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Laboratory in Automatic Control

Lab 3

Analysis of State Variable Models

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Magic square (1/8) Syntax

M = magic(n)

B = sum(M) Matlab code

M = magic(4)

sum(M)

sum(M')

Result

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Magic square (2/8) Syntax

X = diag(v)

Matlab code

M = magic(4)

X = diag(M)

sum(diag(M))

Result

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Magic square (3/8) Subscript

A(i,j)

Matlab code

M = magic(4);

[M(1,1),M(1,2),M(1,3),M(1,4)]

M(1,1)+M(1,2)+M(1,3)+M(1,4)

plot(M)

Result

1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

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Magic square (4/8) Syntax (the colon

operator)

: Matlab code

1:10

100:-7:50

0:pi/4:pi

Result

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Magic square (5/8) Matlab code

M = magic(4)

M(1:3,4)

M(2:4,2:3)

Result

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Magic square (6/8) Matlab code

M = magic(4)

M(:,1:2:4)

M(:,2:3:4)

Result

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Magic square (7/8) Matlab code

M = magic(4)

M(:,:)

M(:,2)

Result

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Magic square (8/8) Matlab code

M = magic(4)

[M(:,:),M(:,2)]

N = M(:,[4 1 2 3])

Result

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Impulse signal Syntax

impulse(sys)

impulse(sys,t)

impulse(sys1,sys2,...,sysN)

impulse(sys1,sys2,...,sysN,t)

impulse(sys1,'PlotStyle1',...,sysN,'PlotStyleN')

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State-space models (1/10) Syntax

sys =

ss(transfer_function) [A,B,C,D] = tf2ss(b,a)

Matlab code

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

sys = ss(H)%[a,b,c,d]=tf2ss([2 8 6],[1 8 16 6])

Example

2

3 2( ) 2 8 6( ) 8 16 6

Y s ss

U s ss s

x( ) = Ax( ) + Bu( )

y( ) = Cx( ) + Du( )

t t t

t t t

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State-space models (2/10)In Matlab 7.4

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State-space models (3/10)2

3 2

( ) 2 8 6

( ) 8 16 6

Y s ss

U s ss s

1

2

3

8 4 1.5 2

x 4 0 0 x 0 ( )

0 1 0 0

( ) 1 1 0.75 0 ( )

u t

x

y t u t x

x

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State-space models (4/10)

1

2

3

8 4 1.5 2

x 4 0 0 x 0 ( )

0 1 0 0

( ) 1 1 0.75 0 ( )

u t

x

y t u t x

x

( )?

( )

Y s

U s

Example

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State-space models (5/10) Syntax

sys = tf(ss(a,b,c,d))

[nums,dens]= ss2tf(a,b,c,d) Matlab code

a=[-8 -4 -1.5;4 0 0;0 1 0];b=[2;0;0];

c=[1 1 0.75];d=[0];sys = tf(ss(a,b,c,d))%[nums,dens]= ss2tf(a,b,c,d)

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State-space models (6/10)

Syntax

Y = expm(X)

Matlab code

a=[0 -2;1 -3];t=0.2;Phi=expm(a*t)

0 2A , 0.2

1 3

( ) exp(A )

( ( ) : fundamental or

state transition matrix)

t

t t

t

Example

Result

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State-space models (7/10)

Syntax

lsim(sys,u,t)

lsim(sys,u,t,x0)

[y,t,x] = lsim(sys,u,t,x0)

n = length(X)

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State-space models (8/10)

Matlab code

a=[0 -2;1 -4];b=[2;0];c=[1 0];d=[0];sys=ss(a,b,c,d); % state-space modelx0=[2 2]; % initial conditionst=[0:0.01:1];u=0*t; % zero input

[y,T,x]=lsim(sys,u,t,x0);

subplot(3,1,1),plot(T,x)subplot(3,1,2),plot(T,x(:,1))subplot(3,1,3),plot(T,x(:,2))dt=1;Xf1=expm(a*dt)*x0'

Result

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State-space models (9/10)

Result

subplot(3,1,1)

subplot(3,1,2)

subplot(3,1,3)

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State-space models (10/10)

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Lab Assignments (1/2)

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Lab Assignments (2/2)Mp3.4 Consider the system