.t/" C

Z t

0

eA.t"!/Bu.!/ d! "Du.t/ D CeAtx.0/:

■ Therefore, we can determine the initial state x.0/ given a finite

observation period (and not use differentiators!).



PROOF: We prove that the above equations are correct by substitution:

W "1o .t1/

Z t1

0

eAT tC T Ny.t/ dt D W "1o .t1/

Z t1

0

eAT tC T CeAt dt x.0/

D W "1o .t1/Wo.t1/x.0/

D x.0/:

■ If a continuous-time system is observable, and if it is also stable, then

Wo D

Z 1

0

eAT !C T CeA! d!

is the unique (positive-definite) solution to the (Lyapunov) equation

AT Wo CWoA D "C T C:

Wo is called the observability Gramian.

■ This relationship can be proven in the same way we proved the

similar relationship for the controllability gramian.

■ If measurement (sensor) noise is IID N .0; *2I / then Wo is a measure

of error covariance in measuring x.0/ from u and y over longer and

longer periods

limt!1

E k Ox.0/ " x.0/k2 D *x.0/T W "1o x.0/:

■ If A is stable, then W "1o > 0 and we can’t estimate the initial state

perfectly even with an infinite number of measurements u.t/ and y.t/

for t ! 0 (since memory of x.0/ fades).

■ If A is not stable then W "1o can have a nonzero nullspace W "1

o x.0/ D 0

which means that the covariance goes to zero as t !1.

■ Wo may be a better indicator of observability than O.



5.6: Discrete-time Gramians

Discrete-time controllability Gramian

■ In discrete-time, if a system is controllable, then

WdcŒn " 1# Dn"1X

mD0

AmBBT .AT /m

is nonsingular. In particular,

Wdc D1X

mD0

AmBBT .AT /m

is called the discrete-time controllability Gramian and is the unique

positive-definite solution to the Lyapunov equation

Wdc " AWdcAT D BBT :

■ As with continuous-time, Wdc measures the minimum energy required

to reach a desired point x1 starting at xŒ0# D 0 (with no limit on m)

min

(mX

kD0

kuŒk#k2

ˇˇˇˇˇ

xŒ0# D 0; xŒm# D x1

)

D xT1 W "1

dc x1:

ASIDE: When considering discrete-time stability, we showed that this

form of equation is indeed a Lyapunov equation.

Discrete-time observability Gramian

■ In discrete-time, if a system is observable, then

WdoŒn " 1# Dn"1X

mD0

.AT /mCC T Am

is nonsingular. In particular,



Wdo D1X

mD0

.AT /mCC T Am

is called the discrete-time observability Gramian and is the unique

positive-definite solution to the Lyapunov equation

Wdo " AT WdoA D C T C:

■ As with continuous-time, if measurement (sensor) noise is IID

N .0; *I / then Wdo is a measure of error covariance in measuring xŒ0#

from u and y over longer and longer periods

limt!1

E k OxŒ0# " xŒ0#k2 D *xŒ0#T W "1do xŒ0#:



5.7: Computing transformation matrices

Transformation to controllability form

■ Given a system


y.t/ D Cx.t/CDu.t/;

can we find a transformation (similarity) matrix T to transform this

system into controllability form? Recall, this looks like:

Pxco.t/ D

2

66664

0 $ $ $ 0 "an

1 ::: "an"1::: : : : :::

0 $ $ $ 1 "a1

3

77775

xco.t/C

2

66664

1

0:::

0

3

77775

u.t/

y.t/ Dh

ˇ1 ˇ2 $ $ $ ˇn

i

xco.t/CDu.t/:

■ Note that Cco D I so it is controllable. Thus, our original system must

be controllable.

■ The transformation is accomplished via

x D T xco:

■ Thus

T "1AT D Aco; T "1B D Bco

C T D Cco; D D Dco:

■ Let’s find T explicitly; let

T D Œt1; : : : ; tn#:



■ Note that B D TBco or

B D T

2

66664

1

0:::

0

3

77775

D t1:

■ So, t1 D B. From AT D TAco we have

AŒt1; : : : ; tn# D Œt1; : : : ; tn#

2

66664

0 $ $ $ 0 "an

1 ::: "an"1::: : : : :::

0 $ $ $ 1 "a1

3

77775

:

■ So

ŒAt1; : : : Atn# D

"

t2; : : : ; tn;"nX

kD1

aktn"kC1

#

:

■ By induction,

At1 D t2 D AB

At2 D t3 D A2B; and so forth : : :

so

T D ŒB AB $ $ $ An"1B# D C:

CONCLUSION: A system fA; B; C; Dg can be transformed to controllability

canonical form if and only if it is controllable, in which case the

change of coordinates is

x D Cxco:

EXTENSION I: If xold D T xnew then T D ColdC"1new. That is, to convert

between any two realizations, T is a combination of the controllability



matrices of the two different realizations.

