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Example 9.5 Obtain using the describing function method the amplitude and frequency of the limit cycle in the output C for the system described by the block diagram of Figure 9.7 if Gs =[r]

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Control Engineering Problems with Solutions

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Derek P Atherton

Control Engineering Problems with

Solutions

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Preface

The purpose of this book is to provide both worked examples and additional problems, with answers

only, which cover the contents of the two Bookboon books ‘Control Engineering: An introduction

with the use of Matlab’ and ‘An Introduction to Nonlinearity in Control Systems’ Although there was considerable emphasis in both books on the use of Matlab/Simulink, such usage may not always be possible, for example for students taking examinations Thus in this book there are a large number of problems solved ‘long hand’ as well as by Matlab/Simulink A major objective is to enable the reader

to develop confidence in analytical work by showing how calculations can be checked using Matlab/Simulink Further by plotting accurate graphs in Matlab the reader can check approximate sketching methods, for say Nyquist and Bode diagrams, and by obtaining simulation results see the value of approximations used in solving some nonlinear control problems

I wish to acknowledge the influence of many former students in shaping my thoughts on many aspects

of control engineering and in relatively recent years on the use of Matlab In particular, Professor Dingyu Xue whose enthusiasm for Matlab began when he was a research student and who has been a great source

of knowledge and advice for me on its use since that time, and to Dr Nusret Tan for his assistance and advice on some Matlab routines I wish to thank the University of Sussex for the facilities they have provided to me in retirement which have been very helpful in writing all three bookboon books and finally to my wife Constance for her love and support over many years

Derek P Atherton

University of Sussex

Brighton

May 2013

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1 Introduction

1.1 Purpose

The purpose of this book is to provide both worked examples and additional problems, with answers only,

which cover the contents of the two Bookboon books Control Engineering: An introduction with the

use of Matlab[1] and An Introduction to Nonlinearity in Control Systems [2], which will be referred to

as references 1 and 2, respectively, throughout this book In reference 1 the emphasis in the book was to show how the use of Matlab together with Simulink could avoid the tedium of doing some calculations, however, there are situations where this may not be possible, such as some student examinations Thus

in this book as well as working out in many cases the examples ‘long hand’, the solutions obtained using Matlab/Simulink are also given Matlab not only allows confirmation of the calculated results but also provides accurate graphs of say Nyquist plots or root locus diagrams where an examination question may ask for a sketch Academics have been known to say they gained significant knowledge of a topic from designing exercises for students Unlike 50 years ago when slide rules and logarithmic tables were used to solve problems designing exercises is now much easier because in most instances results can be checked using appropriate computer programs, such as Matlab Thus with these tools students can build their own exercises and gain confidence in solving them by doing appropriate checks with software The examples and problems have been carefully chosen to try and bring out different aspects and results

of problem solving without, hopefully creating too much repetition, which can ‘turn off’ the most ardent enthusiast Before the examples in each chapter a very brief overview of aspects of the topics covered is given but more details can be found in the relevant chapters of references 1 or 2, which are referred to

in the relevant chapters of this book

The examples and problems are included under the following topic titles

2 Mathematical Models and Block Diagrams

3 Transfer Functions and their Time Domain Responses

4 Frequency Responses and their Plotting

5 Feedback Loop Stability

6 State Space Models and Transformations

7 Control System Design

8 Phase Plane Analysis

9 The Describing Function and Exact Relay Methods

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to be undertaken, which will be our concern here

Transducer conversionVariable

element

Signal processing transmissionSignal Signal

utilization Physical

Figure 2.1 Components of a typical measurement system.

The basic mathematical model of a component with lumped parameters is a differential equation Although all component models are nonlinear one may often be able to approximate them under certain conditions by a linear differential equation Control engineers usually work with two equivalents of a linear differential equation, a transfer function or a state space model, as described in chapter 2 of reference

1 Thus a component model is typically shown by a block and labelled with its transfer function G (s)

as shown in Figure 2.2, where the input to the block is labelled U (s) and the output Y (s) This means that Y(s)=G(s)U(s), whereU (s)is the Laplace transform of the input signal u (t) and Y (s)is the Laplace transform of the output signal y (t) The corresponding relationship in the time domain is the convolution integral, see appendix A reference 1, given byy t = ∫t g t − u d = ∫t g u t − d

0 0

) ( ) ( )

( ) ( )

where g (t) the weighting function, or impulse response, of the block has the Laplace transform G (s)

It is normally understood that when the lower case is used, i.e u, it is a function of t and when the upper cases is used, i.e U it is a function of s.

