continuous & discreate control systems

C o n t i n u o u s a n d D i s c r e t e C o n t r o l S y s t e m s

John Dorsey

Chapter 1

Preliminaries

1.1 Overview

The opening chapter of the book doesn't require the use of Matlab, but we can take a

preliminary look at Matlab and some of its basic commands.

Matlab is, to me, just a versatile, on-line calculator, with some some limited graphical

capabilities that are not user friendly. I use it in preference to a hand held calculator, but if

I want to do some serious number crunching I turn to a real programming language.

Matlab is an interpreted language. That is, it runs along executing each instruction as it

comes to it. In normal use there is no compilation of the code. The latest releases do include

a compiler, however. In normal use, without compilation, `for' loops and `while' loops don't

run extremely fast. However, if we put a semicolon behind each command inside such a loop

things speed up considerably. It is then possible to run simple iterative routines in a couple

of seconds on the current generation of desktop computers. This is plenty fast enough for

the kinds of programs we generally write with Matlab. On average, the specialized programs

you �nd in subsequent chapters have fewer than thirty statements. Thus, Matlab serves as

a `quick and dirty' super calculator.

1.2 The basic Commands

To illustrate the basic commands I will use what I refer to as `in line Matlab dialogues.'

When I type something, Matlab then responds. Hence the dialogue.

To multiply to numbers together we type

EDU>2 * 4

ans =

8

EDU>

We can assign names to numbers and then compute with them as follows.

2 CHAPTER 1. PRELIMINARIES

EDU>a = 2

a =

2

EDU>b = 4

b =

4

EDU>c = a*b

c =

8

EDU>

We can form arrays in the same way.

EDU>a = [1 2]

a =

1 2

EDU>b = [ 1 ; 2 ]

b =

1

2

EDU>c = a*b

c =

5

EDU>

Note that the semicolon separates rows in forming the column vector b.

For the two arrays a and b, if we can also type

1.2. THE BASIC COMMANDS 3

EDU>c = b*a

c =

1 2

2 4

EDU>

Matlab also contains all the usual mathematical functions. For instance,

EDU>y = cos(pi)

y =

-1

EDU>x = atan(0.5)

x =

4.636476090008061e-01

EDU>

Note that the angle x is in radians. If we want to plot the function cos t over two periods we

can do so using program chap1cda, repeated below for convenience.

t1 = 0

t2 = 4*pi

x = linspace(t1,t2)

y = cos(x)

plot(x,y)

print -deps cosx.eps

The resulting plot is shown in Figure 1.1

The Matlab command linspace creates 100 equally spaced points between 0 and 4�. If

we want more, or less, than 100 points we can type

linspace(x,y,npts),

where npts is the number of points desired.

We could also generate this plot using the program chap1cdb, repeated below for conve-

nience.

t1 = 0

t2 = 4*pi

npts = 200

t = t1


0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 1.1: Plot of cos x Generated by chap1cda.m

dt = (t2 - t1)/npts

for i = 1:npts

y(i) = cos(t);

x(i) = t;

t = t + dt;

end

plot(x,y)

print -deps cosx.eps

The commands inside the `for' loop are terminated with a semicolon. This suppresses Mat-

lab's print function and drastically speeds up the execution of the program.

We have given an idea of how to use Matlab. To see all the possible Matlab commands

pull down the Help menu in the Matlab command window and select Help Window. To see

all the commands in the Control Toolbox type

help control

In the Matlab command window. Matlab will respond with

EDU>help control

Control System Toolbox.

Version 4.0 Student Edition 31-Dec-1996

Creation of LTI models.

ss - Create a state-space model.


zpk - Create a zero/pole/gain model.

tf - Create a transfer function model.

dss - Specify a descriptor state-space model.

filt - Specify a digital filter.

set - Set/modify properties of LTI models.

ltiprops - Detailed help for available LTI properties.

Data extraction.

ssdata - Extract state-space matrices.

zpkdata - Extract zero/pole/gain data.

tfdata - Extract numerator(s) and denominator(s).

dssdata - Descriptor version of SSDATA.

get - Access values of LTI model properties.

Model characteristics.

class - Model type ('ss', 'zpk', or 'tf').

size - Input/output dimensions.

isempty - True for empty LTI models.

isct - True for continuous-time models.

isdt - True for discrete-time models.

isproper - True for proper LTI models.

issiso - True for single-input/single-output systems.

isa - Test if LTI model is of given type.

Conversions.

ss - Conversion to state space.

zpk - Conversion to zero/pole/gain.

tf - Conversion to transfer function.

c2d - Continuous to discrete conversion.

d2c - Discrete to continuous conversion.

d2d - Resample discrete system or add input delay(s).

Overloaded arithmetic operations.

+ and - - Add and subtract LTI systems (parallel connection).

* - Multiplication of LTI systems (series connection).

\ - Left divide -- sys1\sys2 means inv(sys1)*sys2.

/ - Right divide -- sys1/sys2 means sys1*inv(sys2).

' - Pertransposition.

.' - Transposition of input/output map.

[..] - Horizontal/vertical concatenation of LTI systems.

inv - Inverse of an LTI system.

Model dynamics.

pole, eig - System poles.

tzero - System transmission zeros.


pzmap - Pole-zero map.

dcgain - D.C. (low frequency) gain.

norm - Norms of LTI systems.

covar - Covariance of response to white noise.

damp - Natural frequency and damping of system poles.

esort - Sort continuous poles by real part.

dsort - Sort discrete poles by magnitude.

pade - Pade approximation of time delays.

State-space models.

rss,drss - Random stable state-space models.

ss2ss - State coordinate transformation.

canon - State-space canonical forms.

ctrb, obsv - Controllability and observability matrices.

gram - Controllability and observability gramians.

ssbal - Diagonal balancing of state-space realizations.

balreal - Gramian-based input/output balancing.

modred - Model state reduction.

minreal - Minimal realization and pole/zero cancellation.

augstate - Augment output by appending states.

Time response.

step - Step response.

impulse - Impulse response.

initial - Response of state-space system with given initial state.

lsim - Response to arbitrary inputs.

ltiview - Response analysis GUI.

gensig - Generate input signal for LSIM.

stepfun - Generate unit-step input.

