Chapter 3: Dynamic Response

1
Chapter 3 Dynamic Response

• Part C Transient-response analysis with MATLAB.

2
Introduction

• The practical procedure for plotting time
  response curves of systems higher than
  second-order is through computer simulation.
• In this part, computational approach to the
  transient-response analysis with MATLAB is
  presented through various examples.

3
Representation of a Linear System

• A linear system can be represented either
• In state-variable form
• with the values of the matrices F, G, H and the
  constant J.
• Or
• By its transfer function
• Either in numerator-denominator polynomial form,
• Or in pole-zero form
• Or in partial expansion form

4
Example 1 Standard State-Variable Form

• Consider a linear system described by

5
Unit-Step Response for a Control System defined
in State-Variable form

• defines state variable matrices
• generates plot of unit-step response (with Time
  (sec) and Amplitude labels on x- and y-axis
  respectively, and Step response title )
• Note time vector is automatically determined
  when t is not explicitly included in the step
  command.

• F 0 10 -0.05
• G 00.001
• H 0 1
• J 0
• step(F,G,H,J)

6
50-Step Response for a system defined in
State-Variable form

• F 0 10 -0.05
• G 00.001
• H 0 1
• J 0
• sys ss(F, 50G, H, J)
• step(sys)

• defines state variable matrices
• defines system by its state-space
• matrices
• generates plot of 50-step response vs t
• Note State-variable form is also called
  state-space form

7
Unit-Step Response on specific time interval for
a system defined in State-Variable form

• F 0 10 -0.05
• G 00.001
• H 0 1
• J 0
• sys ss(F, G, H, J)
• t 00.2100
• ystep(sys,t)
• plot(t,y)

• defines state variable matrices
• defines system by its state-space
• matrices
• setup time vector ( dt 0.2 sec)
• plots unit step response versus time ranging
  from 0 to 100 sec (with x- and y-labels)

8
Impulse Response for a system defined in
State-Variable form

• F 0 10 -0.05
• G 00.001
• H 0 1
• J 0
• sys ss(F, G, H, J)
• impulse(sys)

• defines state variable matrices
• defines system by its state-space
• matrices
• generates plot of impulse response
• (with labels title)
• Note an alternative use of impulse command is
• impulse(F,G,H,J)

9
Example 2 Initial Conditions

• Consider a linear system such as
• In state-variable form, it is described by

10
Initial Condition Response for a system defined
in State-Variable form

• F 0 1-10 -5
• G 00
• H 1 0
• J 0
• t 00.53
• yinitial(F,G,H,J,21,t)
• plot(t,y)

• defines state variable matrices
• set up time vector
• computes initial condition response
• generates plot of response
• Note Initial conditions are defined between
  .

11
Example 3 Transfer function in
numerator-denominator form

• Consider a linear system whose the transfer
  function is

12
Unit-Step Response for a system Transfer Function
defined in num/den polynomial form

• num 0 0 25
• den 1 4 25
• step(num,den)

• defines numerator
• defines denominator
• generates plot of unit-step response (with
  labels and title)

13
50-Step Response for a system Transfer Function
defined in num/den polynomial form

• num 0 0 50
• den 1 0.2 1
• step(num,den)

• defines numerator
• defines denominator
• generates plot of 50-step response (with labels
  and title)

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Unit-Step Responses for system Transfer Functions
defined by

• t 00.210
• zeta 0 0.2 0.4 0.6 0.8 1
• for n 16
• num 0 0 1
• den 1 2zeta(n) 1
• y(151,n),x, t step(num,den,t)
• end
• plot(t,y)

• setup time vector
• defines zeta,
• numerator and
• denominator
• generates 2-D plot of the n unit-step responses
  (on same graph)

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Unit-Step Responses for system Transfer Functions
defined by

• t 00.210
• zeta 0 0.2 0.4 0.6 0.8 1
• for n 16
• num 0 0 1
• den 1 2zeta(n) 1
• y(151,n),x, t step(num,den,t)
• end
• mesh(t,zeta,y)

• setup time vector
• defines zeta,
• numerator and
• denominator
• generates 3-D plot of the n unit-step responses
  (on same graph)

