2 Year System Dynamics Sheet 6
2 Year System Dynamics Sheet 6
2 Year System Dynamics Sheet 6
System Dynamics
Sheet 6
2- In the motor position servo of the following figure, let G(s) = 1/[s(s + 1)] represent the motor and
load and Gc = K, the controller-amplifier. Calculate the steady-state errors of the system for unit step
and unit ramp signals applied in turn to reference input R and disturbance input D. Explain the results
obtained physically.
1
3- For the system in the following figure, with G(s) =(s + 1)(s + 3)/[s(s + 2)(s + 4)]:
a) What is the system type number?
b) What is the gain of the loop gain function?
c) What are the steady-state errors following unit step and unit ramp inputs?
If G(s) = 1/ [(s + 1) (s + 4)]:
d) What is the dominating time constant of the plant?
e) For what value of K will feed back make the dominating system time constant half of
that in part (d)? .
f) Calculate the unit step response for K in part (e) and find the steady-state error.
g) What limits the increase of K desirable to reduce the steady-state error in part (f)?
Find K and the corresponding steady-state error for a system damping ratio of about
0.7.
4- For the unity feedback system shown Figure, specify the gain and pole location of the
compensator so that the overall closed-loop response to a unit-step input has an overshoot of no
more than 25%, and a 0.1 sec settling time.
Find K, z, and p so that the closed-loop system has a 10% overshoot to a step input and a settling
time of 1.5 sec.
2
6- Suppose you are to design a unity feedback controller for a first-order plant, as depicted in Figure.
(The configuration shown here is referred to as a proportional-integral controller). You are to
design the controller so that the closed-loop poles lie within the shaded regions shown in the
second Figure.
a) What values of ωn and ζ correspond to the shaded regions in the Figure (A simple
estimate from the figure is sufficient.)
b) Let Kα = α = 2. Find values for K and KI so that the poles of the closed-loop system lie
within the shaded regions.
7- In the following figure, let G(s) be the open-loop unstable plant G(s) = I/(s - 1). design the simplest
possible controller Gc(s) that will satisfy all the following specifications:
a) The steady-state error for ramp inputs must be less than 0.1.
b) The system settling time must be about 4 sec.
c) The system damping ratio should be 0.5.
3
8- Realize the following dynamic compensators by means of operational amplifiers giving values for
resistors and capacitance in terms of megohms (=106 ohm) and microfarads (= 10-6 farad):
a) Pl controller: (10 + 20 / s).
b) Phase lead compensator: 2 (0.5 s + 1) / (0.1 s + 1).
9- Derive the transfer functions corresponding to the operational amplifier circuits in Figures.
.