Lectures on Multivariable Feedback Controlprofsite.um.ac.ir/~karimpor/multi/Multivariable_chapter5.pdf · 5-3 Canonical Decomposition of a Linear Time-invariant Dynamical Equation

Lectures on Multivariable Feedback Control Ali Karimpour

Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of

Mashhad (September 2009)

Chapter 5: Controllability, Observability and Realization

5-1 Controllability of Linear Dynamical Equations

5-2 Observability of Linear Dynamical Equations

5-3 Canonical Decomposition of a Linear Time-invariant Dynamical Equation

5-4 Realization of Proper Rational Transfer Function Matrices

5-5 Irreducible Realizations

5-5-1 Irreducible realization of proper rational transfer functions

5-5-2 Irreducible Realization of Proper Rational Transfer Function Vectors

5-5-3 Irreducible Realization of Proper Rational Matrices

Chapter 5 Lecture Notes of Multivariable Control

2

System analyses generally consist of two parts: quantitative and qualitative study. In the

quantitative study we are interested in the exact response of the system to certain input and initial

conditions. In the qualitative study we are interested in the general properties of a system. In the

beginning of this chapter we shall introduce two qualitative properties of linear dynamical

equations: controllability and observability.

Network synthesis is one of the important disciplines in electrical engineering. It is mainly

concerned with determining a passive or an active network that has a prescribed impedance or

transfer function. The subject matter we shall introduce in the reminder of this chapter is along

the same line, that is, to determine a linear time invariant dynamical equation (realization) that

has a prescribed rational transfer matrix.

5-1 Controllability of Linear Dynamical Equations

In this section, we shall introduce the concept of controllability of linear dynamical equations. To

be more precise, we study the state controllability of linear state equation. As will be seen

immediately, the state controllability is a property of state equation only, output equations do not

play any role here.

Definition 5-1

The state equation

)()()()()( tutBtxtAtx +=&

is said to be state controllable at time 0t , if there exist a finite 01 tt > such that for any )( 0tx in the

state space Σ and any 1x in Σ , there exist an input ],[ 10 ttu that will transfer the state )( 0tx to the

state 1x at time 1t . Otherwise, the state equation is said to be uncontrollable at time 0t .

This definition requires only that the input u be capable of moving any state in the state space to

any other state in a finite time, what trajectory the state should take is not specified. Furthermore

there is no constraint imposed on the input. From this drawbacks we see clearly that the property

of state controllability may not imply that the system is controllable in a practical sense. This is

because state controllability is concerned only with the value of the states at discrete values of


3

time target hitting, while in most practical cases we want the outputs to remain close to some

desired value or trajectory for all values of time and without using inappropriate control signals.

So now we know that state controllability does not imply that the system is controllable from a

practical point of view. But what about the reverse: If we do not have state controllability is this

an indication that the system is not controllable in a practical sense? In other words should we be

concerned if a system is not state controllable? In many cases the answer is “no” since we may

not be concerned with the behavior of the uncontrollable states which may be outside our system

boundary or of no practical importance.

So is the issue of state controllability of any value at all? Yes because it tells us whether we have

included some states in our model which we have no means of affecting. It also tells us when we

can save on computer time by deleting uncontrollable states which have no effect on the output

for zero initial conditions. In summary state controllability is a system theoretical concept which

is important when it comes to computations and realizations. However its name is somewhat

misleading and most of the above discussion might have been avoided if only Kalman who

originally defined state controllability had used a different terminology. For example better terms

might have been point wise controllability, or, state affect ability, from which it would have been

understood that although all the states could be individually affected we might not be able to

control them independently over a period of time.

Skogestad and Postlethwaite in “Multivariable feedback control” introduce a more practical

concept of controllability which they call “input-output controllability”

Theorem 5-1

The n-dimensional linear time-invariant state equation

BuAxx +=&

is controllable if and only if any of the following equivalent condition is satisfied.

1. The )(npn× controllability matrix

[ ]BABAABBS n 12 ..... −= 5-1

has rank n (full row rank).

2. The nn× controllability grammian


4

∫•∗=

t AAtct deBBeW

0ττ 5-2

is nonsingular for any 0>t .

3. For every eigenvalue λ of A, the )( pnn +× complex matrix [ ]BAI |−λ has rank n(full

row rank).

Proof: See “Linear system theory and design” Chi-Tsong Chen

5-2 Observability of Linear Dynamical Equations

The concept of observability is dual to the controllability. Roughly speaking, controllability

studies the possibility of steering the state from the input, observability studies the possibility of

estimating the state from the output. If a dynamical equation is controllable, all the modes of the

equation can be exited from the input, if a dynamical equation is observable, all the modes of the

equation can be observed at the output.

Definition 5-2

The dynamical equation

)()()()()()()()()()(tutEtxtCtytutBtxtAtx

+=+=&

is said to be state observable at time 0t , if there exist a finite 01 tt > such that for any 0x at time 0t ,

the knowledge of the input ],[ 10 ttu and the output ],[ 10 tty over the time interval ],[ 10 tt suffices to

determine the state 0x . Otherwise the state equation is said to be unobservable at time 0t .

Theorem 5-2

The n-dimensional linear time-invariant dynamical equation

EuCxyBuAxx

+=+=&

is observable if and only if any of the following equivalent condition is satisfied.

1. The nnq ×)( observability matrix


5

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−1

2

.

.

nCA

CACAC

V 5-3

has rank n (full column rank).

