Modern Control Engineering

688 Chapter 9 / Control Systems Analysis in State Space

For system S 1 :

1.A necessary and sufficient condition for complete state controllability is that the

rank of the n*nrmatrix

ben.

2.A necessary and sufficient condition for complete observability is that the rank of

then*nmmatrix

ben.

For system S 2 :

1.A necessary and sufficient condition for complete state controllability is that the

rank of the n*nmmatrix

ben.

2.A necessary and sufficient condition for complete observability is that the rank of

then*nrmatrix

ben.

By comparing these conditions, the truth of this principle is apparent. By use of this

principle, the observability of a given system can be checked by testing the state con-

trollability of its dual.

Detectability. For a partially observable system, if the unobservable modes are

stable and the observable modes are unstable, the system is said to be detectable. Note

that the concept of detectability is dual to the concept of stabilizability.

Example Problems and Solutions

A–9–1. Consider the transfer function system defined by Equation (9–2), rewritten

(9–68)

Derive the following controllable canonical form of the state-space representation for this

transfer-function system:

G (9–69)

x# 1

x# 2







x#n- 1

x#n

W = G

0 0    0

• an

1 0    0

• an- 1

0 1    0

• an- 2

p

p

p

p

0 0    1

• a 1

WG

x 1

x 2







xn- 1

xn

W + G

0 0    0 1

Wu

Y(s)

U(s)

=

b 0 sn+b 1 sn-^1 +p+bn- 1 s+bn

sn+a 1 sn-^1 +p+an- 1 s+an

CB  AB  p  An-^1 BD

CCACp(A)n-^1 C*D

CCACp(A)n-^1 C*D

CB  AB  p  An-^1 BD

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