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Chapter 3: Dynamic Response. Part C: Transient-response analysis with MATLAB. Introduction. The practical procedure for plotting time response curves of systems higher than second-order is through computer simulation .

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Chapter 3 dynamic response

Chapter 3: Dynamic Response

Part C: Transient-response analysis with MATLAB.


Introduction
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.


Representation of a linear system
Representation of a Linear System

A linear system can be representedeither:

  • 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


Example 1 standard state variable form
Example 1: Standard State-Variable Form

  • Consider a linear system described by:


Unit step response for a control system defined in state variable form

F = [0 1;0 -0.05];

G = [0;0.001];

H = [0 1];

J = 0;

step(F,G,H,J)

% defines state variable matrices

% generates plot of unit-step response (with Time (sec)and Amplitudelabels on x- and y-axis respectively, andStep responsetitle )

Note: time vector is automatically determined when t is not explicitly included in the step command.

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


50 step response for a system defined in state variable form

F = [0 1;0 -0.05];

G = [0;0.001];

H = [0 1];

J = 0;

sys = ss(F, 50*G, 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

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


Unit step response on specific time interval for a system defined in state variable form

F = [0 1;0 -0.05];

G = [0;0.001];

H = [0 1];

J = 0;

sys = ss(F, G, H, J);

t = 0:0.2:100;

y=step(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)

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


Impulse response for a system defined in state variable form

F = [0 1;0 -0.05];

G = [0;0.001];

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)

Impulse Response for a system defined in State-Variable form


Example 2 initial conditions
Example 2: Initial Conditions

  • Consider a linear system such as:

  • In state-variable form, it is described by:


Initial condition response for a system defined in state variable form

F = [0 1;-10 -5];

G = [0;0];

H = [1 0];

J = 0;

t = 0:0.5:3;

y=initial(F,G,H,J,[2;1],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 [ ].

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


Example 3 transfer function in numerator denominator form
Example 3: Transfer function in numerator-denominator form

  • Consider a linear system whose the transfer function is:


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)

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


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)

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


Unit step responses for system transfer functions defined by

t = 0:0.2:10;

zeta = [0 0.2 0.4 0.6 0.8 1];

for n = 1:6;

num = [0 0 1];

den = [1 2*zeta(n) 1];

[y(1:51,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)

Unit-Step Responses for system Transfer Functions defined by


Unit step responses for system transfer functions defined by1

t = 0:0.2:10;

zeta = [0 0.2 0.4 0.6 0.8 1];

for n = 1:6;

num = [0 0 1];

den = [1 2*zeta(n) 1];

[y(1:51,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)

Unit-Step Responses for system Transfer Functions defined by


Unit step response for a 3 rd 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)

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


Impulse response for a system transfer function defined in num den polynomial form

num = [0 0 1];

den = [1 0.2 1];

sys=tf(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)

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


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)

Alternative approach to obtain Impulse Response


Example 4 transfer function in standard 2 nd order system
Example 4: Transfer function in standard 2nd order system

  • Consider a standard second order system:

natural

undamped

frequency

damping

ratio


Matlab description of standard second order system

w0 = 5;

damping_ratio = 0.4;

[num0,den] = ord2(w0,damping_ratio);

num = 5^2*num0;

printsys(num,den,’s’)

% defines natural undamped frequency

% defines damping ratio

% defines numerator

% prints num/den as a ratio of s-polynomials

num/den =

MATLAB Description of Standard Second Order System


Example 5 transfer function in pole zero form
Example 5: Transfer function in pole-zero form

  • Consider a linear system whose the transfer function is:


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

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


Example 6 transfer function in partial expansion form
Example 6: Transfer function in Partial Expansion Form

  • Consider a linear system whose the transfer function is:


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’)

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


Convertion

State-variable form

Transfer function:

In num-den polynomial form

In zero-pole form

In partial expansion form

Convertion

[num,den] = ss2tf(F,G,H,J)

[z,p,k]=tf2zp(num,den)

[r,p,K]=residue(num,den)

[z,p,k] = ss2zp(F,G,H,J)


Convertion1

State-variable form

Transfer function:

In num-den polynomial form

In zero-pole form

In partial expansion form

Convertion

[F,G,H,J] = tf2ss(num,den)

[num,den]=zp2tf(z,p,k)

[num,den]=residue(r,p,K)

[F,G,H,J] = zp2ss(z,p,k)



Title grid labels on the graphical screen

title (‘Step-response’);

grid;

sys = …;

t = 0:0.2:100;

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.

