Sheet 1:
Linear discrete systems - Sampling and Aliasing
Q1. Draw a block diagram of digital feedback control system. Show the
type of the signal at each point on the block diagram (whether
continuous, discrete, and ZOH (output of D/A) signals.
Q2. What are the advantages of digital control over analog control
systems?
Q3. Explain how to adjust the sampling interval using timer interrupts
and ballast coding.
Q4. Solve the following difference equation:
(a) x(k  2)  3x(k  1)  2 x(k )  0; x(0)  0, x(1)  1.
(b) x(k  2)  x(k  1)  x(k )  0; x(0)  1, x(1)  1.
(c) x(k  2)  1.3 x(k  1)  0.4 x(k )  0, x(0)  1, x(1)  0.
Q5. Given the difference equation
yk  0.5 yk 1  0.5 yk 2  0.25uk 1
(a) Find the discrete transfer function H(z) =Y(z)/U(z).
(b) Draw a block diagram of this discrete system using delays and gains.
(c) Find the pulse response of the system.
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Q6.Consider a discrete damped sinusoid:
 
ek  e(k )  r k sin k  0.5 k sin  k 
 2
Plot the signal and then find:
(a) The z-transform of ek,
(b) The pole/zero locations.
(c) The oscillation speed (number of samples/cycle).
Q7. Given the two poles s = -1, -2, find the equivalent poles in the zplane. Which of these two poles is faster in response? Which is the more
dominant?
Q8. The standard form of the continuous-domain transfer function for an
underdamped system is
n2
G( s)  2
s  2n s  n2
which has pole locations at
s  n  jn 1   2
Given corresponding z-plane complex poles at z = rexp(±jϴ), find the
equivalent damping ratio ζ and natural frequency ωn of these discrete
system poles in terms of r, ϴ, and sampling period T.
Answer:
n 
1
(ln r ) 2   2 ,  
T
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 ln r
(ln r ) 2   2
Q9. Consider a signal of frequency 10Hz, sampled at a rate of fs = 50Hz.
What are the frequencies that appear at the sampled signal? Does the
aliasing problem occur? Why?
Q10. Consider a signal of frequency 10Hz, sampled at a rate of fs = 15Hz.
What are the frequencies that appear at the sampled signal? Does the
aliasing problem occur? Why? How to solve it?
Q11. Consider a signal of frequency 10Hz, sampled at a rate of fs = 50Hz.
Assume that there is a noise component at 40 Hz. What are the
frequencies that appear at the sampled signal? Does the aliasing problem
occur? Why? How to solve it?
Q12. Using a MATLAB program, verify the ideal reconstruction method:
r (t ) 

 (t  kT )
k 
T
 r (kT )sinc


 r (kT )
k 
sin(  (t  kT ) / T )
( (t  kT ) / T )
For your sampled data, use 6 samples (k = 0...5) of a 1 Hz sine wave with
T = 0.2. Compute a full period of the reconstructed continuous signal at a
much finer time step, such as 0.01 second. Compare the reconstructed
signal using only 6 samples with the original continuous signal. It really
works!
Q13. What are the advantages of zero-order hold (ZOH) reconstruction
over the reconstruction using the ideal low pass filter (LPF)?
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Sheet 2:
The z-Transform
Q1. Obtain the z-transform for the following curve (assume T=1):
Q2. A function y(t) = 2sin(4t) is sampled every T = 0.1 s. Find the ztransform of the resultant number sequence.
Q3. Find the z-transform of the following function, assuming that T =
0.5 sec:
(a) Y ( s) 
1
,
s 2 ( s  1)
e Ts
(c ) Y ( s ) 
,
s( s  1)
( s  1)
(e) Y ( s ) 
,
s( s  2)
(b) Y ( s) 
1
,
s2
( s  3)
,
( s  1)( s  2)
s
( f ) Y (s) 
( s  1) 2
(d ) Y ( s) 
Check your answer using the tables of z-transform.
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Q4. Determine the z-transform of the following time domain functions.
(a) x(k )  k
(b) x(k )  k 2
(c) x(t )  1  e at
(d ) x(t )  teat
Hint: you can check your answer with MATLAB command ztrans. For
example, we can solve (d) using the following commands:
>> syms T a k z;
>> xk=(k*T)*(exp(-a*k*T));
>> xz=ztrans(xk,k,z)
Q5. For the discrete transfer function G(z) below:
G( z) 
1
z  0.5 z  0.5
2
Find:
(a) The unit pulse response.
(b) The unit step response. Verify the DC gain.
Q6. Determine the final value of the sequences whose z-transform is:
X ( z) 
1
1

