Instrumentation and Control/Assignment 2: Control

Intended Learning Outcomes assessed by this work

1. Understand the basic underpinning theory of linear continuous-time dynamical systems and be

able to simulate their behaviour;

2. Manipulate and solve differential equations using Laplace transforms and interpret the system

poles and zeros.

Hand-in date: 18 April 2016

Note that the Hand-in date is a ‘hard’ deadline, in the sense that if the coursework is handed in late

then a zero mark is awarded. You are encouraged to submit this coursework well in advance of the

deadline date.

Marks allocated

Question 1: 10 marks

Question 2: 20 marks

Question 3: 20 marks

Question 4: 20 marks

Question 5: 20 marks

Presentation: 10 marks

Coursework Submission

The completed coursework should be submitted, over the counter, to the Assignment Office before

16.00 hours on the deadline date. The coursework must include a printed bar-coded coversheet. The

bar-coded coversheets can be generated from NOVA, online assessments system, which can be

accessed by all students from https://webapp.coventry.ac.uk/Nova/NovaMain.aspx.

Control Coursework J. E. Trollope

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Please note that the work submitted should be your individual effort. The penalties for plagiarism

are severe. Evidence of plagiarism will be dealt with according to the rules and regulations outlined

in the Student Handbook (Faculty of Engineering and Computing).

Assessment Criteria

There are 100 marks in total for this assignment and a breakdown of these marks is provided in the

assignment details. Of these marks, 90 marks are for accuracy in the mathematics, clear logical

arguments, calculations and use of software packages (if appropriate). The remaining 10 marks are

for the general presentation.

Marks may be deducted for:

1. writing mathematics without suitable explanatory text (as a general rule, all mathematics

should be contained in proper, full sentences);

2. mathematical errors (including errors of notation);

3. poor proof-reading (for example, poor spelling, poor grammar, etc.);

4. poor readability, for example, confusing explanations, etc.;

5. not suitably commenting any computer software package commands.

Items 1 and 2 will normally lead to deductions from the marks for accuracy, whereas items 3-4 relate

to the presentation mark. (Suitable allowance will be made for students with a recognised disability

such as dyslexia.)

Typically, to obtain a first class mark (= 70%), the report should have few mathematical errors and all

work should be suitably explained. Similarly, to obtain a pass mark (= 40%), the report should

typically show evidence that the basic concepts are understood, but several errors and/ or lack of

proper explanations may be present. It should be understood that these are only typical cases and

that the marking scheme will determine the final mark obtained.

Control Coursework J. E. Trollope

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Question 1

a) Suppose ( ) and ( ) are the Laplace transforms of the reference ( ) and system output ( )

respectively. If the blocks 1, 2, , and in Figure 1 are transfer functions in the Laplace variable

, e.g. 1( ), find the closed loop transfer function of the form ( )

( )

.

(10 marks)

1

R(s) Y(s)

+

–

2

+

–

Figure 1: Linear time-invariant system

Control Coursework J. E. Trollope

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Question 2

a) For the closed loop system configuration in Figure 2, obtain an expression for the overall closed

loop transfer function

(4marks)

b) Hence or otherwise write down the closed loop characteristic equation

(2 marks)

c) Use the Routh-Hurwitz stability criterion to find the values of for which the system is stable

(6 marks)

d) Sketch the root locus for this system and indicate the main points

(4 marks)

e) Use MATLAB to confirm your findings in part d)

(4 marks)

–

( + 1)

( – 1)( + 2)

Figure 2: Closed-loop transfer-function

Control Coursework J. E. Trollope

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Question 3

A system is configured as a negative unity feedback control system having a transfer function

( ) =

1

( + 1)( + 4)

which is cascaded with a proportional controller having a gain .

a) With the gain initially set to unity, determine the closed loop transfer function and calculate the

system poles.

(3 marks)

b) With the gain set to unity apply a unit step at time = 0 (assuming zero initial conditions). Use

MATLAB to obtain the step response. Comment on the form of the response i.e. describe the key

features (Hint: relate this to the inverse Laplace transform of ( ).

.(6 marks)

c) In an attempt to reduce the steady state error, the gain is increased to 10. With this value of

repeat steps a) and b). Check your results using MATLAB.

(6 marks)

d) The proportional controller is now replaced by a proportional plus integral controller. You are now

required to produce a closed-loop system response which is critically damped (i.e. fastest response

without overshoot). Determine the controller gains required. Implement the controller in MATLAB

and demonstrate that the response is as required.

(5 marks)

Control Coursework J. E. Trollope

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Question 4

The system given by the transfer function in Question 3 is now configured in state space form

= +

=

where is the system matrix, is the input vector, is the output vector, and are the system

input and output, respectively, and is the system state vector.

(a) By defining 1 = and 2 = write down the corresponding phase variable canonical form.

(4 marks)

(b) The system is now configured for state variable feedback in order to control the dynamic

behaviour of the system, i.e. =

where

is a feedback vector containing controller gains.

Making use of modal control the slow eigenvalue is to be relocated such that its response is now 10

times faster. Determine the feedback vector to achieve this response, and demonstrate the

performance using MATLAB.

(6 marks)

(c) Explain what is meant by observability and controllability. Show that the system is both

observable and controllable using the modal matrix method.

(3 marks)

(d) Write down the Kalman observability and controllability test matrices, hence or otherwise

confirm your results in (c).

(3 marks)

(e) Work back from the state space representation obtained in (a) to show how a transfer function

representation can be obtained. Hence or otherwise write down the open-loop characteristic

equation and show that the eigenvalues of the system matrix are the same as the system poles.

(4 marks)

Control Coursework J. E. Trollope

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Question 5

In the UK, research is being undertaking on the feasibility of railway carriages coupling and

uncoupling whilst on the move. The new railway concept is to increase the railway network capacity,

connectivity and to reduce the cost of travel and carbon emissions.

As an engineer your task is to design a control system for the future frontal structure of the railway

carriage. It is envisaged this new design will be transformed from today’s passive design to an active

design, with a stiffness controller to deal with all possible scenarios. The premise of the future

frontal structure is to absorb the collision energy between the coupling trains in an effective manner

to minimise the jerk (change in acceleration) experienced by the occupants on-board the carriages.

The control system must be designed for the following two scenarios:

1) A train consisting of three carriages coupling onto a train consisting of 1 carriage with a

speed difference of 5kmh (each carriages mass is 43 tonnes)

2) A train consisting of five carriages coupling onto a train consisting of 1 carriage with a speed

different of 7khm (each carriage mass is 43 tonnes)

In both cases the front of the larger train couples up to the rear of the single carriage.

You may wish to consider the following points:

The energy resulting from the coupling action can be determined by using the conservation

of momentum and energy equations

Based on the passive design case, the coupling energy is distributed between the carriages

based on the mass ratios i.e. three carriages mass versus one carriage mass (3:1) as in

scenario one

o Is this desirable No. Thus, the need the active design of the frontal structure

The use of a mass spring damper model for the frontal structure of the railway carriage

o This is used to capture the displacement (y-axis) versus time (x-axis) of the coupling

mechanism and further possible simulation models e.g. force versus deformation

(relating to work done/delta energy)

The use of feedback control to achieve desired outcome

List any assumptions made

The use of MATLAB to simulate the design

This open ended question is designed for you to demonstrate your understanding of the control

lectures by applying your knowledge to a realistic control system design problem. We are not

expecting completed designs – rather a good understanding of the control problem will achieve high

marks.

(20 marks)