HOMEWORK 3: Continuous Signals and Systems
1) Classify each of the following signals as continuous vs. discrete, analog vs. digital, periodic vs. aperiodic,
deterministic vs. random:
(a) The letter grades you take from the courses in the Department.
(b) Your GPA trajectory (or history, if you’d like to call it).
(c) One month temperature reading of outside.
(d) Five years temperature reading of outside.
(e) Solution of an ordinary linear differential equation.
2) Using f(t) = -t{u(t+1)-u(t)} on the RHS, determine the explicit
mathematical expression and sketch the following signals:
(a) f(t/2)
(b) f(t-1)
(c) f(1-t)
3) Evaluate the following integrals (RECALL: u(t) is the unit step function):

(a)
   d
(e)




(b)
1
   d
f   t   d
(c)
1
(f)


5
e
t
5


1
  d
 t  10 dt
(d)



(g)
f    d

 u t  t  3dt

4) For the following scenarios, what are the (input and output) signals and the system, and can you reach a
conclusion about the linearity of the system with the given information? If yes, what is your conclusion? If
no, please explain. Please refer to the linearity rules in your judgement, and do not write long storie. Note
that all the perturbations are in step-change fashion.
(a) At steady state (s.s.), temperature of a reactor (T) is 200 oC and the coolant flow rate is 5 L/s. When
cooling flow rate is increased to 6 L/s, T reaches 150 oC at s.s.
(b) At s.s., distillate concentration is 2.5 mol/L at a reflux rate of 100 L/min and a reboiler heat duty of 2 kW.
When the reflux rate is increased (at constant reboiler heat duty) to 120 L/min, distillate concentration
reaches 2.6 mol/L at s.s. When the reflux rate is decreased (at constant reboiler heat duty) to 60 L/min,
distillate concentration drops down to 2.3 mol/L at s.s.
(c) Identical to part (b), with the following given information. For the first perturbation, it takes 20 min to
(practically) reach the s.s., while it takes 40 min to (practically) reach the s.s. for the second perturbation.
(d) Identical to part (b) and excluding part (c), with the following given information. When the reboiler heat
duty is increased (at constant reflux rate) to 2.3 kW, distillate concentration drops down to 2.2 mol/L at s.s.
When reboiler head duty is decreased to 1.8 kW and reflux is increased to 140 L/min simultaneously,
distillate concentration is increased to 2.9 mol/L at s.s.
5) A LTI system, with all initial conditions equal to zero, is specified by the following equation:
d2y
dy
df
 5  6y 
 f
2
dt
dt
dt
(a) Determine and sketch the impulse response of the system. Is the system causal? Why?
HINT: To plot functions, you may use ezplot on MATLAB, e.g. ezplot(‘exp(-t)’, [0 10]) plots e-t for 0<t<10.
(b) If f(t) = e-tu(t), then determine the zero-state response of the system, using Laplace transforms and
numerical (not graphical!) convolution, separately. Make sure that results are identical. Sketch the response
using MATLAB.
6) Find and sketch the response of the following system (impulse response, IR) to the following input signal
using graphical convolution.
Before starting the graphical convolution, try to answer these questions without any calculation, but only
via analyzing the system.
(a) Is the system causal? Is the system time-invariant? Does the system have a time delay?
(b) What is the smallest time value at which you are going to observe a non-zero response? Why?
(c) Will the response be exactly zero in finite time?
(d) If an impulse at t = -10 min is applied on the system, what will be the system’s response?
(e) Now you may continue with the question, perform the graphical convolution and determine y(t).
7) Plot the impulse and step responses of the following systems using MATLAB (not SIMULINK). Try to explain
their impulse and step responses, e.g. why the initial response or derivative is zero/non zero, why the
responses of 2nd order process are not identical, what might be rationale behind these differences, etc.
1
1
g 2 s  
,
,
5s  110 s  1
5s  1
20 s  1
g 5 s  
5s  110 s  1
g1 s  
g 3 s  
2s  1
,
5s  110 s  1
g 4 s  
8s  1
,
5s  110 s  1
8) Use SIMULINK for this question. For the following process and disturbance transfer functions
1
1
g p s  
and g d s  
, determine
5s  110 s  1(20 s  1)
2s  1
(a) Openloop response of the process given a unit step change only to the manipulated variable.
(b) Openloop response of the process given a step change of magnitude -1 only to the disturbance.
(c) Both perturbation in (a) and (b) are acting on the process simultaneously. Why do you think the initial
response is in the direction of the perturbation?
(d) Regulatory control of the process against a step change of magnitude -1 in the disturbance. Use a P-only
control. Change (play with) the Kc value as you like to obtain a desirable response from the process. Compare
the controlled process and uncontrolled process; what is the benefit of feedback control?
(e) If you swap gp and gd and run the same process in closedloop manner, is the regulatory control
performance of the new process (you may change the Kc value if you like) better or worse compared to the
previous one? What is the reason for the observed difference?