(1  z 1 ) (1  e aT z 1 )
Q7. Find the inverse z-transform of X(z) using both long division and
partial fraction methods. Find also the steady state value of x(n).
10 z  5
,
( z  1)( z  0.2)
(a)
X ( z) 
(c )
X ( z )  1  2 z 1  3z 2  4 z 3 ,
( e)
X ( z) 
(b)
1  2 z  3z 2  4 z 3  5 z 4
,
z4
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(d )
(f)
1
1 z
z2
X ( z) 
.
z ( z  2)
X ( z) 
X ( z) 
10 z  5
.
z  1.2 z  0.2
2
Sheet 3:
Discrete Block diagrams
Q1. Assume that the following plant transfer functions are preceded by a
zero-order hold. Compute the equivalent discrete transfer function G(z)
using a sampling period T = 1. Check you answer with MATLAB.
(a) G ( s ) 
1
s2
1
s ( s  1)
1
(c ) G ( s )  2
s 1
(b) G ( s ) 
Q2. Calculate and plot the pulse response of the following system
assuming that the sampling period T = 1sec.
Solution:
The transfer function of the
ZOH is
G1 ( s) 
1  e Ts
,
s
For this system, we can write
y( z)  G1G2 ( z)u( z).
Now, T=1 and
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G1G2 ( s) 
1  e Ts
,
s 2 ( s  1)
Or by partial fractional expansion we can write
1 
1 1
G1G2 ( s )  (1  e  s ) 2  
.
s s 1
s
From the z-transform tables
 z
z
z  ze 1  1  2e 1

G1G2 ( z )  (1  z )