Frequency response.

bode - Bode plot of the frequency response.

sigma - Singular value frequency plot.

nyquist - Nyquist plot.

nichols - Nichols chart.

ltiview - Response analysis GUI.

evalfr - Evaluate frequency response at given frequency.

freqresp - Frequency response over a frequency grid.

margin - Gain and phase margins.

System interconnections.

append - Group LTI systems by appending inputs and outputs.

parallel - Generalized parallel connection (see also overloaded +).

series - Generalized series connection (see also overloaded *).

feedback - Feedback connection of two systems.


star - Redheffer star product (LFT interconnections).

connect - Derive state-space model from block diagram description.

Classical design tools.

rlocus - Evans root locus.

rlocfind - Interactive root locus gain determination.

acker - SISO pole placement.

place - MIMO pole placement.

estim - Form estimator given estimator gain.

reg - Form regulator given state-feedback and estimator gains.

LQG design tools.

lqr,dlqr - Linear-quadratic (LQ) state-feedback regulator.

lqry - LQ regulator with output weighting.

lqrd - Discrete LQ regulator for continuous plant.

kalman - Kalman estimator.

kalmd - Discrete Kalman estimator for continuous plant.

lqgreg - Form LQG regulator given LQ gain and Kalman estimator.

Matrix equation solvers.

lyap - Solve continuous Lyapunov equations.

dlyap - Solve discrete Lyapunov equations.

care - Solve continuous algebraic Riccati equations.

dare - Solve discrete algebraic Riccati equations.

Demonstrations.

ctrldemo - Introduction to the Control System Toolbox.

jetdemo - Classical design of jet transport yaw damper.

diskdemo - Digital design of hard-disk-drive controller.

milldemo - SISO and MIMO LQG control of steel rolling mill.

kalmdemo - Kalman filter design and simulation.

EDU>

All the commands that we have discussed, plus many, many more are listed. Note that there

are also �ve demonstration programs that can be run.

Chapter 2

The Laplace Transform

2.1 Important Commands

The Matlab commands of most use in this chapter are

1. conv

2. roots

3. residue

The conv command is a quick means of multiplying together polynomials. For instance,suppose in the Matlab command window we type

p = [1 2 1]

This creates a row vector that stands for the polynomial

p(s) = s2 + 2s + 1:

If we next create the polynomial

q(s) = s3 + 2s2 + 3s+ 1

by typing

q = [1 2 3 1]

we can then �nd the multiplication of these two polynomials by typing

r = conv(p,q)

What this would like if you were in the Matlab command window is:

1

2 CHAPTER 2. THE LAPLACE TRANSFORM

EDU>p = [1 2 1]

p =

1 2 1

EDU>q = [1 2 3 1]

q =

1 2 3 1

EDU>r = conv(p,q)

r =

1 4 8 9 5 1

EDU>

Thusr(s) = p(s)q(s) = s5 + 4s4 + 8s3 + 9s2 + 5s+ 1:

The roots command can be used to �nd the roots of a polynomial. For instance, supposewe type:

a = [1 2]

b = [1 4]

c = [1 8]

p = conv{a,b}

p = conv(p,c)

roots(p)

What we have done is create the polynomials

s + 2 ; s+ 4 ; and s+ 8;

and then multiply them together to create the polynomial p(s). We then use the Matlabcommand roots to �nd roots �nd the roots ( which we already know). If we typed thesecommands in the Matlab command window we would get:

EDU>a = [1 2]

a =

1 2

2.1. IMPORTANT COMMANDS 3

EDU>b = [1,4]

b =

1 4

EDU>c = [1,8]

c =

1 8

EDU>p = conv(a,b)

p =

1 6 8

EDU>p= conv(p,c)

p =

1 14 56 64

EDU>roots(p)

ans =

-7.99999999999998

-4.00000000000002

-2.00000000000000

EDU>

Note that we put commas between the entries in `b.' They aren't required, but theysometimes are helpful. Note also that we have some round-o� error when we compute theroots.

The residue command enables us to quickly �nd the residues in a partial fraction ex-pansion. For instance, suppose we have

Y (s) =10(s+ 4)

s(s+ 10)(s+ 20)

4 CHAPTER 2. THE LAPLACE TRANSFORM

We �rst create the two polynomials

a(s) = 10(s+ 4)

b(s) = s(s + 10)(s+ 20);

and then use the Matlab command

[R,P,k} = residue[a,b]

The vector R contains the residues, the vector P the corresponding pole and the vector K thedirect terms. The direct terms would occur if we had

F (s) =C1

s+ p1+

C2

s+ p2+ : : :+

Cn

s + pn+K(s):

We won't often encounter a situation where the term K(s) is present, but Matlab can handleit. For the Y (s) considered here, the Matlab entries and responses are as follows

EDU>a = [1 4]

a =

1 4

EDU>a = 10*a

a =

10 40

EDU>c = [1 0]

c =

1 0

EDU>d = [1 10]

d =

1 10

EDU>e = [1 20]

e =

2.1. IMPORTANT COMMANDS 5

1 20

EDU>b = conv(c,d)

b =

1 10 0

EDU>b = conv(b,e)

b =

1 30 200 0

EDU>[R,P,K] = residue(a,b)

R =

-0.80000000000000

0.60000000000000

0.20000000000000

P =

-20

-10

0

K =

[ ]

EDU>

Thus, the partial fraction expansion is

10(s+ 4)

s(s+ 10)(s+ 20)=

0:2

s+

0:6

s+ 10�

0:8

s + 20:

Chapter 3

The Transfer Function

3.1 Useful Commands

Several useful Matlab commands are

1. mesh

2. view

3. print

4. tf

The command mesh generates a three dimensional plot similar to those inFigures 3.4 and 3.5. The following program was used to generate Figure 3.4.

xt = 3

xb = -6

yt = 3

yb = -3

dx = (xt -xb)/100

dy = (yt -yb)/100

x=xb

y=yb

for l = 1:100

for k = 1:100

s = x + i*y;

a = sqrt( (x + 2)^2 + y^2);

b = sqrt(x^2 + y^2);

Z(l,k) = 20*log( b/a );

1

2 CHAPTER 3. THE TRANSFER FUNCTION

if Z(l,k) > 10000

Z(l,k) = 10000;

end

X(k) = x;

x = x + dx;

end

x = xb;

Y(l) = y;

y = y + dy;

end

mesh(X,Y,Z)

view(37,20)

print tent.eps

clear X

clear Y

clear Z

This program is typical of the kind of Matlab programs I write: crude,short, and e�ective. There are a lot of sophisticated Matlab programs thatcome in handy once in a while, but from my perspective the most e�ectiveway to use Matlab is to write short programs to solve speci�c problems,problems that almost invariably fail to �t nicely into some `canned' routine.

