ELEC4632 Computer Control Systems Final Examination, Term 3, 2022
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School of Electrical Engineering and Telecommunications
ELEC4632 Computer Control Systems
Final Examination, Term 3, 2022
Question 1 (10 marks)
Sample the continuous-time system
˙(x)(t) =
using the sampling interval h = 0.4. Determine the pulse transfer function from u to y.
Question 2 (10 marks)
A discrete-time control system is described by
|
|
0.5 |
0 |
0.1 |
|
1 + a2 |
0 |
-4 |
|
|
|
x(k + 1) = |
1 - a |
-0.1a |
-7 |
|
7 |
a - 3 |
1 |
|
x(k). |
|
|
-0.2 |
0 |
-0.4 |
|
0.4 |
0 |
2 + a2 |
|
|
where the parameter a varies from -1 to 1. Determine for which values of a this system is
(a) observable ;
(b) detectable.
Question 3 (10 marks)
Given the following discrete time system
x(k + 1) = Gx(k) + Hu(k)
y(k) = Cx(k)
where
G = 0 1 0 , H = 1 , C =h 0 0 1 i.
Design
(a) State-feedback controller with the desired closed loop poles are 0.0, 0.8 j0.25.
(b) Observer with the desired observer poles are 0.0, 0.4 j0.4.
(c) Output feedback controller combining the state-feedback controller from (a) and the observer from (b).
Question 4 (10 marks)
The characteristic equation of a discrete-time control system is given by
z2 + Kz - K + 0.4 = 0
where the parameter K varies from -1 to +1. Determine the range of the parameter K for stability.
Question 5 (10 marks)
Consider the following nonlinear discrete-time system
x(k + 1) = cos(x(k)y(k))x(k) + sin(x(k)y(k))y(k),
y(k + 1) = sin(x(k)y(k))x(k) - cos(x(k)y(k))y(k).
(a) Is this system globally asymptotically stable?
(b) Is the singular point (0, 0) of this system asymptotically stable?
(c) Is the singular point (0, 0) of this system stable in the sense of Lyapunov?
(d) Find all singular points of this system.
Question 6 (10 marks)
Consider the optimal control problem for the system
x(k + 1) = 2x(k) + u(k)
with initial condition x(0) = 0.9, the cost function
J = 
x2 (k) ! min
and the constraint
ju(k)j 1.
Determine the optimal control strategy and the optimal value of the cost function.
2023-12-01