Modern Control Engineering

Example Problems and Solutions 689

(9–70)

Solution.Equation (9–68) can be written as

which can be modified to

(9–71)

where

Let us rewrite this last equation in the following form:

From this last equation, the following two equations may be obtained:

(9–72)

(9–73)

Now define state variables as follows:

Then, clearly,

sXn- 1 (s)=Xn(s)







sX 2 (s)=X 3 (s)

sX 1 (s)=X 2 (s)

Xn(s)=sn-^1 Q(s)

Xn- 1 (s)=sn-^2 Q(s)







X 2 (s)=sQ(s)

X 1 (s)=Q(s)

+Abn-an b 0 BQ(s)

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

snQ(s)=-a 1 sn-^1 Q(s)-p-an- 1 sQ(s)-an Q(s)+U(s)

=

U(s)

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

=Q(s)

Yˆ(s)

Ab 1 - a 1 b 0 Bsn-^1 +p+Abn- 1 - an- 1 b 0 Bs+Abn-an b 0 B

Yˆ(s)=

Ab 1 - a 1 b 0 Bsn-^1 +p+Abn- 1 - an- 1 b 0 Bs+Abn-an b 0 B

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

U(s)

Y(s)=b 0 U(s)+Yˆ(s)

Y(s)

U(s)

=b 0 +

Ab 1 - a 1 b 0 Bsn-^1 +p+Abn- 1 - an- 1 b 0 Bs+Abn-an b 0 B

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

y =Cbn-an b 0 bn- 1 - an- 1 b 0 pb 1 - a 1 b 0 DF

x 1

x 2







xn

V +b 0 u