Modern Control Engineering

Example Problems and Solutions 825

The characteristic equation of the system is

The eigenvalues of matrix Aare–1and–2.
It is desired to have eigenvalues at –3and–5by using a state-feedback control u=–Kx.
Determine the necessary feedback gain matrix Kand the control signal u.

Solution.The given system is completely state controllable, since the rank of

is 2. Hence, arbitrary pole placement is possible.
Since the characteristic equation of the original system is

we have

The desired characteristic equation is

Hence,

It is important to point out that the original state equation is not in the controllable canonical
form, because matrix Bis not

Hence, the transformation matrix Tmust be determined.

Hence,

Referring to Equation (10–13), the necessary feedback gain matrix is given by

Thus, the control signal ubecomes

u=-Kx=-[6.5 2.5]B

x 1

x 2

R

= C 15 - 2  8 - 3 DB

0.5

0

0

0.5

R =[6.5 2.5]

K= Ca 2 - a 2 a 1 - a 1 D T-^1

T-^1 = B

0.5

0

0

0.5

R

T=MW=CBABDB

a 1

1

1

0

R = B

0

2

2

- 6

RB

3

1

1

0

R =B

2

0

0

2

R

B

0

1

R

a 1 =8, a 2 = 15

(s+3)(s+5)=s^2 +8s+ 15 =s^2 +a 1 s+a 2 = 0

a 1 =3, a 2 = 2

s^2 +3s+ 2 =s^2 +a 1 s+a 2 = 0

M= CBABD= B

0

2

2

- 6

R

∑s I-A∑=^2

s

2

- 1

s+ 3

(^2) =s^2 +3s+ 2 =(s+1)(s+2)= 0