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MATHEMATICAL
MODELLING& VARIOUSCONTROL SYSTEM MODELS &RESPONSES USING MATLAB
By-Sanjeev
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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
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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.
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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
12
)( 2
ss
s
s H
>>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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Continuous Time Transfer Function(2)
Include delay to continuous time Transfer Function
Example23
12)(
2
2
ss
ses H
s
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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Continuous Time Transfer Function(3)
Function: Use zpk function to create transferfunction of following form:
Example 21
5.02
23
12)(
2
ss
s
ss
ss H
>>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
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Continuous Time State Space Models(1)
State Space Model for dynamic system
DuCxy
BuAxx
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:
0103
0
25
10
2
1
DCBAx x
x
>>A = [0 1;-5 -2];
>>B = [0;3];>>C = [0 1];
>>D = [0];
>>sys=ss(A,B,C,D)
a =
x1 x2x1 0 1
x2 -5 -2
Matlab Output
b =
u1x1 0
x2 3
c =
x1 x2
y1 0 1
d =
u1
y1 0
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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
Building Models for LTI System
Continuous Time Models
Discrete Time Models
Combining ModelsTransient Response Analysis
Frequency Response Analysis
Stability Analysis Based on FrequencyResponse
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
12
)( 2
z z
z z H
>>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
5.0
2)(
z z
z z H
>>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[
nnn
nnn
DuCxy
BuAxx
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
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Discrete Time State Space Models(2)
Function: Use ss function creates state spacemodels. For example:
010
3
0
25
10
][
][][
2
1
DCBAx
n x
n xn
>>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
a =
x1 x2
x1 0 1x2 -5 -2
Matlab Output
b =
u1
x1 0x2 3
c =
x1 x2
y1 0 1
d =
u1
y1 0
Sampling time: 0.4
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Lecture Overview
Building Models for LTI System
Continuous Time Models
Discrete Time Models
Combining ModelsTransient Response Analysis
Frequency Response Analysis
Stability Analysis Based on FrequencyResponse
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
Building Models for LTI System
Continuous Time Models
Discrete Time Models
Combining ModelsTransient Response Analysis
Frequency Response Analysis
Stability Analysis Based on FrequencyResponse
Other Information
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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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Transient Response Analysis(2)
Unit step response of the transfer function system
Consider the system: 254
252 ss
s H
%*****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
grid
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Transient Response Analysis(3)
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
0.2
0.4
0.6
0.8
1
1.2
1.4
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Transient Response Analysis(4)
Alternative way to generate Unit step responseof the transfer function, H(s)
If step input is , then step response isgenerated with the following command:
%*****Numerator & Denominator of H(s)
>>num = [0 0 25];den = [1 4 25];%*****Create Model
>>H=tf(num,den);
>>step(H)
>>step(10*H)
)(10 t u
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Transient Response Analysis(5)
Impulse response of the transfer function system
Consider the system: 254
252
ss
s H
%*****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)
impulse(sys,t)
grid
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Transient Response Analysis(6)
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
-0.5
0
0.5
1
1.5
2
2.5
3
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Transient Response Analysis(7)
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
254
252
ss
s H
2
1)( ssU
254
251
254
251222
ssssssssY
Indicate Step response
NEW H(s)
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Transient Response Analysis(8)
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
grid
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Transient Response Analysis(9)
Unit Ramp response of H(s)
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7Unit Ramp Response Curve of H(s)
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Lecture Overview
Building Models for LTI System
Continuous Time Models
Discrete Time Models
Combining ModelsTransient Response Analysis
Frequency Response Analysis
Stability Analysis Based on FrequencyResponse
Other Information
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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
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Frequency Response Analysis(2)
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‟
%*****Numerator & Denominator of H(s)
num = [0 0 25];den = [1 4 25];%*****Use „bode‟ function
sys=tf(num,den)
bode(sys)
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Frequency Response Analysis(3)
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 )
-60
-50
-40
-30
-20
-10
0
10
20
100
101
102
-180
-135
-90
-45
0
254
252
ss
s H
Bode magnitude plot
Bode phase plot
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Lecture Overview
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
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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
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Stability Analysis Based on FrequencyResponse(2)
Example: Matlab code to draw a Nyquist Plot
Consider the system 18.0
12
ss
s H
%*****Numerator & Denominator of H(s)
num = [0 0 1];
den = [1 0.8 1];
%*****Draw Nyquist Plot
sys=tf(num,den)
nyquist(sys)
grid
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Stability Analysis Based on FrequencyResponse(2)
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
-1
-0.5
0
0.5
1
0 dB
-20 dB
-10 dB
-6 dB
-4 dB
-2 dB
20 dB
10 dB
6 dB
4 dB
2 dB
18.0
12
ss
s H
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Lecture Overview
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
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Other Information
Use help to find out more about theMatlab functions shown in this lecture
Check out Control System Toolbox forother Matlab functions
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ROOT LOCUS PLOT
Example: Matlab code to draw a Root Locus PlotConsider the system
18.0
1
2
sss H
%*****Numerator & Denominator of H(s)num = [0 0 1];den = [1 0.8 1]; %*****Draw Root Locus Plot
sys=(num,den)rlocus(sys)grid
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Root locus plot
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1-4
-3
-2
-1
0
1
2
3
4
0.0950.130.19
0.3
0.55
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
0.0180.040.0650.0950.130.19
0.3
0.55
0.0180.040.065
Root Locus
Real Ax is
I m a g i n a r y A x i s
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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?
end
YES
Revisit the designe.g. Combine model?
NO
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Transient response Specifications
Unit Step Response of G(s)
Time (sec)
A m p l i t u d
e
0 0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1
1.2
1.4
Peak Time
Rise Time
Steady State
Settling Time
0.1
0.5
Delay Time
Mp
Unit Step Response of G(s)
Time (sec)
A m p l i t u d
e
0 0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1
1.2
1.4
Peak Time
Rise Time
Steady State
Settling Time
0.1
0.5
Delay Time
Mp
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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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