Tutorial Sheet 1 Control Systems

Control Systems (EE302)
Tutorial - 1
Note:
If a particular method has been specified in a question, you will only get credit if you use
that method.
Ques num. 1: Find the Laplace transform of the following functions:

(a) 2e−2t sin(2t)1(t)
(b) t2 e−2t 1(t)
(c) (1 − e−10t )1(t)
(d) e2t u(t) + e−3t 1(−t)
(e) (t sin(2t) + e−2t )1(t)
(f) (e4t sin(3t) + 2e4t sin(3t))1(t)
(g) t2 et sin(5t)1(t)
(h) t3 1(t)
Ques num. 2: Derive the Laplace transform of the following (from definition):
(a) t2 1(t)
(b) sin(ω)1(t)
(c) e−at cos(ωt)1(t)
Verify each case using MATLAB.
Ques num. 3: Use initial value theorem to get f (0+ ) and f ′ (0+ )
s2
for F (s) = s2 −4s+9
.
Ques num. 4: Use final value theorem for obtaining f (∞) for F (s) given as:
2s − 7 2s + 7
(a) (b)
s(s2 + 0.1s + 4) s(s2 + 4)
Ques num. 5: Find the final values of x(t), ẋ(t) as t → ∞ for the following Laplace
transforms, X(s), given by the following.
2s+1
(a) s4 +8s3 +16s2 +s
2
(b) s(s2 −s−2)
5(s+2)
(c) s(s2 +4)
(4s2 +3s)
(d) (s2 +as−4)
,a ∈R
(2s−a)
(e) s(s2 +ϵs+4)
,a ∈ R and ϵ > 0
6s−31
(f) s2 (4s+3)
s2 +4s+7
(g) s2 +s+5
2
(h) s2 +4s+7
s2 +3s−2
(i) s(s+a)
,a ∈R
as+5
(j) s2 +3s−2
,a ∈R
Ques num. 6: Find the inverse Laplace transforms of the following transfer functions:
(a). G(s) = s(s5(s+2)
2 +6s+9)
s
(b). G(s) = (s2 +1)2
2
(c). G(s) = s s+1
+s+2
Show the steps required for hand-computation and also write the code required (in the
language of your choice) to verify your computation on a computer.
Ques num. 7: Find the impulse response of the following differential equation:
d2 d d
y(t) + 6 y(t) + 2y(t) = 2 r(t) + r(t)
dt2 dt dt
(Assume zero initial conditions)
Ques num. 8: : Solve following initial value problem using Laplace transform:
ÿ − ẏ − 6y = 2; y(0) = 1, ẏ(0) = 0
Ques num. 9: An input x(t) = e−2t u(t) + δ(t − 6) is applied to an LTI system with
impulse response h(t) = u(t). Find the output y(t) using Laplace transform. Verify your
answer by computing y(t) in time domain.
k(s−z)
Ques num. 10: Consider first order transfer function G(s) = s−p
with p < 0 and k ̸= 0.

(a) Find step response in terms of k, z, p.
(b) Show that the step response of G(s) can be written as

y(t) = y(∞) − (y(∞) − y(0+ ))ept .
(c) Show that when z > 0, there exists some t1 ∈ (0, ∞) such that y(t1 ) = 0.
Ques num. 11: Find inverse Laplace transform to get y1 (t) and y2 (t) for:
16 256
(a) Y1 (s) = (b) Y2 (s) =
(s + 4)2 (s2 + 4s + 16)2
Ques num. 12: A system with input u(t) and output y(t) is governed by the relation:
d d d2
u + u2 = y− y.
dt dt dt2
(a) Is the system linear? Give a brief reason.
(b) Is the system time-invariant? Give a brief reason.
(c) Is the system causal? Give a brief reason.
Ques num. 13: (4 marks) Determine if the following two systems are linear/nonlinear, time
invariant/time-varying, static/dynamic, causal/non-causal. Provide brief justifications for
your answer in each case.
(a) y(t) = 2tr(t) + 3r(t − 3)
(b) ẍ(t) + x(t) = r(t) and y(t) = 7x(t) + 2x(t), t ≥ 0
Ques num. 14: For the system in qn. 1(b) above, assume zero initial conditions. If possible,
design bounded inputs (i.e., ∃M ∈ R s.t. |r(t)| < M ∀t ), such that
(a) y(t) → ∞ as t → ∞
(b) y(t) → 0 as t → ∞
(c) limt→0 y(t) does not exist
Ques num. 15: Design an electrical circuit (with any components of your choice) with the
transfer function from input voltage Vi (s) to output voltage Vo (s) being
Vo (s) 5(s + 1)
=
Vi (s) s+2
Ques num. 16: In the circuit shown in the figure, switch K is moved from position ‘a’ to
position ‘b’ at t = 0 (a steady state existed at position ‘a’ prior to t = 0 ). Solve for
current i(t), using the Laplace transform.
Von (s)
Ques num. 17: (a) Find G(s) = Vin (s)
for the given RLC network (second figure below).
(b) Use series-parallel-based simplification rules for RLC.