Cnew D#

Bnew AnewBnew $ $ $ An"1new Bnew

$

D#

T "1Bold T "1AoldT T "1Bold $ $ $ T"1An"1

old T T "1Bold

$

D T "1Cold;

or

T D ColdC"1new:

EXTENSION II: If xold D T xnew then T D O"1oldOnew. This can be shown in a

similar way.



5.8: Canonical (Kalman) decompositions

■ What happens if fA; Bg not controllable or if fA; C g not observable?

% Is “part” of the system controllable?

% Is “part” of the system observable?

■ Given a system with fA; Bg not controllable, Px.t/ D Ax.t/C Bu.t/,

y.t/ D Cx.t/, let’s try to transform to controllability canonical form.

■ Let t1 D B; t2 D AB; : : : tr D Ar"1B, and suppose t1; t2; : : : ; tr are

independent but

trC1 D ArB D "˛rt1 " $ $ $ " ˛1tr

for some constants ˛1; : : : ; ˛r .

■ Then rank.C/ D r since the vectors ArB; ArC1B; : : : ; An"1B can all be

expressed as a linear combination of t1; t2; : : : ; tr .

■ Let srC1; : : : ; sn be your favorite vectors for which

NCold Dh

t1 $ $ $ tr srC1 $ $ $ sn

i

is invertible.

% If we were able to change coordinates to controllability form via

xold D T xnew, we would use T D ColdC"1new.

% But, the system is not controllable, so we use T D NColdC"1new D NCold.

■ We get (in the new coordinate system)"

Pxc.t/

Px Nc.t/

#

D

"

Ac A12

0 A Nc

#"

xc.t/

x Nc.t/

#

C

"

Bc

0

#

u.t/

y.t/ Dh

Cc C Nc

i"

xc.t/

x Nc.t/

#

CDu.t/:



■ Ac is a right-companion matrix and Bc is of the

controllability-canonical form.

■ We see that the uncontrollable modes x Nc are completely decoupled

from u.t/.

EXAMPLE: Consider1

s C 1D

s " 1

.s C 1/.s " 1/D

s " 1

s2 " 1:

■ In observer-canonical form,

Px.t/ D

"

0 1

1 0

#

x.t/C

"

1

"1

#

u.t/

y.t/ Dh

1 0i

x.t/:

■ So,

t1 D

"

1

"1

#

and t2 D At1 D

"

"1

1

#

D "t1:

■ So, rank.C/ D 1. Let s1 D Œ1 0#T . The converted state-space form is

PNx.t/ D

"

"1 "1

0 1

#

Nx.t/C

"

1

0

#

u.t/

y.t/ Dh

1 1i

Nx.t/:

■ Now suppose that we desire to transform a system that is not

observable. The dual form separates out the unobservable states.

■ Let t1 D C; t2 D CA; : : : tr D CAr"1, and suppose t1; t2; : : : ; tr are

independent but

trC1 D CAr D "˛rt1 " $ $ $ " ˛1tr

for some constants ˛1; : : : ; ˛r .



■ Then, let srC1; : : : ; sn be your favorite row vectors for which

NOold D

2

6666666664

t1:::tr

srC1

:::sn

3

7777777775

is invertible.

% If we were able to change coordinates to observability form via

xold D T xnew, we would use T D O"1oldOnew.

% But, the system is not observable, so we use T D NO"1oldOnew D NO

"1old.

■ Then, we get the Kalman decomposition"

Pxo.t/

Px No.t/

#

D

"

Ao 0

A21 A No

#"

xo.t/

x No.t/

#

C

"

Bo

B No

#

u.t/

y.t/ Dh

Co 0i"

xo.t/

x No.t/

#

CDu.t/:

■ Note: No path from x No to y!

Full Kalman decomposition

■ We can do both transformations in sequence to get full Kalman

decomposition. See text chapter 16.



5.9: Popov–Belevitch–Hautus controllability/observability tests

PBH EIGENVECTOR TEST: fC; Ag is an unobservable pair iff a non-zero

eigenvector v of A satisfies C v D 0. (i.e., C and v are perpendicular).

PROOF:) Suppose Av D $v and C v D 0 for v ¤ 0. Then

CAv D $C v D 0 and so forth up to CAn"1v D $n"1C v D 0. So,

Ov D 0 and since v ¤ 0 this means that O is not full rank and fC; Ag

is not observable.

■( Now suppose that O is not full rank (fC; Ag unobservable). Let’s

extract the unobservable part. That is, find T such that

T "1AT D

"

Ao 0

A21 A No

#

C T Dh

Co 0i

;

where Ao is size r where r D rank.O/ and therefore A No is size n " r .

