Sanjeev Presentation

8/3/2019 Sanjeev Presentation

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By-Sanjeev 1

MATHEMATICAL

MODELLING& VARIOUSCONTROL SYSTEM MODELS &RESPONSES USING MATLAB

By-Sanjeev

By-Sanjeev 2

Presentation Overview

Control Toolbox Introduction

Building Models for LTI System Continuous Time Models

Discrete Time Models

Combining Models

Transient Response Analysis

Frequency Response Analysis

Stability Analysis Based on FrequencyResponse

Other Information

By-Sanjeev 3

Control System Toolbox

This is a toolbox for control system designand analysis. It supports transfer function andstate-space forms (continuous/ discrete time,

frequency domain), as well as functions forstep, impulse, and arbitrary input responses.Functions for Bode, Nyquist,root-locus plots &

many control system designs are included.

By-Sanjeev 4

Building Models for LTI System

Control System Toolbox supportscontinuous time models and discrete timemodels of the following types*:

Transfer Function

Zero-pole-gain

State Space

* Material taken from http://techteach.no/publications/control_system_toolbox/#c1

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Continuous Time Transfer Function(1)

Function: Use tf function create transfer function

of following form:

Example 23

>>num = [2 1];

>>den = [1 3 2];

>>H=tf(num,den)

Transfer function:2 s + 1

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

s^2 + 3 s + 2

Matlab Output

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Include delay to continuous time Transfer Function

Example23

Transfer function:

2 s + 1

exp(-2*s) * -------------

s^2 + 3 s + 2

>>num = [2 1];

>>den = [1 3 2];

>>H=tf(num,den,’inputdelay’,2)

Matlab Output

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Function: Use zpk function to create transferfunction of following form:

Example 21

>>num = [-0.5];

>>den = [-1 -2];>>k = 2;

>>H=zpk(num,den,k)

Zero/pole/gain:2 (s+0.5)

-----------

(s+1) (s+2)

Matlab Output

By-Sanjeev 8

Continuous Time State Space Models(1)

State Space Model for dynamic system

Matrices: A is state matrix; B is input matrix; C isoutput matrix; and D is direct

transmission matrixVectors: x is state vector; u is input vector; and y isoutput vector

Note: Only apply to system that is linear and time invariant

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Continuous Time State Space Models(2)

Function: Use ss function creates state spacemodels. For example:

DCBAx x

>>A = [0 1;-5 -2];

>>B = [0;3];>>C = [0 1];

>>D = [0];

>>sys=ss(A,B,C,D)

x1 x2x1 0 1

x2 -5 -2

Matlab Output

u1x1 0

y1 0 1

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Conversion between different models

Converting From Converting to Matlab function

Transfer Function Zero-pole-gain [z,p,k]=tf2zp(num,den)

Transfer Function State Space [A,B,C,D]=tf2ss(num,den)

Zero-pole-gain Transfer Function [num,den]=zp2tf(z,p,k)

Zero-pole-gain State Space [A,B,C,D]=zp2ss(z,p,k)

State Space Transfer Function [num,den]=ss2tf(A,B,C,D)

State Space Zero-pole-gain [z,p,k]=ss2zp(A,B,C,D)

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% Transfer function model

num = [0 0 25];den = [1 4 25];G = tf(num,den)

% Zero-Pole-Gain model[z,p,k]=tf2zp(num,den)% State-Space model

[A,B,C,D]=tf2ss(num,den)

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Lecture Overview

Continuous Time Models

Combining ModelsTransient Response Analysis

Other Information

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Discrete Time Transfer Function(1)

Function: Use tf function create transfer function

of following form:

Example: with sampling time 0.423

>>num = [2 1];

>>den = [1 3 2];>>Ts=0.4;

>>H=tf(num,den,Ts)

Transfer function:

2 z + 1-------------

z^2 + 3 z + 2

Sampling time: 0.4

Matlab Output

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Discrete Time Transfer Function(2)