(c) Write KCL/KVL in Laplace transformed variables (instead of time-domain variables)
(d) Find VI(s)
in (s)
and VI(s)
in (s)
, I is the current through the resistor where Von is measured
across.
Ques num. 18:

(a) The switch K in fig(a) is initially closed at ‘a’ till steady state is attained. Then at t=0,
it is thrown to position ‘b’. Obtain vout (t) for t > 0 using the Laplace transform suitably.
(b) For the circuit in fig(b), obtain the transfer function VVout (s)
in (s)
.
(c) For the active network in fig(c), write down suitable equations in transformed variables,
using KCL at nodes 1, 2, 3 and 4. Hence, obtain VVout (s)
in (s)
.
Ques num. 19: For the given translational mechanical system in the figure (on next page),
(a) Use a free body diagram to obtain the differential equations in terms of x1 (t), x2 (t)
and their derivatives.
Figure 1: Figure for Q18
X2 (s)
(b) Hence, obtain the transfer function G(s) = F (s)
(c) What are the poles and zeros of G(s)?
Ques num. 20:

Consider the relevant figure for this question.
(a) Find transfer function from f1 to x2 .
(b) Find transfer function from f2 to x1 .
(c) Write units for each of ki ,fi ,xi ,di explicitly.
Ques num. 21: For the rotational mechanical system given in the figure, draw both “series”
and “parallel” analog circuits and explain using dynamical equations for the system.
Ques num. 22: Rotational spring/mass/(moment of)inertia/torque.
(a) List units of k1 , d1 , θ2 , J2 , J1 (SI units).

d
(b) Find transfer function from T2 to dt θ2 (rate of change of θ2 ).
Ques num. 23: Motor/rotational motion example:

Find the transfer function from armature voltage ea (t) to θL (t)(load angle).
Vout (s)
Ques num. 24: For the Op-amp circuit shown in figure below, find transfer function Vin (s)

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Control Systems (EE302)

Ques num. 1: Find the Laplace transform of the following functions:

(b) t2 e−2t 1(t)

(c) (1 − e−10t )1(t)

(d) e2t u(t) + e−3t 1(−t)

(e) (t sin(2t) + e−2t )1(t)

(f) (e4t sin(3t) + 2e4t sin(3t))1(t)

with p < 0 and k ̸= 0.

(b) Show that the step response of G(s) can be written as

(b) Is the system time-invariant? Give a brief reason.

(c) Is the system causal? Give a brief reason.

(a) y(t) = 2tr(t) + 3r(t − 3)

(b) ẍ(t) + x(t) = r(t) and y(t) = 7x(t) + 2x(t), t ≥ 0

(b) Use series-parallel-based simplification rules for RLC.

Ques num. 18:

(c) What are the poles and zeros of G(s)?

Ques num. 20:

Ques num. 22: Rotational spring/mass/(moment of)inertia/torque.

Figure 3: Figure for Q22

(a) List units of k1 , d1 , θ2 , J2 , J1 (SI units).

Ques num. 23: Motor/rotational motion example:

Figure 5: Figure for Q24

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