Download State-space model - Control Engineering - Exam and more Exams Materials science in PDF only on Docsity!

Cork Institute of Technology

Bachelor of Engineering (Honours) in Electronic Engineering -Award

(NFQ – Level 8)

January 2007

CONTROL ENGINEERING

(Time: 3 Hours)

INSTRUCTIONS:

Answer any FIVE questions.

Each question is worth 20 marks.

Examiners: Dr. T O’Mahony

Prof. G. Hurley

Dr. S. Foley

Q1. (a) List the main advantages of using a state-space model to represent a system.

[5 marks]

(b) A dc motor has the transfer function

( )

3 2

+ + +

+

=

s s s

s

Gs

It is desired to control the motor using a state-feedback controller, such that the

closed-loop system has an overshoot of less than 5% and a settling time of

approximately 5sec. Determine the locations of the dominant closed-loop poles to

achieve these specifications and subsequently design a state-feedback controller to

satisfy the design objectives.

[15 marks]

Q2. Consider the system model

( )

(2 1)

s G s e s

−

Design a digital cancellation pole-placement controller for this system. The

objective of the design is to achieve a closed-loop response that is twice as fast as

the open-loop response and the closed-loop system should accurately track step-

like set-point signals and reject step-like disturbances.

[20 marks]

Q3. (a) The process reaction curve and ultimate gain method are two common methods for

tuning PID controllers. Briefly describe the characteristic features of each tuning

method with reference to the limitations (or disadvantages) of the methods. Is it

possible to overcome these limitations and, if so, briefly describe how you would

do so.

[10 marks]

(b) The response of an industrial process to a step input is illustrated in Figure Q3(b).

Determine the process gain ( k s ), time constant, ( τ ), and time-delay, ( τ d ). Design a

PI controller for this system using the tuning rule (Rivera & Jun)

( 2 )

c s d

K

k

τ

τ λ

=

+

T i = τand

f d

T

τλ

τ λ

=

+

. This tuning rule assumes that the PI controller is defined

by ( ) Ts T s

G s K

i f

c c

















= +

1. Clearly explain (i) the purpose of the coefficient

T (^) f and (ii) how you would choose the coefficient^ λ^.

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

1

2

Time (sec.)

Amplitude

Figure Q3(b)

[10 marks]

Q4. (a) Consider the feedback systems illustrated in figures Q4(a) (i) – (iv). Identify which

are linear and which are non-linear feedback systems, giving reasons for your

answer. Briefly explain how you would mathematically analyse [e.g. how would

you determine whether the feedback system was stable or how would you go about

the process of designing a controller?] each of these systems, giving reasons for

your answer.

M=

R(s)

K=3 (^) Y(s) E(s) U(s)

( 1 )( 5 )( 10 )

50

s + s + s +

actuator

process

-

• (^) E

U

(i)

(ii)

Controller (^) ZOH ( 2 )

s +

R(s) (^) Y(s)

T=0.

E(s) e(k) u(k) U(s)

u ( k )= e ( k )+ 0. 5 e ( k − 1 )+ u ( k − 1 )

(iii)

( ) ( )

2

2

sin '( )

d Y t (^) g Y t U t dt L

• = ; g, L constants;

( ) 1

( )

p i

U s K E s T s

= +

(iv)

Figure Q4(a)

[10 marks]

Controlle^ Plant r(t) + u(t)

-

y(t)

( 3 )

+

−

s s

s

e(t) 3

e ( ) t

Gc

R(s) (^) Y

E U U′

saturation

Gp

(b) How can the non-linear actuator shown in Figure Q4 (b) be used to determine the

parameters of a PI controller? Given that the describing function for the non-

linearity is

4 M

D

π a

= , design a PI controller for the system using the Ziegler-

Nichols tuning rules summarised in Table Q4(b).

