CSE-Lab Manual 2015

2015

Subject Code-2141004

Control System EngineeringLaboratory File

B.E. (Electronics and Communication)

Submitted to Department of Electronics and Communication Engineering

Sanjaybhai Rajguru College of Engineering-Rajkot.

CSE Lab Manual E.C. Department Sanjaybhai Rajguru College of Engineering, Rajkot

Sanjaybhai Rajguru College of Engineering, RAJKOT

This is to certify that Mr./Ms. Of semester , Enrollment no.: has satisfactorily completed his term work in For the term ending in 2015.

Date:

Signature of Teacher Head of the Department


Sanjaybhai Rajguru College of Engineering‐Rajkot Department of Electronics & Communication Engineering

Semester‐IV Subject: Control System Engineering (2141004)

Index

Sr. No Title Date Sign 1 Introduction to MATLAB family of programs.

2 Using 'MATLAB base program + Control System

toolbox', for control systems analysis and design.

Brief introduction to Symbolic Math toolbox.

3 Building simple Simulink simulations. Running

Simulink simulation to predict a system's

behaviour.

4 Presenting MATLAB functions to carryout block

diagram manipulations, followed by time-response

analysis. Using Simulink simulation for time-

response analysis.

5 Analysis of the system behaviour to changes in

input reference signals or disturbances using steady-

state and transient performance indices. Brief

introduction to LTI Viewer, and time response

analysis using this GUI tool.

6 Stability analysis in state space. Stability analysis

using root locus plots.

7 Design using root locus plots. Brief introduction to

SISO Design Tool, and root locus design using this

GUI tool.

8 Introducing MATLAB commands for Bode and

Nyquist plots. Stability analysis using these plots.

9 Frequency response measures from Bode plots and

Nichols chart, for unity-feedback systems.

Frequency response measures from Bode plots for

non-unity-feedback systems.

10 Design using Bode plots and Nichols chart.

Frequency response design using SISO Design

Tool.


PRACTICAL - 1

MATLAB Window Environment and the Base Program

Objectives:

Introduction to MATLAB family of programs

Brief introduction to MATLAB base program in an interactive "hands on" tutorial style

MATLAB, an abbreviation of MATrix LABoratory, is a high-level technical computing

environment suitable for solving scientific and engineering problems. The MATLAB family

of programs includes the base program plus a variety of application-specific solutions called

toolboxes. Toolboxes are comprehensive collections of MATLAB functions that extend the

MATLAB environment to solve particular classes of problems. In this module, the

environment provided by the base program is explored.

The MATLAB base program along with the functions from the Control System Toolbox, can

be used to analyze and design control systems problems such as those covered in this course.

Control System Toolbox includes interactive analysis and design tools called LTI Viewer,

and SISO Design Tool. The LTI Viewer uses a Graphical User Interface (GUI) to generate

and view time and frequency response plots of Linear Time-Invariant (LTI) transfer functions

and obtain measurements from these plots. The SISO Design Tool uses a GUI to provide a

convenient and interactive way to design Single-Input-Single-Output (SISO) control systems

by the conventional trial-and-error approach. We start using functions of Control System

Toolbox in module 2, and enhance its usage in later modules. Whenever MATLAB is

referred to in this course, you can interpret that to mean the base program plus the Control

System Toolbox. This, in fact, constitutes the beginning of the study of MATLAB for control

engineering. There is so much more that must be explored for additional functionality and

convenience in designing control systems and analyzing a system's behaviour.

Symbolic Math Toolbox adds symbolic mathematics capability to the MATLAB

environment. Functions and equations can be entered symbolically. Symbolic expressions can

be manipulated algebraically and simplified. Transfer functions can be entered almost as

written. Laplace transforms as well as their inverses can be entered and found in symbolic

form. An introduction to Symbolic Math Toolbox and its capabilities will be provided in

Module 2.

MATLAB's Simulink software is used to simulate systems. It uses a GUI to interact with

blocks that represent subsystems. We can position the blocks, resize the blocks, label the

blocks, specify block parameters, and interconnect blocks to form complete systems from

which simulations can be run. Simulink will be introduced in Module 3. We will enhance

usage of Simulink in later modules to study the effects of nonlinearities, and uncertainties in

model parameters on the performance of closed-loop systems.


PRACTICAL - 2

Control System Toolbox and Symbolic Math Toolbox

Objectives:

Using 'MATLAB base program + Control System toolbox', for control systems analysis

and design.

Brief introduction to Symbolic Math toolbox.

MATLAB is presented as an alternate method of solving control system problems. You are

encouraged to solve problems first by hand and then by MATLAB so that insight is not lost

through mechanized use of computer programs.

In Module 1, we have presented a tutorial on MATLAB window environment and some of its

basic commands. In the present module, our objective is to take a primary look at MATLAB

Control System Toolbox, which expands MATLAB base program to include control-system

specific commands. In addition, presented is a MATLAB enhancement – Symbolic Math

Toolbox, that gives added functionality to MATLAB base program and Control System

Toolbox.

Control System Toolbox

Symbolic Math Toolbox


http://nptel.iitk.ac.in/courses/Webcourse-contents/IIT-Delhi/Control%20system%20design%20n%20principles/matlab/module2/module2-1_middle.htm


PRACTICAL – 2.1


Control System Toolbox

Control System Toolbox is a collection of commands to be used for control systems' analysis

and design. We will be using only some of these commands, because of the limited nature of

course profile. Description of these commands will be distributed in different modules.

In this module, we will present commands related to transfer functions and system responses.

To see all the commands in the Control System Toolbox and their functionalities, type help

control in the MATLAB command window. MATLAB will respond with >> help control

General.

ctrlpref - Set Control System Toolbox preferences.

ltimodels - Detailed help on the various types of LTI models.

ltiprops - Detailed help on available LTI model properties.

Creating linear models.

tf - Create transfer function models.

zpk - Create zero/pole/gain models.

ss, dss - Create state-space models.

frd - Create a frequency response data models.

filt - Specify a digital filter.

lti/set - Set/modify properties of LTI models.

Data extraction.

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

zpkdata - Extract zero/pole/gain data.

ssdata - Extract state-space matrices.

dssdata - Descriptor version of SSDATA.

frdata - Extract frequency response data.

lti/get - Access values of LTI model properties.

Conversions.

tf - Conversion to transfer function.

zpk - Conversion to zero/pole/gain.

ss - Conversion to state space.

frd - Conversion to frequency data.

chgunits - Change units of FRD model frequency points.

c2d - Continuous to discrete conversion.

d2c - Discrete to continuous conversion.

d2d - Resample discrete-time model.


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.

lft - Generalized feedback interconnection (Redheffer star product).

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

System gain and dynamics.

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

bandwidth - System bandwidth.

lti/norm - Norms of LTI systems.

pole, eig - System poles.

zero - System (transmission) zeros.

pzmap - Pole-zero map.

iopzmap - Input/output pole-zero map.

damp - Natural frequency and damping of system poles.

esort - Sort continuous poles by real part.

dsort - Sort discrete poles by magnitude.

stabsep - Stable/unstable decomposition.

modsep - Region-based modal decomposition.

Time-domain analysis.

ltiview - Response analysis GUI (LTI Viewer).

step - Step response.

impulse - Impulse response.

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

lsim - Response to arbitrary inputs.

gensig - Generate input signal for LSIM.

covar - Covariance of response to white noise.

Frequency-domain analysis.

ltiview - Response analysis GUI (LTI Viewer).

bode - Bode diagrams of the frequency response.

bodemag - Bode magnitude diagram only.

sigma - Singular value frequency plot.

nyquist - Nyquist plot.

nichols - Nichols plot.

margin - Gain and phase margins.

allmargin - All crossover frequencies and related gain/phase margins.

freqresp - Frequency response over a frequency grid.

evalfr - Evaluate frequency response at given frequency.

frd/interp - Interpolates frequency response data.

Classical design.

sisotool - SISO design GUI (root locus and loop shaping techniques).

rlocus - Evans root locus.


Pole placement.

place - MIMO pole placement.

acker - SISO pole placement.

estim - Form estimator given estimator gain.

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

LQR/LQG design.

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.

augstate - Augment output by appending states.

State-space models.

rss, drss - Random stable state-space models.

ss2ss - State coordinate transformation.

canon - State-space canonical forms.

ctrb - Controllability matrix.

obsv - Observability matrix.

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.

sminreal - Structurally minimal realization.

Time delays.

hasdelay - True for models with time delays.

totaldelay - Total delay between each input/output pair.

delay2z - Replace delays by poles at z=0 or FRD phase shift.

pade - Pade approximation of time delays.

Model dimensions and characteristics.

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

size - Model sizes and order.

lti/ndims - Number of dimensions.

lti/isempty - True for empty models.

isct - True for continuous-time models.

isdt - True for discrete-time models.

isproper - True for proper models.

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

reshape - Reshape array of linear models.

Overloaded arithmetic operations.

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

* - Multiply systems (series connection).


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

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

^ - Powers of a given system.

' - Pertransposition.

.' - Transposition of input/output map.

[..] - Concatenate models along inputs or outputs.

stack - Stack models/arrays along some array dimension.

lti/inv - Inverse of an LTI system.

conj - Complex conjugation of model coefficients.

Matrix equation solvers.

lyap, dlyap - Solve Lyapunov equations.

lyapchol, dlyapchol - Square-root Lyapunov solvers.

care, dare - Solve algebraic Riccati equations.

gcare, gdare - Generalized Riccati solvers.

bdschur - Block diagonalization of a square matrix.

Demonstrations.

Type "demo" or "help ctrldemos" for a list of available demos.

control is both a directory and a function.

--- help for modeldev/control.m ---

MODELDEV/CONTROL

In this module, we will learn how to represent transfer functions in the MATLAB, partial

fraction expansion of rational expressions, representation of transfer functions as LTI objects,

and to obtain time domain responses of LTI systems. Important commands for this module

are:

roots – Find polynomial roots

poly – Convert roots to polynomial

polyval – Evaluate polynomial value

conv – Convolution and polynomial multiplication

deconv – Deconvolution and polynomial division

residue – Partial-fraction expansion (residues)

tf – Creation of transfer functions or conversion to transfer functions

pole – Compute the poles of LTI models

zero – Transmission zeros of LTI systems

tfdada – Quick access to transfer function data

zpkdata – Quick access to zero-pole-gain data

pzmap – Pole-zero map of LTI models

zpk – Create zero-pole-gain models or convert to zero-pole-gain format

step – Step response of LTI models

impulse – Impulse response of LTI models

lsim – Simulate time response of LTI models to arbitrary inputs

gensig – Periodic signal generator for time response simulations with lsim


Polynomials

Consider a polynomial ‘s 3

+3 s 2 +4', to which we attach the variable name p. MATLAB can

interpret a vector of length n +1 as the coefficients of an n th

-order polynomial. Coefficients

of the polynomial are interpreted in descending powers. Thus, if the polynomial is missing

any coefficient, we must enter zeros in the appropriate places in the vector. For example,

polynomial p can be represented by the vector [1 3 0 4] in MATLAB. For example:

>> p=[1 3 0 4]

p =

1 3 0 4

Roots of the polynomial can be obtained by roots command.

>> r=roots(p)

r =

-3.3553

0.1777 + 1.0773i

0.1777 - 1.0773i

Given roots of a polynomial in a vector, a vector of polynomial coefficients can be obtained

by the command poly.

>> p=poly(r)

p =

1.0000 3.0000 0.0000 4.0000

Use the command polyval(p, s) to evaluate the polynomial represented by vector p at arbitrary value

of s. For example, to evaluate the polynomial ‘s 3 +3 s 2 +4' at , type

>> polyval(p,sqrt(2))

ans =

12.8284

The product of two polynomials is found by taking the convolution of their coefficients. The

function conv will do this for us. Consider an example of multiplying polynomial ‘s 3 +3 s 2

+4' with ‘s +2':

>> p1=[1 3 4];


>> p2=[1 2];

>> p3=conv(p1,p2)

p3 =

1 5 10 8

The function deconv divides two polynomials and returns quotient as well as the remainder

polynomial.

>> [q,r]=deconv(p1,p2)

q =

1 1

r =

0 0 2

where q is the quotient and r is the remainder polynomial.

Exercise 2.1

Verify the deconvolution result given in vectors q and r.

Hint: Check whether p1 = conv(q,p2) + r or not?

Exercise 2.2

Let

and . Obtain .

Consider the rational fractions of the form:

where the coefficients and are real constants, and and are integers.A fraction of

the form G(s) can be expanded into partial fractions. To do this, first of all we factorize the

denominator polynomial D(s) into first-order factors. The roots of D(s) can be real or

complex; distinct or repeated.


Let, vectors N and D specify the coefficients of numerator and denominator polynomials

and respectively. The command [A,p,K]=residue(N,D) returns residues in

column vector A, the roots of the denominator in column vector p, and the direct term in

scalar K. If there are no multiple roots, the fraction can be represented as:

If there are roots of multiplicity mr , i.e. , , then the expansion

includes terms of the form:

If , K is empty (zero).

Supplying 3 arguments A, p, and K to residue converts the partial fraction expansion back to

the polynomial with coefficients in N and D.

Consider the rational fraction:

MATLAB solution to partial fraction problem can be given by:

>> N=[10 40];

>> D=[1 4 3 0];

>> [A,p,K]=residue(N,D)

A =

1.6667

-15.0000

13.3333

p =

-3

-1

0


K =

[ ]

Example 2.1

Consider the function

The following MATLAB session evaluates the residues.

>> N = 13;

>> D = [1 6 22 30 13 0];

>> [A,p,K]=residue(N,D)

A =

0.0200 - 0.0567i

0.0200 + 0.0567i

-1.0400

-1.3000

1.0000

p =

-2.0000 + 3.0000i

-2.0000 - 3.0000i

-1.0000

-1.0000

0

K =

[ ]


Exercise 2.3

Represent the matrices A, p, and K obtained in Example 2.1 in partial fraction form and convert

back to the polynomial form. Counter check your answer with MATLAB.

Exercise 2.4

Consider the rational fraction:

Obtain partial fraction form in terms of K. Solve using MATLAB for and countercheck your

answer.

Exercise 2.5

Consider the rational fraction: .

