CSE 417/421 Digital Control - Sheets

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 equations:

(a) x(k  2)  3x(k  1)  2 x(k )  0; x(0)  0, x(1)  1.
(b) x(k  2)  1.3x(k  1)  0.4 x(k )  0; x(0)  1, x(1)  0.

Q5. Given the difference equation

yk  yk 1  yk 2  0.5uk 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.

Q6.Consider a discrete damped sinusoid:

 
ek  e(k )  r k sin k  0.5k sin k 
 2
Plot the signal and then find:

(a) The z-transform of ek,

(b) The pole/zero locations.
(c) The number of samples per cycle.

Q7. Given the two poles s = -1, -2, find the equivalent poles in the z-
plane. Which of these two poles is faster in response? Which is the more
dominant?

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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 = r exp(±jϴ), find the
equivalent damping ratio ζ and natural frequency ω n of these discrete
system poles in terms of r, ϴ, and sampling period T.

Answer: 1  ln r
n  (ln r ) 2   2 ,  
T (ln r ) 2   2
Q9. The following transfer function is a lead network designed to add
about 60 phase lead at ω = 3 rad/sec:
s 1
H ( s)  10
s  10
Find the discrete simulations of H(s) using each of the following
methods: (i) Forward rectangular (ii) Backward rectangular and (iii)
Trapezoidal rules. Also, compute the amount of phase lead given by the
network at z = ejωT. Let T = 0.25 sec.

Q10. 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?

Q11. 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?

Q12. 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?

Q14. What is the advantage 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 of the discrete sequence obtained by sampling

the following continuous signal with sample period T=1.

Q2. A function y(t) = 2sin(4t) is sampled every T = 0.1 s. Find the z-

transform of the resultant number sequence.

Q3. Find the z-transform of the following function, assuming that T =

0.5 sec:

1 1
(a) Y ( s )  , (b) Y ( s )  ,
s ( s  1)
2
s2
e Ts ( s  3)
(c ) Y ( s )  , (d ) Y ( s )  ,
s ( s  1) ( s  1)( s  2)
( s  1) s
( e) Y ( s )  , ( f ) Y (s) 
s ( s  2) ( s  1) 2

Check your answer using the tables of z-transform.

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

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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:

1
G( z ) 
z 2  0.5 z  0.5
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:

1 1
X ( z)  
(1  z 1 ) (1  e aT z 1 )

Q7. Find the inverse z-transform of X(z) using either long division or
partial fraction method. Find also the steady state value of x(n).

10 z  5 1
(a) X ( z)  , (b) X ( z) 
( z  1)( z  0.2) 1 z
z2
(c ) X ( z )  1  2 z 1  3z 2  4 z 3 , (d ) X ( z)  .
z ( z  2)
1  2 z  3z 2  4 z 3  5 z 4 10 z  5
( e) X ( z)  , (f) X ( z)  .
z4 z  1.2 z  0.2
2

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Sheet 3
Block diagrams of discrete-time systems

Q1. For each of the following plant transfer functions, compute the
equivalent discrete transfer function G(z) using a sampling period T = 1.
Assume that each is preceded by a zero-order hold. Check you answer
with MATLAB.

1 1
(a ) G ( s )  (b) G( s) 
s2 s( s  1)

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 1  e Ts
G1 ( s)  ,
s

For this system, we can write

y( z)  G1G2 ( z )u( z ).
Now, T=1 and

1  e Ts
G1G2 ( s)  2 ,
s ( s  1)

By partial fractional expansion we can write

1 1 1 
G1G2 ( s)  (1  e  s ) 2   .
s s s 1
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From the z-transform tables

 z
1 z z  ze1  1  2e 1

G1G2 ( z )  (1  z )   
1  1
.
 ( z  1) 2
z  1 z  e  ( z  1)( z  e )
0.3678 z  0.2644
 2 .
z  1.3678 z  0.3678

For a pulse input, U(z) = 1. Therefore the pulse response will be given by
0.3678 z  0.2644
y( z )  G1G2 ( z )u ( z )  G1G2 ( z )  .
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 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  K ) n .