2. The nn× observability grammian

∫=t ATtA

ot dCeCeWT

0ττ 5-4

is nonsingular for any 0>t .

3. For every eigenvalue λ of A, the nqn ×+ )( complex matrix ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−−−−

−

C

AIλ has rank n (full

column rank).


5-3 Canonical Decomposition of a Linear Time-invariant Dynamical Equation

Consider the dynamical equation

EuCxyBuAxx

+=+=&

5-5

Where A, B, C and E are nn× , pn× , nq× , and pq× real constant matrices. We introduced in

the previous sections concepts of controllability and observability. The conditions for the

equation to be controllable and observable are also derived. A question that may be raised at this

point is: what can be said if the equation is uncontrollable and /or unobservable? In this section

we shall study this problem. Before proceeding, we review briefly the equivalence

transformation. Let Pxx = , where P is a constant nonsingular matrix .The substitutions of Pxx =

into 5-5 yields


6

uExCEuxCPyuBxAPBuxPAPx

+=+=

+=+=−

−

1

1& 5-6

Where 11 ,, −− === CPCPBBPAPA and EE = . The dynamical equations 5-5 and 5-6 are said

to be equivalent, and the matrix P is called an equivalence transformation. Clearly we have

[ ] [ ][ ] PSBABAABBP

PBPPAPBPPAPBPAPPBBABABABSn

nn

==

==−

−−−−−

12

1112112

...............

Since the rank of a matrix does not change after multiplication of a nonsingular matrix, we have

SrankSrank = . Consequently 5-5 is controllable if and only if 5-6 is controllable. A similar

statement holds for the observability part. So following theorem is established.

Theorem 5-3

The controllability and observability of a linear time-invariant dynamical equation are invariant

under any equivalence transformation.

Theorem 5-4

Consider the n-dimensional linear time –invariant dynamical equation 5-5. If the controllability

matrix of 5-5 has rank 1n (where nn <1 ), then there exists an equivalence transformation Pxx = ,

where P is a constant nonsingular matrix, which transform 5-5 into

[ ] Euxx

CCy

uB

xx

AAA

xx

c

ccc

c

c

c

c

c

c

c

+⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

0012

&

&

5-7

and the 1n -dimensional sub-equation of 5-7

EuxCyuBxAx

cc

cccc

+=

+=& 5-8

is controllable and has the same transfer function matrix as 5-5.


7

Furthermore [ ] 121 .........

1

−= nn qqqqP where 1

,....,, 21 nqqq be any 1n linearly independent

column of S (controllability matrix) and the last 1nn − column of P are entirely arbitrary so long

as the matrix [ ]nn qqqq .........121 is nonsingular.


Example 5-1

Consider the three-dimensional dynamical equation

[ ]xyuxx 111100110

110010011

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=&

Solution: Controllability matrix is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

121110010101121110

S

The rank of S is 2, therefore, we choose P as 1

010001110 −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=P

Now let Pxx = . We compute

[ ]xyuxx 121001001

100011001

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=&

Hence, the reduced controllable equation is

[ ] ccc xyuxx 211001

1101

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=&

Theorem 5-5

Consider the n-dimensional linear time –invariant dynamical equation 5-5. If the obsarvability

matrix of 5-5 has rank 2n (where nn <2 ), then there exists an equivalence transformation Pxx = ,

where P is a constant nonsingular matrix, which transform 5-5 into


8

[ ] Euxx

Cy

uBB

xx

AAA

xx

o

o

o

o

o

o

oo

o

+⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

0

0

0

21

0

&

&

5-9

and the 2n -dimensional sub-equation of 5-9

EuxCyuBxAx

oo

oooo

+=

+=& 5-10

is observable and has the same transfer function matrix as 5-5.

Furthermore the first 2n rows of P are any 2n linearly independent rows of V (observability

matrix) and the last 2nn − row of P is entirely arbitrary so long as the matrix P is nonsingular.


Theorem 5-6 (Canonical decomposition theorem)

Consider the linear time-invariant dynamical equation 5-5. By equivalence transformation, 5-5

can be transformed into the following canonical form

[ ] Euxxx

CCy

uBB

xxx

AAAAAA

xxx

c

co

oc

cco

co

oc

c

co

oc

c

co

oc

c

co

oc

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0000 23

1312

&

&

&

5-11

where the vector ocx is controllable but not observable, cox is controllable and observable, and

cx is not controllable. Furthermore sub-equation of 5-11

EuxCyuBxAx

coco

cocococo

+=

+=& 5-12

is controllable and observable and has the same transfer function matrix as 5-5.


Definition 5-3


9

A linear time-invariant dynamical equation is said to be reducible if and only if there exist a linear

time-invariant dynamical equation of lesser dimension that has the same transfer function matrix.

Otherwise, the equation is irreducible.

Theorem 5-7

A linear time invariant dynamical equation is irreducible if and only if it is controllable and

observable.


Theorem 5-8

Let the dynamical equation },,,{ ECBA be an irreducible realization of a qp× proper rational

matrix G(s). Then },,,{ ECBA is also an irreducible realization of G(s) if and only if

},,,{ ECBA and },,,{ ECBA are equivalent, that is, there exist a nonsingular constant matrix P

such that 11 ,, −− === CPCPBBPAPA and EE =


5-4 Realization of Proper Rational Transfer Function Matrices

Consider a p-input q-output system with the linear dynamical equation (state-space) description

EuCxyBuAxx

+=+=&

5-13

where u is the 1×p input vector, y is the 1×q output vector, A, B, C and E are constant matrices

with suitable dimensions. The input-output description (transfer function matrix) of the system is

EBAsICsG +−= −1)()( 5-14

Clearly G(s) is a pq× rational-function matrix. The inverse problem –to find the state-space

description from the input-output description of a system- is much more complicated. It actually

consists of two problems

1- Is it possible at all to obtain the state-space description from the transfer function matrix of

a system?