Function: Use zpk function to create transferfunction of following form:

Example: with sampling time 0.4 21

>>num = [-0.5];

>>den = [-1 -2];>>k = 2;

>>Ts=0.4;

>>H=zpk(num,den,k,Ts)

Zero/pole/gain:

2 (z+0.5)-----------

(z+1) (z+2)

Sampling time: 0.4

Matlab Output

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Discrete Time State Space Models(1)

State Space Model for dynamic system

][][][

][][]1[

Matrices: A is state matrix; B is input matrix; C isoutput matrix; and D is direct

transmission matrixVectors: x is state vector; u is input vector; and y isoutput vector

n is the discrete-time or time-index

Note: Only apply to system that is linear and time invariant

By-Sanjeev 16

Discrete Time State Space Models(2)

Function: Use ss function creates state spacemodels. For example:

>>A = [0 1;-5 -2];

>>B = [0;3];

>>C = [0 1];

>>D = [0];

>>Ts= [0.4];

>>sys=ss(A,B,C,D,Ts)

Transfer function:

2 z + 1

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

z^2 + 3 z + 2

Sampling time: 0.4

Matlab Output

x1 0 1x2 -5 -2

Matlab Output

x1 0x2 3

y1 0 1

Sampling time: 0.4

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Lecture Overview

Other Information

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Combining Models(1)

A model can be thought of as a block withinputs and outputs (block diagram) andcontaining a transfer function or a state-

space model inside itA symbol for the mathematical operations on

the input signal to the block that produces theoutput

TransferFunction

G(s) Input Output

Elements of a Block Diagram

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Combining Models(2)

The Following Matlab functions can be used toperform basic block diagram manipulation

Combination Matlab Command

sys = series(G1,G2)

sys = parallel(G1,G2)

sys = feedback(G1,G2)

G 1(s) G 2 (s)

+G 1(s)

G 2 (s)

+G 1(s) -

G 2 (s)

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Basic arithmetic operations of Models

Arithmetic Operations Matlab Code

Addition sys = G1+G2;

Multiplicationsys = G1*G2;

Inversionsys = inv(G1);

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Lecture Overview

Other Information

By-Sanjeev 22

Transient Response Analysis(1)

Transient response refers to the processgenerated in going from the initial state tothe final state

Transient responses are used toinvestigate the time domain characteristicsof dynamic systems

Common responses: step response,impulse response, and ramp response

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Unit step response of the transfer function system

Consider the system: 254

252 ss

%*****Numerator & Denominator of H(s)

num = [0 0 25];den = [1 4 25];

%*****Specify the computing time

t=0:0.1:7;

sys=tf(num,den)

step(sys,t)

%*****Add grid

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Unit step response of H(s)

Unit Step Response of H(s)

Time (sec)

A m p l i t u d e

0 1 2 3 4 5 6 70

By-Sanjeev 25

Alternative way to generate Unit step responseof the transfer function, H(s)

If step input is , then step response isgenerated with the following command:

>>num = [0 0 25];den = [1 4 25];%*****Create Model

>>H=tf(num,den);

>>step(H)

>>step(10*H)

)(10 t u

By-Sanjeev 26

Impulse response of the transfer function system

Consider the system: 254

num = [0 0 25];

den = [1 4 25];

%*****Specify the computing time

t=0:0.1:7;

Sys=tf(num,den)

impulse(sys,t)

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Impulse response of H(s)

Impulse Response of H(s)

Time (sec)

A m p l i t u d e

0 1 2 3 4 5 6 7-1

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Ramp response of the transfer function system

There‟s no ramp function in Matlab

To obtain ramp response of H(s), divide H(s) by

“s” and use step function

Consider the system:

For unit-ramp input, . Hence

1)( ssU

251222

ssssssssY

Indicate Step response

NEW H(s)

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Example: Matlab code for Unit Ramp Response

