1
R
r
+
u(t)
L
i(t)
+
C
-
v(t)
-
Figure 1: Circuit of problem 1
Test problems
Laplace Transform
1. Consider the circuit of figure 1.
(a) Define the state vector x(t) = (i(t), v(t)) . Write down the differential equation for x.
(b) Define xk = x(kT ), the sampled value of x. Obtain a difference equation of the form
xk+1 = Axk + Buk ,
by using the approximation ẋ(t)|t=kT =
1
T (xk
− xk−1 ).
2. Consider the LTI system with transfer function H(s) = 1/(s + 2) and suppose we want to
modify the system by adding negative feedback such that the closed-loop transfer function is
G(s) = (s + 1)/(s2 + 3s + 3).
(a) What is the transfer function of the compensator?
(b) Sketch the Bode plots of the open loop, compensator, and closed loop.
3. A stable LTI system has frequency response

0≤ω≤π
 j,
H(ω) =
−j, −π ≤ ω ≤ 0

0,
otherwise
(a) What is the impulse response of the system?
(b) Is it causal?
(c) Does the system give a real output for all real inputs?
2
4. The step response of an LTI system is given by
s(t) =
0,
t≤0
1 − 0.5e−t + 0.5e−2t , t > 0
(a) Find its impulse response, transfer function, and frequency response.
(b) Find its steady state response to the input
∀t,
x(t) = cos(ωt)u(t).
(c) Find its response to the input
∀t,
x(t) = cos(ωt).
5. Determine which of the following transfer functions are stable and for those that are, determine the initial and steady state values to a step input.
s−1
,
s4 + 1
s2 − s + 1
,
s4 + 3s2 + 1
s+1
s4 + 2s3 + 3s2 + 4s + 1
6. The open loop plant has transfer function H(s) = s2 /(s2 + 2s + 1). Place the plant in a
closed loop using a PI controller K1 + K2 /s.
(a) Take K2 = 0, and plot the root locus as K1 varies. For what values of K1 is the system
stable? What is the steady state error to a step input as a function of K1 ?
(b) Select K1 , K2 such that the closed loop system is stable and has zero steady-state error
for step inputs.