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
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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
Chapter 5 Lecture Notes of Multivariable Control
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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
Chapter 5 Lecture Notes of Multivariable Control
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∫•∗=
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
Chapter 5 Lecture Notes of Multivariable Control
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⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
−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).
Proof: See “Linear system theory and design” Chi-Tsong Chen
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
Chapter 5 Lecture Notes of Multivariable Control
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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.
Chapter 5 Lecture Notes of Multivariable Control
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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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
Chapter 5 Lecture Notes of Multivariable Control
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[ ] 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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
Definition 5-3
Chapter 5 Lecture Notes of Multivariable Control
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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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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 =
Proof: See “Linear system theory and design” Chi-Tsong Chen
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?
Chapter 5 Lecture Notes of Multivariable Control
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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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=
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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.
Chapter 5 Lecture Notes of Multivariable Control
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:
Chapter 5 Lecture Notes of Multivariable Control
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.
Proof: See “Linear system theory and design” Chi-Tsong Chen
Now consider the dynamical equation
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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.
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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,
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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).
Chapter 5 Lecture Notes of Multivariable Control
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.
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
Chapter 5 Lecture Notes of Multivariable Control
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
)
Chapter 5 Lecture Notes of Multivariable Control
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
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=&
Chapter 5 Lecture Notes of Multivariable Control
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