OPTIMAL CONTROL PROJECT

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OPTIMAL CONTROL PROJECT
Optoelectronic Automation Assembly
( ECES 690 )
Shubham Bhat
Presented by
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Laser-Diode to Fiber Example
NEAR FIELD COUPLING
Aperture 20um x 20um Fiber Core 4 um Prop.
Distance 10 um
Distance 10um No. of. Peaks 10
Edge Emitting Laser Coupled To a Fiber
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Non-Linear Model
y mx c Y0.0441 x 0.1129
Inverse Model Approach
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Problem Statement
State-Space Representation
B 0 0 0.1250
A 0 1.0000 0 0 0
1.0000 0 -0.3750 -0.8750
C 0.0441 0 0
D 0
The Optimal control problem involves minimization
of the cost function
R1 Q CC
where
The continuous time algebraic Riccati equation is
given as
The Kalman gain is given as
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Continuous LQR solution
K_1 0.0441 0.1029 0.1166 K_2
1.0000 2.2359 2.2073 K_3 3.1623
6.2943 5.2356
Q_1C'C Case 1
Q_2diag(1 0 0) Case 2
S_1 0.1368 0.3138 0.3528 0.3138
0.7289 0.8228 0.3528 0.8228
0.9326 S_2 5.2359 9.2073 8.0000
9.2073 19.2084 17.8872 8.0000 17.8872
17.6584 S_3 29.3913 38.6923 25.2982
38.6923 67.4232 50.3546 25.2982 50.3546
41.8846
As Q increases, all states go to zero but at the
expense of control energy
Q_3diag(10 0 0) Case 3
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Discrete LQR solution
State-Space Representation
B_d 0 0 0.0012
A_d 1.00 0.0100 0.00 0 1.00
0.01 0 -0.0037 0.9913
C 0.0441 0 0
D 0
Sampling time 0.01 sec
The Discrete time algebraic Riccati equation is
given as
R1 Q CC
where
The Kalman gain is given as
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Discrete LQR solution
K_1 0.0441 0.1028 0.1166 K_2
0.9986 2.2337 2.2058 K_3 3.1519
6.2797 5.2270
Q_1C'C Case 1
Q_2diag(1 0 0) Case 2
S_1 0.1368 0.3138 0.3528 0.3138
0.7289 0.8228 0.3528 0.8228
0.9326 S_2 5.2359 9.2073 8.0000
9.2073 19.2084 17.8872 8.0000 17.8872
17.6584 S_3 29.3913 38.6923 25.2982
38.6923 67.4232 50.3546 25.2982 50.3546
41.8846
As Q increases, all states go to zero but at the
expense of control energy
Q_3diag(10 0 0) Case 3
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Summary

• Optimal control for an optoelectronic system was
  achieved.
• Two approaches used were
• Continuous LQR method
• Discrete LQR method
• Results with different Q matrix was shown.
• For Large Q, the states go to zero but at an
  expense in control energy.
• Discretizing the system gives accurate results
  until the sampling frequency satisfies the
  nyquist criterion.