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Chp. 3State Variable Models

• Dynamical Systems

• State Space Representation

• Transfer Function

• Realization

• Examples


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Mathematical relations betweenobserved quantities

SystemINPUT OUTPUT


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SystemINPUT OUTPUT

System

INPUT OUTPUT

Controller


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Controller Plant


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• OPERATING POINT, LINEARIZATION


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A=[0 -3;1 -2]

A =

0 -31 -2

>> B=[2;0]

B =

20

>> C=[0 3];D=0;

sys1=ss(A,B,C,D);>> t=0:0.01:10;

>> u1=zeros(1,1001);>> u1(1:end)=1;>> u2=sin(t);>> u3=sin(10*t);>> u4=0*t;>> x0=[1 1];

>> [y,T,x]=lsim(sys1,u1,t,x0);>> plot(T,x(:,1)grid>> plot(T,x(:,2))>> grid

unit step

sinusoidal

zero inputinitial condition

g=tf(sys1)

Transfer function:6

-------------

s^2 + 2 s + 3


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g=tf([2 8 6],[1 8 16 6])

Transfer function:2 s^2 + 8 s + 6

----------------------s^3 + 8 s^2 + 16 s + 6

>> sys1=ss(g)

a =x1 x2 x3

x1 -8 -4 -1.5x2 4 0 0x3 0 1 0

b =u1

x1 2x2 0x3 0

c =x1 x2 x3

y1 1 1 0.75

d = u1y1 0

Continuous-time model.

t=0:0.01:10;

u1=zeros(1,1001);u1(1:end)=1;u2=sin(t);u3=sin(10*t);u4=0*t;x0=[1 1 0];

[y,T,x]=lsim(sys1,u1,t,x0);plot(T,x(:,1))>> [y,T,x]=lsim(sys1,u3,t,x0);>> plot(T,x(:,1))>> xlabel('t (sec)')>> ylabel('x_1')

>> grid


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