Modern Control Engineering

690 Chapter 9 / Control Systems Analysis in State Space

which may be rewritten as

(9–74)

Noting that we can rewrite Equation (9–72) as

or

(9–75)

Also, from Equations (9–71) and (9–73), we obtain

The inverse Laplace transform of this output equation becomes

(9–76)

Combining Equations (9–74) and (9–75) into one vector–matrix differential equation, we obtain

Equation (9–69). Equation (9–76) can be rewritten as given by Equation (9–70). Equations (9–69)

and (9–70) are said to be in the controllable canonical form. Figure 9–1 shows the block diagram

representation of the system defined by Equations (9–69) and (9–70).

y=Abn-an b 0 Bx 1 +Abn- 1 - an- 1 b 0 Bx 2 +p+Ab 1 - a 1 b 0 Bxn+b 0 u

+Abn-an b 0 BX 1 (s)

=b 0 U(s)+Ab 1 - a 1 b 0 BXn(s)+p+Abn- 1 - an- 1 b 0 BX 2 (s)

+Abn-an b 0 BQ(s)

Y(s)=b 0 U(s)+Ab 1 - a 1 b 0 Bsn-^1 Q(s)+p+Abn- 1 - an- 1 b 0 BsQ(s)

x#n=-an x 1 - an- 1 x 2 - p-a 1 xn+u

sXn(s)=-a 1 Xn(s)-p-an- 1 X 2 (s)-an X 1 (s)+U(s)

snQ(s)=sXn(s),

x#n- 1 =xn







x# 2 =x 3

x# 1 =x 2

b 0

y

u

a 1 a 2 an– 1 an

xn xn– 1 x 2 x 1

b 1 – a 1 b 0 b 2 – a 2 b 0 bn– 1 – an– 1 b 0 bn–anb 0



++ ++ ++ ++

++ ++ ++

Figure 9–1 +–

Block diagram

representation of the

system defined by

Equations (9–69)

and (9–70)

(controllable

canonical form).

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