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Laboratory in Automatic Control

LAB 13

System Design Using Simulink

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The Design of State Variable

Feedback Systems (1/10) Consider the third-order system

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The Design of State Variable

Feedback Systems (2/10)Determine a full-state feedback gain matrix and anobserver gain matrix to place the closed-loop system

poles at and the observerpoles at Construct the statevariable compensator using Figure 11.1 as a guideand simulate the closed-loop system using Simulink.Select several values of initial states and initial stateestimates in the observer and display the trackingresults on an xy-graph.

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The Design of State Variable

Feedback Systems (3/10)The compensator can be represented as

Since , we can write

Similarly, with

we obtain

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The Design of State Variable

Feedback Systems (4/10)In matrix form, we have

With initial conditions

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The Design of State VariableFeedback Systems (5/10)

Matlab code A=[0 1 0;0 0 1;-4.3 -1.7 -6.7]; B=[0;0;0.35]; C=[0 1 0]; D=[0];% Controller Gainsp=[-1.4+1.4*j; -1.4-1.4*j; -2];K=place(A,B,p)% Observer Gainsq=[-18+5*j; -18-5*j; -20];L=acker(A',C',q); L=L'% Simulation of closed-loop system with the observer

Ac=[A -B*K;L*C A-B*K-L*C];Bc=[zeros(6,1)];Cc=eye(6);Dc=zeros(6,1);%Developement of the state-space model

sys=ss(Ac,Bc,Cc,Dc);x0=[1;0;0;0.5;0.1;0.1]; t=[0:0.001:3.5];%initial() plots the undriven response of the state-space model[y,t]=initial(sys,x0,t);subplot(311)plot(t,y(:,1),t,y(:,4),'--'),gridsubplot(312)plot(t,y(:,2),t,y(:,5),'--'),gridsubplot(313)

plot(t,y(:,3),t,y(:,6),'--'),grid

Compute the initialcondition response ofstate-space models

Identity matrix

Pole placement design

for single-input systems:A-BK

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The Design of State Variable

Feedback Systems (6/10)Result

0 0.5 1 1.5 2 2.5 3 3.5-2

0

2

4

0 0.5 1 1.5 2 2.5 3 3.5-1

-0.5

0

0.5

0 0.5 1 1.5 2 2.5 3 3.5-2

-1

0

1

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The Design of State Variable

Feedback Systems (7/10)Simulink | Sources | Clock Simulink | Sinks | XY Graph

Simulink | Continuous | State-Space

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The Design of State Variable

Feedback Systems (8/10)

, x Ax Bu

y Cx

,

x A LC BK x Ly

y u Kx

Block diagram in simulink simulations

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The Design of State Variable

Feedback Systems (9/10)

Ctrl+Rchange

the function

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The Design of State Variable

Feedback Systems (10/10)Result

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Lab Assignments Lab 13:

Lab report should at least contain the

MATLAB code, Simulink model, and plots

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Lab Assignments Consider the third-order system

0 1 0 0 0

0 0 1 0 0,

0 0 0 1 0

2 5 1 13 1

1 0 0 0 0

x x u

y x u

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The Design of State Variable

Feedback Systems (2/10)

1,2 3,418 5 , 20s j s

Determine a full-state feedback gain matrix and anobserver gain matrix to place the closed-loop system

poles at and the observerpoles at Construct the statevariable compensator using Figure 11.1 as a guideand simulate the closed-loop system using Simulink.

Select several values of initial states and initial stateestimates in the observer and display the trackingresults on an xy-graph.

1,2 3,41.4 1.4 , 2s j s j