10

2- If yes, how do we obtain the state space description from the transfer function matrix?

Theorem 5-9

A transfer function matrix G(s) is realizable by a finite dimensional linear time invariant

dynamical equation if and only if G(s) is a proper rational matrix.


5-5 Irreducible Realizations

In the following, we introduce definition of the characteristic polynomial and degree of proper

rational matrix. This definition is similar to the scalar case.

Definition 5-4

Consider a proper rational matrix G(s) factored as )()()()()( 11 sDsNsNsDsG rrll−− == . It is

assumed that )(sDl and )(sNl are left coprime and )(sDr and )(sNr are right coprime. Then the

characteristic polynomial of G(s) is defined as

)(det)(det sDorsD lr

And the degree of G(s) is defined as )(detdeg)(detdeg)(deg sDsDsG lr ==

where deg det stands for the degree of determinant.

Note that the polynomial )(det sDr and )(det sDl differ at most by a nonzero constant.

Theorem 5-10

Let the multivariable linear time-invariant dynamical equation

EuCxyBuAxx

+=+=&

be a realization of the proper rational matrix G(s). Then the state space realization is irreducible

(controllable and observable) if and only if

kAsI =− )det( [characteristic polynomial of G(s)]

or

)(degdim sGA=


11

Before considering the general case (irreducible realization of proper rational matrices) we start

the following parts:

a) Irreducible realization of proper rational transfer functions

b) Irreducible realization of proper rational transfer function vectors

c) Irreducible realization of proper rational matrices

5-5-1 Irreducible realization of proper rational transfer functions

Consider the following scalar proper transfer function

0,............ˆ

)( 0110

110 ≠

++++++

= −

−

ααααβββ

nnn

nnn

ssss

sg )))

))

5-15

where iα) and iβ

) for i=0, 1, 2, ….., n, are real constants. By division, g(s) can be written as

0

01

1

22

11

............

)(αβ

ααβββ

)

)

+++++++

= −

−−

nnn

nnn

ssss

sg 5-16

Since the constant 0

0

αβ)

)

gives immediately the direct transmission part of a realization, so 0

0

αβ)

)

=e

and we need to find A, b and c in state space representation. So we need to consider in the

following only the strictly proper rational function

nnn

nnn

ssss

sDsNsg

ααβββ

++++++

== −

−−

............

)()()( 1

1

22

11) 5-17

Let u) and y) be the input and output of )(sg) in above equation. Then we have

uuuyyy nnn

nnn )))))) βββαα +++=+++ −−− ............ )2(

2)1(

1)1(

1)( 5-18

By )1( −ny) we mean (n-1)th derivative of y) .

• Observable canonical form realization

If we choose the state variables as


12

)()()(

)()(...)()()()(.................................

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

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

)()(

11)1(

2

11)2(

1)2(

1)1(

1

22)1(

122)1(

1)1(

1)2(

2

11)1(

11)1(

1

tutxtx

tutytutytytx

tutxtxtutytutytytx

tutxtxtutytytx

tytx

nnn

nnnnn

nnn

nnn

n

)

)))))

))))))

))))

)

−−

−−−−−

−−

−

−+=

−++−+=

−+=−+−+=

−+=−+=

=

βα

βαβα

βαβαβα

βαβα

5-19

Differentiating 1x in 5-19 once and using 5-18, we obtain

)()()()()()(...)()()()( )1(1

)1(1

)1(1

)1(1

)()1(1 tutxtutytutytutytytx nnnnnnn

nnn )))))))) βαβαβαβα +−=+−=−++−+= −−−−

The foregoing equations can be arranged in matrix form as

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−

n

n

n

n

n

n

n

n

x

xxx

y

u

x

xxx

.10...00ˆ

ˆ.

1...00.......

0...100...010...00

.

3

2

1

1

2

1

1

2

1

3

2

1

β

βββ

α

ααα

&

&

&

&

5-20

So the observable canonical form realization of equation 5-16 is:

[ ] u

x

xxx

y

u

x

xxx

n

n

n

n

n

n

n

n

0

03

2

1

1

2

1

1

2

1

3

2

1

.10...00

.1...00

.......0...100...010...00

.

αβ

β

βββ

α

ααα

)

)

&

&

&

&

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−

5-21

The observability matrix of equation 5-21 is


13

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

××××

−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

− 1..............

1...0010...00

.

.1

1

α

ncA

cAc

V

where × means any possible elements. The matrix V is nonsingular for any iα and iβ , hence 5-

21 is observable no matter D(s) and N(s) in 5-17 are coprime or not, and is thus called observable

canonical form realization.

The dynamical equation 5-21 is controllable as well if D(s) and N(s) in 5-17 are coprime. Indeed

if D(s) and N(s) in 5-17 are coprime, then AsDsg dim)(deg)(deg == , and dynamical equation 5-

21 is controllable and observable (irreducible realization) following theorem 5-10. Furthermore if

D(s) and N(s) in 5-17 are not coprime then according to theorem 5-10, dynamical equation 5-21

are not controllable.

