Classical LQ Regulator Applied to the System with Nonzero Excitations

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Classical LQ Regulator Applied to the System with
Nonzero Excitations

• Ryszard Gessing
• Silesian University of Technology
• Gliwice, Poland

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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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Introduction

• LQR state feedback, zero external excitations
  (including the set point)
• Nonzero external excitations problem of
  optimal tracking (excitatons are known in all the
  future)
• Stochastic approach LQG problem (probability
  distributions of excitations are known)
• LQR with output feedback and nonzero external
  excitations

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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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The CL system
has the poles
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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CL system
steady state
for increaments
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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Reduced order Luenberger observer
-eigenvalues of E
control law
LQ observer based regulator
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CL system
has the poles
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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State transformation
Lemma 1 If
then the observer based LQ regulator is optimal
Lemma 2
Let
then
(suboptimality)
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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Theorem 1
Let
then
(suboptimality for increments from steady state)
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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Plant
Performance index
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Control law gain
Poles of the CL system with state feedback
Observer based LQ regulator
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CL system
has the poles
(of the CL system with LQ regulator and state
feedback)
(of the observer)
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Steady state errors
For set points
we have appropriately
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Outline of Presentation

• Introduction
• LQ Regulator with State Feedback
• State Feedback and Nonzero Set Point
• Observer Based Regulator
• Optimality of the Regulator
• CL system with Nonzero Set Point
• Example
• Final Conclusions

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Classical LQ regulator applied in the system with
nonzero external excitation-

• is optimal for transients in the case of state
  feedback
• is suboptimal for transients in the case of
  output feedback (observer based regulator)
• -faster modes of the observer the transients
  are closer to optimal.
• Only current measurement of excitation is needed
  for implementation (no prediction, no probability
  distributions).