Modern Control Engineering

Section 9–2 / State-Space Representations of Transfer-Function Systems 649

systems, controllability, and observability are presented in this chapter. Useful design

methods based on state-feedback control are given in Chapter 10.

Outline of the Chapter. Section 9–1 has presented an introduction to state-space

analysis of control systems. Section 9–2 deals with the state-space representation of

transfer-function systems. Here we present various canonical forms of state-space equa-

tions. Section 9–3 discusses the transformation of system models (such as from transfer-

function to state-space models, and vice versa) with MATLAB. Section 9–4 presents

the solution of time-invariant state equations. Section 9–5 gives some useful results in

vector-matrix analysis that are necessary in studying the state-space analysis of control

systems. Section 9–6 discusses the controllability of control systems and Section 9–7

treats the observability of control systems.

9–2 STATE-SPACE REPRESENTATIONS OF

TRANSFER-FUNCTION SYSTEMS

Many techniques are available for obtaining state-space representations of

transfer-function systems. In Chapter 2 we presented a few such methods. This section

presents state-space representations in the controllable, observable, diagonal, or Jordan

canonical forms. (Methods for obtaining such state-space representations from transfer

functions are discussed in detail in Problems A–9–1throughA–9–4.)

State-Space Representations in Canonical Forms. Consider a system defined

by

(9–1)

whereuis the input and yis the output. This equation can also be written as

(9–2)

In what follows we shall present state-space representations of the system defined by

Equation (9–1) or (9–2) in controllable canonical form, observable canonical form, and

diagonal (or Jordan) canonical form.

Controllable Canonical Form. The following state-space representation is called

a controllable canonical form:

G (9–3)

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

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

y

(n)

+a 1 y

(n-1)

+p+an- 1 y

+an y=b 0 u