EG3311

All candidates

Semester 1 Examinations 2019

DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY

THE CHIEF INVIGILATOR

Department ENGINEERING

Module Code EG3311

Module Title STATE VARIABLE CONTROL

Exam Duration TWO HOURS

CHECK YOU HAVE THE CORRECT QUESTION PAPER

Number of Pages 7

Number of Questions Part A: 8 (multiple choice); Part B: 3

Instructions to Candidates You should answer all questions from Part A on the answer sheet

provided.

You should attempt only two (2) questions from Part B.

Attempted solutions which the candidate does not wish to submit

should be crossed out by the candidate. If you do attempt more

than two questions from Part B, and do not identify which two you

want marked, only the first two in the answer book will be marked.

For each question, the distribution of marks out of 20 is indicated in

brackets.

FOR THIS EXAM YOU ARE ALLOWED TO USE THE FOLLOWING:

Calculators Permitted calculators are the Casio FX83 and FX85 models

Books/Statutes provided by

the University

Engineering Data Book

Are students permitted to

bring their own

Books/Statutes/Notes?

No

Additional Stationery Answer sheet (provided by the Department)

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EG3311

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PART A

For each question in Part A there is only one correct answer. [Each correct answer awards 2.5 marks.]

a) A MIMO LTI system of the form

x˙(t) = Ax(t)+Bu(t)

y(t) =Cx(t)+Du(t)

has n= 20 states, m= 4 inputs, and p= 10 outputs. What are the dimensions of the B matrix of the system?

[a1] 10×4

[a2] 10×20

[a3] 20×4

[a4] 20×20

b) If p is position and v is velocity, then the LTI system[

v˙(t)

p˙(t)

]

= A

[

v(t)

p(t)

]

+

[

1

m

0

]

F(t)

can be considered a correct state-space representation of Newton’s second law of motion F = mv˙ if

[b1] A=

[

0 0

1 0

]

[b2] A=

[

1 0

0 1

]

[b3] A=

[

1 1

0 0

]

[b4] A=

[

1 0

0 0

]

c) Consider the LTI system

x˙(t) =

[

0 0

0 −1

]

x(t)+

[

1

0

]

u(t)

y(t) =

[

1 −1]x(t)

with initial condition x(0) =

[

0

1

]

and constant input u(t) = 1, t ≥ 0. The time evolution of the output y(t) for

t ≥ 0 is

[c1] y(t) = e−t

[c2] y(t) = 1− e−t

[c3] y(t) = t

[c4] y(t) = t− e−t

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EG3311

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d) Consider the LTI system

x˙(t) =

[

0 0

0 1

]

x(t)+

[

0

1

]

u(t)

y(t) =

[

1 1

]

x(t)

If the initial state x(0) has both elements different from 0 then the corresponding free response of the output y

[d1] diverges

[d2] converges to 0 without oscillations

[d3] converges to 0 with oscillations

[d4] does not converge to 0 and does not diverge

e) The LTI system introduced in question d) is

[e1] fully controllable and fully observable

[e2] fully controllable but not fully observable

[e3] fully observable but not fully controllable

[e4] neither fully controllable nor fully observable

f) The LTI system

x˙(t) =

[

2 −3

0 −1

]

x(t)+

[

2

0

]

u(t)

y(t) =

[

1 1

]

x(t)

is open-loop unstable and not fully controllable. Given the state-feedback law u(t) = [k1,k2]x(t), which of the

following statements is correct:

[f1] for any k1, k2 the closed-loop system remains unstable

[f2] for any k1, k2 the closed-loop eigenvalues are real

[f3] the behaviour of the closed-loop system is not affected by the value of k1

[f4] for any k1, k2 the closed-loop system has a pole in 0

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EG3311

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g) With regards to output feedback design, where a state observer and a state feedback law are used, which of

the following statements is true

[g1] the feedback control action is computed using the states of the state observer

[g2] the dynamics of the closed-loop system is independent of the dynamics of the state observer

[g3] the dynamics of the closed-loop system is independent of the design of the state feedback law