Title, Grid & Labels on the graphical screen


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 x=3.4, y=-0.06.

% writes  at x=4.1, y=1.86

% waits until the cursor is positioned (using the mouse) at the desired position in the screen and then writes on the plot at the cursor’s location the text enclosed in simple quotes.

Note: any number of gtext command can be used in a plot.

Writing Text on the Graphical Screen


Use of symbols in graph

num = [0 0 25];

den = [1 6 25];

t = 0:0.5:5;

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.

Use of Symbols in graph


Use of symbols in graph cont d

num = [0 0 25];

den = [1 6 25];

t = 0:0.5:5;

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

Use of Symbols in graph (cont’d)


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

Additional Convenient MATLAB Commands


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

Computing Roots


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

Stability Analysis by Computing Roots

2 poles are in the RHP


Plotting poles and zero in the s domain

num=[0 2 1];

den= [2 3 2];

zmap(num,den)

Plotting Poles and Zero in the s-domain

poles as crosses

zero as circle


Step response versus a parameter range

xlabel('Time (sec)');

ylabel('Amplitude');

Title('Step-Response

versus K parameter');

grid;

text(7.1,3.8,'K=6.5');

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');

t=0:0.1:10;

K=[6.5 7 7.5 8 10 12.5];

for n=1:6

num=[K(n) K(n)];

den=[1 5 K(n)-6 K(n)];

[y(1:101,n),x,t]=step(num,den,t);

end

plot(t,y);

Step-Response versus a Parameter Range


Reminder rise time
Reminder: Rise Time

  • The rise time is the time requiredfor the responseto rise from 0% to 100% of its final value.

1

t

d

0.5

0

t

r

Note: for overdamped systems, the 10%

to 90% rise time is commonly used.


Computing rise time using m atlab

num= [0 0 25];

den=[1 6 25];

t=0:0.001:5;

[y,x,t]=step(num,den,t);

r=1;

while y(r) <1.0001;

r=r+1;

end;

rise_time=(r-1)*0.001

rise_time =

0.5540

Computing Rise Time using MATLAB

No ;


Reminder peak time
Reminder: Peak Time

  • The peak time is the time requiredfor the responseto reach the first peak of the overshoot.

t

p

1

t

d

0.5

0

t

r


Computing peak time using matlab

num= [0 0 25];

den=[1 6 25];

t=0:0.001:5;

[y,x,t]=step(num,den,t);

[ymax,tp]=max(y);

peak_time=(tp-1)*0.001

peak_time =

0.7850

Computing Peak Time using MATLAB

No ;


Reminder maximum overshoot
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.


Computing maximum overshoot using matlab

num= [0 0 25];

den=[1 6 25];

t=0:0.001:5;

[y,x,t]=step(num,den,t);

[ymax,tp]=max(y);

peak_time=(tp-1)*0.001

max_overshoot=ymax-1

peak_time =

0.7850

max_overshoot =

0.0948

Computing Maximum Overshoot using MATLAB


Reminder settling time
Reminder: Settling Time

  • The settling time is the time required for the response curve to reach and stay within a range about1% 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


Computing settling time using matlab based on 2

num= [0 0 25];

den=[1 6 25];

t=0:0.001:5;

[y,x,t]=step(num,den,t);

s=5001;

while y(s)>0.98 & y(s)<1.02;

s=s-1;

end;

settling_time=(s-1)*0.001

settling_time =

1.1880

Computing Settling Time using MATLAB(based on +/-2%)


Chapter 3 dynamic response

Results given by MATLAB are:

rise_time = 0.5540; peak_time = 0.7850,

max_overshoot = 0.0948, settling_time = 1.1880