Identify the points where:

1. and

2.

Note: The roots of the numerator polynomial, i.e. , , are known as the zeros of and

the roots of the denominator polynomial, i.e. , , are known as the poles of

Transfer Functions

Transfer functions can be represented in MATLAB as LTI (Linear Time Invariant) objects

using numerator and denominator polynomials. Consider the transfer function given by

It can be represented in MATLAB as:

>> num = [1 1];

>> den = [1 3 1];

>> G = tf(num,den)


Transfer function:

..s + 1

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

s^2 + 3 s + 1

Example 2.2

The function conv has been used to multiply polynomials in the following MATLAB session

for the transfer function

>> n1 = [5 1];

>> n2 = [15 1];

>> d1 = [1 0];

>> d2 = [3 1];

>> d3 = [10 1];

>> num = 100*conv(n1,n2);

>> den = conv(d1,conv(d2,d3));

>> GH = tf(num,den)

Transfer function:

7500 s^2 + 2000 s + 100

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

....30 s^3 + 13 s^2 + s

To learn more about LTI objects given by tf, type ltimodels tf in MATLAB command

window. Type ltiprops tf on MATLAB prompt to learn the properties associated with an LTI

object represented by tf.

Transfer functions can also be entered directly in polynomial form as we enter them in the

notebook using LTI objects. For example, observe the following MATLAB session.

>> s=tf('s') %Define 's' as an LTI object in polynomial form

Transfer function:


s

>> G1=150*(s^2+2*s+7)/[s*(s^2+5*s+4)] % Form G1(s) as an LTI transfer function

% in polynomial form.

Transfer function:

150 s^2 + 300 s + 1050

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

......s^3 + 5 s^2 + 4s

>> G2=20*(s+2)*(s+4)/[(s+7)*(s+8)*(s+9)] % Form G2(s) as an LTI transfer

% function in polynomial form.

Transfer function:

.....20 s^2 +120 s + 160

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

s^3 + 24 s^2 + 191s + 504

The commands pole and zero calculate the poles and zeros of LTI models.

>> pole(G1)

ans =

0

-4

-1

>> zero(G1)

ans =

-1.0000 + 2.4495i

-1.0000 - 2.4495i

To extract numerator and denominator polynomials, use the function tfdata .

>> [num,den]=tfdata(G,'v')

num =

0 1 1

den =

1 3 1


To extract zeros and poles of transfer function simultaneously, use the function zpkdata .

>> [z,p]=zpkdata(G,'v')

z =

-1

p =

-2.6180

-0.3820

If we know zeros and poles of the system with gain constant, the transfer function of LTI

system can be constructed by zpk command. For example, to create a unity gain transfer

function G3(s) with zero at -1 and two poles at -2.618 and -0.382, follow the MATLAB

session given below.

>> G3=zpk(-1,[-2.618 -0.382],1)

Zero/pole/gain:

(s+1)

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

(s+2.618)(s+0.382)

The polynomial transfer function created with tf can be converted to zero-pole-gain model by

the command zpk and vice versa . The following MATLAB session gives the zero-pole-gain

format of LTI system represented by G(s).

>> zpk(G)

Zero/pole/gain:

(s+1)

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

(s+2.618)(s+0.382)

To observe the polynomial form of the transfer function G3(s), enter

>> tf(G3)

Transfer function:

s+1

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

s^2 + 3s +1


To learn more about LTI objects given by zpk, type ltimodels zpk in MATLAB command

window. Type ltiprops zpk on MATLAB prompt to learn the properties associated with an

LTI object represented by zpk.

The function pzmap(G) plots the poles and zeros of the transfer function G(s) on complex

plane. When used with two left hand side arguments, [p,z] = pzmap(G), the function returns

the poles and zeros of the system in two column vectors p and z. For example:

>> [p,z]=pzmap(G)

p =

-2.6180

-0.3820

z =

-1

System Response

Step and impulse responses of LTI objects can be obtained by the commands step and

impulse . For example, to obtain the step response of the system represented in LTI object G,

enter

>> step(G)

The MATLAB response to this command is shown in Fig. 2.1.

Fig. 2.1

To obtain the impulse response, enter

>> impulse(G)


The MATLAB response to this command is shown in Fig. 2.2.

Fig. 2.2

Step and impulse response data can be collected into MATLAB variables by using two left

hand arguments. For example, the following commands will collect step and impulse

response amplitudes in yt and time samples in t.

[yt, t] = step(G)

[yt, t] = impulse(G)

Response of LTI systems to arbitrary inputs can be obtained by the command lsim. The

command lsim(G,u,t) plots the time response of the LTI model G to the input signal

described by u and t. The time vector t consists of regularly spaced time samples and u is a

matrix with as many columns as inputs and whose ith

-row specifies the input value at time

t(i). Observe the following MATLAB session to obtain the time response of LTI system G to

sinusoidal input of unity magnitude.

>> t=0:0.01:7;

>> u=sin(t);

>> lsim(G,u,t)


The response is shown in Fig. 2.3.

Fig. 2.3

Exercise 2.6

i. Obtain the response of to ramp and parabolic inputs using

lsim command.

ii. Obtain the response of to ramp and parabolic inputs using

step command.

The function gensig generates periodic signals for time response simulation with lsim

function. It can generate sine, square, and periodic pulses. All generated signals have unit

amplitude. Observe the following MATLAB session to simulate G ( s ) for 20 seconds with a

sine wave of period 5 seconds.

>> [u,t]=gensig( 'sin' ,5,20); %Sine wave with period 5 sec and duration 20 sec

>> lsim(G,u,t) %Simulate G(s) with u and t.


The response is shown in Fig. 2.4.

Fig. 2.4

Exercise 2.7

Generate square and pulse signals with the period of 4 seconds and obtain time

response of for a duration of 30 seconds.

Example 2.3

The following MATLAB script calculates the step response of second-order system

with and various values of

t=[0:0.1:12]; num=[1];

zeta1=0.1; den1=[1 2*zeta1 1];


zeta2=0.2; den2=[1 2*zeta2 1];

zeta3=0.4; den3=[1 2*zeta3 1];

zeta4=0.7; den4=[1 2*zeta4 1];

zeta5=1.0; den5=[1 2*zeta5 1];

zeta6=2.0; den6=[1 2*zeta6 1];

[y1,x]=step(num,den1,t); [y2,x]=step(num,den2,t);



plot(t,y1,t,y2,t,y3,t,y4,t,y5,t,y6)

xlabel( 't' ), ylabel( 'y(t)' )

grid

Response through the above MATLAB script is shown in Fig 2.5.

Fig. 2.5

The following MATLAB script calculates the impulse response of second-order system

with and various values of

t=[0:0.1:10]; num=[1];

zeta1=0.1; den1=[1 2*zeta1 1];


zeta2=0.25; den2=[1 2*zeta2 1];

zeta3=0.5; den3=[1 2*zeta3 1];

zeta4=1.0; den4=[1 2*zeta4 1];

[y1,x,t]=impulse(num,den1,t);




plot(t,y1,t,y2,t,y3,t,y4)

xlabel( 't' ), ylabel( 'y(t)' )

grid

Response through the above MATLAB script is shown in Fig 2.6.

Fig. 2.6

Right-clicking away from the curves obtained by step, impulse, and lsim commands brings

up a menu. From this menu, various time-response characteristics can be obtained and plotted

on the graph.


PRACTICAL – 2.2


Symbolic Math Toolbox

MATLAB's Symbolic Math (Symbolic Mathematics) toolbox allows users to perform

symbolic mathematical computations using MATLAB. The only basic requirement is

to declare symbolic variables before they are used. For control systems analysis and

design, symbolic math toolbox is particularly important because of the following:

1. Transfer functions and other expressions can be entered in symbolic form as

we write them in the notebook.

2. Symbolic expressions can be manipulated algebraically and simplified.

3. It is straightforward to convert symbolic polynomials to the vectors of

corresponding power-term coefficients.

4. Expressions can be ‘pretty printed' for clarity in the MATLAB command

window.

Type help symbolic on the MATLAB command prompt to see all the functionalities

available with Symbolic Math toolbox. In this section, we will learn commands of

Symbolic Math toolbox particularly useful for control engineering.

First we demonstrate the power of Symbolic Math toolbox by calculating inverse

Laplace transform. The following MATLAB session gives the steps performed during

the calculation of inverse Laplace transform of using Symbolic

Math.


Exercise 2.8

Initialize s and T as symbols. Using Symbolic Math tools, find the response of the

first- order system ,with the step excitation of strength A .

Hint: Find the inverse Laplace transform of .

Exercise 2.9

Find manually the Laplace transform of . Using Symbolic

Math tools, declare t as symbolic variable and countercheck your result. Use the

function laplace(g) to calculate the Laplace transform.

The function simple(G) finds simplest form of G(s) with the least number of terms.

simplify(G) can be used to combine partial fractions. Symbolic Math toolbox also

contains other commands that can change the look of the displayed result for

readability and form. Some of these commands are: collect(G) - collects common-

coefficient terms of G(s); expand(G) - expands product of factors of G(s); factor(G)-

factors G(s); vpa(G, places) - stands for variable precision arithmetic (this command

converts fractional symbolic terms into decimal terms with a specified number of

decimal places).

Consider the function . MATLAB response to the

simplification of using the function simplify is shown below.

>> syms s

>> G = (1/s)+(1/3)*(1/(s+4))-(4/3)*(1/(s+1))

G =

1/s+1/3/(s+4)-4/3/(s+1)

>> pretty(G)

...................1 ....................1

1/s + 1/3 ------- ...- 4/3...-------

................s + 4...............s + 1

>> pretty(simplify(G))


.............4

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

s (s + 4) (s + 1)

In the above example, the symbolic fractions 1/3 and 4/3 will be converted to 0.333

and 1.33 if the argument places in vap(G, places) is set to 3.

>> pretty(vpa(G,3))

............0.333 ....1.33

1/s + ---------- - ----------

........... s + 4. ....s + 1.

Exercise 2.10

Consider the function . Show using Symbolic Math

tools that the simplified form of its Laplace transform is .

Basic mathematical operators +, , , and are applicable to symbolic objects also.

For example, simplification of the closed-loop system with forward gain forward-

path transfer function , and feedback-path transfer function

, is given by . This can be done using

Symbolic Math toolbox as follows

>> syms s K;

>> G = 2.5/(s*(s+5));

>> H = 1/(0.1*s+1);

>> M = K*G/(1+G*H)

M =

5/2*K/s/(s+5)/(1+5/2/s/(s+5)/(1/10*s+1))

>> pretty(M)

K


5/2 ------------------------------------------

/ 1 \

s (s + 5)| 1 + 5/2 ---------------------- |

\ s (s + 5) (1/10 s + 1) /

>> pretty(simplify(M))

(s + 10) K

5/2 -------------------------

3 2

s + 15 s + 50 s + 25

With Symbolic Math toolbox, symbolic transfer functions can easily be converted to

the LTI (Linear Time-Invariant) transfer function objects. This conversion is done in

two steps. The first step uses the command numden to extract the symbolic numerator

and denominator of G (s). The second step converts, separately, the numerator and

denominator to vectors using the command sym2poly . The last step consists of

forming the LTI transfer function object by using the vector representation of the

transfer function's numerator and denominator. The command sym2poly doesn't work

on symbolic expressions with more than one variable.

As an example, we form the LTI object from obtained in the above example

with .

The command poly2sym converts polynomial coefficient vectors to symbolic

polynomial.


PRACTICAL - 3

Simulink

Objectives:

Building simple Simulink simulations.

Running Simulink simulation to predict a system's behaviour.

The MATLAB Control System Toolbox offers functions for finding the transfer functions

from a given block diagram. However, as we shall shortly see, the simulation environment

provided by MATLAB‟s Simulink Toolkit obviates the need for block diagram reduction.

The Simulink model mimics the block diagram of a feedback control system and is used to

evaluate the response of controlled variable to any test input. It also provides the response of

any internal variable of the control system (output variable of a subsystem block) without the

need for block diagram reduction.

Let us reiterate the fact we have emphasized earlier: a good plant/process model is the

backbone of any realistic control design. A Simulink model based on the structure and

parameters of the system model is constructed. The responses of the actual system and its

Simulink model are obtained using a set of test inputs. If the actual responses to the test

inputs were significantly different from the Simulink responses, certain model parameters

would have to be revised, and/or the model structure would have to be refined to better reflect

the actual system behaviour. Once satisfactory model performance has been achieved,

various control schemes can be designed and implemented.

In practice, it is best to test a control scheme off-line by evaluating the system performance in

the “safety” of the Simulink environment. The key components of a control system include

actuators, sensors, and the plant/process itself. A decision to include all aspects such as

amplifier saturation, friction in the motor, backlash in gears, dynamics of all the devices, etc.,

may improve the model, but the complexity of the model may result in a more complicated

controller design, which will ultimately increase the cost and sophistication of the system.

The design is usually carried out using an approximated model; the evaluation of the design

is done on the “true” model, which includes nonlinearities, and other aspects neglected in the

approximate model. Simulink is an excellent tool for this evaluation.

SIMULINK (SIMUlation LINK) is an extension of MATLAB for modeling, simulating, and

analyzing dynamic, linear/nonlinear, complex control systems. Graphical User Interface

(GUI) and visual representation of simulation process by simulation block diagrams are two

key features which make SIMULINK one of the most successful software packages,

particularly suitable for control system design and analysis.

Simulation block diagrams are nothing but the same block diagrams we are using to describe

control system structures and signal flow graphs. SIMULINK offers a large variety of ready-

to-use building blocks to build the mathematical models and system structures in terms of


block diagrams. Block parameters should be supplied by the user. Once the system structure

is defined, some additional simulation parameters must also be set to govern how the

numerical computation will be carried out and how the output data will be displayed.

Because SIMULINK is graphical and interactive, we encourage you to jump right in and try

it. To help you start using SIMULINK quickly, we describe here the simulation process

through a demonstration example with MATLAB version 7.0, SIMULINK version 6.0.

To start SIMULINK, enter simulink command at the MATLAB prompt. Alternatively one

can also click on SIMULINK icon shown in Fig. 3.1.