• Controllable canonical form realization

We shall now introduce a different realization, called the controllable canonical form realization,

of g(s) in equation 5-15. Again change g(s) to its strictly proper counterpart )(sg) as:

nnn

nnn

ssss

sDsNsg

ααβββ

++++++

== −

−−

............

)()()( 1

1

22

11)

Let u) and y) be the input and output of )(sg) in above equation. Let us introduce a new variable

)(sv) defined by )()()( 1 susDsv )) −= . Then we have

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

svsNsysusvsD

))

)

==

5-22

5-23

We may define the state variable as:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

− )(....

)()(

)(....

)()(

)(

)1(

2

1

tv

tvtv

tx

txtx

tx

nn

)

&)

)

5-24

Clearly nn xxxxxx === −13221 ,.....,, &&& . From 5-22 we have


14

)(...)()()()(...)()()()( 1211)1(

1)1(

1)( txtxtxtutvtvtvtutvx nnn

nnn

nn αααααα −−−−=−−−−== −

−−

))))))&

Equation 5-23 can be written as

)(...)()()(...)()()()()( 1211)1(

1)1(

1 sxsxsxsvsvsvsvsNsy nnnn

nn ββββββ +++=+++== −−

−)))))

These equations can be arranged in matrix form as

[ ] u

x

xxx

y

u

x

xxx

n

nnn

nnnn

0

03

2

1

121

121

3

2

1

....

1.000

...1...000.......0...1000...010

.

αβ

ββββ

αααα

)

)

&

&

&

&

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−− 5-25

The controllability matrix of equation 5-25 is

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

××−××

×== −

...1

...10.......

1...0010...00

.....

1

1

α

bAAbbS n

where × means any possible elements. The matrix S is nonsingular for any iα and iβ , hence 5-25

is controllable no matter D(s) and N(s) are coprime or not, and is thus called controllable

canonical form realization.

The dynamical equation in 5-25 is observable as well if D(s) and N(s) in 5-17 are coprime. Indeed

if D(s) and N(s) in 5-17 are coprime, then AsDsg dim)(deg)(deg == , and dynamical equation 5-

25 is controllable and observable (irreducible realization) following theorem 5-10. Furthermore if

D(s) and N(s) are not coprime then according to theorem 5-10, dynamical equation 5-25 are not

observable.

Example 5-2

Derive controllable and observable canonical realization for following system.


15

61163248182)( 23

23

++++++

=sssssssg

Solution: By division g(s) can be written as:

26116

20266)( 23

2

++++

++=

ssssssg

Hence its observable canonical form realization is:

[ ] uxxx

y

uxxx

xxx

2100

62620

6101101600

3

2

1

3

2

1

3

2

1

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

&

&

&

Its controllable canonical form realization is:

[ ] uxy

uxx

262620100

6116100010

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=&

Example 5-3

Derive irreducible realization for following transfer function.

61163248182)( 23

23

++++++

=sssssssg

Solution: To derive irreducible realization for g(s), N(s) and D(s) must be coprime so we have:

265

2066532162

61163248182)( 22

2

23

23

+++

+=

++++

=++++++

=ss

sssss

sssssssg

Hence its observable canonical form of irreducible realization is:

[ ] uxx

y

uxx

xx

210

620

5160

2

1

2

1

2

1

+⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡&

&

Its controllable canonical form realization is:


16

[ ] uxy

uxx

262010

5610

+=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡−−

=&

• Realization from the Hankel matrix

Consider the following scalar proper transfer function

nnn

nnnn

sssss

sgαα

ββββ+++

++++= −

−−

............

)( 11

22

110 5-26

where iα and iβ for i=0, 1, 2, ….., n, are real constants. We expand it into an infinite power

series of descending power of s as

......)3()2()1()0()( 321 ++++= −−− shshshhsg 5-27

The coefficients { }.....,2,1,0,)( =iih will be called Markov parameters and are obtained by

following equation:

......))3()2()1()0()(......(...... 32111

22

110 +++++++=++++ −−−−−− shshshhsssss n

nnn

nnn ααββββ

We form the βα × matrix

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−+++

++

=

)1(...)2()1()(..............

)2(...)5()4()3()1(...)4()3()2(

)(...)3()2()1(

),(

βαααα

βββ

βα

hhhh

hhhhhhhh

hhhh

H 5-28

It is called a Hankel matrix of order βα × . Note that the coefficient h(0) is not involved in

),( βαH .

Theorem 5-11

The proper transfer function g(s) in 5-26 has degree n if and only if

....,3,2,1,),(),( =++= lkeveryforlnknHnnH ρρ

where ρ denotes rank.


Now consider the dynamical equation


17

eucxybuAxx

+=+=&

Its transfer function is clearly equal to

ebAsIcsebAsIcsg +−=+−= −−−− 1111 )()()( 5-29

This can be extended by Taylor series as

.....)( 3221 ++++= −−− bscAcAbscbsesg 5-30

Hence we conclude that {A, b, c, e} is a realization of g(s) if and only if

......,3,2,1)( 1 == − ibcAih i 5-31

With this background we are ready to introduce a different realization. Consider a proper transfer

function )(/)()( sDsNsg = with nsD =)(deg . Here we do not assume that D(s) and N(s) are

coprime, hence the degree of g(s) may be less than n. First we form the Hankel matrix

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++−+

+

=+

)2(.....)2()1()12(.....)1()(

........

