Transcript Lu Liu

Applied Control and Computing Laboratory
Global Robust Output Regulation of
Lower Triangular Systems
with Unknown High-Frequency Gain Sign
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
result gives:
 When the sign of
propose:
is known, the existing
is unknown, we
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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 Barbalat’s 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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