matlab and simulink for control system

MATLAB and Simulink for Control Systems

V.SitaramGupta

Control Systems

• Control means to regulate, direct, command, or govern. A system is a

collection, set, or arrangement of elements (subsystems).

• A control system is an interconnection of components forming a system

configuration that will provide a desired system response

• In order to identify, delineate, or define a control system, we introduce

two terms: input and output here

• The input is the stimulus, excitation, or command applied to a control

system, and the output is the actual response resulting from a control

system

Control Systems• Control systems can have more than one input or output

• The input and the output represent the desired response and the actual

response respectively. A control system provides an output or response

for a given input or stimulus, as shown in Fig

• Control system applications are found in robotics, space-vehicle

systems, aircraft autopilots and controls, ship and marine control

systems, intercontinental missile guidance systems, automatic control

systems for hydrofoils, surface-effect ships, and high-speed rail systems

including the magnetic levitation systems

Examples of Control Systems• Residential heating and air-conditioning systems controlled by a thermostat

• The cruise (speed) control of an automobile

• Manual control:

– Opening or closing of a window for regulating air temperature or air quality

– Activation of a light switch to regulate the illumination in a room

– Human controlling the speed of an automobile by regulating the gas supply to

the engine

• Automatic traffic control (signal) system at roadway intersections

• Control system which automatically turns on a room lamp at dusk, and turns it off in

Daylight

• Automatic hot water heater

Control System Configuration

• Block: -A block is a set of elements that can be grouped together, with

overall characteristics described by an input/output relationship as

shown in Fig

• Transfer Function: - The transfer function is a property of the system

elements only, and is not dependent on the excitation and initial

conditions. The transfer function of a system (or a block) is defined as

the ratio of output to input as shown in Fig

Control System Configuration• Open-loop Control System: - Open-loop control systems represent the

simplest form of controlling devices. A general block diagram of open-

loop system is shown in Fig

• Closed-loop (Feedback Control) System: - Closed-loop control systems

derive their valuable accurate reproduction of the input from feedback

comparison.

Control System Terminology

An Introduction to MATLAB

and the Control Systems toolbox

MATLAB• MATLAB is essentially a programming interface that can be used for a

variety of scientific calculations, programming and graphical visualization

• Its basic data element is an array, and its computations are optimized for

this data type, which makes it ideal for problems with matrix and vector

formulations

• MATLAB is also extendible by means of add-on script packages called

toolboxes, which provide application-specific functions for use with

MATLAB

• For this course, we will mostly be using MATLAB’s basic matrix/vector

operations and graphing capabilities in conjunction with the control

system toolbox

Interface

See the text book Analysis and Design of Control System using MATLAB

Examples an MATLAB and Basic Control Systems

Control System Toolbox• Control System Toolbox is a package for MATLAB consisting of tools

specifically developed for control applications

• The package offers data structures to describe common system

representations such as state space models, zero-pole gain and transfer

functions, as well as tools for analysis and design of control systems

• It has a collection of algorithms, written mostly in M-files, that

implements common control system design, analysis, and modeling

techniques

• Here you will get to know the basic commands of Control System

Toolbox. When you have completed this exercise, you should be able to

understand and use Control Systems Toolbox

Control System Toolbox• The most commonly used functions are presented below. You are

encouraged to look through the help files for other functions and options

that you might find useful in future

• Using MATLAB to Create Models: -

• Why Model?

– Represent

– Analyze

• What kind of systems are we interested?

– Single-Input-Single-Output (SISO)

– Linear Time Invariant (LTI)

– Continuous

Control System Toolbox• Three Basic types of model representation for continuous LTI systems

– Transfer Function Representation (TF)

– Zero-Pole-Gain Representation (ZPK)

– State Space Representation (SS)

• Transfer Function Models are created using the function

tf( numerator, denominator) where numerator and denominator are

two vectors containing the coefficients of the polynomials in the

numerator and denominator of the transfer function

Control System Toolbox• State space models are created using the function ss(A,B,C,D) where

A,B,C and D are the matrices forming the state space system

• Zero-Pole-Gain forms of transfer functions, ie

• can be specified directly as zpk(z,p,K) where z and p are the vectors of

zero and pole values in the complex plane (z = [z1 z2 …] and p = [p1 p2

…]), and K is the gain

Control System Toolbox

• The models thus generated can be used in conjunction with the operators

+,- and * to generate new models. sys1 * sys2 produces a series

interconnection from input through sys2 and sys1 (in that order) to the

output. sys1 ± sys2 represent parallel interconnections

• Converting Models:- Control system models can be converted from

one form to the other. The three functions presented above (ss(), tf(),

zpk()) are overloaded to perform arbitrary system conversions. For

example, if sys1 is a state-space model, we can generate an equivalent

transfer function model sys2 by issuing the command

>> sys2 = tf(sys1);

• In addition to these, there are other functions that are used for converting

elements of one form(say, the matrices of a state-space model) to the

elements of another form(say, the numerator and denominator of a

transfer function model: tf2ss, ss2tf, tf2zp, zp2tf, zp2ss, ss2zp


Control System ToolboxSystem Analysis: -

• Once a model has been introduced in MATLAB, we can use a series of

functions to analyze the system.

