State-space models 9 use of MATLAB

J A Rossiter

1

Slides by Anthony Rossiter

Introduction

1. The previous videos have demonstrated numerous mechanisms for creating state space models to represent systems.

2. Matrix handling is not, in general, a paper and pen exercise, so this video gives a number of worked examples using the MATLAB tool for the number crunching.

3. Code is in the file statespace9.m

Slides by Anthony Rossiter

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CxyBuAxx ;

STATE SPACE OBJECTS IN MATLAB

Slides by Anthony Rossiter

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ss.m

MATLAB has a special format for storing state space systems.

Once in this format, the user can deploy a number of other MATLAB functions for system analysis and control design.

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step.m impulse.m

pzmap.m feedback.m

bode.m nyquist.m

AND MORE!

See Use of MATLAB chapter for illustrations

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Examples using MATLAB real time

A=[1 2;3 2];

B=[1;0]; C=[1 -1];

D=0; G=ss(A,B,C,D)

A=[0 1 2;-1 3 0;0 2 0];

B=[1;0;-2]; C=[1 -1 0];

D=0; G=ss(A,B,C,D)

Slides by Anthony Rossiter

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A=[-1 0.1;0.2 -0.5]; B=[1;2]; C=[1 0];

D=0; G=ss(A,B,C,D)

step(G)

pzmap(G)

Gc=feedback(G,1)

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-1

-0.5

0

0.5

1Pole-Zero Map

Real Axis (seconds-1)

Imagin

ary

Axis

(seconds

-1)

0 5 10 15 200

0.5

1

1.5Step Response

Time (seconds)

Am

plit

ude

TRANSFER FUNCTION TO STATE SPACE

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)(

)(

)()( sU

sd

snsY

Cxy

BuAxx

Slides by Anthony Rossiter

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Examples transfer function

parameters to state space parameters

n=[1 2];

d=[1 4 -2]; [A,B,C,D]=tf2ss(n,d)

n=[-9 0 2];

d=[1 3 2 -2]; [A,B,C,D]=tf2ss(n,d)

STATE SPACE TO TRANSFER FUNCTION NOTE: THIS IS DONE FOR SISO ONLY AS ONE NEEDS MORE CARE FOR MIM0 PROCESSES AND THAT DETAIL IS BEYOND THIS VIDEO SERIES

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)(

)(

)()( sU

sd

snsY

Cxy

BuAxx

Slides by Anthony Rossiter

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Examples state space to transfer

function

A=[1 2;3 2]; B=[1;0];C=[1 -1]; D=0; [n,d]=ss2tf(A,B,C,D,1)

A=[0 1 2;-1 3 0;0 2 0];

B=[1;0;-2];C=[1 -1 0]; D=0; [n,d]=ss2tf(A,B,C,D,1)

SIMILARITY TRANSFORM: INVARIANCE OF BEHAVIOUR AND EIGENVALUES

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Txz

zCTy

uTBzTATz

Cxy

BuAxx

C

BA

ˆ

1

ˆˆ

1

CCTBTBATAT ˆ;ˆ;ˆ 11

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Similarity transform and

behaviour A=[-1 0 0.2;-1 -3 0;0 0.1 -0.5];

B=[1;0;-2];C=[1 -1 0]; D=0; T=[2 1 1;0 1 -1; -3 0 2];

Ti=inv(T); Ahat=T*A*Ti;

Bhat=T*B;Chat=C*Ti; G=ss(A,B,C,D);

Ghat=ss(Ahat,Bhat,Chat,D); [y,t]=step(G);

[yhat]=step(Ghat,t); plot(t,y,'r',t,yhat,'o');

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

DIAGONAL CANONICAL FORM

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zCWyuBWzAWWzCBA ˆˆ

1

ˆ

1 ;

1;; WWACxyBuAxx

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Examples eigenvalue/

vector decomposition

A=[-1 0 0.2;-1 -3 0;0 0.1 -0.5];

[W,lambda]=eig(A); Ahat=inv(W)*A*W

lambda

USING MATLAB TO FIND TRANSFORMATIONS WHICH GIVE CANONICAL FORMS

The key command is canon.m

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For transformation z = Tx to be

determined.

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For transformation z = Tx to be

determined.

Companion form where the characteristic

polynomial appears in the rightmost column.

Summary

Demonstrated the use of MATLAB for definition and analysis of state space systems.

Easy to plot behaviours, form closed-loop systems, find poles, do state transformations, etc.

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© 2016 University of Sheffield This work is licensed under the Creative Commons Attribution 2.0 UK: England & Wales Licence. To view a copy of this licence, visit http://creativecommons.org/licenses/by/2.0/uk/ or send a letter to: Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. It should be noted that some of the materials contained within this resource are subject to third party rights and any copyright notices must remain with these materials in the event of reuse or repurposing. If there are third party images within the resource please do not remove or alter any of the copyright notices or website details shown below the image. (Please list details of the third party rights contained within this work. If you include your institutions logo on the cover please include reference to the fact that it is a trade mark and all copyright in that image is reserved.)

Anthony Rossiter Department of Automatic Control and

Systems Engineering University of Sheffield www.shef.ac.uk/acse