........)1(.....)2(

)(.....)2()1(

),1(

nhnhnhnhnhnh

nhhnhhh

nnH 5-32

Note that there is one more row than column, and the Markov parameters up to h(2n) are used in

forming H(n+1,n). Let the first σ rows be linearly independent and the )1( +σ th row of H(n+1,n)

be linearly dependent on its previous rows. So we can find { }σaaa ,...,, 21 such that

0),1(]0.....01.....[ 21 =+ nnHaaa σ 5-33

We claim that the σ -dimensional dynamical equation


18

[ ] uhxy

u

hh

hhh

x

aaaaa

x

)0(00.....001)(

)1(..

)3()2()1(

.....10.....000....................00.....00000.....10000.....010

.

1321

+=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

+

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−

=

− σσ

σσ

& 5-34

is a controllable and observable (irreducible realization) of g(s). Because of 5-33 we have

......,3,2,1)1(.....)1()()( 21 =−+−−+−−=+ iihaihaihaih σσ σ

Using this we can easily show

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

++

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

=

)(......

)2()1(

.....,

)2(......

)4()3(

,

)1(......

)3()2(

2

kh

khkh

bA

h

hh

bA

h

hh

Ab k

σσσ

5-35

Clearly we have

.....,)3(,)2(,)1( 2 hbcAhcAbhcb === 5-36

This show that 5-34 is indeed a realization of g(s). The controllability matrix of 5-34 is

[ ] ),(..... 1 σσHbAAbbS n == −

The Hankel matrix ),( σσH has rank σ , hence {A, b} is controllable. The observability matrix of

5-34 is

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

1...000..............0...1000...0100...001

.1

2

ncA

cAcAc

V

clearly {A, c} is observable. Hence 5-34 is an irreducible realization.

Example 5-4

Derive irreducible realization for following transfer function.


19

61163248182)( 23

23

++++++

=sssssssg

Solution: To derive irreducible realization for g(s), we derive Markov parameters as

.....2303410141062)( 654321 ++−−+−+= −−−−−− sssssssg

We form the Hankel matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

−

=

2303410341014101410

14106

)3,4(H

We can show that the rank of )3,4(H is 2. Hence we have

[ ] 0)3,4(0156 =H

Hence an irreducible realization of g(s) is

[ ] uxy

uxx

201106

5610

+=

⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−−

=&

5-5-2 Irreducible Realization of Proper Rational Transfer Function Vectors

In this section realizations of vector rational transfer functions ( p×1 or 1×q rational function

matrices) will be studied. Consider the 1×q rational function matrix

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

)(..

)()(

)(2

1

sg

sgsg

sG

q

5-37

It is assumed that every )(sgi is irreducible. We first expand G(s) to


20

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

)(..

)()(

.

.)(2

1

2

1

sg

sgsg

e

ee

sG

qq)

)

)

5-38

where )(∞= ii ge and )(sgi) is a strictly proper rational function. We compute the least common

denominator of )(sgi) and then express G(s) as

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+++

++++++

++++

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−

−−

−−

−

qnn

qn

q

nnn

nnn

nnn

q ss

ssss

ss

e

ee

sG

βββ

ββββββ

αα

......

....

....

.....1

.

.)(

22

11

22

221

21

12

121

11

11

2

1

5-39

It is claimed that the dynamical equation

u

e

ee

x

y

yy

uxx

qqnqnqqn

nnn

nnn

q

nnn

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

−−

−−

−−

−−

.

.

.................

...

...

.

.

10..00

...1...000..............0...1000...010

2

1

1)2()1(

21)2(2)1(22

11)2(1)1(11

2

1

121

ββββ

ββββββββ

αααα

&

5-40

is a realizable of 5-39. This can be proved by using the controllable form realization of g(s) in 5-

25. By comparing 5-40 with 5-25, we see that the transfer function from u to iy is equal to

nnn

inn

in

ii ss

sse

ααβββ

++++++

+ −

−−

.........

11

22

11

which is the ith component of G(s). This proves the assertion. Since )(sgi for qi ,...,2,1= are

assumed to irreducible, the degree of G(s) is equal to n. The dynamical equation 5-40 has


21

dimension n, hence it is a minimal dimensional realization of G(s). We note that if some of

)(sgi are not irreducible, then 5-40 is not observable, although it remains to be controllable.

For column rational functions it is not possible to have the observable form realization (Explain

why?) .

By the same procedure we can derive observable form realization for p×1 proper rational

function matrices. Similarly for row rational functions it is not possible to have the controllable

form realization (Explain why?)

Example 5-5

Derive an irreducible realization for the following column rational function.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

+++

=

34

)2)(1(3

)(

ss

sss

sG

Solution: To derive irreducible realization for G(s), we have

⎥⎦

⎤⎢⎣

⎡

++++

++++⎥

⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡

+++

++++⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

+++

=

2396

61161

10

)2)(1()3(

)3)(2)(1(1

10

34

)2)(1(3

)(

2

2

23

2

ssss

sss

sss

sssss

sss

sG

Hence a minimal dimensional realization of G(s) is given by

uxy

uxx

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

10

132169

100

6116100010

&

5-5-3 Irreducible Realization of Proper Rational Matrices

There are many approaches to find irreducible realizations for pq× proper rational matrices. One

approach is to first find a reducible realization and then apply the reduction procedure to reduce it

to an irreducible one. Reduction procedure is done by removing first any uncontrollable modes,


22

and then any unobservable modes. The theoretical basis of the algorithm is the staircase algorithm

(details in “Multivariable Feedback Design” in J. M. Maciejowski). We discuss four methods

according to this approach.

