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Problem 7.60PP
Compute the controller transfer function [from Y(s) to tyfsj] in Example. What is the prominent 
feature of the controller that allows tracking and disturbance rejection?
Example Steady-State Tracking and Disturbance Rejection of Motor Speed by Extended 
Estimator
Construct an estimattN' to ccxitFol the state and cancel a constant bias at the 
output and track a constant reference in the rootof speed system described
x = - 3 i + h. (7.256a)
y = x + w, (7.256b)
w = 0» (7.256c)
f = 0. (7.256d)
Place the control pole at 5 = —5 and the two extended estimator poles at 
5 = - 1 5 .
Solutioii. To besin. we destitn die control law by iiniorins the equivalent 
Place the control pole at s = —5 and the two extended estimattMr poles 
s = -1 5 .
Solution. To begin, we design die ctmtiol law by ignoring the equivalrat 
disturbance. Rather, we notice by inspectitMi that a gain c f —2 will move 
the single pole from —3 to the desired —5, Therefore, K = 2. The sys­
tem augmented with equivalent external input p, which replaces the actual 
disturbance w and the le fn race r , is ^ v en 1^
p = 0 ,
i = -2 x + u + pt
- 4 r 
-5
0 02 04 a s 04 1.0 U 1.4 14 1.8 2 
Time (fee)
W>
Figure 7.72
Motor Speed system with extended esQmaton (a) block diagram; (b) command step response and 
(fistufbMce step response
The extended estimator equatims are 
^ = / i( e -x ) ,
1 = -3S-I-« + p + fe (e - i) .
The estimator error gain is found to be L ^ [ 225 27 from the 
characteristic equadon
* • [ 1 * + +
A block diagram of the system is given in Fig. 7 .7 ^ ^ , and the step responses 
to input at the command r (applied at t = 0 sec) and at the disturbance w 
(applied at / w 0.5 sec) are shewn in Rg. 7.72(bX
Step-by-step solution
step 1 of 1
The related equation
p = /i ( c - x ) 
x = -3 x + p + 4 + i3 
u = - K x - p
p l ^ r o - I , ] [ p
j J [ o - 3 - i T - i . J
« = [ - ! - 4 ?
X
The controller transfer function is
r{S) S{S+3+K+l,)
- 2 7 9 ( £ f+ 4 .0323) 
5 ( 5 + 3 2 )