Fig. 3.1 MATLAB Desktop main menu and SIMULINK icon

A SIMULINK Library Browser (Fig. 3.2) appears which displays tree-structured view of the

SIMULINK block libraries. It contains several nodes; each of these nodes represents a library

of subsystem blocks that is used to construct simulation block diagrams. You can

expand/collapse the tree by clicking on the boxes beside each node and block in the

block set pan.

Fig. 3.2 SIMULINK Library Browser


Expand the node labeled Simulink. Subnodes of this node ( Commonly Used Blocks,

Continuous, Discontinuities, Discrete, Logic and Bit Operations, etc…) are displayed.

Now for example, expanding the Sources subnode displays a long list of Sources library

blocks. Simply click on any block to learn about its functionality in the description box (see

Fig. 3.3).

Fig. 3.3 Blocks in Sources subnode

You may now collapse the Sources subnode, and expand the Sinks subnode. A list of Sinks

library block appears (Fig.3.4). Learn the purpose of various blocks in Sinks subnode by

clicking on the blocks.


Fig. 3.4 Blocks in Sinks subnode

Exercise 3.1

Expand the Continuous, Discontinuities, Discrete, and Math Operations subnodes. Study the

purpose of various blocks in these subnodes in description box.

We have described some of the subsystem libraries available that contain the basic building

blocks of simulation diagrams. The reader is encouraged to explore the other libraries as well.

You can also customize and create your own blocks. For information on creating your own

blocks, see the MATLAB documentation on “Writing S- Functions”.

We are now ready to proceed to the next step, which is the construction of a simulation

diagram. In the SIMULINK library browser, follow File New Model or hit Ctrl+N to


open an „untitled' workspace (Fig.3.5) to build up an interconnection of SIMULINK blocks

from the subsystem libraries.

Fig. 3.5 Untitled workspace

Let us take a simple example. The block diagram of a dc motor (armature-controlled) system

is shown in Fig. 3.6

Fig. 3.6 Block diagram of a dc motor (armature-controlled) system


where

is the resistance of the motor armature (ohms) = 1.75

is the inductance of the motor armature (H) = 2.83

is the torque constant (Nm/A) = 0.0924

is the back emf constant (V sec/rad) = 0.0924

is the inertia seen by the motor (includes inertia of the load) (kg-m2 ) =

is the mechanical damping coefficient associated with rotation (Nm/(rad/sec))= 5.0

is the applied voltage (volts) = 5 volts

We will implement the model shown in Fig. 3.6 in the untitled work space (Fig. 3.5).

Let us first identify the SIMULINK blocks required to implement the block diagram of Fig.

3.6. This is given in Fig. 3.7.

Fig. 3.7 SIMULINK blocks required for implementation

Identifying the block(s) required for simulation purpose is in fact the first step of the

construction of simulation diagram in SIMULINK. The next step is to “drag and drop ” the

required blocks from SIMULINK block libraries to untitled workspace. Let us put the very

first block for applied voltage (Ea) to workspace.


Expand the Sources subnode, move the pointer and click the block labeled Constant, and

while keeping the mouse button pressed down, drag the block and drop it inside the

Simulation Window; then release the mouse button (Fig. 3.8).

Right clicking on the block will provide various options to users from which one can cut,

copy, delete, format (submenu provides facilities for rotation of the block, flipping, changing

the font of block name,...), etc...

Exercise 3.2

Drag and drop all the blocks we have identified (Fig. 3.7) from the Library Browser to the

untitled Workspace and place them as shown in Fig. 3.9.

Fig. 3.8 Drag and drop blocks to Workspace from Library Browser


Fig. 3.9 Unconnected blocks in Workspace

It is visible that all the block parameters are in their default settings. For example, the default

transfer function of Transfer Fcn block is and default signs of Sum block are + +. We

need to configure these block parameters to meet our modeling requirements. It is

straightforward. Double click the block to set up its parameters. For example, double clicking

the Transfer Fcn block opens the window titled Block Parameters: Transfer Fcn, shown in

Fig. 3.10.

Fig. 3.10 Transfer function block parameters window


For armature circuit transfer function, no need to change the numerator parameter. For

denominator parameters, enter for , which will be interpreted by

SIMULINK as .

To enhance the interpretability of simulation diagram, we can also change the block

identification name. Simply click on the text Transfer Fcn to activate the small window

around the text to change the block name. For our simulation block diagram, the suitable text

for Transfer Fcn block may be Armature circuit.

Before we move to the last step of interconnecting the blocks as per the desired structure, just

finish Exercise 3.3. Note that the Decimation parameter value by default is 1. Increasing this

value reduces the number of data samples taken over the simulation time. We have used the

default value of 1.

Exercise 3.3

Modify all the block parameters as per system parameters given for Fig. 3.7, and give

appropriate names to the blocks.

Lines are drawn to interconnect these blocks as per the desired structure. A line can connect

output port of one block to the input port of another block. A line can also connect the output

port of one block with input ports of many blocks by using branch lines. We suggest readers

to perform the following line/block operations on blocks dragged in workspace to get hands

on practice.

To connect the output port of one block to the input port of another block:

1. Position the pointer on the first block's output port; note that the cursor shape

changes to cross hair.

2. Press and hold down the left mouse button.

3. Drag the pointer to second block's input port.

4. Position the pointer on or near the port; the pointer changes to a double cross

hair.

5. Release the mouse button. SIMULINK replaces the port symbol by a connecting

line with an arrow showing the direction of signal flow.


Another simple methodology doesn't require dragging the line. Block1 output port is required

to be connected to Block2 input port.

1. select Block1 by clicking anywhere inside the block.

2. Hold down the Ctrl key.

3. Click on block2; both the blocks will be connected.

To connect the output port of one block with the input ports of several blocks, we can use

branch lines. Both the existing line and the branch line carry the same signal. To add a

branch line, do the following:

1. Position the pointer on the line where you want the branch line to start.

2. While holding down the Ctrl key, left click on the line segment; note that

the cursor shape changes to cross hair.

3. Release the control key, while pressing down the left mouse button; drag the

pointer to the input port of the target block.

4. Release the mouse button; target block will be connected to the line

segment.

Some of the important line-segment and block operations are as follows:

1. To move a line segment, position the pointer on the segment you want to

move. Press and hold down the left mouse button. Drag the pointer to the

desired location and release. Note that this operation is valid with line

segments only, not with the dedicated connecting lines between two blocks.

2. To disconnect a block from its connecting lines, hold down the shift key, then

drag the block to a new location. Incoming and outgoing lines will be

converted to red colored dotted lines. One can insert a different block between

these lines.


Exercise M3.4

Connect all the blocks appropriately as per the block diagram given in Fig. 3.7. Make

use of the block interconnection points discussed above.

Now let us give a name to the untitled workspace. Hit Ctrl + S to save the developed

simulation diagram to the disk with an appropriate name. The file will be saved with the

extension .mdl , an abbreviation for the „model'.

We save the file using the name armature_dcmotor.mdl; the complete simulation diagram

is shown in Fig. 3.11.

Finally, we need to set the parameters for the simulation run. Press Ctrl + E to open the

simulation parameter configuration window. Left panel of the window (Fig. 3.12) displays a

tree structured view of parameter submenu. In the Solver submenu, enter the start and stop

time of the simulation (Fig. 3.13).

Fig. 3.11 Final simulation diagram


Fig. 3.12 Parameter configuration submenu

Fig. 3.13 Enter simulation time

Now we are ready to simulate our block diagram of armature-controlled dc motor. Press

icon to start the simulation. Note that the icon changes to ; pressing this icon, one can stop

the simulation before stop time. After simulation is done, double click the Scope block to

display the angular velocity variation with time. Click the autoscale icon in the display

window to scale the axes as per the variable ranges. Autoscaled scope display is shown in

Fig. 3.14. With zoom facility, try zooming the portion of graph between 0.5 to 1 sec, and 20

to 25 unit angular velocity to identify the numerical value of angular velocity at 0.8 seconds.


Fig. 3.14 Scope display of angular velocity

Set y-axis limits by right-clicking the axis and choosing Axes Properties. In Y-min, enter the

minimum value for the y-axis. In Y-max, enter the maximum value for the y-axis. In Title,

enter the title of the plot. See Fig. 3.15.

Fig. 3.15 Scope axis properties editor


Click the icon shown on the icon bar of Fig. 3.14 to open scope parameter editor (Fig.

3.16). General parameters include Number of axes, Time range, Tick labels, and

Sampling.

Click on the Data history button. If you want input-output data from this scope to be

available to MATLAB workspace for further analysis, check the button Save data to

workspace. In the box Variable name, enter the variable name for saving the data. By

default it will save the data with variable name ScopeData. With the pop-down menu

Format, select the format in which you want to save the data.

Three specific formats for saving the data are as follows:

1. Structure with time: Data will be saved in structured format with time steps. Type

the following commands in your MATLAB prompt and observe the outputs.

Fig. 3.16 Scope parameter setting window

>> ScopeData

ScopeData =

time: [4663x1 double]

signals: [1x1 struct]

blockName: 'armature_dcmotor/Angular Velocity'


Structures are used in MATLAB to store mixed mode data types, and individual fields of the

structure can be accessed by „dot ' operator. To see the information stored in the field

signals, type:

>> ScopeData.signals

ans =

values: [4663x1 double]

dimensions: 1

label: ''

title: ''

plotStyle: 0

It indicates that the field signals contains subfield values, which is of 4663 x 1 size vector

containing the values of angular velocity. Try accessing the field time of ScopeData.

Exercise 3.5

Plot the angular velocity against time using the plot command. Give suitable title to the

plot and labels to x and y axes.

Hint: You need to plot ScopeData.signals.values against ScopeData.time.

2. Structure : This is the same as Structure with time; the only difference is that the time

steps will not be saved.

Exercise 3.6

Run the simulation with scope data to be saved as Structure format. Verify that the time

field of ScopeData structure is actually an empty matrix.


3. Array : Array format is simply a two column matrix with number of data points being

equal to number of rows. The maximum number of data points limits to the number entered

in the box Limit data points to last. In Fig. 3.16, the limit is 5000 data points.

Exercise 3.7

Run the simulation with scope data to be saved as Array format. Repeat Exercise 3.5 with

saved data matrix.

We have used an example to show how to build the simulink diagram, how to enter data and

carry out a simulation in the SIMULINK environment. The reader will agree that this is a

very simple process.

Solving the following exercises will make the readers more confident in solving the control

system design and analysis problems through SIMULINK.

Exercise 3.8

This problem requires some modification in the above considered armature-controlled d.c.

motor SIMULINK example.

1. The angular position is obtained by integrating the angular velocity. Implement

this in the model and display angular position in a different scope.

2. Remove the constant applied voltage block. Obtain the Step, Ramp, and Sinusoidal

responses of the system.

3. Simulate a closed-loop position control system assuming a proportional

controller of gain KP .

Hint: Add a reference input signal for angular position and add a summing block which

calculates the error between the angular position reference and the measured angular

position. Multiply this error by the gain KP and let this signal be the applied voltage Ea .

Exercise 3.9

Consider a dynamic system consisting of a single output , and two inputs r and w :

where


Model the system in SIMULINK.

For and both step signals, obtain the system output . Display the output on scope.

Also, using a SIMULINK block, etch the output to the workspace.

Hint: To implement deadtime, use the block Transport delay from Continuous, and

identify suitable block from Sinks library to fetch the variable directly to the workspace.

Exercise 3.10

Assign 0.5, , and to , respectively.

Simulate for 10 seconds.

Display plots for on scope, and also save to the

MATLAB file Yt.mat.

Hint: Use Math Function, Add, Divide, and Trigonometric Function from Math

Operations block libraries.

Exercise 3.11

This problem is to study the effects of Proportional (P), Proportional + Integral (PI), and

Proportional + Derivative (PD) control schemes on the temperature control system. A

temperature control system has the block diagram given in Fig.3.17. The input signal is a

voltage and represents the desired temperature Simulate the control system using

SIMULINK and find the steady-state error of the system when is a unit-step function

and (i) (ii) and (iii) .


Fig. 3.17

Hint: Use PID Controller block from Simulink Extras Additional Linear to

implement the PID controller.

Exercise 3.12

The block diagram in Fig. 3.18 shows the effect of a simple on-off controller on second-

order process with deadtime.

Fig. 3.18

Implement and test the model in SIMULINK for step inputs of 5.0 and 10.0. Display the

control signal u( t ) and output the on separate scopes and also fetch both the signals

with time information to MATLAB workspace. Using MATLAB plot function, plot the

control signal and the output; both against time on single graph.

Hint: To implement on-off controller, use Relay from Discontinuous block library.

Exercise 3.13

Simulate the Van der Pol oscillator, described by the following nonlinear differential

equation: ,

where is the disturbance (forcing function) given by . Assign

Hint: Rewrite the second-order Van der Pol differential equation as a system of coupled


first-order differential equations. Let,

The SIMULINK block diagram is to be built using Signal Generator, Gain, Integrator,

Add, and Scope. The output of Add block is . Integrating it once will lead you to

and second integration will lead you to Double click the Integrator block to add the

initial conditions .


PRACTICAL - 4

Feedback System Simulation

Objectives:

Presenting MATLAB functions to carryout block diagram manipulations,

followed by time-response analysis.

Using Simulink simulation for time-response analysis.

Transfer Function Manipulation

System Response

Simulink Simulation





PRACTICAL – 4.1


Transfer Function Manipulation

Suppose we have developed mathematical models in the form of transfer functions for the

plant, represented by G(s), and the controller, represented by D(s), and possibly many other

system components such as sensors and actuators. Our objective is to interconnect these

components to form block diagram representation of a feedback control system. MATLAB

offers several functions to carry out block diagram manipulations.

Two methods are available:

1. Solution via series, parallel, and feedback commands:

series(G,D) for a cascade connection of G(s) and D(s); parallel(G1,G2) for a parallel

connection of G1(s) and G2(s); feedback(G,H, sign) for a closed-loop connection with G(s)

in the forward path and H(s) in the feedback path; and sign is -1 for negative feedback or +1

for positive feedback (the sign is optional for negative feedback); and cloop(G,sign) for a

unity feedback system with G(s) in the forward path, and sign is -1 for negative feedback or

+1 for positive feedback (the sign is optional for negative feedback).