In the second approach irreducible realization will yield directly. We discuss one method

according to this approach.

Method I: Given a pq× proper rational matrix G(s), if we first find an irreducible realization for

every element )(sgij of G(s) as

jijjiiji

jijijijij

uexcy

ubxAx

+=

+=&

In order to avoid cumbersome notation, we assumed that G(s) is a 22× matrix, and then the

composite dynamical equation is

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

2

1

2221

1211

22

21

12

11

2221

1211

2

1

2

1

22

21

12

11

22

21

12

11

22

21

12

11

22

21

12

11

0000

00

00

000000000000

uu

eeee

xxxx

cccc

yy

uu

bb

bb

xxxx

AA

AA

xxxx

&

&

&

&

5-41

Clearly the transfer function of 5-41 is

)()()()()(

)()()()(

00

00

)(0000)(0000)(0000)(

0000

2221

1211

22221

222221211

2121

12121

121211111

1111

2221

1211

22

21

12

11

122

121

112

111

2221

1211

sGsgsgsgsg

ebAsIcebAsIcebAsIcebAsIc

eeee

bb

bb

AsIAsI

AsIAsI

cccc

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

+−+−+−+−

=

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

⎥⎦

⎤⎢⎣

⎡

−−

−−

−

−

−

−

5-

42

Clearly 5-41 is generally not controllable and not observable. To reduce this realization to

irreducible one requires the application of the reduction procedure twice (theorems 5-4 and 5-5).

Method II: Given a pq× proper rational matrix G(s), if we find the controllable canonical-form

realization for the ith column, Gi(s), of G(s), say


23

iiiii

iiiii

uexCyubxAx

+=+=&

where iii CbA ,, and ie are of the form shown in 5-40, iu is the ith component of u and iy is the

1×q output vector due to the input iu , then the composite dynamical equation

[ ] [ ]ueeexCCCy

u

uu

b

bb

x

xx

A

AA

x

xx

pp

ppppp

........

....00

......0...00...0

......00

.......0....00....0

....

2121

2

1

2

1

2

1

2

1

2

1

+=⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

&

&

&

5-43

Is a realization of G(s). Clearly the transfer function of 5-43 is

[ ] [ ]

[ ] )(

)(...)()(......

)(...)()()(...)()(

)(.....)(

....

...00......0...00...0

)(...00......0...)(00...0)(

....

21

22221

11211

111

111

212

1

1

12

11

21

sG

sgsgsg

sgsgsgsgsgsg

ebAsICebAsIC

eee

b

bb

AsI

AsIAsI

CCC

qpqq

p

p

pppp

p

pp

p

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=+−+−=

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−

−

−

−

5-

44

Because of the structure of iii CbA ,, it can readily verified that the realization is always

controllable. It is however generally not observable. To reduce the realization to an irreducible

realization, one requires the application of the reduction procedure only once (theorems 5-5).

Method III: It is possible to obtain different controllable realization of a proper G(s). Let a

pq× proper rational matrix G(s), where )()()( ∞+= GsGsG)

. Let )(sψ be a monic least common

denominator of G(s), say and of the form

mmmm ssss αααψ ++++= −− ...)( 2

21

1 5-45

Then we can write G(s) as

[ ] )(...)(

1)( 121 ∞++++= − GRsRsR

ssG m

mm

ψ 5-46

where iR are pq× constant matrix. Then the dynamical equation


24

[ ] uGxRRRRy

u

I

x

IIIII

II

x

mmm

p

p

p

p

ppmpmpm

pppp

pppp

pppp

)(...

0..

00

...

...000.......

0...000...00

121

121

∞+=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

−−

−− αααα

& 5-47

is a realization of G(s). To show this it is sufficient to show )()()( 1 sGGBAsIC =∞+− − . Define

BAsIsV 1)()( −−= or BsVAsI =− )()( . V(s) is a mmp× matrix. If we partition it as

[ ])(...)()()( 21 sVsVsVsV Tm

TTT = where the prime denotes the transpose and )(sV Ti is a mm×

matrix, than BsVAsI =− )()( or BsAVssV += )()( implies

pmmmm

mmm

IsVsVsVssVsVssVssV

sVssVssV

sVssV

+−−−−===

==

=

−

−−

)(..........)()()()()()(

.....................................)()()(

)()(

1211

11

1

12

32

21

ααα

These equations imply

pmmm IsVssVss ==+++ − )()()()....( 11

11 ψαα

and

misIs

sV pi

i ,...,2,1)(

)(1

==−

ψ 5-48

Consider

)(...)()()()()()( 12111 sVRsVRsVRGsCVGBAsIC mmm +++=∞+=∞+− −

−

After substituting 5-48 we have

)()()(...

)()(...)()()()()()(1

11

12111

sGGs

sRsRR

GsVRsVRsVRGsCVGBAsICm

mm

mmm

=∞++++

=

∞++++=∞+=∞+−−

−

−−

ψ

Because of the forms of A and B, it is easy to verify that the realization is controllable. It is,

however, generally not observable. To reduce the realization to an irreducible realization, one

requires the application of the reduction procedure only once (theorems 5-5).


