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Problem 5.33PP
Consider the plant transfer function 
bs + k
C(i) =' fiimMfi + (Af +m)te + (M + m)k]
to be put in the unity feedback loop of Fig. This is the transfer function relating the input force u(t) 
and the position y(t) of mass M in the noncoilocated sensor and actuator problem. In this 
problem, we will use root-locus techniques to design a controller Dc(s) so that the closed-loop 
step response has a rise time of less than 0.1 sec and an overshoot of less than 10%. You may 
use Matlab for any of the following questions;
(a) Approximate G(s) by assuming that m ^ 0, and let M = 1. k = 1. b = 0.1, and Dc(s) = K. Can K 
be chosen to satisfy the performance specifications? Why or why not?
(b) Repeat part (a) assuming that Dc(s) = K(s + z), and show that K and z can be chosen to meet
(b) Repeat part (a) assuming that Dc(s) = K(s + z). and show that K and z can be chosen to meet 
the specifications.
(c) Repeat part (b), but with a practical controller given by the transfer function
Dc{s)=K*- s + p
Pick p so that the values for K and z computed in part (b) remain more or less valid.
(d) Now suppose that the small mass m is not negligible, but is given by m = MA 0. Check to see 
if the controller you designed in part (c) still meets the given specifications. If not, adjust the 
controller parameters so that the specifications are met.
Figure Unity feedback system
O— — » G (i) I O
Step-by-step solution
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