Q4. Sketch the step response y(t) of the system shown below for the three
samples k = 0, 1, 2. Use sample period T = 1.

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Q5. Calculate the output response of the following sampled-data system

to a unit step input. Assume that T = 1 s.

Answer:

The transfer function of the closed-loop system is

G( z) G( z) z
Y ( z)  R( z )  . ,
1  G( z) 1  G( z) z  1
1 z (1  e T )
G(s)   G( z)  .
s( s  1) ( z  1)( z  e T )
Substituting in y(z) gives
z 2 (1  e T )
Y ( z)  T T
.
( z  1)( z  2e z  e )
2

Since T = 1,

0.632 z 2
y( z )   0.632 z 1  1.096 z 2  1.25 z 3  
z  1.736 z  1.104 z  0.368
3 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.

K (0.368 z  0.264)
G( z )  .
( z  0.368)( z  1)

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.

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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
T=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

10
G( s)  .
s( s  10)
Assume that T = 1 s and design a controller so that the system response to
a unit step input is

y(kT) = 0 , 0.4, 1, 1, …

Note that G(s) is to be preceded by a ZOH.

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Q5. The open-loop transfer function of a system together with a zero-

order hold is given by
0.2( z  0.8)
HG( z )  .
z 2  1.5 z  0.5
Design a digital controller so that the closed-loop system will have ζ =
0.6 and wn = 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 closed-
loop 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.

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.

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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.

Q10. The open-loop unit step responses of two systems are shown below.
Obtain the transfer function of these systems and use the Ziegler–Nichols
tuning algorithm to design discrete-time PI controller.

Q11. The open-loop unit step responses of two systems are shown below.
Obtain the transfer function of these systems and use the Ziegler–Nichols
tuning algorithm to design discrete-time PID controller.

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Q12. 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: 2 z 1
s .
T z 1
Q12. 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.

Q13. 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.

Q14. 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) Deduce the matrix Φ.
(b) Calculate the least squares estimate of a1, a2 and b.
(c) Calculate the model output.
(d) Calculate the residuals. Comment on the values of the residuals.
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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 the
following state variables:

x   y, y , z, z 
T

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
Q4. Write the transfer function of the following state space description:

0 1  1
x (t )    x(t )   u (t ),
 3  2  0
y (t )  1 0x(t ).

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Q5. Given the following state space description:

0 1  1
x (t )    x(t )   u (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 continuous-

time system (using a sampling period of 1 sec). Check with MATLAB.

 1 0  1
x (t )    x(t )   u (t ),
 0  2 1
y (t )  1 0x(t ).
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).

Q8. Given the following discrete-time system:

  0.5 1 1
x(k  1)    x(k )   u (k )
 0 1  0
y (k )  1 0x(k )

(a) Find the eigenvalues of the system.

(b) Find the transfer function of the system.
1
 z  0.5  1   1  z 1
1 0
1  1 
H ( z )  C ( zI  ) 1   1 0
1
     
 0 z  1  0  ( z  0.5)( z  1)  0 z  0.5  0 

z  1 1   ( z  1)
1 1

( z  0.5)( z  1)  0  ( z  0.5)( z  1)
1

( z  0.5)

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Q9. Check the controllability of the following discrete-time systems

1 0  0
x(k  1)    x(k )   u (k )
 0 2 1
Q10. Given the following discrete-time state space model:

 1 1  0
x(k  1)    x(k )   u (k ),
 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.

Q11. Given the following discrete-time state space model:

1 0  1
x(k  1)    x(k )   u (k ),
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.

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(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.

Q13. Check the observability of the following discrete-time systems

1 0 0
(a) x(k  1)    x(k )   u (k )
0 2 1
y (k )  0 1x(k )
1 1 1
(b) x(k  1)    x(k )   u (k )
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  1)    x(k )   u (k )
 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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