Note in this problem that I set up the range of the variables x and y,namely xb, xt, yb, and yt, and then de�ned the increments dx and dy

by which x and y would change in the nested `for' loop. The nested `for'loops in this program are not very eÆcient because Matlab is an interpretedlanguage, that is it is not compiled and then executed, although that ischanging because the latest edition comes with a compiler. However, forroutine use it is still an interpreted language.

This proves to be inconsequntial because the current generation of desk-top computers are so fast that even ineÆcient programs run in a second ortwo. Therefore, it is not worth spending an hour writing an eÆcient pro-

gram when you can write a crude one in �ve minutes that will execute in

two seconds or less. This is my view of Matlab. With rare exceptions, theprograms I write are programs that I will use to solve a speci�c problemand then discard. But they are very, very useful, because they let me doanalysis, like simple optimizations, that I would not even attempt withouta tool like Matlab.

Returning to the code, once the nested `for' loops have calculated allthe values of the functions, and values of x, y, and z have been stored

3.1. USEFUL COMMANDS 3

in the approprtiate vectors, then two commands mesh and view are used.The command mesh draws the three dimensional plot. The command view

determines the angle at which you view the plot. The command has twoarguments, so that when we use it we tupe

view(AZ,EL),

where the �rst argument is the azimuth from which the plot is viewed andsecond argument is the elevation from which the plot is viewed.

The azimuth and elevation chosen, combined with the order in whichthe data was stored in the vectors X, Y, and Z, yields what I consider the`natural' view of the plot, with values on the x-axis increasing from rightto left and values on the y-axis increasing from front to back. If you don'tview this as `natural,' type:

help view

to �nd your own preferred perspective.The command print tent.eps stored the �gure as a postscript �le

called `tent.eps.' You could also use

print -deps tent.eps,

to store the �gure as encapsulated postscript. Indeed if you type

help print

you will �nd that there are more ways to store the �gure than I care toenumerate here.

The command

tf(a,b)

creates a transfer function from the the two polynomials a(s) and b(s), wherea(s) is the numerator polynomial and b(s) is the denominator polynomial.For instance, suppose

G(s) =10(s+ 1)

s(s+ 2)(s+ 10):

Then the following Matlab `dialogue' creates this transfer function.

4 CHAPTER 3. THE TRANSFER FUNCTION

EDU>a = 10*[1 1]

a =

10 10

EDU>c = [1 0]

c =

1 0

EDU>d = [1 2]

d =

1 2

EDU>e = [1 10]

e =

1 10

EDU>b = conv(c,d)

b =

1 2 0

EDU>b = conv(b,e)

b =

1 12 20 0

EDU>g = tf(a,b)

Transfer function:

10 s + 10


-------------------

s^3 + 12 s^2 + 20 s

Chapter 4

Introducing Feedback

4.1 Useful Commands

Typing

help control

will generate all the commands in the Matlab control tool box. There are agreat many commands. Several that are worth introducing at this junctureare:

1. pole

2. damp

3. pzmap

4. step

The command \verb+pole+ finds the poles of a transfer function. For instance,

consider the following Matlab dialogue.

EDU>a = [5 1]

a =

5 1

EDU>b =[1 13 44 32]

1

2 CHAPTER 4. INTRODUCING FEEDBACK

b =

1 13 44 32

EDU>g = tf(a,b)

Transfer function:

5 s + 1

------------------------

s^3 + 13 s^2 + 44 s + 32

EDU>pole(g)

ans =

-7.999999999999991e+00

-4.000000000000006e+00

-9.999999999999998e-01

EDU>

The pole are ats = �1; �4; and � 8:

Note that there is some round o� error.The command damp �nds the damping ratio and natural frequency of

the poles of a transfer function. To see how this command works considerthe following Matlab dialogue.

EDU>a = 18

a =

18

EDU>b = [1 7 24 18]

b =

1 7 24 18


EDU>g = tf(a,b)

Transfer function:

18

-----------------------

s^3 + 7 s^2 + 24 s + 18

EDU>damp(g)

Eigenvalue Damping Freq. (rad/s)

-1.00e+00 1.00e+00 1.00e+00

-3.00e+00 + 3.00e+00i 7.07e-01 4.24e+00

-3.00e+00 - 3.00e+00i 7.07e-01 4.24e+00

EDU>

Matlab uses the term eigenvalue for poles of the transfer function. We willencounter the terminology eigenvalue in Chapter 18. when we begin talkingabout state models. We will see there that the eigenvalues of a state modelcorrespond to the poles of a transfer function.

The command pzmap plots the poles of a transfer function. For

g =18

s3 + 7s2 + 24s+ 18;

typing

pzmap(g)

yields Figure 4.1The command step generates the unit step response of a transfer func-

tion. The following Matlab dialogue generates the step response of the sim-ple transfer function and saves the reponse as an encapsulated postscript�le.

G(s) =5

s+ 5:

EDU>a = 5

a =

5


-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-3

-2

-1

0

1

2

3

Real Axis

Imag

Axi

s

Pole-zero map

Figure 4.1: Pole/zero Map Created bu Command pzmap(g)


EDU>b = [1 5]

b =

1 5

EDU>g = tf(a,b)

Transfer function:

5

-----

s + 5

EDU>step(g)

EDU>print -deps stepg.eps

EDU>

The response is shown in Figure 4.2There are many more commands in the Matlab controls toolbox worth

investigating. We will discuss some of these in subsequent chapters, but thestudent is encouraged to investigate these commands.