2. Solution via algebraic operations:

G*D for a cascade connection of G(s) and D(s); G1+G2 for a parallel connection of G1(s)

and G2(s); G/(1+G*H) for a closed-loop negative feedback connection with G(s) in the

forward path and H(s) in the feedback path; and G/(1-G*H) for positive feedback systems.


PRACTICAL – 4.2


System Response

The transfer function manipulations give us a transfer function model M(s) between

command input R(s) and output Y(s); model Mw(s) between disturbance input W(s) and output

Y(s); a model between command input R(s) and control U(s), etc. It is now easy to use some

of the control analysis commands available from the Control System Toolbox. impulse(M)

and step(M) commands represent common control analysis operations that we meet in this

book. Also frequently used in the book are frequency-response plots.

Example 4.1

Consider the block diagram in Fig. 4.1.

Fig. 4.1

For value of gain KA = 80, the following two MATLAB sessions evaluate the step responses

with respect to reference input R(s) and disturbance signal W(s) for

>> %Step response with respect to R(s)

>> s = tf('s');

>> KA = 80;

>> G1 = 5000/(s+1000);

>> G2 = 1/(s*(s+20));

>> M = feedback(series(KA*G1,G2),1)

>> step(M)


The MATLAB responds with

Transfer function:

.......................400000

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

s^3 + 1020 s^2 + 20000 s + 400000

and step response plot shown in Fig. 4.2. The grid has been introduced in the plot by right

clicking on the plot and selecting Grid option.

Fig. 4.2

>> %Step response with respect to W(s)

>> s = tf('s');

>> KA = 80;

>> G1 = 5000/(s+1000);

>> G2 = 1/(s*(s+20));

>> Mw = (-1) * feedback(G2, KA*G1)

>> step(Mw)

MATLAB responds with

Transfer function:

.......................s-1000

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

s^3 + 1020 s^2 + 20000 s + 400000


and step response plot shown in Fig. 4.3.

Fig. 4.3

Example 4.2

Let us examine the sensitivity of the feedback system represented by the transfer function

The system sensitivity to parameter K is

Figure 4.4 shows the magnitudes of and versus frequency for K =

0.25; generated using the following MATLAB script. Text arrows have been introduced in

the plot by following Insert from the main menu and selecting the option Text Arrow.

Note that the sensitivity is small for lower frequencies, while the transfer function primarily

passes low frequencies.

w = 0.1:0.1:10;

M = abs(0.25./((j*w).^2+j*w+0.25));

SMK = abs((j*w .* (j*w + 1))./((j*w).^2 + j*w +0.25));

plot(w,M,'r',w,SMK,'b');

xlabel('Frequency (rad/sec)');

ylabel('Magnitude');


Fig. 4.4

Of course, the sensitivity S only represents robustness for small changes in gain K. If K

changes from 1/4 within the range K = 1/16 to K = 1, the resulting range of step responses,

generated by the following MATLAB script, is shown in Fig. 4.5. This system, with an

expected wide range of K, may not be considered adequately robust. A robust system would

be expected to yield essentially the same (within an agreed-upon variation) response to

selected inputs.

s = tf('s');

S = (s*(s+1))/(s^2+s+0.25);

M1 = 0.0625/(s^2+s+0.0625);

M2 = 0.25/(s^2+s+0.25);

M3 = 1/(s^2+s+1);

step(M1);

hold on;

step(M2);

step(M3);


Fig. 4.5


PRACTICAL – 4.3


Simulink Simulation

Simulink simulation is an alternative to block diagram manipulation followed by time-

response analysis. From the Simulink model of a control system, output y in response to

command r, output y in response to disturbance w, control u in response to command r, and

all other desired internal variables can be directly obtained.

Example 4.3

Control system design methods discussed in this course are based on the assumption of

availability of linear time invariant (LTI) models for all the devices in the control loop.

Consider a speed control system. The actuator for the motor is a power amplifier. An

amplifier gives a saturating behaviour if the error signal input to the amplifier exceeds linear

range value.

MATLAB simulink is a powerful tool to simulate the effects of nonlinearities in a feedback

loop. After carrying out a design using LTI models, we must test the design using simulation

of the actual control system, which includes the nonlinearities of the devices in the feedback

loop.

Figure 4.6 is the simulation diagram of a feedback control system: the amplifier gain is 100

and the transfer function of the motor is 0.2083/(s +1.71). We assume the amplifier of gain

100 saturates at +5 or -5volts. The result of the simulation is shown in Fig. 4.7.

The readers are encouraged to construct the simulink model using the procedure described in

Module 3. All the parameter settings can be set/seen by double clicking on related blocks.

Fig. 4.6

Time and Output response data have been transferred to workspace using To Workspace

block from Sinks main block menu. Clock block is available in Sources main menu. These

variables are stored in the structure Output and Time in the workspace, along with the

information regarding simulink model name. For example,


>> Output

Output =

time: [ ]


blockName: 'M4_3/To workspace Output'

>> Time

Time =

time: [ ]


blockName: 'M4_3/To workspace Time'

To access output and time values, one needs to access Output.signals.values and

Time.signals.values . The step response plot has been generated by the following MATLAB

script.

>> plot(Time.signals.values,Output.signals.values)

>> xlabel('Time (sec)');

>> ylabel('Output');

>> title('Step Response');

Fig. 4.7


Example 4.4

In this example, we simulate a temperature control system with measurement noise added to

the feedback signal. The process transfer function is

The deadtime minutes. The measurement noise parameters we have used are: mean

of 0, variance of 0.5, initial seed of 61233, and sample time of 0. The simulink inputs a step

of 30 to the system (Fig. 4.8). Deadtime block in this figure is Transport Delay block from

Continuous library, and Random Number block is from Sources library.

Fig. 4.8

The data has been transferred to the workspace using To Workspace block. The step

response, generated using the following MATLAB script is shown in Fig. 4.9.

>> plot(Time.signals.values,Y.signals.values);

>> ylabel('Output (Y)');

>> xlabel('Time(min)');

>> title('Step Response');

Fig. 4.9


The performance of the system with measurement noise removed, is shown in Fig. 4.10. To

remove the effect of noise, simply disconnect the Random number block from the Sum

block in the feedback path.

Fig. 4.10


PRACTICAL - 5

Time Response Characteristics and LTI Viewer

Objectives:

Analysis of the system behaviour to changes in input reference signals or

disturbances using steady-state and transient performance indices.

Brief introduction to LTI Viewer, and time response analysis using this GUI tool.

Time response characteristics of interest to us in analysis and design of control systems are:

rise time (tr), settling time (ts), peak time (tp) , and peak overshoot (Mp) of step response of a

system. These parameters can directly be obtained from the plot generated by the command

step. Right-clicking away from the curve generated by step brings up a menu. From this

menu, various time-response characteristics can be selected.

MATLAB has a powerful GUI tool, called LTI Viewer, for obtaining both the time and

frequency response characteristics of a given Linear Time Invariant system. In this module,

we give the basic features of this GUI tool and use it for time response analysis. Later in

Module 9, this tool will be exploited for frequency response analysis.

LTI Viewer

Time response characteristics in Matlab window

Time response characteristics in Simulink window





PRACTICAL – 5.1


LTI Viewer

The LTI ( Linear Time Invariant) Viewer is an interactive graphical user interface

(GUI) for analyzing the time and frequency responses of linear systems and comparing

such systems. In particular, some of the graphs the LTI Viewer can create are step and

impulse responses, Bode, Nyquist, Nichols, pole-zero plots, and responses with

arbitrary inputs. In addition, the values of critical measurements on these plots can be

displayed with a click of the mouse. Table 5.1 shows the critical measurements that are

available for each plot.

Table 5.1

-

Peak time

or Peak

frequency

Settling

time

Rise

time

Steady

state

value

Gain/phase

margins;

zero

dB/1800

frequencies

Pole-

zero

value

Step

- -

Impulse

- - - -

Bode

- - -

-

Nyquist

- - -

-

Nichols

- - -

-

Pole-

zero - - - - -

Here we provide steps you may follow to use the LTI Viewer to plot time and

frequency responses.

1. Access the LTI Viewer: The LTI Viewer window, shown in Fig. 5.1, can be

accessed by typing ltiview in the MATLAB Command Window or by

executing this command in an M-file.


Fig. 5.1 LTI Viewer Window

2. Create LTI transfer functions: Create LTI transfer functions for which you

want to obtain responses. The transfer functions can be created in an M-file or

in the MATLAB Command Window. Run the M-file or MATLAB Command

Window statements to place the transfer function in the MATLAB workspace.

All LTI objects in the MATLAB workspace can be exported to the LTI

Viewer.

The following MATLAB commands create the transfer function .

>> s = tf('s');

>> G = 1/(s+1);

3. Select LTI transfer functions for the LTI Viewer: Choose Import… under

the File menu in the LTI Viewer window and select all LTI objects whose

responses you wish to display in the LTI Viewer sometime during your current

session. Selecting G results in Fig. 5.3.


Fig. 5.2 Import System Data Window

4. Select the LTI objects for the next response plot: Right-click anywhere in

the LTI Viewer plot area to produce a pop-up menu. Under Systems, select or

deselect the objects whose plots you want or do not want to show in the LTI

Viewer. More than one LTI transfer function may be selected.

5. Select the plot type: Right-click anywhere in the LTI Viewer plot area to

produce a pop-up menu. Under Plot Types, select the type of plot you want to

show in the LTI Viewer (Fig. 5.3).

Fig. 5.3


Example 5.1

Take a typical underdamped second-order transfer function and import it to the LTI

viewer window.

6. Select the characteristics: Right-click anywhere in the LTI Viewer plot area

to produce a pop-up menu. Under Characteristics, select the characteristics of

the plot you want displayed. More than one characteristic may be selected. For

each characteristic selected, a point will be placed on the plot at the appropriate

location.

The characteristics menu and the submenu are displayed in Fig. 5.4 obtained by

importing a typical second-order transfer function.

Note the difference in the definition of rise time employed in the LTI Viewer with

respect to the one used in the text.

Fig. 5.4 LTI Viewer with step response plot and characteristics


7. Interact with the plot:

o Zoom In (Zoom Out): Select the Zoom In (Zoom Out) button (with

the + ( ) sign) on the tool bar. Hold the mouse button down and drag a

rectangle on the plot over the area you want to enlarge (deluge), and

release the mouse button.

o Original View: Select Reset to Original View in the menu popped up

by right-click to return to original view of your plot after zooming.

o Grid: Select Grid in the right-click menu to on or off the grid. The

right click menu will not work if any zoom button on the tool bar is

selected.

o Normalize: Select Normalize in the right-click menu to normalize all

curves in view.

o Characteristics: Read the values of the characteristics by placing the

mouse on the characteristics point on the plot. Left-click the mouse to

keep the values displayed (Fig. 5.5).

o Properties: Select Properties in the right-click menu or double-click

anywhere (except the curve) on the graph sheet to change the

appearance of the graph. You can change the title, axes labels, and

limits, font sizes and styles, colours, and response characteristics

definitions.

o Coordinates and curve: Left-click the mouse at any point on the plot

to read the system identification and the coordinates (Fig. 5.5, shown

by square block).

o Add text and graphics: Under the File menu, choose Print to Figure.

An additional figure with toolbar will open (Fig. 5.6).The toolbar of

this figure has additional tools for adding text, arrows, lines, legends,

and to rotate the figure.

o Additional plot-edit capabilities: The Edit menu of the LTI Viewer

and the figures created by selecting Print to Figure, offer a wide

variety of controls over the plot presentation.

8. Viewer preferences: Choose Viewer Preferences… in the Edit menu. LTI

Viewer Preferences window (Fig. 5.7) will open. This allows users to set

parameters such as units (Units), style of labels (Style), response

characteristics definitions (Characteristics), and also provide a control over

the time and frequency vector generation (Parameters).


Fig. 5.5 Coordinate values in step response curve

Fig. 5.6 LTI response with text and graphics editor toolbar


Fig. 5.7 LTI Viewer Preferences window


PRACTICAL - 5.2


Time Response Characteristics in MATLAB window

Example 5.1

Consider the position control system shown in Fig. 5.8.

.

Fig. 5.8

(transfer function of motor, gears, and load)

(transfer function of power amplifier)

KA = preamplifier gain

sensitivity of input and output potentiometers

Time response characteristics of the system are generated by the following MATLAB script.

s = tf('s');

G1 = 100/(s+100);

G2 = 0.2083/(s*(s+1.71));

K = 1000;

G = series(K*G1,G2);

H = 1/pi;

M = series(feedback(G,H),H);

ltiview(M);


MATLAB response is shown in Fig. 5.9.

Fig. 5.9

In the following we compare the time response characteristics of this system with the one

obtained by replacing the power amplifier with a transfer function of unity (Fig. 5.10).

s = tf('s');

G1 = 1;

G2 = 0.2083/(s*(s+1.71));

K = 1000;

G = series(K*G1,G2);

H = 1/pi;

M1 = series(feedback(G,H),H);

ltiview(M1);


Fig. 5.10

The second-order approximation seems to be quite reasonable.

The two responses in Figs 5.9 and 5.10 can be obtained in a single plot using the command

ltiview(M,M1). Alternatively, choose Import... under the File menu in the LTI Viewer

window and select M.

Example 5.2

Consider a unity feedback system shown in Fig. 5.11.

Fig. 5.11


We will consider the following three controllers:

D1(s) = 3; D2(s)=3+(15/ s ); and D3(s)=(3+(15/ s )+0.3 s )

The following MATLAB code generates the step responses shown in Fig. 5.12.

s = tf('s');

G = 59.292/(s^2+6.9779*s+15.12);

D1 = 3;

D2 = 3 + 15/s;

D3 = 3 + 15/s +0.3*s;

M1 = feedback(D1*G,1);



step(M1);

hold;

step(M2);

step(M3);

We need not access LTI viewer to find out the time response characteristics of the closed-

loop system. The step command in MATLAB has this feature inbuilt in it as is seen in Fig.

5.12.


Fig. 5.12

Note that adding the integral term increases the oscillatory behaviour but eliminates the

steady-state error, and that adding the derivative term reduces the oscillations, while

maintaining zero steady-state error.


