State Space Models312

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STATE SPACE MODELS

MATLAB Tutorial


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Why State Space Models

The state space model represents a physical

system as nfirst order differentialequations. This form is better suited for

computer simulation than an nthorder input-

output differential equation.


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Basics

Vector matrix format generally is given by

!here yis the output equation" and xis the state vector


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PARTS OF A STATE SPACE

REPRESENTATION

State Vaia!les a subset of system variables !hich

if #no!n at an initial time t$along !ith subsequent

inputs are determined for all time t%t$&

State E"#atio$s n linearly independent first order

differential equations relating the first derivatives of

the state variables to functions of the state variables

and the inputs.

O#tp#t e"#atio$s algebraic equations relating the

state variables to the system outputs.


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E%AMPLE

The equation gathered from the free body diagram is

mx"& bx'& kx- f't( ) $

*ubstituting the definitions of the states into the

equation results in

mv'& bv& kx- f't( ) $

*olving forv' gives the state equation

v') '-b/m)v& '-k/m)x & f't(+m

The desired output is for the position"x" so

y)x


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Co$t&

,o! the derivatives of the state variables are

in terms of the state variables" the inputs"and constants.

x' = v

v'' ()k/m*x + ()b/m* v+ f(t*,m

y)x


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P-TTIN. INTO VECTOR)MATRI%

FORM

ur state vector consists of t!o variables"x

and vso our vector-matrix !ill be in the form


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Expla$atio$

The first ro! of Aand the first ro! of Bare

the coefficients of the first state equation forx'. Li#e!ise the second ro! of Aand the

second ro! of Bare the coefficients of the

second state equation for v'. Cand Dare the

coefficients of the output equation for y.


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E%ACT REPRESENTATION


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/OW TO INP-T T/E STATE

SPACE MODEL INTO MATLAB

n order to enter a state space model into MATLAB"

enter the coefficient matrices A" B" C" and Dinto

MATLAB. The syntax for defining a state space

model in MATLAB is

statespace ) ss'A" B" /" 0(

!here A" B" C" and Dare from the standard vector-

matrix form of a state space model.


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Exa0ple

1or the sa#e of example" lets ta#e m ) 2" b ) 3" and # ) 4.

%% m ) 25

%% b ) 35

%% # ) 45

%% A ) 6 $ 7 5 -#+m -b+m 85

%% B ) 6 $ 5 7+m 85

%% / ) 6 7 $ 85 %% 0 ) $5

%% statespace9ss ) ss'A" B" /" 0(


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O#tp#t

This assigns the state space model under

the name statespace9ss and output thefollo!ing

a =

x1 x2

x1 0 1 x2 -1.5 -2.5


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Co$t&

b =

u1

x1 0 x2 0.5

c =

x1 x2

y1 1 0


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Co$t&

d =

u1 y1 0

Continuous-time mode.


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E%TRACTIN. A1 B1 C1 DMATRICES FROM A STATE SPACEMODEL n order to extract the A" B" /" and 0

matrices from a previously defined statespace model" use MATLAB:s ssdata

command.

6A" B" /" 08 ) ssdata'statespace(

!here statespace is the name of the statespace system.


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Exa0ple

%% 6A" B" /" 08 ) ssdata'statespace9ss(

The MATLAB output !ill be! =

-2.5000 -0.#50 $.0000 0


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Co$t&

% =

0.2500 0

C =

0 0.5000

& =

0


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STEP RESPONSE -SIN. T/E

STATE SPACE MODEL

nce the state space model is entered into MATLAB it is easy

to calculate the response to a step input. To calculate the

response to a unit step input" use step'statespace(

!here statespace is the name of the state space system.

1or steps !ith magnitude other than one" calculate the step

response using

step'u ; statespace(

!here u is the magnitude of the step and statespace is the

name of the state space system.


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State Space Models312

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