Time (sec.)

Am

plitu

de

Step Response

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.2: Step Response to Command step(g)

Chapter 1

Root Locus Analysis

4.1 Useful Commands

We introduce here the commands

1. rlocus,

2. rlocfind,

The simplest way to draw a root locus is to use the rlocus command in programchap5rld,m, repeated below for convenience.

%Root locus drawn using rlocus(g,gain) where gain is vector of gains that

%user generates.

gainmax = 150

gainmin = 0

npts = 400

dgain = (gainmax - gainmin)/npts

newgain = gainmin

%

% generate gain vector

%

gain = linspace(0,400,npts)

%

% generate transfer function

%

a = [1 3]

b = [1 19 18 0]

g = tf(a,b)

%

% generate root locus

%

rlocus(g)

1

2 CHAPTER 4. ROOT LOCUS ANALYSIS

-15 -10 -5 0 5-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

Imag

Axi

s

Figure 4.1: Root Locus Generated by Use of rlocus(g) Command

print -deps chap5rld.eps

The result of this program is shown in Figure 4.1 Note the stray line from the zero up to onelimb of the root locus. This program is not very good, and I seldom use it.

We can get around the problem of stray lines by using the Matlab program chap5rla.m,repeated below for convenience that draws the root locus shown in Figure 4.2.

%Root locus drawn using rlocus(g,gain) where gain is vector of gains that

%user generates.

gainmax = 150

gainmin = 0

npts = 400

dgain = (gainmax - gainmin)/npts

newgain = gainmin

%

% generate gain vector

%

for i = 1:npts

gain(i) = newgain;

newgain = newgain + dgain;

end

%



-18 -16 -14 -12 -10 -8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Real Axis

Imag

Axi

s

Figure 4.2: Root Locus Generated by Use of rlocus(g,gain) Command

%

a = [1 3]

b = [1 19 18 0]

g = tf(a,b)

%

% generate root locus

%

rlocus(g,gain)

print -deps chap5rla.eps

The command

rlocus(g,gain)

prints an x at each point of the root locus. Figure 4.2 the x's are so close together they forma thick solid line. We can spread the x's out by plotting fewer points.

We can also write our own root locus program. Suppose for unity feedback we have

Gp(s) =(s2 + 12s+ 72)

s4 + 12s3 + 72s2 + 120s+ 100

=(s+ 6� j6)(s+ 6 + j6)

(s+ 1� j)(s+ 1 + j)(s+ 5� j5)(s+ 5 + j5);


and

Gc(s) =160(s+ 1)2

s(s + 24:71)

Then the Matlab program chap5rl.m listed below will plot the root locus for 0 < K < 160

%Root Locus Program chap5rl.m

gaininc = 0.25

gainmin = 0

gainmax = 160

gain = gainmin;

kount = 1;

while (gain <= gainmax);

a = [1,2,1];

b= [1,12,72];

top = conv(a,b);

c = [1,0];

d = [1,24.71];

e =[1,10,50];

f = [1,2,2];

bot = conv(c,d);

bot = conv(bot,e);

bot = conv(bot,f);

roots(top);

roots(bot);

top = top * gain;

top=[0,0,top];

p= bot + top;

x = roots(p);

a1(kount) = real ( x(1) );

b1(kount) = imag ( x(1) );

c1(kount) = real( x(2) );

d1(kount) = imag ( x(2) );

e1(kount) = real( x(3) );

f1(kount) = imag( x(3) );

g1(kount) = real( x(4) );

h1(kount) = imag( x(4) );

i1(kount) = real( x(5) );

j1(kount) = imag( x(5) );

k1(kount) = real( x(6) );

l1(kount) = imag( x(6) );

gain = gain + gaininc;

kount = kount + 1;

end

save real1.dat a1 -ascii


-25 -20 -15 -10 -5 0-8

-6

-4

-2

0

2

4

6

8

Figure 4.3: Root Locus Generated by Program chap5rl.m

save imag1.dat b1 -ascii

save real2.dat c1 -ascii

save imag2.dat d1 -ascii

save real3.dat e1 -ascii

save imag3.dat f1 -ascii

save real4.dat g1 -ascii

save imag4.dat h1 -ascii

save real5.dat i1 -ascii

save imag5.dat j1 -ascii

save real6.dat k1 -ascii

save imag6.dat l1 -ascii

plot(a1,b1,'k.',c1,d1,'k.',e1,f1,'k.',g1,h1,'k.',i1,j1,'k.',k1,l1,'k.')

print -deps chap5rl.eps

In the plot command 'k.' places a '.' at each point that is on the root locus. The command

print -deps chap5rl.eps

saves the root locus plot as an encapsulated postscript �le. The root locus is shown inFigure 4.3

The command rlocfind can be used to �nd the gain associated with a speci�c set ofpoles. Suppose we have displayed in the Matlab command window the root locus shown inFigure 4.2. Then the following Matlab diaglogue allows us to �nd the gain associated with aparticular set of poles.


EDU>[K,poles]=rlocfind(g)

Select a point in the graphics window

selected_point =

-6.146464646464647e+00 + 3.820512820512819e+00i

K =

1.167208589859281e+02

poles =

-6.160840653110633e+00 + 3.804837293076535e+00i

-6.160840653110633e+00 - 3.804837293076535e+00i

-6.678318693778726e+00

When we enter the command

[K,poles]=rlocfind(g),

a set of crosshairs appear, the intersection of which can be controlled by the computer mouse.When we have the maneuvered the crosshairs to the desired position, clicking the mousecauses the value of K and the poles locations shown above to be displayed. In Figure 4.4 thelocation of the three poles corresponding to the gain 162.72 are marked by the + signs.

In general, I �nd the Matlab plotting routines to be diÆcult to use and inferior to thequality of graph that can be obtained with a software package like Deltagraph. Even Excelis superior to Matlab for graphing data. So the user has to make the choice whether to useMatlab to generate the plot, or to use Matlab to generate the data, and then plot the datausing some other software package.