PRACTICAL - 5.3


Time Response Characteristics in SIMULINK window

Example 5.3

Consider a unity feedback, PID controlled system with following parameters:

Plant transfer function:

PID controller transfer function:

Reference input:

A Simulink block diagram is shown in Fig. 5.13.

Fig. 5.13

The Simulink model inputs a step of 30 to the system.

Time response characteristics of the model developed in Simulink can be obtained by

invoking the LTI Viewer directly from the Simulink. LTI Viewer for Simulink can be

invoked by following Tools -- Control Design -- Linear Analysis as shown in Fig. 5.14.


Fig. 5.14

This opens Control and Estimation Tools Manager shown in Fig. 5.15.

Fig. 5.15


Select linearization input-output points by right clicking on the desired line and selecting

Linearization Points in your Simulink model. Fig. 5.16 explains the selection of input point.

Similarly output point has been selected. Once input-output linearization points appear in the

Control and Estimation Tools Manager window, click on the Linearize Model at the

bottom of the window (Fig. 5.15). The type of response can be selected from the drop down

menu just near the Linearize Model button. Other linear analysis plots available are: Bode,

Impulse, Nyquist, Nichols, Bode magnitude plot, and pole-zero map.

Fig. 5.16

The step response with the characteristics is shown in Fig. 5.17.

Fig. 5.17


Example 5.4

Consider a unity feedback, PI controlled system with following parameters:

Plant transfer function:

Plant deat-time: = 0.15 minutes

PI controller transfer function:

Actuator saturation characteristic: Unit slope; maximum control signal =1.7.

Reference input:

A Simulink block diagram is shown in Fig. 5.18. PID controller block is available in

Additional Linear block set in Simulink Extras block library. The parameters of PID

controller block have been set as : P = 0.09, I = 0.95, and D = 0.

Fig. 5.18

We can invoke LTI Viewer to determine the time response characteristics of the system of

Fig. 5.18. However, LTI Viewer will linearize the system and then plot the characteristics.

An alternative is to transfer the output response data to workspace using To Workspace

block and then generate the time response using the plot command. A MATLAB code can

then be written to determine the required characteristics. The following code gives settling

time, peak overshoot, and peak time for the output response data of Fig. 5.18.

time = t.signals.values;

y = Y.signals.values;

plot(time,y);

title('Step response');

xlabel('Time (min)');


ylabel('Output');

ymax = max(y);

step_size = 10;

peak_overshoot = ((ymax-step_size)/step_size)*100

index_peak = find(y == ymax);

peak_time = time(index_peak)

s = length(time);

while((y(s)>=0.95*step_size) & (y(s)<=1.05*step_size))

s = s-1;

end

settling_time = time(s)

The MATLAB responds with Fig. 5.19 and the following characteristics.

peak_overshoot =

39.0058

peak_time =

1.9339

settling_time =

5.9739

Fig. 5.19


PRACTICAL - 6

Stability Analysis on Root Locus Plots

Objectives:

Stability analysis in state space.

Stability analysis using root locus plots.

Stability Analysis in State Space

Root Locus Analysis




PRACTICAL – 6.1


Stability Analysis in State Space

There are several useful MATLAB commands related to analysis and design in state space.

Commands useful for stability analysis in state space are:

ss; step; initial; eig; tf; minreal

The MATLAB program below forms the matrices A,b,c and d for the system

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

>> b = [0; 0; 1];

>> c = [1 0 0 ];

>> d = 0;

The output generated in the MATLAB Command Window by executing these commands is

A =

0 1 0

0 0 1

0 -2 -3

b =

0

0

1

c =

1 0 0

d =

0

The command ss (A,b,c,d) creates the state space model; let us call it stmod . The MATLAB

dialogue is given below.

>> stmod = ss(A,b,c,d)


a =

......x1 .x2 x3

x1 ..0 ..1 ...0

x2 ..0 ..0 ...1

x3 ..0 .-2 ..-3

b =

.......u1

x1 ...0

x2 ...0

x3 ...1

c =

......x1 x2 x3

y1 ..1 ..0 ...0

d =

.......u 1

y1 ....0

Continuous-time model.

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

have already discussed. For instance, the command step(stmod) generates the plot of unit-

step response y(t) (with zero initial conditions).

The command initial(stmod, x0) creates the free response (with input u(t) = 0) to initial

condition vector x0.

Typing eig(stmod) in the command window produces the dialogue

>> eig(stmod)

ans =


0

-1

-2

Thus the eigenvalues of the plant matrix A are

MATLAB has the means to perform model conversions. Given the state model stmod, the

syntax for conversion to transfer function model is

tfmod = tf(stmod)

Common pole-zero factors of the transfer function model must be cancelled before we can

claim that we have the transfer function representation of the system. To assist us in pole-zero

cancellation, MATLAB has mineral (tfmod) function.

Asymptotic stability is given by the command eig(A). If conversion of the state model to

transfer function gives a model with no pole-zero cancellations, the system is both

controllable and observable; asymptotic stability implies BIBO stability and vice versa.

Stability analysis can be carried out using transfer functions under the assumption of

controllability and observability of the corresponding dynamic system.

The command pole(tfmod) finds the poles of a transfer function. The command

pzmap(tfmod) plots the poles and zeros of the transfer function. A MATLAB code for the

Routh stability criterion can easily be written for stability analysis. However, our focus in this

course is on root locus for stability analysis. The focus in this MATLAB module will

therefore be on root locus analysis.


PRACTICAL - 6.2


Root Locus Analysis

In the following, we provide brief description of powerful MATLAB commands for root

locus analysis. The reader may wonder why the instructors emphasize learning of hand

calculations when powerful MATLAB commands are available. For a given set of open-loop

poles and zeros, MATLAB immediately plots the root loci. Any changes made in the poles

and zeros, immediately result in new root loci and so on.

Depending on our background and aptitude, we may, after a while, begin to make some sense

of the patterns. May be we finally begin to formulate a set of rules that enables us to quickly

make a mental sketch of the root locus the moment the poles and zeros appear. In other

words, by trial-and-error, we find the rules of the root locus.

Through the systematic formulation of set of rules of the root locus, we look for the clearest,

and simplest explanation of the dynamic phenomena of the system. The rules of the root

locus give us a clear and precise understanding of the endless patterns that can be created by

an infinite set of characteristic equations. We could eventually learn to design without these

rules, but our level of skill would never be as high or our understanding as great. This is true

of other analysis techniques also such as Bode plots, Nyquist plots, Nichols charts, and so on,

covered later in the course.

MATLAB allows root locus for the characteristic equation

1 + G (s)H(s) = 0

to be plotted with the rlocus(GH) command. Points on the root loci can be selected

interactively (placing the cross-hair at the appropriate place) using the [K,p] = rlocfind(GH)

command. MATLAB then yields the gain K at that point as well as all the poles p that have

that gain. The root locus can be drawn over a grid generated using the sgrid (zeta, wn)

command, that allows constant damping ratio zeta and constant natural frequency wn curves.

The command rlocus (GH, K) allows us to specify the range of gain K for plotting the root

locus. Also study the commands [p,K]=rlocus(GH) and [p]=rlocus(GH,K) using MATLAB

online help.

Example 6.1

Consider the system shown in the block diagram of Fig. 6.1.

Fig. 6.1


The characteristic equation of the system is

1 + G(s) = 0

with

The following MATLAB script plots the root loci for

s = tf('s');

G = 1/(s*(s+7)*(s+11));

rlocus(G);

axis equal;

Clicking at the point of intersection of the root locus with the imaginary axis gives the data

shown in Fig. 6.2. We find that the closed-loop system is stable for K < 1360; and unstable

for K > 1360.

Fig. 6.2

Fig. 6.3 shows step responses for two values of K .

>> K = 860;


>> step(feedback(K*G,1),5)

>> hold;

% Current plot held

>> K = 1460;

>> step(feedback(K*G,1),5)

Fig. 6.3

Example 6.2

Consider the system shown in Fig. 6.4.

Fig 6.4

The plant transfer function G(s) is given as as


The following MATLAB script plots the root locus for the closed-loop system.

clear all;

close all;

s = tf('s');

G = (s+1)/(s*(0.1*s-1));

rlocus(G);

axis equal;

sgrid;

title('Root locus for (s+1)/s(0.1s-1)');

[K,p]=rlocfind(G)

Fig 6.5

selected_point =

-2.2204 + 3.0099i

K =

1.4494


p =

-2.2468 + 3.0734i

-2.2468 - 3.0734i

Example 6.3

For a unity feedback system with open-loop transfer function

a root locus plot shown in Fig. M6.6 has been generated using the following MATLAB code.

s = tf('s');

G = (s^2-4*s+20)/((s+2)*(s+4));

rlocus(G);

zeta = 0.45;

wn = 0;

sgrid(zeta,wn);

Properly redefine the axes of the root locus using Right click --> Properties --> Limits.

Fig. 6.6


Clicking on the intersection of the root locus with zeta=0.45 line gives the system gain K =

0.415 that corresponds to closed-loop poles with Clicking on the intersection of the

root locus with the real axis gives the breakaway point and the gain at that point.


PRACTICAL - 7

Root Locus Design and SISO Design Tools

Objectives:

Design using root locus plots.

Brief introduction to SISO Deisgn Tool, and root locus design using this GUI

tool.

We will carry out Control system design using MATLAB dialogues. MATLAB is an

interpreted language. That is, it runs along executing each instruction as it comes to it. This

interactive feature of MATLAB will be exploited in our design exercise. The designer will be

in the loop.

The MATLAB SISO Design Tool, developed on the MATLAB dialogue pattern, is a

powerful GUI tool wherein the designer is always in the design loop. We will outline this

tool, with an example, at the end of this module.

Design using Matlab Dialogues

MATLAB's SISO Design tool




PRACTICAL - 7.1


Design using MATLAB dialogues

Example 7.1

A unity feedback system has forward path transfer function

Our goal is to design a cascade PID compensator that improves the transient as well as

steady-state performance of the closed-loop system (refer Nise(2004)).

We want to achieve a transient response that has no more than 20% overshoot. This, as we

know, can be achieved by simple gain adjustment. We therefore first evaluate the

uncompensated system operating at 20% overshoot. The following MATLAB dialogue is

helpful for this evaluation.

s = tf('s');

G1 = (s+8)/((s+3)*(s+6)*(s+10));

zeta = 0.456;

rlocus(G1);

sgrid(zeta,0);

Fig. 7.1


Selecting the point of intersection of the root locus with 20% overshoot line in

Fig. 7.1 using rlocfind(G) command, MATLAB yields the gain at that point, as well as all

the poles that have that gain . The MATLAB session follows:

>> [K, p] = rlocfind(G1)

Select a point in the graphics window

selected_point =

-5.4171 +10.4814i

K =

119.6589

p =

-5.4142 +10.4814i

-5.4142 -10.4814i

-8.1716

The unit step response of the uncompensated system with K =119.6589, is given by the

following dialogue:

>> M = feedback(K*G1,1)

Transfer function:

.............119.7 s + 957.3

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

s^3 + 19 s^2 + 227.7 s + 1137

>> step(M)


Fig. 7.2

The step response is shown in Fig. 7.2; the peak time t p =0.296 sec.

The position error constant is

Therefore

Let us now fix specifications for design. Say, we want to design a PID controller so that the

closed-loop system operates with a peak time that is two-thirds that of the uncompensated

system at 20% overshoot, with zero steady-state error for a step input.

To compensate the system to reduce the peak time to two-thirds of the uncompensated

system, we must first find the compensated system's desired dominant pole location. The

imaginary part of the compensated dominant pole is


and the real part is

Next we design a PD compensator. We search for compensator's zero location so that the root

locus of the compensator system passes through the desired dominant pole location. If we are

using SISO Design Tool (described later in this module), the trial-and-error search for the

compensator zero is straightforward. Otherwise, we may proceed as follows.

Evaluate the angle contributed by all the poles and zeros of G(s) at s d = -8.1571+ j 15.9202

using the following MATLAB command:

>> Sd = -8.1571+15.9202i

Sd =

-8.1571 +15.9202i

angle_at_dominant_pole=(180/pi)*(angle(polyval([1,8],Sd))-(angle(polyval([1,3],Sd))...

+angle(polyval([1,6],Sd))+angle(polyval([1,10],Sd))))

angle_at_dominant_pole =

-198.4967

We find the sum of angles from uncompensated system's poles and zero to the desired

compensated dominant pole to be –198.4967o. Thus the contribution required from the

compensator zero is 198.4967 180 = 18.4967o. Assuming that compensator zero is located

at as shown in Fig. 7.3, we obtain

This gives

zc = 55.7467.


Fig. 7.3

Thus the PD controller is

The complete root locus for the PD-compensated system is sketched in Fig. 7.4. Using

rlocfind command, the gain at the design point is 5.2410. Complete analysis of PD-

compensated system is as follows:

>> D1 = s+55.7467;

>> rlocus(D1*G1);

>> sgrid(zeta,0);

>> [K,p]=rlocfind(D1*G1)


selected_point =

-8.1043 +15.6832i

K =

5.2410

p =

-8.0799 +15.6914i


-8.0799 -15.6914i

-8.0812

The zeros are at -8,-55.7467. The effect of third closed-loop pole is nearly cancelled by a

zero.

Fig. 7.4

The PD-compensated system has settling time 0.454 sec., peak time 0.178 sec., and steady-

state error 0.072, as seen in simulation shown in Fig. 7.5.

>> M=feedback(K*D1*G1,1);

>> step(M)

We see the reduction in peak time and improvement in steady-state error over the

uncompensated system.

We now design a PI compensator to reduce the steady-state error to zero for a step input. Any

PI compensator will work as long as the zero is placed close to the origin. This ensures that

PI compensator will not change the transient response obtained with the PD compensator

(The placement of the zero of the PI compensator is not entirely arbitrary. The location of the

zero influences the magnitude of the relevant error constants. In the case of example under

consideration, the placement of the zero influences the magnitude of Kv ).

Choosing


we sketch the root locus for the PID-compensated system.

Fig. 7.5

The following session generates the root locus and simulates the PID-compensated system.