25

Method IV:It is possible to obtain observable realization of a proper G(s). Let

...........)2()1()0()( 21 +++= −− sHsHHsG 5-49

where H(i) are pq× constant matrices. Let )(sψ be the monic least common denominator of G(s)

and of the form shown in 5-45. Then after deriving H(i) one can simply show

)(...)2()1()( 21 iHimHimHimH mααα −−−+−−+−=+ 5-50

This is the key equation in the following development. Let },,,{ ECBA be a realization of G(s)

in 5-49. Then we have similar to 5-30

...........)()( 32211 ++++=−+= −−−− BsCACABsCBsEBAsICEsG 5-51

From 5-49 and 5-51, we may conclude, similar to 5-36, that },,,{ ECBA is a realization of G(s)

in 5-49 if and only if )0(HE = and

,....2,1,0)1( ==+ iBCAiH i 5-52

Now we claim that the dynamical equation

[ ] uHxIy

u

mHmH

HH

x

IIIII

II

x

q

qqmqmqm

qqqq

qqqq

qqqq

)0(0...00

)()1(

..)2()1(

...

...000.......

0...000...00

121

+=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

−− αααα

&5-53

Is a qm-dimensional realization of G(s). Indeed using 5-50 , we can readily verify that

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

++

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

=

)(...

)2()1(

,...,

)2(...

)4()3(

,

)1(...

)3()2(

2

imH

iHiH

BA

mH

HH

BA

mH

HH

AB i 5-54

Consequently we have )1( += iHBCAi . This establishes the assertion. The observability matrix of

5-53 is unit matrix of order qm, hence 5-53 is always observable. It is, however, generally not

controllable. To reduce the realization to an irreducible realization, one requires the application of

the reduction procedure only once (theorems 5-4).

Now we shall discuss in the following a method which will yield directly irreducible realizations.

This method is based on the Hankel matrices.


26

Consider the pq× proper rational matrix G(s) given in 5-49. Let

mmmm ssss αααψ ++++= −− ...)( 2

21

1 be the least common denominator of all elements of G(s).

Define the qmqm× and pmpm× matrices

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

−− qqmqmqm

qqqq

qqqq

qqqq

IIIII

II

M

121 ......000

.......0...000...00

αααα

5-55

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−−

= −

−

pppp

pmppp

pmppp

pmppp

II

IIII

I

N

1

2

1

...00.........

.0.....00...00...00

α

ααα

5-56

we also define the two following Hankel matrices

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

+=

)12()1()(...

)1()3()2()()2()1(

mHmHmH

mHHHmHHH

T 5-57

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

++

=

)2()2()1(...

)2()4()3()1()3()2(

~

mHmHmH

mHHHmHHH

T 5-58

Since T and T~ consists of m block rows and m block columns, and since H(i) are pq× matrices, T

and T~ are of order pmqm× . Using 5-50, it can be readily verified that

TNMTT ==~ 5-59

and, in general,

,.....2,1,0== iTNTM ii 5-60


27

Note that the left-upper-corner of ii TNTM = is )1( +iH . Let lkI , be a )( kllk >× constant matrix

of the form [ ]0, klk II = , where kI is the unit matrix of order k, and 0 is the )( klk −× zero

matrix. Then the corner element )1( +iH can be selected from ii TNTM = as

....,2,1,0)1( ,,,, ===+ iITNITIMIiH Tpmp

iqmq

Tpmp

iqmq 5-61

where the superscript T denotes the transpose. From this equation and 5-52, we conclude that

{ })0(,,, ,, HEICTIBMA qmqT

pmp ==== is a qm-dimensional realization of G(s) in 5-49. Note that

the realization is the one in 5-53 and is observable but not necessarily controllable. Similarly, the

dynamical equation

[ ][ ]

)0(

)(...)2()1(

0...0

,

,

HE

mHHHTIC

IIBNA

qmq

pT

pmp

=

==

==

=

5-62

is a pm-dimensional realization of G(s). The controllability matrix of this realization is a unit

matrix; hence the realization is always controllable. The realization however is generally not

observable.

Now we shall use the singular value decomposition, to find an irreducible realization directly

from T and T~ .Singular value decomposition implies the existence of qmqm× and pmpm×

unitary matrix U and V such that HVUT Σ= 5-63

where ⎥⎦

⎤⎢⎣

⎡=Σ

000S

, },......,,{ 21 rdiagS σσσ= with 0........21 >≥≥≥ rσσσ and },min{ pmqmr ≤ ,

and HV is complex conjugate transpose of V. Clearly r is the rank of T. Let rU and rV be the first

r column of U and V, then we can write T as

VUVSSUSVUT Hrr

Hrr

))=== 2/12/1 5-64

where 2/1SUU r=)

is a rqm× and HrVSV 2/1=

) is a pmr × matrix. Define the pseudo inverse of U

)

and V)

as H

rUSU 2/1† −=)

and 2/1† −= SVV r

) 5-65


28

We can now establish the following theorem.

Theorem 5-12

Consider a pq× proper rational matrix G(s) expanded as ∑∞

=−=

0)()(

iisiHsG , we form T and T~ as

in 5-57 and 5-58, and factor T as VUT))

= , by singular value decomposition as shown in 5-64.

Then the { }ECBA ,,, defined by

)0(

)(

)(

~

,

,

††

HE

UofrowsqfirstUIC

VofcolumnspfirstIVB

VTUA

qmq

Tpmp

=

=

=

=

))

))

))

5-66

is an irreducible realization of G(s).