%*****Numerator & Denominator of NEW H(s)

num = [0 0 0 25];den = [1 4 25 0];%*****Specify the computing time

t=0:0.1:7;

sys=tf(num,den)

step(sys,t)

%*****Add grid & title of plot

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Unit Ramp response of H(s)

0 1 2 3 4 5 6 70

7Unit Ramp Response Curve of H(s)

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Lecture Overview

Other Information

By-Sanjeev 32

Frequency Response Analysis(1)

For Transient response analysis - hard todetermine accurate model (due to noise orlimited input signal size)

Alternative: Use frequency response approachto characterize how the system behaves in thefrequency domain

Can adjust the frequency responsecharacteristic of the system by tuning relevantparameters (design criteria) to obtain acceptabletransient response characteristics of the system

By-Sanjeev 33

Bode Diagram Representation of Frequency Response

Consists of two graphs:

Log-magnitude plot of the transfer function

Phase-angle plot (degree) of the transfer functionMatlab function is known as „bode‟

num = [0 0 25];den = [1 4 25];%*****Use „bode‟ function

sys=tf(num,den)

bode(sys)

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Example: Bode Diagram for

Bode plot of H(s)

Frequency (rad/sec)

P h a s e ( d e g )

M a g n i t u d e ( d B )

Bode magnitude plot

Bode phase plot

By-Sanjeev 35

Lecture Overview

Combining Models

Other Information

By-Sanjeev 36

Stability Analysis Based on FrequencyResponse(1)

Stability analysis can also be performedusing a Nyquist plot

From Nyquist plot – determine if system is

stable and also the degree of stability of asystem

Using the information to determine how

stability may be improvedStability is determined based on the

Nyquist Stability Criterion

By-Sanjeev 37

Example: Matlab code to draw a Nyquist Plot

Consider the system 18.0

num = [0 0 1];

den = [1 0.8 1];

%*****Draw Nyquist Plot

sys=tf(num,den)

nyquist(sys)

By-Sanjeev 38

The Nyquist Plot for

Nyquist plot of H(s)

Real Axis

I m a g i n a r y A x i s

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-20 dB

-10 dB

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Lecture Overview

Combining Models

Other Information

By-Sanjeev 40

Other Information

Use help to find out more about theMatlab functions shown in this lecture

Check out Control System Toolbox forother Matlab functions

By-Sanjeev 41

ROOT LOCUS PLOT

Example: Matlab code to draw a Root Locus PlotConsider the system

%*****Numerator & Denominator of H(s)num = [0 0 1];den = [1 0.8 1]; %*****Draw Root Locus Plot

sys=(num,den)rlocus(sys)grid

By-Sanjeev 42

Root locus plot

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1-4

0.0950.130.19

0.0180.040.0650.0950.130.19

0.0180.040.065

Root Locus

Real Ax is

I m a g i n a r y A x i s

By-Sanjeev 43

Procedure of Designing a Control System

System & Required Design Specifications Mathematical Model

Test the System

1. Fulfill the Required Design Specification ?• Transient Response Analysis• Frequency Response Analysis

2. How stable or robust ? Is your system stable?• Stability Analysis Based on Frequency Response

Are (1) & (2) satisfy?

Revisit the designe.g. Combine model?

By-Sanjeev 44

Transient response Specifications

Unit Step Response of G(s)

Time (sec)

A m p l i t u d

0 0.5 1 1.5 2 2.5 3

Peak Time

Rise Time

Steady State

Settling Time

Delay Time

Unit Step Response of G(s)

Time (sec)

A m p l i t u d

0 0.5 1 1.5 2 2.5 3

Peak Time

Rise Time

Steady State

Settling Time

Delay Time

By-Sanjeev 45

Frequency Domain Characteristics

What is the bandwidth of the system?

What is the cutoff frequencies?

What is the cutoff rate? Is the system sensitive to disturbance?

How the system behave in frequency domain?

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