>> D2 = (s+0.5)/s

Transfer function:

s + 0.5

----------

.....s

>> D = D1*D2;

>> rlocus(D*G1)

>> sgrid(zeta,0)

>> [K,p]=rlocfind(D*G1)



selected_point =

-7.2512 +14.3478i

K =

4.3928

p =

-7.4116 +14.3004i

-7.4116 -14.3004i

-8.1037

-0.4660

>> step(feedback(K*D*G1,1))

The zeros are at -0.5,-8,-55.7467. The effects of third and fourth closed-loop poles are nearly

cancelled by zeros.

Fig. 7.6


Fig. 7.7

Searching the 0.456 damping ratio line (Fig. 7.6), we find the dominant second-order poles

to be , with an associated gain of 4.3928. Simulation in Fig. 7.7 gives

settling time 2.67 sec, peak time 0.184 sec, and zero steady state error.

PD compensator improved the transient response by decreasing the time required to reach the

first peak as well as yielding some improvement in the steady-state error. The complete PID

controller further improved the steady-state error without appreciably changing the transient

response designed with the PD controller.

Example 7.2

Given a unity feedback position control system with forward path transfer function:

The goal is to design a cascade compensator to meet the following requirements (refer

Nise(2004)).

................(i) 25% overshoot, (ii) settling time: 2 sec, and (iii) Kv = 20.

The 25% overshoot corresponds to a damping ratio of 0.404. Consider the following

MATLAB session.

s = tf('s');

G1 = 6.63/((s)*(s+1.71)*(s+100));


zeta = 0.404;

rlocus(G1);

sgrid(zeta,0);


[K,p] = rlocfind(G1);


selected_point =

-0.8057 + 1.8944i

>> K

K =

64.6665

>> p

p =

1.0e+002 *

-1.0004

-0.0083 + 0.0190i

-0.0083 - 0.0190i

Fig. .7.8


The intersection of 25% overshoot line with the root locus (Fig. 7.8) locates the system's

dominant second-order poles at and the gain at the dominant poles is

64.6665. The location of the third closed-loop pole is at 100.04; second-order

approximation is thus valid.

The simulation of the closed-loop system's step response is given by the following

commands:

M = feedback(K*G1,1);

step(M);

Fig. 7.9

Figure 7.9 shows that the design requirement of 25% overshoot is met.

The settling time is 4.07 sec. and

Comparing these values with the design requirements, we want to improve the settling time

by a factor of two and we want approximately eight-fold improvement in Kv .


We first attempt lead compensator design to improve transient response. To obtain a settling

time ts of 2 secs and 25% overshoot, the real part of dominant closed-loop poles should be at

and the imaginary part at

We now assume a lead compensator zero and find the compensator pole location so that lead-

compensated root locus passes through Let us assume compensator zero

at Angular contribution at the design point by open-loop poles and

compensator zero is obtained as follows:

Sd = -2+4.5285i;

angle_at_dominant_pole=(180/pi)*(angle(polyval([1,2],Sd))-(angle(polyval([1,0],Sd))...

+angle(polyval([1,1.71],Sd))+angle(polyval([1,100],Sd))))

angle_at_dominant_pole =

-120.1384

Compensator pole must contribute 120.1384 -180 = -59.8616o for the design point to be on

the compensated system's root locus. From the geometry shown in Fig. 7.10,

This gives pc = 4.6291.

Fig. 7.10

Thus the lead compensator is


Root locus plot and its analysis for the lead-compensated system is given by the following

MATLAB session.

>> D1 = (s+2)/(s+4.6291);

>> rlocus(D1*G1);

>> sgrid(zeta,0)

Properly redefine the axes of the root locus using Right click --> Properties --> Limits .

>> [K,p] = rlocfind(D1*G1)


selected_point =

-1.9621 + 4.4410i

K =

372.8743

p =

1.0e+002 *

-1.0026

-0.0200 + 0.0444i

-0.0200 - 0.0444i

-0.0208

>> step(feedback(K*D1*G1,1));


Fig. 7.11

Fig. 7.12


The transient response specifications are satisfied, with gain = 372.8743 (Figs. 7.11-7.12).

The lead-compensated system Kv becomes

Since we want Kv=20, the amount of improvement required over the lead-compensated

system is 20/6.2462=3.2. Choose pc =0.01 and calculate zc =0.0032, which is 3.2 times larger

than pc, that is we choose the lag compensator as

Root locus plot and its analysis for the complete lag-lead compensated system is given by the

following session:

D2 = (s+0.032)/(s+0.01);

rlocus(D1*D2*G1);

sgrid(zeta,0);


>> [K,p] = rlocfind(D1*D2*G1)


selected_point =

-1.9621 + 4.4410i

K =

373.5528

p =

1.0e+002 *

-1.0026

-0.0199 + 0.0444i

-0.0199 - 0.0444i

-0.0208


-0.0003

>> step(feedback(K*D1*D2*G1,1))

Fig. 7.13

Fig. 7.14

The design point has not moved with the addition of the lag compensator, and the gain at this

point is 373.5528 (Fig. 7.13). We also see from Fig. 7.14 that peak overshoot is higher than

specified. The system may be redesigned to reduce the peak overshoot.


Example 7.3

Consider a unity feedback system with the plant

The plant varies significantly:

It is desired to achieve robust behaviour (refer Dorf and Bishop (1998)). A design carried on

the nominal plant with K =1, gives the following cascade compensator.

For qualitative robustness analysis, we have obtained step response for the four conditions:

and . The results obtained using the

following MATLAB code, are summarized in Fig. 7.15, and Table 7.1.

s = tf('s');

G1 = 1/((s+1)^2);

G2 = 1/((0.5*s+1)^2);

G3 = 2/((s+1)^2);

G4 = 2/((0.5*s+1)^2);

D = (1+0.16*s)*(72.54+12/s);

M1 = feedback(G1*D,1);




step(M1);

hold;

step(M2);

step(M3);


step(M4);

The scale has been modified using Right click --> Properties -->Limits.

Fig. 7.15

Table 7.1

Plant

conditions

M1 (nominal

system)

M2 M3 M4

Percent

overshoot

11.9 2.4 8.69 1.63

Settling time

(sec)

0.551 0.155 0.40 0.0351

As is seen from this robustness analysis, the deviations from the nominal design performance

do not take the system outside the acceptable range; the design is therefore robust.

Example 7.4

A control system has the structure shown in Fig 7.16. The parameter variations are

and with the nominal conditions Km = 2 and p =4. Furthermore,

a third pole at s= 50 has been omitted from the model (refer Dorf and Bishop(1998)).


Fig. 7.16

A design carried out on the nominal system gives the following result.

We examine the performance of the system with nominal parameters Km =2, p =4; and worst-

case parameters Km =1.5, p =3. We also examine the nominal system with the third-pole

added, so that the control plant is

The robustness analysis results, obtained using the following MATLAB code, are

summarized in Fig. 7.17 and Table 7.2.

s = tf('s');

G1 = 2/(s*(s+4));

G2 = 1.5/(s*(s+3));

G3 = 2*50/(s*(s+4)*(s+50));

D = (1+0.06*s)*(294.69+4453/s);




step(M1);

hold;

step(M2);

step(M3);



Fig 7.17

Table 7.2

Plant

conditions

M1 (nominal

system)

M2 M3

Percent

overshoot

31.9 39.2 81.6

Settling time

(sec)

0.266 0.394 0.847

The system does not offer robust performance to parameter variations. The response of the

system with added pole shows that again the system fails the requirement of robust

performance. Since the results are not robust, it is possible to iterate on the design until an

acceptable performance is achieved. The interactive capability of MATLAB allows us to

check the robustness by simulation.


PRACTICAL - 7.2


MATLAB's SISO Design Tool

The SISO ( Single- Input/ Single- Output) Design Tool is a graphical user interface

which allows one to design single-input/single-output compensators by interacting

with the root locus plots and Bode plots of the closed-loop/open-loop systems. The

tool also has the option of using a Nichols chart, which can be selected under the

View menu. After the tool produces these plots, one can adjust the closed-loop poles

along the root locus and read gain, damping ratio, natural frequency, and pole

locations. These changes immediately get reflected to Bode plots and immediate

changes in the system's closed-loop response can be viewed in the LTI Viewer for

SISO Design Tool window.

One can add poles, zeros, and compensators, which can be interactively changed to

see the immediate effects on the root locus, Bode plots, and time response.

With Bode plots, one can affect the gain change by shifting the Bode magnitude

curve up and down; and gain, gain margin, gain crossover frequency, phase margin,

phase crossover frequency, and whether the loop is stable or unstable can be checked.

These changes immediately get reflected to root locus plots and immediate changes

in the system's closed-loop response can be viewed in the LTI Viewer for SISO

Design Tool window.

The following steps are required to use the SISO Design Tool.

1. Access the SISO Design Tool: The SISO Design Tool window, shown in Fig.

7.18, can be accessed by typing sisotool in the MATLAB Command Window or by

executing this command in an M-file. In the block diagram given in the top-right

corner of SISO Design Tool window, C represents the compensator.

...............An interactive tutorial on SISO Design Tool can be invoked by selecting

SISO Design Tool Help under the SISO Design Tool window Help menu.

2. Create LTI transfer functions: Create open-loop LTI transfer functions for which

you want to analyze closed-loop characteristics or design compensators. The transfer

functions can be created in an M-file or in the MATLAB Command Window. Run

the M-file or MATLAB Command Window statements to place the transfer function

in the MATLAB workspace. All LTI objects in the MATLAB workspace can be

exported to the SISO Design Tool .

The following MATLAB commands create the transfer function .

num=4500;


den=[1 361.2 0];

G=tf(num,den)

3. Create the closed-loop model for the SISO Design Tool: Choose Import… under

the File menu in the SISO Design Tool window to display the window shown in Fig.

7.19.

LTI objects can be selected from the SISO Models list and can be exported

to one of the blocks of the system by pressing the right-facing arrow next to the

selected block ( G, H, F, or C ) located in the section of the window labeled System

Data. Press the Other… button to rotate through a selection of feedback structures

and select the desired configuration. Alternatively, this can also be done by pressing

the FS button on the bottom right corner of closed-loop block diagram in the SISO

Design Tool window (Fig. 7.18). Root locus and Bode diagrams will change

immediately to reflect the changes in the feedback structure (Fig. 7.20).

LTI transfer function generating command, tf(num,den), can also be supplied

directly into the spaces for transfer functions in Fig. 7.19.

Fig. 7.18 SISO Design Tool window

4. Interact with the SISO Design Tool: After the Import System Data window

closes, the SISO Design Tool window now contains the root locus and Bode plots

for the system as shown in Fig. 7.20. In this example, we have considered the open-

loop system given by


.

Under the Analysis menu, select the desired response to open the LTI Viewer for

SISO Design Tool window (Fig. 7.21). Right-click on the LTI Viewer for SISO

Design Tool and choose desired Plot Types, Systems, and Characteristics.

Fig. 7.19 Import System Data Window

Loop gain can be changed in three different ways:

i. In Root Locus Editor: Keep the mouse pointer on a closed-loop pole

(squares) on the root locus. The arrow cursor changes to a hand. Hold

down the left mouse button and drag the closed-loop pole along the

root locus. Bode plot and the closed-loop response in the LTI Viewer

will immediately change to reflect the gain change. The value of the

gain will be displayed in the Current Compensator section of the


ii. In Open-Loop Bode Editor: Keep the mouse pointer anywhere on

the Bode magnitude curve. The arrow cursor changes to a hand. Hold

down the left mouse button and shift the curve up or down. Root locus

and closed-loop response in the LTI Viewer will immediately change

to reflect the gain change.

iii. In the Current Compensator Window: Type the desired gain value

in the C(s)= box in the SISO Design Tool window.

With gain changes, you can read the gain and phase margins and gain

crossover and phase crossover frequencies at the bottom of the Bode

magnitude and phase plots. Also, at the bottom of the Bode magnitude plot,

you are told whether or not the closed-loop system is stable.


Fig. 7.20 Root locus and Bode plots of G in SISO Design Tool window

Fig. 7.21 LTI Viewer for SISO Design Tool window


5. Design constraints: Design constraints may be added to your plots. These

constraints may be selected by right-clicking a respective plot and selecting

Design Constraints. To put new constraints, choose New… and to edit

existing constraints, choose Edit… . For example, Fig.7.22 shows the

selection of design constraint: damping ratio=0.5. On pressing OK, indicators

appear identifying portions of the root locus where the damping ratio is less

than 0.5(shaded gray), equal to 0.5(damping line), and greater than 0.5(

Fig.7.23 ). Note the change made in axes limits in this figure with respect to

Fig.7.20 using the Property Editor

Constraints may also be edited on the plots. Two black squares appear

on the constraint. You can drag these with your mouse anywhere in the plot

region. Point the mouse at the boundary of the constraint. When it changes to

four-pointed arrow, you can drag the boundary to a new position. The values

of the constraints are displayed in the Status Bar at the bottom of the plots.

Fig. 7.22 Adding design constraints

Fig.7.23 Adding design constraints


6. Properties: Right-clicking in a plot's window and selecting Properties…

displays the Property Editor window. From this window, some of the

properties of the plot such as axes labels and limits can be controlled. Try

exploring various options available with root locus and Bode plot properties

editor.

7. Add poles, zeros, and compensators: Poles and zeros may be added from

the SISO Design Tool window toolbar shown in Fig. 7.23. Let your mouse

pointer rest on the button for a few seconds to see the functionality of the

button in the form of screen tips. Add real pole; Add real zero; Add

complex pole; Add complex zero; Delete pole/zero; ..... functions are

available.

Use the SISO Design Tool toolbar and select the desired real/complex

pole/zero compensator. Move the mouse on the plots; your cursor shows that

a compensator was selected.. Place the cursor arrow to the point on the root

locus or Bode plot where you want to add the compensator, and click. The

compensator will be updated in the Current Compensator section of the

SISO Design Tool window. Compensator addition will be reflected

immediately in the root locus, Bode plots, and LTI Viewer for SISO Design

Tool window.

Go to SISO Tool Preferences… --> Options under the Edit menu to change

the way the compensator is represented.