Table Q4(b): PID tuning using Ziegler-Nichols

ultimate gain method

Controller Type K P TI TD

Proportional-only 0. 5 KU

Proportional +

Integral

0. 45 KU

T U

PID 0. 6 KU

T U

T U

M=

R(s)

Y(s) E(s) U(s)

( 1 )( 5 )( 10 )

s + s + s +

Non-linear actuator

Linear process

-

+

E

U

Figure Q4 (b)

[10 marks]

Q5. (a) The frequency response of an industrial process is illustrated in Figure Q5(a).

Identify a model for the system from this data.

[10 marks]

(b) A flow-chart for a typical system identification procedure is illustrated in Figure

Q5(b). With reference to your laboratory practice, describe how you would (or

did) apply this system identification procedure to an industrial or real-life

situation.

[10 marks]

Figure Q5(a)

Final

goal

A priori

knowledge

Design of experiments

Sampling time selection Input signals

Signal filtering

Application of identification algorithm

Process model

Model verification

Final model

Model order

determination

Assumption of model structure

Task Economy

Physical laws

Pre-measurements Operating conditions

Yes

No

• -44..55ddBB

-1- 1442 2ddeegg

**-

-**

0

Magnitude (dB)

10

- 10 - 10 - 10 - 10

0

-54 0

-36 0

-18 0

0

Phase (deg)

Bode Diag ra m

Frequency (rad/se c)

Figure Q5(b)

Q6. (a) In a liquid-level control problem, the output volumetric flow rate is proportional

to the square root of the height of liquid in the tank. Considering the balance

questions yields the following differential equation –

( ) ( )

( )

dt

dht k (^) iu t = ks ht + A

where A is constant and represents the cross-sectional area of the tank, ki is a

constant associated with the input value and ks is a constant associate with the

output-flow rate. In this equation u(t) represents the input (% opening of valve)

and h(t) is the output (liquid height). How could this system be linearised?

Briefly describe the key concepts associated with the linearisation method and

subsequently linearise the system.

[7 marks]

(b) Figure Q6(b) illustrates the open-loop step response of a liquid-level control

problem. To generate this response the input valve was opened to 30% and at

time t = 1100sec the valve opening was increased to 4.62% in a step-like fashion.

Identify a model for this system and subsequently, design a controller to achieve

accurate control of the liquid height. For this problem, you may use any controller

design method, but you must outline reasons for your choice and describe the

rational behind your design.

If the cross-sectional area of the tank is 196cm

2

, use the response of Figure Q6(b)

to determine the constants ki and k s.

Figure Q6(b)

[13 marks]

Q7. (a) The most commonly implemented PID algorithm is the series structure defined by

( (^) D )

I

P sT sT

G s K + 















= + 1

() 1

Sketch a block diagram illustrating how you would implement this algorithm in

Simulink.

[5 marks]

(b) Derive the transfer function in the Z-domain for the series PID algorithm defined

in part (a) and subsequently write down the difference equation that defines this

control law. Sketch a block diagram implementation of the discrete difference

equation using summing junctions, delays and gain blocks.

[10 marks]

(c) Briefly describe one limitation of the PID controller algorithm presented in part (a)

and discuss how you would overcome that limitation.

[5 marks]

FORMULAS

2

2 2 ln^ 100

β PO ζ β β π

 

= =  

+  

n

ts

ζω

≈ ;

d

tr

ω

π

=

95 95 ... 10 100

s

T T

T =

...

T s

τ τ

2 ω d (^) = ω n 1 −ζ

d d

T

π

ω

=

2

( ) (^2 ) 2

ss n s d

n n

k G s e s s

ω τ

ζω ω

−

1 |^0 |

log 20

ss

G dB k

− ^ ω=  = (^)    

2

20log

2 1

ss p

k M

ζ ζ

 

= ^ 

 

 − 

2

ω p = ω n 1 − 2 ζ

x x u

y x u

= +

= +

A B

C D

&

[ ]

( ) 1

( ) ( )

( )

Y s s s U s

− = C Φ s B + D Φ = I − A

ˆ 1

[ ]

T T

θ

− = Φ Φ Φ Y