Example 5-6

Derive an irreducible realization for the following proper rational function.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−

+++

−−−

=

153

154

1)1(

232

)(2

2

ss

ss

ssss

sG

Solution: Least common denominator of G(s), is 2)1()( += sssψ , so m=3. G(s) can be shown by

.....2106

2105

.2104

2103

2102

2111

3402

)( 654321 −−−−−−⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−

−= sssssssG

T and T~ are

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−−−−−

−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

212121050403212121040302212121

030211

)5()4()3()4()3()2()3()2()1(

HHHHHHHHH

T


29

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−

−−−−

−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

212121060504212121

050403212121040302

)6()5()4()5()4()3()4()3()2(

~

HHHHHHHHH

T

Non-zero singular values of T are 10.23, 5.7852, 0.8995 and 0.2254. So 4=r , rU and rV are

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

=

0.18880.38750.5432 0.13820.4522 0.2949-0.23110.6989- 0.1888-3875.00.5432-0.1382-

0.10260.0978-0.1057-0.54960.5306-0.6022- 0.58720.10490.6574- 0.4905 0.0196-0.4003-

0071.00581.00.5238-0.2357-5392.04316.00.26270.6738-

0071.00581.0052380.23570.8264-0.1054-0.2078-0.51270.0071-0.0581-0.5238-0.2357-0.1621-0.8902-0.25450.3413-

rr VU

U)

and V)

are

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

==

0.0896 0.2147 0.0896- 0.0487 0.2519- 0.3121- 0.3675 0.2797- 0.3675- 0.0927- 0.5711- 0.4652

1.3066 0.5557 1.3066- 0.2543- 1.4124 0.0471- 0.4421 2.2356- 0.4421- 1.7579 0.3355 1.2803-

0.0034- 0.0551- 1.2598- 0.7539- 0.2560- 0.4093 0.6317 2.1553-

0.0034 0.0551 1.2598 0.7539 0.3923- 0.1000- 0.4999- 1.6398 0.0034- 0.0551- 1.2598- 0.7539-

0.0770- 0.8443- 0.6121 1.0915-

2/1

2/1

Hr

r

VSV

SUU

)

)

The pseudo inverse of U)

and V)

are

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

== −

0.0149- 1.1356- 0.0149 1.7406- 0.0149- 0.3415- 0.0613- 0.4551 0.0613 0.1112- 0.0613- 0.9386- 0.2178- 0.1092 0.2178 0.0864- 0.2178- 0.1058 0.0737- 0.2107- 0.0737 0.1603 0.0737- 0.1067-

2/1† HrUSU

)


30

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

== −

0.3976 0.4086 0.2259 0.0432 0.9525 0.3109- 0.0961 0.2185-

0.3976- 0.4086- 0.2259- 0.0432- 0.2161 0.1031- 0.0440- 0.1718 1.1176- 0.6349- 0.2441 0.0328 1.3847- 0.5171 0.0081- 0.1251-

2/1† SVV r

)

Now according theorem 5-12 the irreducible realization of G(s) is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

0.4476- 0.1354 0.1181- 0.1246 0.8076 0.2888- 0.1800- 0.2227-

0.0772 0.1604- 1.0139- 0.1588 0.1904- 0.2155 0.0369 1.2497-

~ †† VTUA))

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

0.2519- 0.3121- 0.5711- 0.4652 1.4124 0.0471- 0.3355 1.2803-

)(, VofcolumnspfirstIVB Tpmp

))

⎥⎦

⎤⎢⎣

⎡==

0.0034- 0.0551- 1.2598- 0.7539- 0.0770- 0.8443- 0.6121 1.0915-

)(, UofrowsqfirstUIC qmq

))

⎥⎦

⎤⎢⎣

⎡−

−==

3402

)0(HE

Clearly this realization is controllable and observable. Since it is irreducible by theorem 5-12.

Exercises

5-1 Show that the 33× system defined in example 4-1 has the same transfer function as the

reduced controllable equation derived in that example.

5-2 Check the controllability and observability of the following dynamical equation.

a.

xy

uxx

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

121110

111001

342100010

&

b.

[ ]xy

uxx

031031

2025016200340

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=&


31

5-3 Find the degrees and the characteristic polynomials of the following proper rational matrices.

a.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

+

+++

+

sss

s

sss

s1

41

)3(1

51

23

)1(1

2

2

b.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+++

+++

)1)(2(1

21

)1)(2(1

)1(1

2

sss

sss

5-4 Find irreducible controllable or observable canonical-form realizations for the matrices

a.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

+++

)4()1(22

)3)(1)(2(2

2

2

sssss

ssss

b. ⎥⎦

⎤⎢⎣

⎡+

++++

+3

2

2 )1(22

)2()1(32

ssss

sss

5-5 Find irreducible realizations of the rational matrices

a.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+++

+++

)1)(2(1

21

)1)(2(1

)1(1

2

sss

sss b.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

++

sss

ss

ss

23

121

2

33

2

References

Skogestad Sigurd and Postlethwaite Ian. (2005) Multivariable Feedback Control: England, John

Wiley & Sons, Ltd.

Maciejowski J.M. (1989). Multivariable Feedback Design: Adison-Wesley.

Chi-Tsong Chen (1999). Linear system theory and design: Oxford University Press

Documents

Lectures on Multivariable Feedback Controlprofsite.um.ac.ir/~karimpor/multi/Multivariable_chapter5.pdf · 5-3 Canonical Decomposition of a Linear Time-invariant Dynamical Equation