8. Editing compensators and prefilters: The pole and zero values of the

compensators and prefilters can be edited in several ways. The most

convenient is to click on C or F blocks in the block diagram representation in

top-right corner of the SISO Design Tool window (Fig. 7.23).This operation

will open prefilter or compensator editor window shown in Fig. 7.24. Desired

real/complex zero/pole locations can be edited here. The same windows can

also be opened by following Compensators Edit C or F from the


In control systems design, we use compensators of the form

that alter the roots of the characteristic equation of the closed-loop system.

However, the closed-loop transfer function, M(s), will contain the zero of D(s)

as a zero of M(s).This zero may significantly affect the response of the system

M(s). We may use a prefilter F(s) to reduce the effect of this zero on M(s). For

example, if we select


we cancel the effect of the zero without changing the dc gain. Prefilters

may be employed with lead/PD compensation.

Fig. 7.24 Editing compensators and prefilters

Example 7.4

Consider a plant with the following transfer function:

We will design a cascade controller using SISO Design Tool in interactive

mode, so that the unity feedback closed-loop system meets the following

criteria (refer Kuo and Golnaraghi(2003)):

Steady-state error due to unit-ramp input 0.000443

Maximum overshoot 5 percent

Rise time 0.002 sec

Settling time sec

The first step is to import the model into SISO Design Tool. System transfer


functions can be imported in SISO Design Tool by clicking on File and then

going to Import… Before executing this sequence, create transfer function G

in MATLAB Command Window:

num=4500;

den=[1 361.2 0];

G=tf(num,den)

In order to examine the system performance, we start by using a proportional

controller. The system root locus can be obtained by clicking on View in the

main menu and then selecting Root Locus only. Fig. 7.25 shows the root

locus of the system. The plot is for K =1(by default).

Fig. 7.25

In order to see the poles and zeros of G and H , go to the View menu and

select System Data, or alternatively, double click the block G or H in the top-

right corner of the block diagram in the SISO Design Tool window. The

System Data window is shown in Fig.7.26.


Fig. 7.26

You may obtain the closed-loop system poles by selecting Closed-Loop

Poles from the View menu. Closed-loop poles are given in Fig.7.27.

Fig. M7.27

In order to see the closed-loop system time response to a unit-step input,

select the Response to Step Command in the Analysis main menu. With

specific selections made in Systems and Characteristics submenus, we

generate Fig. 7.28, which shows the unit step response of the closed-loop

system with unity gain controller, i.e. , K= 1.


Fig. 7.28

As a first step to design a controller, we use the built-in design criteria option

within the SISO Design Tool to establish the desired closed-loop poles

regions on the root locus. To add the design constraints, use the Edit menu

and choose the Root Locus option. Select New to enter the design constraints.

The Design Constraints option allows the user to investigate the effect of the

following:

o Settling time

o Percent overshoot

o Damping ratio

o Natural frequency

We will use the settling time and the percent overshoot as primary constraints.

After designing a controller based on these constraints, we will determine

whether the system complies with the rise time constraint or not. Figure 7.29

shows the addition of settling time constraint. Peak overshoot constraint is

added on similar lines.


Fig. 7.29

Figure 7.30 shows the desired closed-loop system pole locations on the root

locus after inclusion of the design constraints (Note that the scale has been

modified by following Right click --> Properties --> Limits. Left-clicking

anywhere inside plot will remove the black squares on the zeta lines).

Obviously, closed-loop poles of the system for K = 1 are not in the desired

area. Note the definition of the desired area: the vertical gray bar signifies the

boundary for that portion of the root locus where the settling time requirement

is not met; and the gray bars on damping lines signify the boundary for that

portion of the root locus where the peak overshoot requirement is not met.

Changing K will affect the pole locations. In the Root Locus window, C(s)

represents the controller transfer function. Fig.7.30 corresponds to C(s) = K =

1 . Hence, if C(s) is increased, the effective value of K increases, forcing the

closed-loop poles to move together on the real axis, and then ultimately to

move apart to become complex. See Fig. 7.31 wherein K= 16.8 gives closed-

loop poles on the damping lines. However, the settling time requirement is not

met.

Fig. 7.30


Fig. 7.31

The closed-loop poles of the system must lie to the left of the boundary

imposed by the settling time(Fig. 7.31). Obviously, it is impossible to use the

proportional controller (for any value of K ) to move the poles of the closed-

loop system farther to the left-hand plane. However, a PD controller C(s)= K

(1+ sTD ) may be used to accomplish this task. The zero s = - 1/ TD of the

compensator has to be placed far into the left half plane to move the root-

locus plot to the left. We have tried various values:

s = - 1/0.0005; - 1/0.001; - 1/0.0015; - 1/0.002,...

The value s = -1/0.00095 gives satisfactory results.

To add a zero to the controller, click the C block in the block diagram in top-

right corner of Fig. 7.31. Fig. 7.32 shows the Edit Compensator window and

how the PD controller is added:

Fig. 7.32

Figure 7.33 shows the plot for s= - 1/0.00095. Note that the closed-loop poles


have been dragged to the desired locations. The value of K that achieves the

desired dominent closed-loop poles is 282. This value of K forces the closed-

loop poles to the desired region. The system closed-loop poles are shown in

Fig. 7.34.

Fig. 7.33

Fig. 7.34

The step response of the controlled system in Fig. 7.35 shows that the system

has now complied with all design criteria.


Fig. 7.35


PRACTICAL - 8

Stability Analysis on Bode / Nyquist Plots

Objectives:

Introducing MATLAB commands for Bode and Nyquist plots.

Stability analysis using these plots .

As with the root locus plots, it will be tempting to rely exclusively on MATLAB to obtain

Bode, Nyquist, and other frequency-response plots. We strongly recommend that MATLAB

should be treated as one tool in the tool kit that can be used to design and analyze control

systems. There is no substitute for a clear understanding of the underlying theory.

MATLAB font does not permit the frequency symbol , which will appear as w in our

MATLAB codes. Also note that the default unit for frequency is rad/sec. This can be altered,

if required, to hertz by editing the plot axes; but we will continue to use rad/sec

Bode Plots

Nyquist Plots

Stability Margins





PRACTICAL – 8.1


Bode Plots

For a given transfer function G(s), Bode plot can be produced in different ways:

1. By simply using the command

bode (G)

We will get the Bode plot of G(s) in the current window for a default frequency range set by

the MATLAB package. MATLAB will automatically choose the frequency values by placing

more points in regions where the frequency response is changing rapidly. Fig. 8.1 shows the

Bode diagram for the transfer function

generated by the following code.

s = tf('s');

G = (s+1)/(0.1*s+1);

bode(G);

Fig. 8.1


2. To produce a Bode plot over a specified frequency range wmin to wmax, we use the

command

bode (G, {wmin, wmax})

For example, bode(G,{0.1,10}) will produce a Bode plot in the range 0.1 to 10 rad/sec,

shown in Fig. 8.2.

Fig. 8.2

If we supply the vector

w = [0.1; 0.3; 0.5; 0.7; 0.9; 1.0; 3.0; 5.0; 7.0; 9.0; 10];

and type

bode (G,w)

MATLAB will compute the magnitude and phase at the frequency points supplied in the

frequency vector.

If you choose to specify the frequencies explicitly, it is desirable to generate the vector w

using the logspace function. logspace (d1,d2) generates a vector of 50 points logarithmically

equally spaced between decades 10d1

and 10d2

. (50 points include both endpoints. There are

48 points between the endpoints). To generate 50 points between 0.1 rad/sec and 100 rad/sec,

enter the command


w = logspace (-1,2)

logspace(d1,d2,n) generates n points logarithmically equally spaced between decades 10d1

and 10d2

( n points include both endpoints). For example, to generate 100 points between 1

rad/sec and 1000 rad/sec, enter the command

w = logspace (0,3,100)

3. When invoked with left-hand arguments, such as

[mag,phase,w] = bode (G)

bode returns the frequency response of the system in variables mag, phase and w. No plot is

drawn on the screen. The magnitude and phase characteristics are placed in the workspace

through the matrices mag and phase. The vector w contains the values of the frequency in

rad/sec at which the magnitude and phase will be calculated.

If we supply the vector w and type

[mag, phase, w] = bode (G,w)

MATLAB will use the frequency vector supplied and compute the magnitude and phase at

those frequency points. This is clear from the following MATLAB session.

>> w = logspace(-1, 1, 5)

w = 0.1000 0.3162 1.0000 3.1623 10.0000

>> [mag,phase,w]=bode(G,w)

mag(:,:,1) =

1.0049

mag(:,:,2) =

1.0483

mag(:,:,3) =

1.4072

mag(:,:,4) =

3.1623

mag(:,:,5) =

7.1063


phase(:,:,1) =

5.1377

phase(:,:,2) =

15.7372

phase(:,:,3) =

39.2894

phase(:,:,4) =

54.9032

phase(:,:,5) =

39.2894

w =

0.1000

0.3162

1.0000

3.1623

10.0000

MATLAB does not assume that the system is a SISO system, but allows greater flexibility for

dealing with more complex systems. MATLAB stores mag and phase variables as matrices

of dimension p x q x n, where p is the number of inputs, q is the number of outputs, and n is

the number of frequency points. For SISO systems, the dimension of mag and phase will be

To access the magnitude and phase points for pth

input and qth

output, we have to

write mag(p, q, :) or phase(p, q, :). For example,

>> mag(1,1,3)

ans =

1.4072

>> phase(1,1,3)

ans =

39.2894


For SISO systems, mag(:,:,3) and phase(:,:,3) will also give the same answer.

We use mag(:,:)', phase(:,:)' to convert arrays to column vectors, where the apostrophe

signifies matrix transpose. For example, the following command stores points of the Bode

plot in matrix form with magnitude in dB, phase in degrees, and frequency in rad/sec.

bode_points = [w, 20*log10( mag(:,:)' ), phase(:,:)' ]

Information about the plots obtained with bode can be found by left-clicking the mouse on

the curve. You can find the curve's label, as well as the coordinates of the point on which you

clicked. Right-clicking away from a curve brings up a menu. Form this menu, you can select

(i) system responses to be displayed (magnitude plot, phase plot, or both), and (ii)

characteristics, such as peak response and stability margins. When selected, a dot appears on

the curve at the appropriate point. Let your mouse rest on the point to read the value of the

characteristic. You may also (iii) select the frequency units in Hz., magnitude in absolute,

phase in radians, (iv) set axis limits, and (v) grid etc.

A Bode plot with grid selected is shown in Fig. 8.3.

Fig. 8.3


PRACTICAL - 8.2


Nyquist Plots

We can use MATLAB to make Nyquist plots using the command nyquist(G). Information

about the plots obtained with this command can be found by left-clicking the mouse on the

curve. You can find the curve's label, as well as the coordinates of the point on which you

have clicked and the frequency. Right-clicking away from a curve brings up a menu. From

this menu, you can select (i) system responses to be displayed (with and without negative

frequencies), and (ii) characteristics, such as peak response and stability margins. When

selected, a dot appears on the curve at the appropriate point. Let your mouse rest on the point

to read the value of the characteristic. You may also (iii) select the frequency units in Hz.,

(iv) set axis limits, and (v) grid etc.

We can obtain points on the plot by using [re im w] = nyquist(G), where the real part,

imaginary part, and frequency are stored in re, im, and w, respectively. re and im are 3-

dimensional arrays. We use re(:,:)' and im(:,:)' to convert the arrays to column vectors. We

can specify the range of w by using [re,im] = nyquist(G,w).

A sample Nyquist plot generation is illustrated below.

s = tf('s');

G = 1/(s^2+0.8*s+1);

nyquist(G);

axis equal;

Fig. 8.4


Axis used should be ' equal ', otherwise circles appear squashed.

Note that when a MATLAB operation involves ' Divide by zero ', the resulting Nyquist plot

may be erroneous. For example, Nyquist plot of

is shown in Fig. 8.5.

G=1/((s)*(s+1));

nyquist(G)

Fig. 8.5

If such an erroneous Nyquist plot appears, then it can be corrected by manipulating the axes.

For example, manipulating the x-axis to [-2,2] and y-axis to [-5,5] in Fig. 8.5, results in the

Nyquist plot shown in Fig. 8.6.


Fig. 8.6

Sometimes in the course of using the nyquist function, we may find that a Nyquist plot looks

nontraditional or that some information appears to be missing. It may be necessary in these

cases to adjust the axes and override the automatic scaling, and/or to use the nyquist function

with left-hand arguments with specified frequency range in conjunction with the plot

function. In this way we can focus on the critical region for our stability analysis.


PRACTICAL – 8.3


Stability Margins

We use [GM, PM] = margin(G) to find gain margin (GM) and phase margin (PM).

We use [GM, PM, wg, wphi] = margin(G) to find gain margin (GM), phase margin (PM),

gain crossover frequency (wg), and phase crossover frequency (wphi).

The margin function is invoked in conjunction with the bode function to compute gain and

phase margins. If the margin(G) is invoked without left-hand arguments, the Bode plot is

automatically generated with gain margin, phase margin, gain crossover frequency, and phase

crossover frequency labeled on the plot. This is illustrated in Fig. 8.7.

s = tf('s');

G = 1/(s^2+0.8*s+1);

margin(G);

Fig. 8.7

Stability margins, and gain and phase crossover frequencies are also given by the right-

clicking feature on the plots generated using the MATLAB functions nyquist and bode .


PRACTICAL - 9

Frequency Response Characteristics

Objectives:

Frequency response measures from Bode plots and Nichols chart, for unity-

feedback systems.

Frequency response measures from Bode plots for nonunity-feedback systems.

MATLAB has a powerful GUI tool, called LTI Viewer, for obtaining both the time and

frequency response characteristics of a given Linear Time Invariant system. The basic

features of this GUI tool and its use for time response analysis have already been given in

MATLAB Module 5. The use of LTI Viewer for frequency response analysis is similar.

The frequency response parameters can also be obtained from the plots generated by the

commands nyquist, bode, and nichols. Right-clicking away from the curves generated by

these commands brings up a menu. From the menu, various frequency-response

characteristics can be selected.

Important measures of performance of control systems are the resonance peak Mr , the

resonance frequency , the bandwidth of the closed-loop frequency response, phase

margin , gain margin GM , gain crossover frequency , and phase crossover frequency

.

