780 Chapter 10 / Control Systems Design in State SpaceMATLAB Program 10–12
% Determination of transfer function of observer controller
A = [0 1 0;0 0 1;0 -24 -10];
B = [0;10;-80];
Aaa = 0; Aab = [1 0]; Aba = [0;0]; Abb = [0 1;-24 -10];
Ba = 0; Bb = [10;-80];
Ka = 1.25; Kb = [1.25 0.19375];
Ke = [10;-24];
Ahat = Abb - Ke*Aab;
Bhat = AhatKe + Aba - KeAaa;
Fhat = Bb - Ke*Ba;
Atilde = Ahat - Fhat*Kb;
Btilde = Bhat - Fhat(Ka + KbKe);
Ctilde = -Kb;
Dtilde = -(Ka + Kb*Ke);
[num,den] = ss2tf(Atilde, Btilde, -Ctilde, -Dtilde)
num =
9.1000 73.5000 125.0000
den =
1.0000 17.0000 -30.0000
In the program, matrices JandLrepresent the desired closed-loop poles for pole place-
ment and the desired poles for the observer, respectively. The matrices KandKeare
obtained as
Design step 4: We shall determine the transfer function of the observer controller.
Referring to Equation (10–108), the transfer function of the observer controller can be
given by
We shall use MATLAB to calculate the transfer function of the observer controller.
MATLAB Program 10–12 produces this transfer function. The result is
Define the system with this observer controller as System 1. Figure 10–20 shows the
block diagram of System 1.
=
9.1(s+5.6425)(s+2.4344)
(s+18.6119)(s-1.6119)
Gc(s)=
9.1s^2 +73.5s+ 125
s^2 +17s- 30
Gc(s)=
U(s)
- Y(s)
=
num
den
=-CC
As I-A
B-^1 B
+D
D
Ke= B
10
- 24
R
K=[1.25 1.25 0.19375]
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