Lu LIU and Jie HUANG

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Global Robust Output Regulation of Lower
Triangular Systems with Unknown High-Frequency
Gain Sign
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• Lu LIU and Jie HUANG
• Department of Mechanics Automation Engineering
• The Chinese University of Hong Kong
• 9 December, 2006

2006 Systems Workshop on Autonomous Networks
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Outline

• Introduction
• Problem Formulation
• Main Result
• An Example
• Conclusion

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1. Introduction
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Background

• The global robust output regulation problem for
  nonlinear systems in lower triangular form is
  considered with various solvability conditions.
• A basic assumption is that the sign of the
  high-frequency gain, i.e., the control direction,
  is known.
• The knowledge of the high-frequency gain sign
  makes control design much more tractable.

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Objective

• Solve the global robust output regulation problem
  for nonlinear systems in lower triangular form
  without knowing the high-frequency gain sign.
• Remark
• Nonlinear systems in lower triangular form
  is an important class of nonlinear systems, and
  many systems can be converted into lower
  triangular form by coordinate transformation.

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Related Work

• When the high-frequency gain sign is known
• The global robust output regulation problem
  (GRORP) for nonlinear systems in lower triangular
  form has been solved by using the robust control
  method.
• When the high-frequency gain sign is unknown
• The same problem has been rarely considered in
  the existing literature.

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Proposed Approach

• Approach
• Integrate the robust control approach and
  the adaptive control approach to develop a
  Lyapunov direct method.

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2. Problem Formulation
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Problem Formulation

• Nonlinear systems in lower triangular form

• Remark
• The high-frequency gain sign, i.e., the sign
  of

, is unknown.
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Problem Formulation

• Global Robust Output Regulation Problem (GRORP)
• Design a control law such that, for all
  bounded exogenous signal and any
  uncertain parameter , the
  trajectories of the closed-loop system starting
  from all initial states are bounded, and the
  tracking error e converges to zero asymptotically.

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Remark

• Output regulation problem is more challenging
  than stabilization and the conventional tracking
  and disturbance rejection problem.
• Requires more than stabilization.
• The class of reference signals and disturbances
  are generated by some autonomous differential
  equation.

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Remark

• A general framework has been established to
  convert the output regulation problem for a
  nonlinear system into a stabilization problem for
  an appropriately augmented system (Huang and
  Chen, 2004).
• The GRORP for the original system can be
  converted into a GRSP for an augmented system
  composed of the original plant and the internal
  model
• The solvability of the GRSP for the augmented
  system implies the solvability of the GRORP for
  the original system.

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Solvability of the Problem

• Using the existing framework, the GRORP for the
  original plant can be converted into the GRSP for
  the augmented system
• Remark
• The augmented system is not in the lower
  triangular form as system (1). Some standard
  assumptions are needed to solve the stabilization
  problem for the augmented system.

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Standard Assumptions
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3. Main Result
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Control Strategy

• When the sign of is known, the
  existing result gives

• When the sign of is unknown, we
  propose

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Remark

• Apply change supply rate technique to handle the
  dynamic uncertainty, and Nussbaum gain technique
  for the unknown high-frequency gain sign.
• is introduced to estimate the unknown control
  coefficient b(w).
• N(k) is a type of dynamically generated gain
  which oscillates to ensure that both positive and
  negative control directions are tried (Nussbaum,
  1983).

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Main Theorem

• Theorem

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Idea of the Proof

• Use a recursive approach to design virtual
  control
• Define

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Idea of the Proof
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Outline of the Proof

• By appropriately selecting the design functions,
  we obtain
• then applying a lemma by Ye and Jiang and
  Barbalats lemma gives,

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4. An Example
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An Example

• Plant
• Exosystem

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• Controller

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Simulation
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Simulation
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Simulation
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5. Conclusion
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Conclusion

• Solved the GRORP for nonlinear systems in lower
  triangular form without knowing the
  high-frequency gain sign.
• Obtained the control law by integrating the
  robust control method and the adaptive control
  method.

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Thank you!
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