-18 -16 -14 -12 -10 -8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Real Axis

Imag

Axi

s

Figure 4.4: Root Locus Generated by Use of rloc�nd Command

Chapter 6

Specifying Performance

6.1 Useful Commands

Some useful commands that we can use with this chapter are

1. damp

2. pole

3. zpk

4. zpkdata

5. tfdata

6. ltiview

The Matlab command damp �nds the natural frequency !n and the damping ratio � forall the poles of transfer function. For instance, let

Tc(s) =250

(s+ 4� j3)(s+ 4 + j3)(s+ 10)

The following Matlab dialogue shows how to use zpk and damp.

EDU>g = zpk([],[-4+j*3,-4-j*3,-10],250)

Zero/pole/gain:

250

----------------------

(s+10) (s^2 + 8s + 25)

EDU>damp(g)

Eigenvalue Damping Freq. (rad/s)

1

2 CHAPTER 6. SPECIFYING PERFORMANCE

-4.00e+00 + 3.00e+00i 8.00e-01 5.00e+00

-4.00e+00 - 3.00e+00i 8.00e-01 5.00e+00

-1.00e+01 1.00e+00 1.00e+01

EDU>

The arguments of the command zpk are:

1. [ ] indicating there are no zeros in the transfer function

2. [-4j*3,-4-j*3,-10]+ de�ning the poles of the system

3. 250 de�ning the gain of the system.

The command

damp(g)


The extracts the damping ratio and naturalfrequency of all poles of the system.The matlab command tfdata returns the zeros, poles and gain of the system. For

instance, for the system we just de�ned we have

EDU>[Z,P,K] = zpkdata(g,'v')

Z =

Empty matrix: 0-by-1

P =

-4.000000000000000e+00 + 3.000000000000000e+00i

-4.000000000000000e+00 - 3.000000000000000e+00i

-1.000000000000000e+01

K =

250

EDU>

The Matlab command tfdata works in a similar way, as demonstrated by the followingMatlab dialogue.

EDU>num = 250

num =

250

EDU>p1 = [1 4-j*3]

p1 =

Column 1

1.000000000000000e+00

Column 2

4.000000000000000e+00 - 3.000000000000000e+00i

4 CHAPTER 6. SPECIFYING PERFORMANCE

EDU>p2 = [1 4+j*3]

p2 =

Column 1

1.000000000000000e+00

Column 2

4.000000000000000e+00 + 3.000000000000000e+00i

EDU>p3 = [1 10]

p3 =

1 10

EDU>den = conv(p1,p2)

den =

1 8 25

EDU>den = conv(den,p3)

den =

1 18 105 250

EDU>g1 = tf(num,den)

Transfer function:

250

--------------------------

s^3 + 18 s^2 + 105 s + 250

EDU>[n,d] = tfdata(g1,'v')

n =

0 0 0 250


d =

1 18 105 250

EDU>

One additional command of some use is ltiview. This command opens a graphical userinterface (GUI) window that allows the user to select any system currently de�ned in theworkspace and make various plots, such as the step response, Bode magnitude and phaseplot, nyquist plots, and so on. At this juncture in the book the only plots that the reader`oÆcially' knows about are the step and impulse responses. One of the features of this GUIwindow is that the reader can select various characterstics of the plot and have them appearon the plot. Of course, this could be done almost as easily using the command plot, andeach user has to decide which display method is best. It is really a matter of taste. Thiscommand is probably best used in the early stages of a design process to compare two orthree preliminary designs.

Chapter 7

Root Locus Design

7.1 Useful Commands

Most of the commands that pertain to root locus design were discussed in Chapters 5 and 6.One command of great use is the command

feedback.

The following Matlab dialogue shows how it is used.

1

2 CHAPTER 7. ROOT LOCUS DESIGN

EDU>clear all

EDU>g = zpk([-3],[0 -1 -18], 78)

Zero/pole/gain:

78 (s+3)

--------------

s (s+1) (s+18)

EDU>h = 1

h =

1

EDU>tc = feedback(g,h)

Zero/pole/gain:

78 (s+3)

----------------------

(s+13) (s^2 + 6s + 18)

EDU>step(tc,3)

EDU>print -deps stepresp.eps

The feedback command lets us determine the closed loop transfer function for outputfeedback. Here we have chosen unity feedback with the command

h = 1

We can then �nd the unit step response, shown in Figure 7.1, with the command

step(tc,3)

One additional command is of some use, namely ltiview. This command opens a graphicaluser interface (GUI) window that allows the user to select any system currently de�ned inthe workspace and make various plots, such as the step response, Bode magnitude and phaseplots, Nyquist plots, and so on. At this juncture in the book the only plots that the reader`oÆcially' knows about are the step and impulse responses.

One of the features of this GUI window is that the reader can select various characteristicsof the plot and have them appear on the plot. Of course, this could be done almost as easilyusing the command plot, and each user has to decide which display method is best. It isreally a matter of taste. This command is probably best used in the early stages of a designprocess to compare two or three preliminary designs.


Time (sec.)

Am

plitu

de

Step Response

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 7.1: Step Response of Closed Loop System

Chapter 8

Speed Control

8.1 Data Description

All the data �les in this chapter are in column form with the �rst column

being the time and the second column being the system step respoonse. For

the step responses of the d c motors the armature voltage response is also

given.

The motor data is contained in a folder called Motordata. The data

for transfer function identi�cation in problems 8,9, and 10 is contained in a

folder called xferid. For problems 8,9, and 10, the inputis a unit step.

1

Chapter 9

Frequency Response

9.1 Useful Commands

Several useful Matlab commands are

1. bode,

2. axis,

3. subplot

The Matlab program chap9bode1, repeated below for convenience, creates the Bode magni-tude and phase plots shown in Figure 9.1.

a = 100*[1 5]

b = [1 19 18 0]

g = tf(a,b)

grid on

bode(g)

print -deps chap9bode1.eps

The transfer function in this case is

Gp(s) =100(s+ 5)

s(s+ 1)(s+ 18):

The disadvantage of this plot is that I would prefer to have the scaling of the magnitudeplot be in increments of 20 dB because in the transfer indenti�cation process we look forslopes of � 20, 40 and 60 db/dec.

We can write our own program to compute the Bode magnitude and phase plots. Anexample is program bodeplot.m, listed below for convenience.

