Modern Control Engineering

Section 10–6 / Design of Regulator Systems with Observers 779

Also, is defined by

where and

[See Equation (2–35) for the calculation of b’s.] Then the state-space equation and out-

put equation can be obtained as

Design step 2: As the first trial, let us choose the desired closed-loop poles at

s=–1+j2, s=–1-j2, s=–5

and choose the desired observer poles at

s=–10, s=–10

Design step 3: We shall use MATLAB to compute the state feedback gain matrix K

and the observer gain matrix Ke.MATLAB Program 10–11 produces matrices KandKe.

y =[1 0 0]C

x 1

x 2

x 3

S +[0]u

C

x

1

x

2

x

3

S = C

0

0

0

1

0

- 24

0

1

- 10

SC

x 1

x 2

x 3

S +C

0

10

- 80

Su

b 0 =0,b 1 =0,b 2 =10, b 3 =-80.

=- 24 x 2 - 10 x 3 +b 3 u

x

3 =-a 3 x 1 - a 2 x 2 - a 1 x 3 +b 3 u

x

3

MATLAB Program 10–11

% Obtaining the state feedback gain matrix K

A = [0 1 0;0 0 1;0 -24 -10];

B = [0;10;-80];

C = [1 0 0];

J = [-1+j2 -1-j2 -5];

K = acker(A,B,J)

K =

1.2500 1.2500 0.19375

% Obtaining the observer gain matrix Ke

Aaa = 0; Aab = [1 0]; Aba = [0;0]; Abb = [0 1;-24 -10];Ba = 0; Bb = [10;-80];

L = [-10 -10];

Ke = acker(Abb',Aab',L)'

Ke =

10

-24