Modern Control Engineering

692 Chapter 9 / Control Systems Analysis in State Space

Then Equation (9–80) can be written as

(9–82)

By substituting Equation (9–82) into Equation (9–81) and multiplying both sides of the equations

bys, we obtain

Taking the inverse Laplace transforms of the preceding nequations and writing them in the

reverse order, we get

Also, the inverse Laplace transform of Equation (9–82) gives

Rewriting the state and output equations in the standard vector-matrix forms gives Equations

(9–78) and (9–79). Figure 9–2 shows a block diagram representation of the system defined by

Equations (9–78) and (9–79).

y=xn+b 0 u

x#n=xn- 1 - a 1 xn+Ab 1 - a 1 b 0 Bu

x#n- 1 =xn- 2 - a 2 xn+Ab 2 - a 2 b 0 Bu







x# 2 =x 1 - an- 1 xn+Abn- 1 - an- 1 b 0 Bu

x# 1 =-an xn+Abn-an b 0 Bu

sX 1 (s)=-an Xn(s)+Abn-an b 0 BU(s)

sX 2 (s)=X 1 (s)-an- 1 Xn(s)+Abn- 1 - an- 1 b 0 BU(s)







sXn- 1 (s)=Xn- 2 (s)-a 2 Xn(s)+Ab 2 - a 2 b 0 BU(s)

sXn(s)=Xn- 1 (s)-a 1 Xn(s)+Ab 1 - a 1 b 0 BU(s)

Y(s)=b 0 U(s)+Xn(s)

y

u

an an– 1 a 1

x 1 x^2 xn–^1 xn

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



+

+

+

++

Figure 9–2

Block diagram

representation of the

system defined by

Equations (9–78)

and (9–79)

(observable

canonical form).

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