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The first set of examples will be concerned with model representations for a single block The transfer function of a component, assumed to behave linearly, is the Laplace transform of its linear constant parameter differential equation model, assuming all initial conditions are zero This transfer function,

typically denoted by, G(s), will be the ratio of two rational polynomials with real coefficients, that is

)(/

)

(

)

G = The roots of A(s) and B(s) respectively are the poles and zeros of G(s) A transfer

function is strictly proper when it has more poles than zeros When the number of poles is equal to the number of zeros the transfer function is said to be proper The transfer function is stable if all its poles have negative real parts In Matlab the transfer function is typically entered by declaring the coefficients

of the polynomials A(s) and B(s) or in the zero-pole-gain form A state space model represents an n th order differential equation by a set of n first order differential equations represented by four matrices A,

B, C and D For a single-input single-output system (SISO) the dimensions are nxn; 1xn, an n column

vector; nx1, an n row vector, and 1x1, a scalar Whilst a state representation has a unique transfer

function the reverse is not true Some simple aspects of state space representations will be covered here with more in chapter 6

The interconnection of model blocks is typically shown in a block diagram or signal flow graph where only

the former will be considered here Often the 's' is dropped in the block diagram so that the relationship for Figure 2.2 is typically denoted by Y = GU

G(s)

Figure 2.2 Single block representation.

In connecting block diagrams it is assumed that the connection of one block G2 to the output of another

G1 does not load the former so that if X = G1U and Y = G2X then Y1 = G2G1X as shown in Figure 2.3

U

Figure 2.3 Series connection of blocks.

For two blocks in parallel with Y1 = G1U, Y2 = G2U and Y = Y1 +Y2 then Y = (G1 + G2)U In Matlab the

series connection notation is G1 * G2 and the parallel one G1 + G2 Figure 2.4 shows a simple feedback

loop connection for which the relationships for the two blocks are C = GX and Y= HC with X = R – Y Eliminating X to get the closed loop transfer function, T, between the input R and output C gives

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G +

H Y

Figure 2.4 Closed loop block diagram

The required command in Matlab is T=feedback(G,H) If the positive feedback configuration is required

then the required statement is T=feedback(G,H,sign) where the sign = 1 This can also be used for

the negative feedback with sign = -1 Block diagrams and signal flow graphs, an alternative graphical

representation which will not concern us here, simply describe sets of simultaneous equations Often

textbooks give sets of rules for manipulating block diagrams and obtaining relationships between the

variables involved but in many engineering problems there are not many interconnections between blocks

and one can work from first principles writing out expressions and eliminating variables as done above

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The standard single-input single-output feedback control loop is typically assumed to be of the form shown in Figure 2.5 G, Gc and H are respectively the transfer functions of the plant, controller and measurement transducer, and the input signals R, D and N are respectively the reference or command input, a disturbance and measurement noise U is the control signal to the plant and C the output or controlled variable The open loop transfer function, * RO V , is the transfer function around the loop

with the negative feedback assumed, that is with ‘s’ omitted, * RO * F *+ The closed loop transfer function C/R is often denoted by T The error is the difference between the demanded output and the actual output C Normally the units of R and C will be different, for example C might be a speed and

R a voltage with the transducer H having units of V/rads/s Typically, the feedback loop is designed to achieve zero error between R and HC, which will be a voltage The error in speed will be C –R/H, which with no voltage error will only be the demanded speed if H is known exactly The transfer function from the input to the error at the input to Gc is 1-TH

_

U

Figure 2.5 Basic feedback control loop

The first two examples deal with transfer functions and their zeros and poles, and are followed by three examples dealing with the interconnection of transfer functions and their evaluation in Matlab Mathematical models can also be entered and their responses to different inputs found using Simulink The ‘Continuous’ category of Simulink includes the following model forms, transfer function blocks for either polynomial or zero pole form of entry, a state space block, an integrator block The ‘Math operations’ category, includes a gain block and a sumer The next example covers a few basic aspects of using these blocks in Simulink

2.2 Examples

Example 2.1

Find the poles and zeros of the transfer function

233

1)

+++

+

=

s s s

s s

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To find the poles one needs to find the roots of the denominator polynomial

023

Note the complex roots are returned as a second order polynomial

Alternatively the transfer function could have been entered in zero-pole-gain form as below and the transfer function in polynomial form found

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V V V

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1

(

)2()

s s

V V

Note that in the product G1G2the zero at s=−2 from G1cancels the pole at s=−2 of G2 giving:-

)4(

)

1

(

42 3 2

+

=

s s s

s G

The first transfer function G1 can be entered by making use of the convolution instruction ‘conv’ as follows:-

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... 26

[0, -2 , -4 , -0 .5±0.866j; -1 ,-3 ; stable, all poles negative real part]

Problem 2.2

Find the poles and zeros of the transfer function... Gc is 1-TH

_

U

Figure 2.5 Basic feedback control loop

The first two examples deal with transfer functions and their zeros and poles, and are... frequency) and its response to a unit step input

The d.c gain is obtained by putting s=0 so that G ( ) = / x x = / 6 The poles are -1 , -3

and -4 and

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