1

2 CHAPTER 9. FREQUENCY RESPONSE

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

Bode Diagrams

-50

0

50

100

10-2

10-1

100

101

102

-180

-160

-140

-120

-100

-80

Figure 9.1: Bode Plots Generated by Program chap9bode1.m

fmin =0.01

fmax = 1000

ftop = log10(fmax)

fbot = log10(fmin)

npts = 100

dw = ( ftop - fbot) / npts

for i = 1:npts

w = fmin*( 10^(i*dw) );

s = j* w;

mag(i) = 20*log10(abs(100*(s + 3)/( s * (s + 10)*(s + 100))));

phase(i) =( atan(w/3)-pi/2 -atan(w/10) -atan(w/100) )*180/pi;

radfreq(i) = w;

end

subplot(2,1,1),semilogx(radfreq,mag)

grid on

axis([0.1 1000 -100 20])

subplot(2,1,2), semilogx(radfreq,phase)

grid on

axis([0.1 1000 -180 0])

print -deps chap9bode2.eps

The result is shown in Figure 9.2 Note we have used the subplot command here to get both


10-1

100

101

102

103

-100

-80

-60

-40

-20

0

20

10-1

100

101

102

103

-150

-100

-50

0

Figure 9.2: Bode Plots Generated by Program bodeplot.m

the magnitude and phase plots.This is a reasonable looking plot, but if you want easy control of the axes, tick marks and

line widths your best bet is to generate the data using Matlab and then transfer the data toa good graphics package.

Chapter 1

The Nyquist Criterion

9.1 Useful Commands?

We discuss brie y the Matlab commands

1. nyquist

2. nichols

and conclude they are worthless. The program chap10nyq.m, repeated below for convenience

draws the Nyquist plot shown in Figure 9.1. As can be seen it is of very little help because

of the scaling.

%Program to draw Nyquist Plot for

% G(s) = 100/[s (s + 10)(s + 40)]

%user generates.

%


%

a = 100

b = [1 50 400 0]

roots(b)

g = tf(a,b)

%

% generate Nyquist plot

%

nyquist(g)

print -deps chap10nyq.eps

The command nichols is no better. The program chap10nic.m, repeated below for

convenience, generates the plot shown in Figure 9.2.

%Program to draw Nichols Plot for

% G(s) = 100/[s (s + 10)(s + 40)]

%user generates.

%

1

2 CHAPTER 9. THE NYQUIST CRITERION

Chapter 11

Bode Design

11.1 Useful Commands

We have already discussed the basic command for frequency response analysis, namely bode.

Three other closely related commands are

1. evalfr,

2. freqresp,

3. margin

The �rst two are very minor variations of bode. The third is not much more than that, but we

illustrate it with the short Matlab program chap11margin.m, repeated below for convenience.

This program creates the Bode magnitude and phase plots shown in Figure 11.1.

a = 100000*[1 5]

c = [1 0]

d = [1 1]

e = [1 50]

f = [1 100]

b = conv(c,d)

b = conv(b,e)

b = conv(b,f)

g = tf(a,b)

grid on

margin(g)

print -deps chap11mar.eps

As usual Matlab has chosen very inconvenient scaling for the Bode magnitude plot. This

command to me is only of marginal worth, as are evalfr and freqresp.

1

2 CHAPTER 11. BODE DESIGN

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

Bode Diagrams

-100

-50

0

50

100Gm=16.5 dB (Wcg=66.3); Pm=46.7 deg. (Wcp=19.0)

10-2

10-1

100

101

102

103

-300

-250

-200

-150

-100

-50

Figure 11.1: Bode Plots Generated by Program chap11margin,m

Chapter 12

Robust Control

There is a robust control tool box, but it is devoted to multi-input, multi-output models. A

useful exercise is to write a program to �nd the line of closest approach to the point �1 in

the GH-plane.

1

Chapter 13

Position Control

13.1 Impulse Data

The data is stored in column form. The �rst column is the time in seconds the second columnthe impulse response of the postioning table. The armature voltage is also provided for eachturntable. The �lename

ptkixx

is decoded as: postioning table `k' impulse of xx miliseconds. The armature voltage �lenameis similar but with the suÆx `av.'

13.2 Frequency Response Data

The data is stored in column form as shown in Figure 13.1

Freq.(Rad/sec) Arm Voltage Heliopot Voltage Mag.(dB) Phase(deg)

vvv www xxx yyy zzz...

......

......

Figure 13.1: Data Format for Frequency Response Data

1

Chapter 14

Discrete Systems


We introduce here the commands

1. c2d,

2. step,

3. zpk

4. bode

We have talked about the continuous versions of the last three instruc-tions already.

The following Matlab dialogue forms the tranfer function

G(s) =a

s + a;

and then converts it to the corresponding zero order hold digital equivalentfor f = 10 hz.

EDU>z = [ ]

z =

[ ]

EDU>p = -5

1

2 CHAPTER 14. DISCRETE SYSTEMS

p =

-5

EDU>K = 5

K =

5

EDU>g = zpk(z,p,K)

Zero/pole/gain:

5

-----

(s+5)

EDU>gz = c2d(g,0.1)

Zero/pole/gain:

0.39347

----------

(z-0.6065)

Sampling time: 0.1

EDU>

In getting gz we have used the command zpk to form the continuoustransfer function before converting it to the discrete version. The commandc2d can form several types of discrete equivalents, two of which are discussedin Chapter 15. We have used the default version of zpk which does the zeroorder hold equivalent. We could have typed

gz = c2d(g,0.1,\zoh')

and obtained the same result.The use of the the Matlab command step for discrete systems is demon-

strated by the following dialogue, where we use the function gz obtainedabove.

EDU>step(gz,0:0.1:4)


Time (sec.)

Am

plitu

deStep Response

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 14.1: Step Response Generated by step(gz,0:0.1:4)

EDU>print -deps dstep.eps

EDU>

The notation `0:0.1:4' sets the starting time (t = 0), the ending time (t= 4) and the intersample period, T = 0.1. The second command storesthe time response as an encapsulated postscript �le. That �le is shown inFigure 14.1

The Matlab command zpk can be used with discrete systems as demon-strated by the following dialogue.

