Modern Control Engineering

Example Problems and Solutions 823

The right-hand side of Equation (10–144) is

Clearly, Equation (10–144) holds true. Thus, we have shown that

Next, we shall show that

(10–145)

Note that Equation (10–145) can be written as

Noting that

we have

The derivation shown here can be easily extended to the general case of any positive integer n.

A–10–4. Consider the state equation

where

The rank of the controllability matrix M,

is 2. Thus, the system is completely state controllable. Transform the given state equation into the

controllable canonical form.

Solution.Since

=s^2 +2s+ 1 =s^2 +a 1 s+a 2

∑s I-A∑=^2

s- 1

4

- 1

s+ 3

(^2) =(s-1)(s+3)+ 4
M= CBABD= B

0

2

2

- 6

R

A= B

1

- 4

1

- 3

R, B= B

0

2

R

x

=Ax+Bu

T-^1 B= C

0

0

1

S

TC

0

0

1

S =CBABA^2 BDC

a 2

a 1

1

a 1

1

0

1

0

0

SC

0

0

1

S =CBABA^2 BDC

1

0

0

S =B

B=TC

0

0

1

S =MWC

0

0

1

S

T-^1 B= C

0

0

1

S

T-^1 AT= C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

C

a 2

a 1

1

a 1

1

0

1

0

0

SC

0

0

• a 3

1

0

• a 2

0

1

• a 1

S = C

• a 3
  0
  0

0

a 1

1

0

1

0

S