Laboratory in Automatic Control Lab13
Transcript of Laboratory in Automatic Control Lab13
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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