Final Assessment for Advanced Control Systems

(EEET 2100 and 1368)

Question One Multiple Choices (30 marks)

1. The transfer function of a system is given by

G(s) =

1

s(s+ 1)(s+ 2)

For a unit step input signal, the steady-state value of the step response

is

(a) 0.5;

(b) 0;

(c) ∞;

(d) −1;

(e) none of the above.

2. The transfer function of a system is given by G(s) = 1

(s+2)2

and the pro-

portional controller is K = 1. The complementary sensitivity function

T (s) is calculated as

(a) T (s) = 4

s2+4s+4

;

(b) T (s) = 1

s2+4s+5

;

(c) T (s) = s

2+4s+4

s2+4s+5

;

(d) T (s) = 1

s2+4s+4

;

(e) none of the above.

3. The transfer function of a system is given by G(s) = 1

s+1

. The PI con-

troller is chosen to have the structure C(s) = Kc

s+1

s

where proportional

control gain was chosen to be Kc = 10. In order to reduce the effect of

disturbance, you would choose

(a) Kc = 3;

(b) Kc = 10;

(c) Kc = 30;

(d) Kc = 6;

(e) all the above.

4. The transfer function of a double integrating system is given by G(s) =

10

s2−1

. To design a proportional plus derivative controller (PD) without

filter, we could select the desired closed-loop poles as

(a) −10, −12, −13, −14;

(b) 10, 12, 13, 14;

(c) −10, −10,−100;

(d) −10, −10.

(e) none of the above.

5. The transfer function of a system is given by G(s) = 1

s

. The input

disturbance for this system is known to be a combination of a constant

and a sinusoidal signal with frequency ω0 = 1 Hz. To reject this dis-

turbance without steady-state error, we choose the resonant controller

to have the structure:

(a) C(s) = c3s

3+c2s2+c1s+c0

s(s2+1)

;

(b) C(s) = c2s

2+c1s+c0

(s2−1)

;

(c) C(s) = c2s

2+c1s+c0

(s2+2pi)

;

(d) C(s) = c3s

3+c2s2+c1s+c0

s(s2+2pi)

;

(e) none of the above.

Question Two (20 marks)

The transfer function of a process unit in a resource company is described

by:

G(s) =

1

(s+ 1)(s+ 8)2

e−6s

Design a PID controller with filter to control this time delay system.

1. (10 marks) You may use the MATLAB program pidplace.m to find

the PID controller parameters or you may choose to find the PID con-

troller parameters analytically. Present the PID controller parameters

for the three cases where all the desired closed-loop poles are at −1,

−0.2 and −0.1, respectively.

2. (10 marks) Assuming that the reference signal to the system is 1, sim-

ulate the closed-loop responses for all the three cases. In the simulation

of the PID control system, the derivative control is implemented on the

output only. Present the control signals and output signals graphically.

State your sampling interval and simulation time.

Question Three (30 marks)

A complex system is controlled by the cascade PID control system as illus-

trated in the following figure, where the inner-loop controller is a proportional

controller with gain of K to stabilize the unstable system and the outer-loop

is controlled by a PID controller.

1. (10 marks) Choose the inner-loop controller gain K such that the

inner-loop system has a closed-loop pole at −10 with kp = 1 and find

the controller parameters c2, c1, c0 in the outer-loop PID controller

C(s), where

C(s) =

c2s

2 + c1s+ c0

s

; G2(s) =

2

(s+ 1)(s+ 8)

All desired closed-loop poles for the outer-loop system are chosen to

be −1. (Hint: you may use pole-zero cancelation technique to simplify

the computation)

2. (10 marks) The parameter kp is varying and depending on the op-

erating conditions. In order to guarantee the closed-loop stability for

the variation of kp, the maximum and minimum values of kp need to

be determined. Use Routh-Hurwitz stability criterion to determine the

maximum and minimum values of kp allowed for the closed-loop control

system.

3. (10 marks) With a step reference signal r = 1, simulate the cascade

control system for the nominal case where kp = 1, and the cases with the

maximum and minimum values of kp determined by the Routh-Hurwitz

stability criterion. You need to choose an appropriate sampling interval

∆t and simulation time Ts. You may include a derivative filter for the

outer-loop PID controller implementation.

- m

6

-C(s) - m- K - kps−10 -G2(s) -

6

+ +

- -

R(s) U(s)∗ Y (s)U(s)

Question Four (20 marks)

Use the disturbance-observer based approach to design a PID controller for

the following system:

G(s) =

2

s2

1. (5 marks) Choose the desired closed-loop characteristic polynomial

for the proportional plus derivative controller as s2+2ξwns+w

2

n where

ξ = 0.707 and wn = 3, while the pole for the estimator is −4. What

are the values of K1, K2 and K3?

2. (8 marks) Simulate the closed-loop step response and input distur-

bance rejection where the derivative filter time constant τf = 0.1τD. In

the simulation, the reference signal r = 1 and the input disturbance

has an amplitude of −3 entering the simulation at half of the simula-

tion time. The sampling interval ∆t = 0.001 and the simulation time

Tsim = 3. What are the maximum and minimum values of the control

signal?

3. (7 marks)Evaluate the effect of constraints on the control signal where

the constraint parameters umax and umin are chosen to be 90 percent

of the control signal’s maximum and minimum amplitude from the

previous step.

Submit your MATLAB/Simulink programs together with your solutions,

simulation results (control signal and output plots) and discussions.