16
Unit Step Response for a 3rd order system defined
by its Transfer Function

• num 0 0 0 1
• den 1 1 1 0
• step(num,den)

• defines numerator
• defines denominator
• generates plot of unit-step response (with x-
  and y-labels)

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Impulse Response for a system Transfer Function
defined in num/den polynomial form

• num 0 0 1
• den 1 0.2 1
• systf(num,den)
• impulse(sys)

• defines numerator
• defines denominator
• defines system by its transfer function
• generates plot of impulse response
• Note an alternative use of impulse command is
• impulse(num,den)

18
Alternative approach to obtain Impulse Response

• num 0 1 0
• den 1 0.2 1
• step(num,den)

• defines numerator of sG(s)
• defines denominator
• generates plot of impulse response (with x- and
  y-labels)

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Example 4 Transfer function in standard 2nd
order system

• Consider a standard second order system

natural undamped frequency
damping ratio
20
MATLAB Description of Standard Second Order System

• w0 5
• damping_ratio 0.4
• num0,den ord2(w0,damping_ratio)
• num 52num0
• printsys(num,den,s)

• defines natural undamped frequency
• defines damping ratio
• defines numerator
• prints num/den as a ratio of s-polynomials
• num/den

21
Example 5 Transfer function in pole-zero form

• Consider a linear system whose the transfer
  function is

22
Unit-Step Response for a system Transfer Function
defined in pole-zero form

• num conv(1 2,1 4)
• den conv(1 1 0,1 3)
• step(num,den)

• defines zero ratios
• defines pole ratios
• plots unit-step response

23
Example 6 Transfer function in Partial Expansion
Form

• Consider a linear system whose the transfer
  function is

24
Unit-Step Response for a system Transfer Function
defined in partial expansion form

• r 8/3 -3/2 -1/6
• p 0 -1 -3
• K
• num,den residue(r,p,K)
• step(num,den)

• defines residues
• defines poles
• define additive constant
• convert partial expansion form to polynomial
  form
• plots unit-step response
• Note to see ratio use
• printsys(num,den,s)

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Convertion

• Transfer function
• In num-den polynomial form
• In zero-pole form
• In partial expansion form

• State-variable form

num,den ss2tf(F,G,H,J)
z,p,ktf2zp(num,den)
r,p,Kresidue(num,den)
z,p,k ss2zp(F,G,H,J)
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Convertion

• Transfer function
• In num-den polynomial form
• In zero-pole form
• In partial expansion form

• State-variable form

F,G,H,J tf2ss(num,den)
num,denzp2tf(z,p,k)
num,denresidue(r,p,K)
F,G,H,J zp2ss(z,p,k)
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Cosmetic

• Title, grid, labels, text on graphical screen,
• symbols,

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Title, Grid Labels on the graphical screen

• title (Step-response)
• grid
• sys
• t 00.2100
• y step(sys,t)
• plot (t,y)
• xlabel(t (sec))
• ylabel(response)

• writes the title Step-response
• draws a grid between ticks
• defines system by
• setup time vector ( dt 0.2 sec)
• computes step response
• plots step response
• writes label t (sec) on x-axis.
• writes label response on y-axis.

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Writing Text on the Graphical Screen

• text(3.4, -0.06, Y111)
• text(4.1,1.86,\zeta)
• gtext(blabla)

• writes Y111 beginning at the coordinates x3.4,
  y-0.06.
• writes ? at x4.1, y1.86
• waits until the cursor is positioned (using the
  mouse) at the desired position in the screen and
  then writes on the plot at the cursors location
  the text enclosed in simple quotes.
• Note any number of gtext command can be used in
  a plot.

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Use of Symbols in graph

• num 0 0 25
• den 1 6 25
• t 00.55
• y step(num,den,t)
• plot(t,y,o,t,1,-)

• defines numerator
• defines denominator
• defines time vector
• computes unit-step response
• plot of unit step response y and unit step
  input 1 using oooo and ---- symbols respectively.
  