EDU>gz1 = zpk([ ],-0.5,2,0.1)

Zero/pole/gain:

2

-------

(z+0.5)

Sampling time: 0.1


Time (sec.)

Am

plitu

de

Step Response

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

Figure 14.2: Step Response of gz1

EDU>step(gz1,4)

EDU>print -deps dstepgz1.eps

EDU>

Note that we did not have to specify the sampling rate since when we createdgz1 we speci�ed the sampling rate of 10 hz. The step response is shown inFigure 14.2.

The command bode can be used for discrete systems as well. In thefollowing Matlab dialogue we �nd the frequency response of

G(z) =0:39347

z � 0:6065:

EDU>bode(gz)

EDU>print -deps dbode.eps

EDU>

The Bode phase and magnitude plots are shown in Figure 14.3 In this par-


Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

Bode Diagrams

-15

-10

-5

0

100

101

102

-200

-150

-100

-50

0

Figure 14.3: Bode Phase and Magnitude Plots


ticular case the sampling rate is 10 Hz so that the maximum frequency thatcan be sampled without aliasing is

! = �=T = 31:4 rad/sec:

The vertical dark lines denote the maximum sampling frequency.

Chapter 15

Digital Control


In Chapter 14 we discussed most of the commands of use in analyzing digital control systems.The command c2d discussed in chapter 14 has several alternative modes. Two of importanceto us are the bilinear and ad hoc mappings, both of which can be done with c2d. In additionthe commands

1. d2d

2. damp, and

3. filt

4. ltiview

are also of some use.The command c2d �rst discussed in Chapter 14, can be used to determine the bilinear

and ad hoc digital equivalents of continuous transfer functions.Consider the transfer function

G(s) =10(s+ 2)

(s+ 1)(s+ 4);

discussed in Sections 15.2.3 and 15.2.4 of the text. Then the following Matlab dialoguecreates the equivalent discrete transfer function under the bilinear (Tustin's) mapping.

EDU>g = zpk(-2,[-1 -4],10)

Zero/pole/gain:

10 (s+2)

-----------

(s+1) (s+4)

1

2 CHAPTER 15. DIGITAL CONTROL

EDU>gz = c2d(g,0.1,'tustin')

Zero/pole/gain:

0.43651 (z-0.8182) (z+1)

------------------------

(z-0.9048) (z-0.6667)

Sampling time: 0.1

The following dialogue generates the ad hoc version of G(s).

EDU>gzah = c2d(g,0.1,'matched')

Zero/pole/gain:

0.86538 (z-0.8187)

---------------------

(z-0.9048) (z-0.6703)

Sampling time: 0.1

The command d2d can be used to change an existing discrete transfer function at onesampling frequency to an equivalent discrete transfer function at a new sampling frequency.For instance, we found the discrete equivalent of

G(s) =10(s+ 2)

(s+ 1)(s+ 4)

using the bilinear mapping and a sampling frequency of 10 Hz. The following Matlab dialogueshows how to change to a sampling rate of 20 Hz.

EDU>gz20 = d2d(gz,0.05)

Zero/pole/gain:

0.43651 (z+0.0758) (z-0.9046)

-----------------------------

(z-0.9512) (z-0.8165)

Sampling time: 0.05

The Matlab command damp can be used to �nd the equivalent natural frequency ofdamping ratio of complex poles in the z-plane. The following Matlab dialogue �rst forms thecontinuous transfer function

G(s) =25

(s+ 4� j3)(s+ 4 + j3);

�nds the discrete, zero order hold equivalent, and then uses damp to �nd !n and �.


EDU>gs = zpk([],[-4+j*3,-4-j*3],25)

Zero/pole/gain:

25

---------------

(s^2 + 8s + 25)

EDU>gzc = c2d(gs,0.1,'zoh')

Zero/pole/gain:

0.095495 (z+0.7652)

-----------------------

(z^2 - 1.281z + 0.4493)

Sampling time: 0.1

EDU>damp(gzc)

Eigenvalue Magnitude Equiv. Damping Equiv. Freq. (rad/s)

6.40e-01 + 1.98e-01i 6.70e-01 8.00e-01 5.00e+00

6.40e-01 - 1.98e-01i 6.70e-01 8.00e-01 5.00e+00

The matlab command filt creates a digital �lter using the digital signal processingpreference for powers of z�1. The followng Matlab dialogue creates such a �lter.

EDU>a = [1 2 2 1]

a =

1 2 2 1

EDU>b = [1 2 4 2 ]

b =

1 2 4 2

EDU>gdsp = filt(a,b)

Transfer function:

1 + 2 z^-1 + 2 z^-2 + z^-3

----------------------------

1 + 2 z^-1 + 4 z^-2 + 2 z^-3

4 CHAPTER 15. DIGITAL CONTROL

Sampling time: unspecified

EDU>gdsp10 = filt(a,b,0.1)

Transfer function:

1 + 2 z^-1 + 2 z^-2 + z^-3

----------------------------

1 + 2 z^-1 + 4 z^-2 + 2 z^-3

Sampling time: 0.1

The Matlab command ltiview was discussed in Chapter 7. It is equally applicable todiscrete systems.

Chapter 16

Aircraft Pitch Control

16.1 Impulse Data

The data is stored in column form. The �rst column is the time in seconds the second columnthe impulse response of the postioning table. The armature voltage is also provided for eachturntable. The �lename

ptkixx

is decoded as: postioning table `k' impulse of xx miliseconds. The armature voltage �lenameis similar but with the suÆx `av.'

16.2 Frequency Response Data

The data is stored in column form as shown in Figure 16.1

Freq.(Hz) Freq.(Rad/sec) Arm Voltage Heliopot Voltage Phase(deg)

vvv www xxx yyy zzz...

......

......

Figure 16.1: Data Format for Frequency Response Data

1

Chapter 17

The Transportation Lag


There is not much in the Matlab tool boxes that will help us with a system that has atransportation lag. You can either use Simulink or one of the two Matlab programs ufeedbakor ufeedbakd.