.
2
z  1 z  e 1  ( z  1)( z  e 1 )
 ( z  1)
0.3678 z  0.2644
 2
.
z  1.3678 z  0.3678
1
For a pulse input, u(z) = 1. Therefore the pulse response will be given by
y ( z )  G1G2 ( z )u ( z )  G1G2 ( z ) 
0.3678 z  0.2644
.
z  1.3678 z  0.3678
2
After long division, we obtain the time response
y (n)  0.3678 (n  1)  0.7675 (n  2)  0.9145 (n  3)  
Q3. Obtain the output C(z) for the discrete-time control system whose
block diagram is shown below:
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Q4. Obtain the output sequence c(kT) if the input r(t) is a unit step and
T= 1sec for the discrete-time control system whose block diagram is
shown below:
Final answer:
c(nT )  1  (1  k1 ) n .
Q5. Sketch the step response y(t) of the system shown below for the three
samples k = 0, 1, 2. Use sample period T = 1.
Q6. Obtain the output sequence c(kT) if the input r(t) is a unit step, T=
0.2 sec and K1 = 1 for the discrete-time control system whose block
diagram is shown below:
Final answer:
1
c(nT )  [1  (0.6374) n ]  {0,0.181,0.297,0.371,0.418,...}
2
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Sheet 4:
Stability
Q1. Check the stability of the discrete-time control systems having the
following characteristic equations:
(a) z 2  z  2  0
(b) z 2  0.81  0
(c) z 3  1.3z 2  0.08 z  0.24  0 ( stable)
(d ) z 3  1.2 z 2  1.44 z  0.32  0
(e) z 3  1.1z 2  0.1z  0.2  0
( f ) z 4  1.2 z 3  0.07 z 2  0.3z  0.8  0 (unstable)
Q2. Consider the following closed-loop discrete-time unity feedback
control system with sampling period T = 1 sec. Determine the range for K
for stability by use of Jury's test and also determine the frequency of
oscillations.
G( z ) 
K (0.368 z  0.264)
.
( z  0.368)( z  1)
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Q3. Consider the discrete-time control system described by the difference
equation,
y (k )  0.6 y (k  1)  0.81y (k  2)  0.6 y (k  3)  0.12 y (k  4)  x(k ),
where, x(k) is the input of system and y(k) is the output of system.
Determine the stability of the system.
Q4. A second-order discrete-time system has the following characteristic
equation:
F ( z )  z 2  a1 z  a0  0.
Show the region in the given a0–a1 plane for which the given system is
stable.
Page 10/20
Sheet 5:
Controller Design
Q1. A second-order continuous-time system is required to have a
damping ratio of 0.7, and a settling time of about 1 s.
(a) Find the system poles in the s-plane.
(b) Find the corresponding poles in the z-plane, using the mapping z =
esT, if the input and the output of the system are sampled every 0.1
sec.
Q2. The open-loop transfer function of a plant is given by:
e 4 s
G ( s) 
2s  1
Which is to be preceded by a ZOH circuit.
(a) Design a dead-beat digital controller for the system. Assume that =
1 s.
(b) Draw the block diagram of the system together with the controller.
(c) Plot the time response of the system.
Q3. Repeat Q2 using a Dahlin controller in order to achieve a closed-loop
first-order response with a time constant 1 sec. Plot the response and
compare with the results obtained from the dead-beat controller.
Q4. The open-loop transfer function of a unity feedback system is
G( s) 
10
.
s( s  10)
Assume that T = 1 s and design a controller so that the system response to
a unit step input is
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y(kT) = 0 , 0.4, 1, 1, …
Note that G(s) is to be preceded by a ZOH.
Q5. The open-loop transfer function of a system together with a zeroorder hold is given by
HG ( z ) 
0.2( z  0.8)
.
2
z  1.5 z  0.5
Design a digital controller so that the closed-loop system will have ζ =
0.6 and wd = 3 rad/s. The steady-state error to a step input should be zero.
Also, the steady state error to a ramp input should be 0.5. Assume that T
= 0.2 s.
Q6. The open-loop transfer function of a system is
e 0.16s
G( s) 
0.2s  1
The system is preceded by a sampler and a zero-order hold. The closedloop system is required to have a time constant of 0.4 s. assume the
sample period is 0.04.
(a) Determine the digital controller D(z).
(b) Plot the unit step time response of the system with the controller.
Q7. Design a digital controller, D(z), such that the poles of the following
closed-loop system are placed at z1,2 = 0.4 ± j0.4 in the z-plane. The
steady-state error in the step response must be zero.
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Q8. It is required to design a controller for the following system to
achieve percent overshoot (PO) less than 20 %, settling time ts ≤ 10 s
(2% criterion), and zero steady state error for step and ramp inputs.
Assume that the sampling time is, T = 0.1 s and K = 0.4.
Derive the transfer function of the required digital controller.
Q9. It is required to design a controller for the following closed-loop
system such that percent overshoot (PO) less than 15 % and settling time
ts ≤ 10 s (2% criterion), and zero steady state error for step and ramp
inputs. Assume that the sampling time is, T = 0.2 s and K = 0.4.
Derive the transfer function of the required digital controller.
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Q10. The open-loop unit step response of a system is shown below.
Obtain the transfer function of this system and use the Ziegler–Nichols
tuning algorithm to design discrete-time:
(a) proportional controller;
(b) PI controller;
(c) PID controller.
Q11. Explain what integral wind-up is when a PID controller is used.
How can integral wind-up be avoided?
Q12. 14. Explain what derivative kick is when a PID controller is used.
How can derivative kick be avoided?
Q13. The continuous-time PI controller has the transfer function
U ( s) K p s  K i

.
E ( s)
s
Derive the equivalent discrete-time controller transfer function using the
bilinear transformation:
s
2 z 1
.
T z 1
Page 14/20
Q14. A commonly used compensator in the s-plane is the lead lag, or lag
lead with transfer function
U (s) s  a

.
E (s) s  b
Find the equivalent discrete-time controller using the bilinear
transformation.
Q15. Consider the following model
y (k )  ay (k  1)  bu (k  1)  e(k )
where y(k) is the system output at instant k, and e(k) is an equation error.
Given the following input output data,
(a) deduce the matrix Φ.
(b) calculate the least squares estimate of a and b.
(c) calculate the model output.
(d) calculate the residuals.
Q16. Consider the following model
y(k )  a1 y(k  1)  a2 y(k  2)  bu(k  1)  e(k )
where y(k) is the system output at instant k, and e(k) is an equation error.
Given the same input output data in the previous question,
a)
b)
c)
d)
deduce the matrix Φ.
calculate the least squares estimate of a1, a2 and b.
calculate the model output.
calculate the residuals. Comment on the values of the residuals.
Page 15/20
Sheet 6:
State Space Design
Q1. Consider the following two degree-of-freedom system:
Let m = 1 and k = 1. Write state equations for this system using
x  x1 ,
x1 ,
x2 ,
T
x2 
Q2. Find the eigenvalues of the following matrix:
1 2 1