% Let v2 ¤ 0 be an eigenvector of A No. Then

T "1AT

"

0

v2

#

D

"

Ao 0

A21 A No

#"

0

v2

#

D $

"

0

v2

#

so we have found an eigenvector ´ of A

A´ D $´ where ´ D T

"

0

v2

#

:

% Now, we just need to show that C ´ D 0 (note: ´ ¤ 0).

C ´ D C T„ƒ‚…

NC

"

0

v2

#

Dh

Co 0i"

0

v2

#

D 0

and we are done.



DUAL: fA; Bg is uncontrollable iff there is a left eigenvector wT of A such

that wT B D 0.

INTERPRETATION: In modal coordinates, homogeneous response is

x.t/ D

nX

iD1

e$i tvi.wTi x.0// and y.t/ D Cx.t/:

Or,

y.t/ D

nX

iD1

e$i tC vi.wTi x.0//:

If fC; Ag is unobservable, then it has an unobservable mode, where

Avi D $ivi and C vi D 0.

■ If fA; Bg is uncontrollable, then it has an uncontrollable mode, namely

the coefficients of the state along that mode is independent of the

input u.t/.

% The coefficients of x in the mode associated with $ are wT x.

d

dt.wT x/ D wT .Ax C Bu/ D $.wT x/

or

wT x.t/ D e$t .wT x.0//

regardless of the input u.t/!

PBH RANK TESTS: The following two tests are often easier to perform

1. fA; Bg controllable iff rankh

sI " A Bi

D n for all s 2 C.

) If rankh

sI " A Bi

D n for all s 2 C then there can be no nonzero

vector v such that vTh

sI " A Bi

Dh

vT .sI " A/ vT Bi

D 0:



■ Consequently, there is no nonzero vector v such that vT s D vT A

and vT B D 0. By the PBH eigenvector test, the system will

therefore be controllable

( If the system is controllable, then there is no nonzero vector v

such that vT s D vT A and vT B D 0 by the PBH eigenvector test.

■ Therefore, rankh

sI " A Bi

D n for all s 2 C.

2. fC; Ag observable iff rank

"

C

sI " A

#

D n for all s 2 C. Proof similar.

COMMENTS: rankh

sI " A Bi

D n for all s not eigenvalues of A, so the

test is really fA; Bg controllable iff rankh

$iI " A Bi

D n for $i ,

i D 1; : : : ; n the eigenvalues of A. (Dual argument for observability).

■ Ifh

sI " A Bi

drops rank at s D $ then there is an uncontrollable

mode with exponent (frequency) $.

■ If

"

C

sI " A

#

drops rank at s D $ then there is an unobservable

mode with exponent (frequency) $.

Summary

■ Therefore, we can label individual modes of a system as either

controllable or not, or observable or not.

■ The overall picture is:



replacements Controllableand

observable

Controllablebut not

observable

Observablebut not

controllable

Neitherobservable nor

controllable

u.t/ y.t/

■ Some other definitions:

STABILIZABLE: A system whose unstable modes are controllable.

DETECTABLE: A system whose unstable modes are observable.



5.10: Minimal realizations—Why a system isn’t control/observ-able

■ To realize H.s/ D1

s C 1we could use

1. Px.t/ D "x.t/C u.t/, y.t/ D x.t/. This gives

A D Œ"1#, B D Œ1#, C D Œ1#, D D Œ0#.

C adj.sI " A/B

det.sI " A/D

1

s C 1:

This realization is both controllable and observable.

2. Observer realization of1

s C 1

s " 1

s " 1D

s " 1

s2 " 1. This gives

A D

"

0 1

1 0

#

, B D

"

1

"1

#

, C Dh

1 0i

, D D Œ0#.

C adj.sI " A/B

det.sI " A/D

s " 1

s2 " 1D

1

s C 1:

O D

"

C

CA

#

D

"

1 0

0 1

#

. Observable.

C Dh

B ABi

D

"

1 "1

"1 1

#

. Not controllable.

3. Controller realization of1

s C 1

s C 10

s C 10D

s C 10

s2C 11s C 10.

A D

"

"11 "10

1 0

#

, B D

"

1

0

#

, C Dh

1 10i

, D D Œ0#.

C adj.sI " A/B

det.sI " A/D

s C 10

s2 C 11s C 10D

1

s C 1:

C Dh

B ABi

D

"

1 "11

0 1

#

. Controllable.



O D

"

C

CA

#

D

"

1 10

"1 "10

#

. Not observable.

TREND: Non-minimal realizations of transfer functions will either be

uncontrollable, unobservable, or both.

■ Four equivalent statements:

I: There exist common roots of C adj.sI " A/B and det.sI " A/.