[g4] the feedback control action is computed using the states of the controlled system

h) Given a MIMO LTI system of the form

x˙(t) = Ax(t)+Bu(t)

y(t) =Cx(t)+Du(t)

the Matlab command Z=eig(A) is entered and the following answer is obtained

Z =

-1.1623

5.1623

1.0000

From this, we can conclude that the system is

[h1] observable and controllable

[h2] asymptotically stable

[h3] unstable

[h4] none of the above

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EG3311

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PART B

1. Using Newton’s Laws the differential equation describing the motion of a pendulum may be written as

mlθ¨(t) =−mgsinθ(t)− klθ˙(t)+u(t)

where θ(t) is the angular displacement of the pendulum, m is the mass of the pendulum bob, l is the

length of the massless rod connecting the bob to its pivot, g is the acceleration due to gravity and k is the

frictional force which opposes the motion. The term u(t) is an applied torque.

i) Derive a state equation for the nonlinear system using the angular displacement and its time deriva-

tive as state variables, and the applied torque as the input. [2 marks]

ii) Calculate the equilibrium of the system corresponding to θ= 0. [2 marks]

iii) Derive a linearised state-space model of the nonlinear system, using a suitable equilibrium point

and the Taylor Series expansion. Take θ(t) as the output and give expressions for the A,B,C and

D matrices. Marks will be awarded for the clarity of the derivation. [6 marks]

iv) Find an expression for the poles of the linearised system. What can be said about the stability of

the system? [4 marks]

v) Show that, if the damping ratio is known to be greater than unity, the following condition must be

satisfied

k > m

√

g

l

[4 marks]

vi) If a controller was designed for the linearised system using the model derived in part iii), what

behaviour might one expect if the controller was implemented on the original nonlinear system?

Explain your answer. [2 marks]

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EG3311

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2. Consider the system

Σ∼

{

x˙(t) = Ax(t)+Bu(t)

y(t) = Cx(t)

i) How would one check that the system Σ is fully controllable? [2 marks]

ii) An engineer states that “the controllability criterion only applies when the D matrix is singular”. Is

the engineer correct? Explain your reasoning. [2 marks]

iii) A second order system modelled as the system Σ has state-space matrices

A=

[ −4 2

0 4

]

B=

[

0

1

]

C = [1 0]

Is the system completely controllable? Explain your answer. [4 marks]

iv) In order to improve the stability of the above system, a state feedback control law must be designed

so that the closed-loop system has a natural frequency of ωn= 4 radians per second and a damping

ratio of ζ= 0.8. Calculate a feedback matrix F = [ f1 f2] which will achieve this. [8 marks]

v) Due to a fault in the system, the matrices abruptly change to

A=

[ −4 2

0 −4

]

B=

[

0

0

]

C = [1 0]

Is the system after the fault occurs controllable? Is it stabilisable? Justify your answers.

[4 marks]

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EG3311

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3. i) Describe two advantages of the state-space approach over the transfer function approach to system

modelling. [2 marks]

ii) Consider the mass-spring-damper system below. The vertical displacement of the mass is x(t).

The spring and damper constants are K and B respectively. When the applied force, u(t) = 0, the

vertical displacement of the mass is x(t) = xss (due to gravity).

Mass-spring-damper system

For the general case, when u(t) 6= 0, derive a state-space model of the system using the the dis-

placement from equilibrium z(t) = x(t)− xss as one of the states, and a further suitable second

state. Take the time derivative of z(t) as the output and give expressions for the state-space matri-

ces. [6 marks]

Hint—Derive the equations of motion using Newton’s Laws and then derive an expression for xss

when u(t) = 0 and the system is at rest

iii) It is known that M = 1 kg, K = 5 Nm−1 and B = 2 Nm−1s−1. Based on your answer to part

ii), compute the state-transition matrix, Φ(t) of the system. Express Φ(t) in as simple a form as

possible. [8 marks]

iv) Using your answer to part iii) or otherwise, derive an expression for the forced response of the

system, assuming that the input is a unit step function. The expression should be as simple as

possible. [4 marks]

Version 2 END OF PAPER Page 7 of 7

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