COURSE: AEEC433
DISCRETE-TIME CONTROL SYSTEMS
TAKEHOME EXAM
DUE DATE: 5th MAY 2010
INSTRUCTIONS TO CANDIDATES
Please prepare a report answering all questions. All questions carry equal marks. All
necessary working must be shown. You are reminded of the necessity for proper English
and orderly presentation of your answers. Calculators may be used.
QUESTION 1
By making use of the definition of the z-transform, determine the z-transform of the following
functions:
(a) f (t )  e  at for t  0 (b) f (t )  sin  t for t  0 (c) f (t )  cos  t for t  0
QUESTION 2
Consider the system described by the following equations:
Y ( z)
z 1
 2
U ( z ) z  1.3 z  0.4
Derive the state-space representation of the system in:
(a) Controllable canonical form
(b) Observable canonical form and
(c) Diagonal canonical form
QUESTION 3
Consider the continuous-time system given by:
G( s) 
Y ( s)
1

U ( s) s( s  2)
(a)Show that a continuous-time state-space representation of the system is given by:
 x1  0 1   x1  0
 x   0  2  x   1 u , and
  2  
 2 
x 
y (t )  1 0  1 
 x2 
(b) Derive a discrete-time state-space representation of the above system when the
sampling period T = 1.
(c )Obtain the pulse transfer function of the system using the equation:
F ( z )  C ( zI  G) 1 H
AEEC433.TAKEHOME-TEST-APRIL2010
1
(d) Also, obtain the pulse transfer function of the system by taking the z-transform of
the transfer function G (s ) (when is preceded by a sampler and a zero-order hold).
QUESTION 4
Consider the following pulse-transfer-function system:
Y ( z)
z 1 (1  0.8z 1 )

U ( z ) 1  1.3z 1  0.4 z 2
(a)A state-space representation for this system can be given by:
1   x1 (k )  0
 x1 (k  1)   0
 x (k  1)   0.4  1.3  x (k )  1 u (k )
 2   
 2
 
 x (k ) 
y(k )  0.8 1  1 
 x 2 (k )
Show that this system representation gives a system that is state controllable but not
observable.
(b) A different state-space representation for the same system can be is given by:
 x1 (k  1)  0  0.4  x1 (k )  0.8
 x (k  1)  1  1.3   x (k )   1  u(k )
  2   
 2
 
 x (k ) 
y(k )  0 1  1 
 x 2 (k )
Show that this system representation gives a system that is not completely state
controllable but is observable.
(c) Explain what causes the apparent difference in the controllability and observability of
the same system.
QUESTION 5
Obtain the state transition matrix of the following discrete-time system:
x(k  1)  G x(k )  H u (k )
y (k )  C x(k )
where
1
 0
G
 ,

0
.
16

1


1
H  ,
1
C  1 0
Use two methods to determine a suitable state feedback gain matrix K such that the
system will have the closed-loop poles at:
z  0.5  j 0.5 and z  0.5  j 0.5
QUESTION 6
Consider the following discrete-time system:
AEEC433.TAKEHOME-TEST-APRIL2010
2
x(k  1)  G x(k )  H u (k )
y (k )  C x(k )
where
1
 0
G
 ,
 0.16  1
1
H  ,
1
C  0 1
(a) Design a full-order state observer with desired eigenvalues for the observer matrix:
z  0.5  j 0.5 and z  0.5  j 0.5
QUESTION 7
Consider the control system defined by:
1   x1 (k )   1 
 x1 (k  1)   0
 x (k  1)   0.16  1  x (k )  0.5 u (k )
  2   
 2
 
with initial conditions:
 x1 (0)   1 
 x (0)   1
 2   
(a) Show that the system is completely controllable and
(b) Determine a sequence of control signals u (0) and u (1) such that the state

vector x ( 2) becomes:
 x1 (2)   3
 x (2)   1 
 2   
AEEC433.TAKEHOME-TEST-APRIL2010
3