We observed in module 8 that the measures of performance of a closed-loop system can be

obtained from the Bode plots of its open-loop transfer function. In this module, we will use

Nichols chart that gives closed-loop performance measures from the open-loop frequency

response. However, this technique is applicable only for unity feedback systems. For non-

unity feedback systems, we construct the closed-loop frequency response and therefrom

obtain the measures of performance.

Non-unity Feedback Systems

Unity Feedback Systems on Nichols Chart




PRACTICAL - 10

Frequency Response Design and SISO Design Tool

Objectives:

Design using Bode plots and Nichols chart.

Frequency response design using SISO Design Tool.

We will carry out Control system design using MATLAB dialogues. MATLAB is an

interpreted language. That is, it runs along executing each instruction as it comes to it. This

interactive feature of MATLAB will be exploited in our design exercise. The designer will

always be in the loop.

The MATLAB SISO Design Tool, developed on the MATLAB dialogue pattern, is a

powerful GUI tool, wherein the designer is always in the design loop. We have outlined this

tool earlier in MATLAB Module 7. An example of frequency response design using this tool

will be given at the end of this module.

Design Using MATLAB Dialogues

Robustness Analysis

Design Using SISO Design Tool


http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-Delhi/Control%20system%20design%20n%20principles/matlab/module10/module10-1_middle.htm



PRACTICAL - 10.1


Design Using MATLAB Dialogues

Example 10.1

A unity feedback system has forward path transfer function

Our goal is to design a cascade PD compensator so that the closed-loop system meets the

following specifications (refer Kuo and Golnaraghi (2003)).

Steady-state error due to unit-ramp input 0.000443

Phase angle

Resonance peak

Bandwidth

We want to achieve a steady-state response that has no more than 0.000443 error due to unit-

ramp input. This, as we know, can be achieved by simple gain adjustment. We therefore first

evaluate the ‘uncompensated system' that meets the steady-state accuracy requirement. K

should be set at 181.17.

We now examine the performance of the uncompensated system. The following MATLAB

dialogue is helpful:

s = tf('s');

G = (4500*181.17)/(s*(s+361.2));

figure(1);

margin(G);

grid;

figure(2);

nichols(G);

ngrid;


Fig. 10.1

Fig. 10.2


Since the plot in Nichols coordinate system does not touch any of the M-contours

available by default in the ngrid command, we may generate Bode plot of uncompensated

feedback system to obtain resonance peak Mr.

Fig. 10.3

The uncompensated system with K = 181.17 has (Figs 10.1-10.3) a phase margin

gain crossover frequency resonance peak Mr = 2.5527 (

Mr=108.14/20

), and bandwidth

A PD compensator with transfer function

has asymptotic Bode plot of the form shown in Fig. 10.4.


Fig. 10.4

The logical way to approach the design problem is to first examine how much additional

phase is needed to realize a of 80o. Since the uncompensated system with the gain set to

meet the steady-state requirement has of only 22.6o, the PD compensator must provide

an additional phase of 80 22.6=57.4o. The additional phase must be placed at the desired

gain crossover frequency of the compensated system in order to realize a of 80o.

Referring to the asymptotic Bode plot of PD compensator shown in Fig. 10.4, we see that the

additional phase at frequencies > 1/ TD is always accompanied by a gain in the magnitude

curve. As a result, the gain crossover of the compensated system will be pushed to a higher

frequency. At new gain crossover, the phase of the uncompensated system would correspond

to smaller . Thus we may run into the problem of diminishing returns.

Simple trial-and-error placement of corner frequency of the compensator around the gain

crossover frequency can meet the design requirements. The performance measures in the

frequency domain for the compensated system with the compensator parameters: TD =

0.0005, 0.001, 0.0015, 0.002, and 0.0017, are tabulated in Table 10.1. With TD = 0.0017, the

performance requirements in frequency domain are all satisfied.


Table 10.1

T D

M r =10 dB/20

0.0005 46.1 913 1330 1.3630

0.001 65.5 1060 1590 1.1311

0.0015 78.5 1320 1600 1.0464

0.002 85.5 1660 2050 1.0125

0.0017 81.9 1450 1650 1.0292

The following MATLAB dialogue gives one result of Table 10.1:

Td = 0.0017;

D = 1+Td*s;

figure(1);

margin(D*G);

grid;

figure(2);

nichols(D*G);

ngrid;

figure(3);

bode(feedback(D*G,1));

Example 10.2

Let us reconsider the system of Example 10.1 (refer Kuo and Golnaraghi (2003)). Initially

we take gain K = 181.17 simply because the value was used in Example 10.1. We will tune

the total loop gain later to meet the steady-state performance requirement.

The uncompensated system with K =181.17 has =22.6o and gain crossover frequency

(refer Fig. 10.1). Let us specify to be atleast 64o , and ess (parabolic

input)

We know that PI compensator is an approximation for the lag compensator:


The asymptotic Bode plot of a PI compensator is shown in Fig. 10.5.

Fig. 10.5

From the Bode plot of uncompensated system (Fig. 10.6), we find that the new gain

crossover frequency at which the phase margin is 64o, is 173 rad/sec. The magnitude of

at this frequency is 21.4 dB. Thus the PI compensator should provide an attenuation of

21.4 dB at = 173 rad/sec.

s = tf('s');

G = (4500*181.17)/(s*(s+361.2));

margin(G);


Fig. 10.6

A simple trial-and-error for values of that are sufficiently small (low-

frequency range) can meet the design requirements. The performance measures in the

frequency domain for the compensated system with the compensator parameters: KP =

and KI= 0.00851, 0.0851, 0.851, and 1.702, are tabulated in Table 10.2. Note

that for the values of KP / KI = that are sufficiently small, all vary

little. We choose KI = 0.0851. With this value of KI, the performance requirements in

frequency domain are all satisfied.

Table 10.2

KI

Mr=10

dB/20

0.00851 64.3 173 279 1.0022

0.0851 64.1 173 285 1.0070

0.851 61.1 173 324 1.0572

1.702 57.7 174 323 1.1171

The following MATLAB dialogue gives one of the results of Table 10.2.

KI = 0.0851;

D = 0.0851+(KI/s);

figure(1);


margin(D*G);

grid;

figure(2);

nichols(D*G);

ngrid;

figure(3);

bode(feedback(D*G,1));

Let us now evaluate the steady-state performance.

Steady-state requirement is thus satisfied.

It should be noted that of the system can be improved further by increasing the value of

above 1/0.0851. However, the bandwidth of the system will be reduced. For example, for

is increased to 75.7o but is reduced to 127 rad/sec.

Example 10.3

Consider the unity-feedback system whose open-loop transfer function is

The system is to be compensated to meet the following specifications:

1. Velocity error constant K v = 30.

2. Phase margin .

3. Bandwidth .


It easily follows that K =30 satisfies the specification on KV. The Bode plot of with

K =30 is shown in Fig. 10.7 from which it is found that the uncompensated system has a gain

crossover frequency and phase margin = 17.2o .

s = tf('s');

G = 1/(s*(0.1*s+1)*(0.2*s+1));

K = 30;

margin(K*G);

Fig. 10.7

If lag compensation is employed for this system, the bandwidth will decrease sufficiently so

as to fall short of the specified value of 12 rad/sec, resulting in a sluggish system. This fact

can be verified by designing a lag compensator. If on the other hand, lead compensation is

attempted, the bandwidth of the resulting system will be much higher than the specified

value; the closed-loop system will be sensitive to noise, which is undesirable. This fact can

also be verified by designing a lead compensation scheme.

Let us design a lag-lead compensator to overcome the difficulties mentioned above. Since a

full lag compensator would reduce the system bandwidth excessively, the lag section of the

lag-lead compensator must be designed to provide partial compensation only. The lag section

design therefore proceeds by making a choice of the new gain crossover frequency, which

must be higher than the crossover frequency if the system were fully lag compensated. Full

lag compensation demands that the gain crossover frequency should be shifted to a point

where the phase angle of the uncompensated system is:


+specified phase margin +

From Fig. 10.7, we find that this requirement is met at 2.1 rad/sec. For the design of lag

section of lag-lead compensator, the selected gain crossover frequency should be higher than

2.1. The choice is made as to start with.

From Fig. 10.7, we find that the gain of the uncompensated system at is 20.3 dB.

Therefore to bring the magnitude curve down to 0 dB at , the lag section must provide an

attenuation of 20.3 dB. This gives the parameter of the lag section as

Let us now place the upper corner frequency of the lag section at 1/0.8 rad/sec. This gives the

lag-section transfer function

Bode plot of lag-section compensated system is shown in Fig. 10.8; the phase margin is

23.8o.

D1 = (0.8*s+1)/(8*s+1);

margin(D1*K*G);

Fig. 10.8


We now proceed to design the lead section. We choose The maximum phase

lead provided by the lead section is therefore

The lag-compensated system has a gain of at 5.6 rad/sec. Setting

(the frequency at which the lead section has maximum phase ) = 5.6 rad/sec, we get

.

Take the lead section as

From the Bode plot and Nichols chart of the lag-lead compensated system (Figs 10.9 and

10.10), we find that the phase margin is 55.7 o

and the bandwidth is 12.5 rad/sec. The design

therefore does meet the specifications laid down.

D1 = (0.8*s+1)/(8*s+1);

D2 = (0.565*s+1)/(0.0565*s+1);

figure(1);

margin(D1*D2*K*G);

figure(2);

nichols(D1*D2*K*G);

grid;


Fig.10.9

Fig.10.10


PREACTICAL - 10.4


Robustness Analysis

Example 10.4

A unity feedback system has the plant

It is desired to achieve robust performance. A design carried out on nominal plant with K =1

and p0 = 1, gives the following cascade compensator (refer Dorf and Bishop (1998)).

We examine here the robustness properties of this controller for a range of plant parameter

variations of with K =5. We also examine the robustness for variations in K

with p0 fixed at 1. The results generated using the following MATLAB code are displayed in

Figs 10.11 and 10.12.

s = tf('s');

G11 = 5/((s+0.1)^2);

G21 = 5/((s+1)^2);

G31 = 5/((s+10)^2);

D = (1+0.137*s)*(4222+26300/s);




figure(1);

step(M11);

hold;

step(M21);

step(M31);


G12 = 1/((s+1)^2);

G22 = 2/((s+1)^2);

G32 = 5/((s+1)^2);




figure(2);

step(M12);

hold;

step(M22);

step(M32);


Fig. 10.11 Variations in p0


Fig. 10.12 Variations in K

The simulation results indicate that the PID design is robust with respect to changes in p0 and

K. The differences in the step response for are barely discernible on the plot.

Example 10.5

A unity feedback system has the nominal plant

A cascade design on the nominal system:

We examine here the robustness of the system with respect to deadtime in the loop (refer

Dorf and Bishop (1998)). Simulations were carried out with

with and 0.4 secs. The results are summarized in Fig. 10.13 and Table 10.3. The

MATLAB script is given below.

s = tf('s');

G = 3/(3.1*s+1);


D = (1+0.06*s)*(3.22+2.8/s);

M1 = feedback(G*D,1);

[numd1,dend1]=pade(0.1,2);


del_1 = tf(numd1,dend1);


Gd1 = G*del_1;

Gd2 = G*del_2;

M2 = feedback(Gd1*D,1);


step(M1);

hold;

step(M2);

step(M3);


Fig. 10.13


Table 10.3

Plant

conditions

M1

(zero

deadtime)

M2

(deadtime=0.1)

M3

(deadtime =

0.4)

Percent

overshoot

6.78 8.57 89.4

Settling

time (sec)

3.3 2.89 7.03

Robustness analysis shows that the design is acceptably robust if deadtime is less than 0.1

sec. However, for large deadtimes of the order of 0.4 sec., the design is not robust. It is

possible to iterate on the design until an acceptable performance is achieved.

The interactive capability of MATLAB allows us to check the robustness by simulation.


PRACTICAL - 10.3


Robustness Analysis

Example 10.4

A unity feedback system has the plant

It is desired to achieve robust performance. A design carried out on nominal plant with K =1

and p0 = 1, gives the following cascade compensator (refer Dorf and Bishop (1998)).

We examine here the robustness properties of this controller for a range of plant parameter

variations of with K =5. We also examine the robustness for variations in K

with p0 fixed at 1. The results generated using the following MATLAB code are displayed in

Figs 10.11 and 10.12.

s = tf('s');

G11 = 5/((s+0.1)^2);

G21 = 5/((s+1)^2);

G31 = 5/((s+10)^2);

D = (1+0.137*s)*(4222+26300/s);




figure(1);

step(M11);

hold;

step(M21);

step(M31);


G12 = 1/((s+1)^2);

G22 = 2/((s+1)^2);

G32 = 5/((s+1)^2);




figure(2);

step(M12);

hold;

step(M22);

step(M32);


Fig. 10.11 Variations in p0


Fig. 10.12 Variations in K

The simulation results indicate that the PID design is robust with respect to changes in p0 and

K. The differences in the step response for are barely discernible on the plot.

Example 10.5

A unity feedback system has the nominal plant

A cascade design on the nominal system:

We examine here the robustness of the system with respect to deadtime in the loop (refer

Dorf and Bishop (1998)). Simulations were carried out with

with and 0.4 secs. The results are summarized in Fig. 10.13 and Table 10.3. The

MATLAB script is given below.

s = tf('s');

G = 3/(3.1*s+1);


D = (1+0.06*s)*(3.22+2.8/s);

M1 = feedback(G*D,1);





Gd1 = G*del_1;

Gd2 = G*del_2;



step(M1);

hold;

step(M2);

step(M3);


Fig. 10.13


Table 10.3

Plant

conditions

M1

(zero

deadtime)

M2

(deadtime=0.1)

M3

(deadtime =

0.4)

Percent

overshoot

6.78 8.57 89.4

Settling

time (sec)

3.3 2.89 7.03

Robustness analysis shows that the design is acceptably robust if deadtime is less than 0.1

sec. However, for large deadtimes of the order of 0.4 sec., the design is not robust. It is

possible to iterate on the design until an acceptable performance is achieved.

The interactive capability of MATLAB allows us to check the robustness by simulation.


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CSE-Lab Manual 2015