II: There exist eigenvalues of A which are not poles of G.s/, counting

multiplicities.

III: The system is either unobservable or uncontrollable.

IV: There exist extra (unnecessary) states—non minimal.

DEFINITION: We say a system is minimal if no system with the same

transfer function has fewer states.

PROOF: I”II

IH)II: The transfer function

G.s/ D C.sI " A/"1B CD

DC adj.sI " A/B CD det.sI " A/

det.sI " A/:

If there are common roots in C adj.sI " A/B and det.sI " A/ they will

cancel out of the transfer function. But, eigenvalues of A are

det.sI " A/ D 0, so poles of G.s/ will not contain all eigenvalues of A.

IIH)I: Eigenvalues of A are det.sI " A/ D 0. Poles of G.s/, from above,

are det.sI " A/ D 0 unless canceled. The only way to cancel a pole is

to have common root in C adj.sI " A/B.



PROOF: I”IV fA; B; C; Dg is minimal iff C adj.sI " A/B and det.sI " A/

are coprime (have no common roots).

IH)IV: Suppose C adj.sI " A/B and det.sI " A/ have common roots.

■ Cancel to get

G.s/ DC adj.sI " A/B

det.sI " A/CD D

br.s/

ar.s/

where br.s/ and ar.s/ are coprime (“r” means “reduced”).

■ Because of cancellation, k D deg.ar/ < deg.det.sI " A// D n.

■ Consider controller canonical form realization of br.s/=ar.s/, for

example.

■ It has k states, but same transfer function as fA; B; C; Dg,

contradicting that fA; B; C; Dg minimal.

IVH)I: If there are extra states (non-minimal) then n > k (n D# states,

k D# poles in G.s/).

G.s/ D C.sI " A/"1B CD

DC adj.sI " A/B CD det.sI " A/

det.sI " A/

G.s/ has n poles unless some cancel with C adj.sI " A/B. Therefore,

if n > k, C adj.sI " A/B and det.sI " A/ are not coprime.

PROOF: III”IV Controllable and observable iff minimal.

IIIH)IV: Uncontrollable or unobservable H) not minimal.

Perform Kalman decomposition to split system into co, c No, Nco and Nc No

parts.



u.t/

y.t/

xc.t/

xc.t/

Ac

A Nc

A12

Bc Cc

C Nc

R

RUncontrollablepart ofrealization

NA D

"

Ac A12

0 A Nc

#

, NB D

"

Bc

0

#

, NC Dh

Cc C Nc

i

, ND D Œ0#.

Same transfer function using Ac, Bc, Cc as NA, NB, NC . Therefore

uncontrollable and/or unobservable means not minimal.

IVH)III: Non-minimal means uncontrollable or unobservable.

■ Suppose fA; B; C; Dg is non-minimal.

C.sI " A/"1B CD„ ƒ‚ …

n states

D NC .sI " NA/"1 NB C ND„ ƒ‚ …

r<n states

■

CB

sC

CAB

s2C

CA2B

s3C $ $ $ D

NC NB

sCNC NA NB

s2CNC NA2 NB

s3C $ $ $

■ Consider

OC D

2

6666664

C

CA

CA2

:::

CAn"1

3

7777775

h

B AB A2B $ $ $ An"1Bi



D

2

64

CB $ $ $ CAn"1B::: :::

CAn"1B $ $ $ CA2n"2B

3

75 D

2

64

NC NB $ $ $ NC NAn"1 NB::: :::

NC NAn"1 NB $ $ $ NC NA2n"2 NB

3

75

D n

x??????????y

2

6666664

NC

NC NA::: 0

NC NAn"2

NC NAn"1

3

7777775

""""!r

""""!n"r

2

6666664

NB NA NB $ $ $ NAn"1 NB

0

3

7777775

""""""""""""""""""""""""""!n

x??????y

r

x??????y

n"r

■ Therefore det.O/ det.C/ D 0, so the system is either unobservable,

or uncontrollable, or both.

■ The four equivalences have now been proven.

FACT: All minimal realization of G.s/ are related by a unique change of

coordinates T . Can you prove this?

Where to from here?

■ Have seen important topics of observability and controllability.

■ Now understand how a system could be unobservable or

uncontrollable.

% If physically true, may need to add sensors or actuators.

■ Important to work with minimal realizations, when possible.

■ Now, time to consider: How do I build a controller?

% Very powerful tools in next chapter.


Documents

OBSERVABILITY AND CONTROLLABILITYmocha-java.uccs.edu/ECE5520/ECE5520-CH05.pdfEXAMPLE: Controllability canonical form: x.t/P D 2 6 4 00"a 3 10"a 2 01"a 1 3 7 5x.t/C 2 6 4 1 0 0 3 7