A  2 1 1
 1 0 2


Q3. Write a state space model for the following transfer function:
Y ( s)
s 1
 2
U ( s ) s  5s  6
Page 16/20
Q4. Write the transfer function of the following state space description:
0 1 
1
 x(t )   u (t ),
x (t )  
 3  2
 0
y (t )  1 0x(t ).
Q5. Given the following state space description:
0 1 
1
 x(t )   u (t ),
x (t )  
 3  2
 0
y (t )  1 0x(t ).
Define a new state z1 = 2x1 and z2 = 3x2. Write the state space model in
terms of the new states z1 and z2.
Q6. Find a discrete-time state space model of the following continuoustime system (using a sampling period of 1 sec). Check with MATLAB.
 1 0 
1
 x(k )   u (k ),
x(k  1)  
 0  2
1
y (k )  1 0x(k ).
Q7. Find a discrete-time state space description of a dc motor shown in
the following block diagram (using a sampling period of 1 sec).
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Q8. Given the following discrete-time system:
  0.5
x(k  1)  
 0
y (k )  1 0x(k )
1
1
 x(k )   u (k )
1
0
(a) Find the eigenvalues of the system.
(b) Find the transfer function of the system.
1
z 1
1  1 
 z  0.5  1   1 
1
   
 
1 0
H ( z )  C ( zI   ) 1   1 0 
z  1  0  ( z  0.5)( z  1)
z  0.5  0 
 0
 0
1
1
( z  1)
z  1 1  

( z  0.5)( z  1)
 0  ( z  0.5)( z  1)

1
( z  0.5)
Q9. Check the controllability of the following discrete-time systems
1 0
 0
 x(k )   u (k )
x(k  1)  
 0 2
1
Q10. Given the following discrete-time state space model:
 1 1
0
 x(k )   u (k ),
x(k  1)  
 0 1
1
y (k )  1 0x(k )
(a) Check if the system is controllable.
(b) If the system is controllable, design a feedback controller to obtain
the eigenvalues {0.2, 0.4}. Check using MATLAB.
(c) Design a set point gain such that the steady state error to a step
input is zero.
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Q11. Given the following discrete-time state space model:
1 0 
1
 x(k )   u (k ),
x(k  1)  
1 2 
0
y (k )  1 0x(k )
(a) Check if the system is controllable.
(b) If the system is controllable, design a feedback controller to obtain
the eigenvalues {0.3±j0.3}. Check using MATLAB.
(c) Design a set point gain such that the steady state error to a step
input is zero.
Q12. Given the following discrete-time state space model:
 2 1 2
1


 
x(k  1)   1 0 0  x(k )   0 u (k ),
 0 1 0
0


 
y (k )  1 0 0x(k )
(a) Check if the following pair is controllable.
(b) If the system is controllable, design a feedback controller to obtain
the eigenvalues {0.2, 0.3±j0.3}. Check using MATLAB.
(c) Repeat (b) to locate the eigenvalues at {0, 0.1, 0.2}. Check using
MATLAB.
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Q13. Check the observability of the following discrete-time systems
1 0
0
 x(k )   u (k )
(a ) x(k  1)  
0 2
1
y (k )  0 1x(k )
1 1
1
 x(k )   u (k )
(b) x(k  1)  
0 2
1
y (k )  1 0 x(k )
In each case, if the system is observable, design a state observer for the
system. The observer eigenvalues are to be located at 0.1, 0.2. Check
your answer with MATLAB.
Q14. Given the following state space model:
 3 0
1
 x(k )   u (k )
x(k  1)  
 0 2
1
y (k )  1 1x(k )
Where the following observer gain matrix
 6.875 

L  
2.625


Find the eigenvalues of the observer.
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