Section 10–6 / Design of Regulator Systems with Observers 781MATLAB Program 10–13
% Obtaining the characteristic equation
[num1,den1] = ss2tf(A-B*K,eye(3),eye(3),eye(3),1);
[num2,den2] = ss2tf(Abb-Ke*Aab,eye(2),eye(2),eye(2),1);
charact_eq = conv(den1,den2)
charact_eq =
1.0e+003*
0.0010 0.0270 0.2550 1.0250 2.0000 2.5000
r= 0 u y
+– 9.1s(^2) + 73.5s+ 125
s^2 + 17 s– 30
10(s+ 2)
s(s+ 4) (s+ 6)
Observer controller Plant
The observer controller has a pole in the right-half splane(s=1.6119).The exis-
tence of an open-loop right-half splane pole in the observer controller means that the
system is open-loop unstable, although the closed-loop system is stable. The latter can
be seen from the characteristic equation for the system:
(See MATLAB Program 10–13 for the calculation of the characteristic equation.)
A disadvantage of using an unstable controller is that the system becomes unstable
if the dc gain of the system becomes small. Such a control system is neither desirable nor
acceptable. Hence, to get a satisfactory system, we need to modify the closed-loop pole
location and/or observer pole location.
=(s+ 1 +j2)(s+ 1 - j2)(s+5)(s+10)(s+10)= 0
=s^5 +27s^4 +255s^3 +1025s^2 +2000s+ 2500
∑s I-A+BK∑@s I-Abb+Ke Aab@
Figure 10–20
Block diagram of
System 1.
Second trial: Let us keep the desired closed-loop poles for pole placement as before,
but modify the observer pole locations as follows:
s=–4.5, s=–4.5
Thus,
L=[–4.5 –4.5]
Using MATLAB, we find the new Keto be
Ke= B
- 1
6.25
R