• Key analyses at our disposal:

– Stability analysis

e.g. pole placement

– Time domain analysis

e.g. response to different inputs

– Frequency domain analysis

e.g. bode plot

Control System ToolboxStability analysis: -

• Stability of a linear system is determined by the location of its poles in

the complex plane. (What is the condition for stability?) Use the

commands ssdata and tfdata to extract the necessary data from the

models, and eig and roots to determine stability of the system


When a system becomes unstable, the output of the system approaches infinity

(or negative infinity) Example: - An example to illustrate the importance of

stability is the control of a nuclear reactor


• Time domain analysis:- in the form of time response of a system to a

specified input can be obtained using the following commands:

– step(sys) plots the step response of a system sys.

– impulse(sys) plots the impulse response of sys

– lsim(sys,input_vector,time_vector,initial_state_vector) plots the response

of sys to an arbitrary input. The elements of input_vector define the values of

the input at times corresponding to the elements of time_vector. The initial

state vector can be specified for state-space models only

– Once a model has been inserted in MATLAB, the step response can be

obtained directly from: step(sys) and etc.,


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

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1.2

1.4Step Response

Time (sec)

Am

plitu

de

• MATLAB also caries other useful functions for time domain analysis:– Impulse response impulse(sys)– Response to an arbitrary input e.g. t = [0:0.01:10]; u = cos(t); lsim(sys,u,t)


0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

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0.5

1

1.5Linear Simulation Results

Time (sec)

Am

plitu

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0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

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2.5

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3.5Impulse Response

Time (sec)

Am

plitu

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Control System Toolbox• Frequency domain analysis tools: -

– rlocus(sys) plots the root locus of the system sys with variations in gain

– bode(sys) plots the magnitude and phase angle Bode plots

– [Gm,Pm,Wg,Wp] = margin(sys) calculates the gain margin Gm, the phase

margin Pm, and the frequencies corresponding to their occurrence. Issuing

the command margin(sys) alone plots the Bode diagram and marks the

margins on it

– nyquist(sys) plots the Nyquist plot of the system. Note that the loop at

infinity is not represented on the plot

– sisotool(plant,compensator) opens an interactive mode of the root locus

and bode plots, which can be used to modify the compensator and gain to

achieve the desired system characteristics


-60

-40

-20

0

20

Mag

nitu

de (

dB)

10-1

100

101

102

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-5

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System: sysPole : -2 + 4.58iDamping: 0.4Overshoot (%): 25.4Frequency (rad/sec): 5

System: sysPole : -2 - 4.58iDamping: 0.4Overshoot (%): 25.4Frequency (rad/sec): 5

Control System Toolbox• Extra: Partial Fraction Expansion: -


State – space tools: -

• ctrb(sys) returns the controllability matrix of a state-space system sys

• obsv(sys) returns the observability matrix of a state-space system sys

• [K,S,E] = lqr(A,B,Q,R) : Linear quadratic regulator design – calculates

optimal gain K for a state space system, based on the optimizing

matrices Q and R provided by the designer

Control System ToolboxSome useful MATLAB commands

SIMULINK Control System• Simulink is a simulation program based upon MATLAB

• There are several ways to define a model. One can work graphically

and connect block diagrams with pre-define blocks

• Alternatively one can give the mathematical description in forms of

differential equations in an m file (the format for programs written in the

MATLAB programming language)

• MATLAB/SIMULINK supports both these representations as well as

combinations. Furthermore one can use descriptions that include a

hierarchy of connected subsystems

SIMULINK Control SystemHow to Start Simulink: -

• Start Matlab 9. Then give the command SIMULINK in MATLAB. This

gives a window with blocks as in Figure

SIMULINK Control SystemA Simple System

• Click on File in the Simulink window and choose New->Model. Click on

the block Continous and move a Transfer Fcn to the new window called

“Untitled”. Do the same with Source->Step Fcn and Sinks->Scope.

Draw arrows (left mouse button) and connect the ports on the block. You

should now have a block diagram as in Figure.

• Choose Simulation->Parameters in the window called “Untitled”. Set

Stop time to 5. Open the window Scope by double clicking on it. Put

Horizontal Range to 6. Start a simulation by Simulation->Start (or by

pressing Ctrl t in the window called “Untitled’)

SIMULINK Control System

How to Change a System To change the system to

you double click on the block Transfer Fcn and change Denominator to [1

0.5 2]. Simulate the new system (Simulation->Start or Ctrl t). Change

parameters in the Simulation menu and the scales in the block Scope until

you are satisfied

SIMULINK Control System• How to Change an Input Signal : -To change the input signal, start

with removing the block Step Fcn by clicking on it and delete it by

using Edit->Cut (or pressing Ctrl x). Replace it by a Sources->Signal

Gen block. Double click on Signal Gen and select signal, amplitude and

frequency. Also change Simulation-> Start->Stop Time to 99999 and

press Simulation->Start. This gives an “infi nite” simulation that can be

stopped by pressing Simulation->Stop (or Ctrl t). Can the amplitude of

the input signal be changed during simulation? Also try to change the

block Transfer Fcn during simulation

SIMULINK Control System• How to Use Matlab Variables in Blocks :- Note that variables defined in the

MATLAB environment can be used in Simulink. Define numerator and

denominator by writing the following in the MATLAB window

– num=[1 1]

– den=[1 2 3 4]

• Change Transfer Fcn->Numerator to num and Denominator to den

• How to Save Results to MATLAB variables To save input and output move

two copies of the block Sinks->To Workspace. Make sure that the “save

format” option is set to “Array”

matlab and simulink for control system

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