31
Use of Symbols in graph (contd)

• num 0 0 25
• den 1 6 25
• t 00.55
• y step(num,den,t)
• plot(t,y,x,t,y,-)

• defines numerator
• defines denominator
• defines time vector
• computes step response
• plot of unit step response y using -x-x-x-x-
  symbols

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Additional Convenient MATLAB Commands

• Computing roots using MATLAB
• Plotting pole(s) and zero(s) in the s-plane using
  MATLAB
• Plotting Step-response versus a parameter range
• Obtaining rise time, peak time, maximum overshoot
  and settling time using MATLAB

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Computing Roots

• pol 1 4 3 2 1 4 4
• roots(pol)

• ans
• -3.2644
• -0.6046 0.9935i
• -0.6046 - 0.9935i
• 0.6797 0.7488i
• 0.6797 - 0.7488i
• -0.8858

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Stability Analysis by Computing Roots

• den 1 5 11 23 28 12
• roots(den)

• ans
• -3.0000
• 0.0000 2.0000i
• 0.0000 - 2.0000i
• -1.0000 0.0000i
• -1.0000 - 0.0000i

2 poles are in the RHP
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Plotting Poles and Zero in the s-domain
poles as crosses

• num0 2 1
• den 2 3 2
• zmap(num,den)

zero as circle
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Step-Response versus a Parameter Range

• xlabel('Time (sec)')
• ylabel('Amplitude')
• Title('Step-Response
• versus K parameter')
• grid
• text(7.1,3.8,'K6.5')
• text(7.5,3.15,'7')
• text(7.15,2.65,'7.5')
• text(7.1,2.3,'8')
• text(6.65,1.37,'10')
• text(6.4,0.75,'12.5')

• t00.110
• K6.5 7 7.5 8 10 12.5
• for n16
• numK(n) K(n)
• den1 5 K(n)-6 K(n)
• y(1101,n),x,tstep(num,den,t)
• end
• plot(t,y)

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Reminder Rise Time

• The rise time is the time required for the
  response to rise from 0 to 100 of its final
  value.

1
t
d
0.5
0
t
Note for overdamped systems, the 10 to 90
rise time is commonly used.
r
40
Computing Rise Time using MATLAB

• num 0 0 25
• den1 6 25
• t00.0015
• y,x,tstep(num,den,t)
• r1
• while y(r) lt1.0001
• rr1
• end
• rise_time(r-1)0.001

• rise_time
• 0.5540

No
41
Reminder Peak Time

• The peak time is the time required for the
  response to reach the first peak of the overshoot.

t
p
1
t
d
0.5
0
t
r
42
Computing Peak Time using MATLAB

• num 0 0 25
• den1 6 25
• t00.0015
• y,x,tstep(num,den,t)
• ymax,tpmax(y)
• peak_time(tp-1)0.001

• peak_time
• 0.7850

No
43
Reminder Maximum Overshoot

• The maximum overshoot is the relative maximum
  peak value of the response curve measured from
  the final value.

t
p
M
p
1
t
d
0.5
0
t
r
Note the maximum overshoot directly indicates
the relative stability of the system.
44
Computing Maximum Overshoot using MATLAB

• num 0 0 25
• den1 6 25
• t00.0015
• y,x,tstep(num,den,t)
• ymax,tpmax(y)
• peak_time(tp-1)0.001
• max_overshootymax-1

• peak_time
• 0.7850
• max_overshoot
• 0.0948

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Reminder Settling Time

• The settling time is the time required for the
  response curve to reach and stay within a range
  about 1 or 2 of the final steady-state value.

t
p
M
p
1
1
t
d
0.5
0
t
t
r
s
Note t is the time it takes the system
transients to decay.
s
46
Computing Settling Time using MATLAB(based on
/-2)

• num 0 0 25
• den1 6 25
• t00.0015
• y,x,tstep(num,den,t)
• s5001
• while y(s)gt0.98 y(s)lt1.02
• ss-1
• end
• settling_time(s-1)0.001

• settling_time
• 1.1880

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• Results given by MATLAB are
• rise_time 0.5540 peak_time
  0.7850,
• max_overshoot 0.0948,
  settling_time 1.1880