The program ufeedbak is an interactive program that requires you to enter the data fromthe keyboard. It simulates the response of a continuous system to inputs of the form

u(t) = at3 + bt

2 + ct+ d:

The program ufeedbakd does the same thing for sampled data systems.If you do not like the interactive nature of these programs, then I encourage you to modify

them to your own satisfaction. The program rwtest.m that uses the data �le called data isshort introduction to how to read and write data in Matlab.

1

Chapter 18

The State Model


Several useful Matlab commands related to the analysis of state models. Five are listed

below.

1. ss,

2. ss2ss,

3. canon

4. eig

5. inv

The Matlab program chap18ss, repeated below for convenience, creates the state space model

called stmod

A = [0 1 0;0 0 1;0 -2 -3]

B = [0;0;1]

C = [1 0 0]

D = 0

stmod = ss(A,B,C,D)

The output generated in the Matlab command window by executing this program is

A =

0 1 0

0 0 1

0 -2 -3

1

2 CHAPTER 18. THE STATE MODEL

B =

0

0

1

C =

1 0 0

D =

0

a =

x1 x2 x3

x1 0 1.00000 0

x2 0 0 1.00000

x3 0 -2.00000 -3.00000

b =

u1

x1 0

x2 0

x3 1.00000

c =

x1 x2 x3

y1 1.00000 0 0

d =

u1

y1 0

Continuous-time system.

The system stmod can then be used as an argument for many of the Matlab commands

we have already discussed. For instance, the command


Time (sec.)

Am

plitu

de

Step Response

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

Figure 18.1: Step Response of State Model stmod

step(stmod)

creates the step response shown in Figure 18.1

As another example, typing

eig(stmod)

in the command window produces the dialogue

EDU>eig(stmod)

ans =

0

-1

-2

EDU>

Thus the eigenvalues of the plant matrix A are

� = 0 : � = �1 ; and� = �2:

If we type

[T,F] = eig(A)

4 CHAPTER 18. THE STATE MODEL

we obtain

T =

1.000000000000000e+00 -5.773502691896258e-01 2.182178902359923e-01

0 5.773502691896258e-01 -4.364357804719847e-01

0 -5.773502691896258e-01 8.728715609439694e-01

F =

0 0 0

0 -1 0

0 0 -2

If we then type

tinv = inv(T)

we obtain

tinv =

1.000000000000000e+00 1.500000000000000e+00 4.999999999999999e-01

0 3.464101615137753e+00 1.732050807568877e+00

0 2.291287847477920e+00 2.291287847477920e+00

If we next type

cstmod = ss2ss(stmod,tinv)

we obtain

a =

x1 x2 x3

x1 1.82407e-33 -6.40988e-17 1.93816e-16

x2 2.16731e-16 -1.00000 3.05716e-16

x3 0 0 -2.00000

b =

u1

x1 0.50000

x2 1.73205

x3 2.29129

c =


x1 x2 x3

y1 1.00000 -0.57735 0.21822

d =

u1

y1 0


Except for the round o� error, we now have the original system stmod in Jordan canonical

form.

We can accomplish the same thing with the canon command. That is, if we type

cstmod = canon(stmod)

we obtain

a =

x1 x2 x3

x1 0 0 0

x2 0 -1.00000 0

x3 0 0 -2.00000

b =

u1

x1 0.50000

x2 1.73205

x3 2.29129

c =

x1 x2 x3

y1 1.00000 -0.57735 0.21822

d =

u1

y1 0


This is the same model, sans the round o� error.

Chapter 19

Observability and Controllability


The following commands are of use in studying controllability and observability, and inconverting continuous transfer functions to both continuous and discrete state models.

1. ctrb

2. obsv

The Matlab program dialogue below forms the matrices A and b for the system

_x(t) = Ax(t) + bu(t);

and then uses the Matlab commands ctrb and rank to from the controllability matrix and�nd its rank.

EDU>A = [0 1; 0 0]

A =

0 1

0 0

EDU>b = [0;1]

b =

0

1

EDU>co = ctrb(A,b)

1

2 CHAPTER 19. OBSERVABILITY AND CONTROLLABILITY

co =

0 1

1 0

EDU>rank(co)

ans =

2

EDU>

Thus, we see that even though A does not have full rank the system is controllable.

The Matlab dialogue below forms the matrices A and c for the system

_x(t) = Ax(t)

y(t) = cTx

and then forms the observability matrix and checks its rank.

A = [0 1;-2 -3]

A =

0 1

-2 -3

EDU>c = [0 1]

c =

0 1

EDU>ob = obsv(A,c)

ob =

0 1

-2 -3

EDU>rank(ob)

ans =


2

EDU>

The command c2d can be useful here in �nding the discrete state model equivalent of atransfer function, or continuous state model. For instance, the following dialogue shows howwe can turn a transfer function �rst into a continuous state model and then into the stepinvariant discrete state model.

EDU>g = zpk([],[0 -2],2)

Zero/pole/gain:

2

-------

s (s+2)

EDU>stmod = ss(g)

a =

x1 x2

x1 -2.00000 0

x2 1.00000 0

b =

u1

x1 1.41421

x2 0

c =

x1 x2

y1 0 1.41421

d =

u1

y1 0


EDU>stmodd = c2d(stmod,0.1,'zoh')

a =

4 CHAPTER 19. OBSERVABILITY AND CONTROLLABILITY

x1 x2

x1 0.81873 0

x2 0.09063 1.00000

b =

u1

x1 0.12818

x2 0.00662

c =

x1 x2

y1 0 1.41421

d =

u1

y1 0

Sampling time: 0.1

Discrete-time system.

EDU>

Chapter 20

The Controller/Observer

I do not �nd much in the Matlab control toolbox that will help you with the design ofcontroller/observers. I have included some programs that I have written to support this

activity. They are a start. They are by no means everything you would want. and that isintentional. These programs are crude and cumbersome, but with some work you can turnthem into useful tools. The programs that begin with `build' compute the controller/observer

matrices for some useful choices of dominant poles. The programs that begin with `sim' aresimulation routines. These routines are far from perfect, but they work, and by studying

them you should be able to improve them and make them your own. Good Luck.

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continuous & discreate control systems

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