DUE DATE: 1st Week May 2010
HOMEWORK ASSIGNMENT No.3
COURSE: AEEC433
DISCRETE-TIME CONTROL SYSTEMS
Dr. Tassos Andronikou
INSTRUCTIONS TO CANDIDATES
Prepare a presentable report answering ALL questions. 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
PART A
(a) By selecting the proper state-space variables convert the Initial Value Problem:
y(t )  2 y (t )  u (t ) ;
y (0)  0,
y (0)  0
into a state-space matrix equation of the form:
x (t )  A x(t )  B u (t )
x(t 0 )  c
(b) By using the Eigenvalue Method compute the state-transition matrix,  (t)= e At
(c) Using the equation for the general solution:
x(t )  e A(t t0 ) x(t 0 )  e At
t
e
 A
B( )u ( )d
t0
(i)
Find the solution of the Initial Value Problem, assuming that the
forcing function u(t) is the unit step function.
(d) Determine the closed-loop poles of the system and commend on its stability.
(e) By using the Ackerman’s Method determine the state-variable feedback control gain
matrix K, so that the closed-loop system poles are placed at s  2 and s  3 .
PART B
(a)Using the state-space matrix equation of PART A (a) obtain a discrete-time
representation of the form:
x[( k  1)T ]  G (T ) x(kT )  H (T ) u (kT )
y (kT )  C x(kT )
(b) Assuming that the period is T  1 sec write the discrete equations for the system and
the component states x1 (kT ) and x2 (kT ) , explicitly, for k  1,2,3, and 4 .
.
FUCAEEC433.HOMEWORK.ASSIGNMENT.No3.SPRING2010
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HOMEWORK ASSIGNMENT No.3
COURSE: AEEC433-DISCRETE-TIME CONTROL SYSTEMS
(c) By using the Ackerman’s Method for discrete systems determine the state-variable
feedback control gain matrix K, so that the closed-loop system poles are placed at
s  2 and s  3 .
(d) Assuming that the period is T  1 sec write the discrete equations for the system and
the component states x1 (kT ) and x2 (kT ) , explicitly, for k  1,2,3, and 4 .
(e) Plot and compare the results of PART B (b) and (d) and commend on the effect of
the “new” poles on the system response.
QUESTION 2
(a) By selecting the proper state-space variables convert the Initial Value Problem:
y(t )  5 y (t )  6 y (t )  u (t ) ;
y (0)  0,
y (0)  0
into a state-space matrix equation of the form:
x (t )  A x(t )  B u (t )
x(t 0 )  c
(b) The state-transition matrix,  (t), is given by the relation:

(t )  e At  L1 (sI  A) 1

By using the above relationship and Laplace Transforms:
(i)
(ii)
calculate the state-transition matrix,  (t), and
obtain the inverse of the state transition matrix,  1 (t ) .
(c) Using the equation for the general solution:
x(t )  e A(t t0 ) x(t 0 )  e At
t
e
 A
B( )u ( )d
t0
(i)
Find the solution of the Initial Value Problem assuming that the forcing
function u(t) is the unit step function.
(d) Determine the closed-loop poles of the system and commend on its stability.
(e) Plot and compare the results of QUESTION 1-PART A (c) and QUESTION 2-(c)
and commend on the effect of the “new” poles on the system response.
(f) Plot and compare the results of QUESTION 1-PART B (d) and QUESTION 1-(c)
and commend on the similarity or difference of the discrete and continuous systems.
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