Dealing with Nonlinear Parameterization in Nonlinear Adaptive Control An MMAC Approach

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Dealing with Nonlinear Parameterization in
Nonlinear Adaptive Control ---- An MMAC Approach

• Weitian Chen
• Joint Work with Prof. Brian Anderson

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Outline

• Introduction
• MMAC
• Nonlinear adaptive control with nonlinear
  parameterization
• Systems of interest and problem formulation
• A general dwell-time-switching based MMAC scheme
• Application of the MMAC scheme to a special class
  of nonlinear systems
• Examples and simulation results
• Conclusions and future research topics

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Introduction---- A brief introduction of MMAC (1)

• What is MMAC?
• A short name for multiple model adaptive control
• Main idea To use multiple models to solve
  adaptive control problems
• How old is MMAC?
• Over 30 years, see Lainiotis(1976) and Athans et
  al(1977)

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Introduction---- A brief introduction of MMAC (2)

• What are the motivations for MMAC?
• A single (linear) model is not sufficient for
  complex systems
• NASA report 16 linear models were developed for
  F-8C
• The change of environment leads naturally to
  multiple models.
• Sensor or actuator faults, multiple models are a
  natural choice for fault accommodation

F-15 Eagle from Wikipedia
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Introduction---- A brief introduction of MMAC (3)

• What are the motivations for MMAC? (Continued)
• Motivation from linear adaptive control
• Four standard assumptions in linear adaptive
  control
• System order
• Relative degree
• High-frequency gain sign
• Minimum phase
• In 1980s and 1990s, the strong desire to relax or
  remove as many of them as possible has led the
  use of MMAC in linear adaptive control
• Performance improvement
• Attempt to use MMAC to improve control
  performance.

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Introduction---- A brief introduction of MMAC (4)

• Blending or Switching?
• MMAC schemes based on candidate controller
  blending
• All candidate controllers are used
• The overall controller is obtained as a weighted
  sum of all candidate controllers
• Examples Kosut and Anderson (1988), Fekri et al.
  (2004a,b, 2006)
• MMAC schemes based on candidate controller
  switching
• Only one candidate controller is used at one
  particular time
• The overall controller is determined through a
  switching mechanism
• Examples Middleton et al (1988), Morse et al
  (1992), Narendra et al (1997), Morse (1996,1997),
  Anderson et al (2001), Hespanha et al (2001,
  2002, 2003)
• The difference between the two types of schemes
• The way how the overall control law is formed
• Which is better
• No obvious answer

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Introduction---- A brief introduction of MMAC (5)

• Some difficulties in MMAC
• Local model selection the type of models, the
  number of models, and how to construct the models
• Stability analysis
• Transient instability due to switching
• Design for more complex systems
• Fast time-varying systems
• General nonlinear systems

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Introduction---- A brief introduction of
nonlinear adaptive control

• Nonlinear adaptive control with linear
  parameterization
• Significant progress has been made by a working
  horse called backstepping since 1991
• Main assumptions
• Triangular nonlinear systems
• Linear parameterization
• Nonlinear adaptive control with nonlinear
  parameterization
• Very challenging
• Existing results
• Convex/Concave parameterization Annaswamy and
  her collaborators (1998, 1999,2002)
• Linear parameterization in bounding functions for
  triangular systems Marino and Tomei (1993), Lin
  and Qian (2002)

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Introduction---- The purposes of our research

• To extend Prof Morses dwell-time switching from
  linear systems to nonlinear systems
• To derive sufficient conditions for closed-loop
  stability and carry out stability analysis for
  our proposed MMAC scheme
• To propose an MMAC approach to deal with
  nonlinear parameterization for more general
  nonlinear systems

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Systems of Interest
unknown parameter vector
System state
Control input
Nonparametric uncertainty
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Problem formulation

• Multiple Model Adaptive Control (MMAC) Problem
• Design a state feedback multiple model adaptive
  controller for the system under consideration
• To ensure that, for any x(0), all the closed-loop
  system signals are bounded
• To make x(t) enter asymptotically a neighborhood
  of zero

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Assumption A1
An integer, finite, and known
A subset, known and bounded
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Assumption A2 ---- Relevant to candidate
controllers
1. It actually requires that at least one
candidate controller can do the job 2. It is
necessary for implementing MMAC 3. Since we do
not know which one will work, the job of MMAC is
to find out the right one
Solution exists and is unique for any
nonnegative and bounded
Positive definite and continuous
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Assumption A3 ---- Relevant to state estimators
Multiple state estimators
The modelled part of
H free design Hurwitz matrix
Free design function
It actually requires that at least one state
estimator can provide a good estimate of the
state.
nonnegative and bounded
Positive definite and continuous
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Design of monitoring signals (1)

• Suppose
• Candidate controllers are designed and A.2 is
  satisfied
• Multiple estimators are designed and A.3 is
  satisfied
• Now we are ready to design monitoring signals
• Needed in the switching logic to generate the
  switching sequence

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Design of monitoring signals (2)
positive design constant
signals
signals
A nice property
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Switching Control Mechanism (SCM) ---- Searching
for the right controller
Step 1 Choose a dwell time

• Summary
• If the current candidate controller leads to the
  best monitoring signals. i.e.
  Do not
  switch
• Otherwise, switch to another controller that also
  satisfies

Step 2 Initialization
Step 3
yes
No
yes
Step 4
No
yes
No
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The structure of our MMAC scheme
Monitoring signals
estimators
Switching logic
models
Candidate Controllers
system
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Main results---- Characterization of number of
switchings
system
estimators
candidate controllers
monitoring signals
Defined as on slide 16
A1,A2,A3 satisfied
The no. of switchings is less than
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Main results---- Stability of the closed-loop
system
system
estimators
monitoring signals
candidate controllers
A1,A2,A3 satisfied
Defined as on slide 16
Apply SCM to the considered system
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Applicability of SCM ---- From general systems
to special systems (1)

• To apply SCM
• You have several blocks to design
• Model selection block
• Candidate controller block
• Estimator block
• Monitoring signal block
• Switching logic block
• You have to ensure that SCM satisfies a couple of
  assumptions
• A1, A2, A3
• An additional assumption
• Questions you may want to ask
• Is there any nonlinear system that SCM can be
  designed to satisfy all those assumptions?
• If yes, how do you design it?

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Applicability of SCM ---- From general systems
to special systems (2)

• The answer to the first question is
• Yes!!
• To answer the second question, we consider the
  following systems

Unknown functions but with known triangular
bounding functions
Known functions and triangular form
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Applicability of SCM ---- From general systems
to special systems (3)
Adaptive control
Linear parameterization
Used to be challenging
Solved by backstepping in 1991
What can we do if linear parameterization assumpt
ion does not hold??
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Applicability of SCM ---- From general systems
to special systems (4)

• Our solution

SCM
Backstepping
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Applicability of SCM ---- From general systems
to special systems (5)

• A list of what we have designed
• Multiple estimators
• Candidate controllers
• Monitoring signals
• Switching logic
• Under two mild assumptions, the designed SCM is
  proved to satisfy all conditions required to
  obtain those main results

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Examples
Inverse of the Stribeck parameter
Example 1 friction compensation system
Example 2 a single-link robot with one revolute
elastic joint
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Simulation results---- Example 1 model
selection and initial condition
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Simulation results---- Example 1 switching
properties
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Simulation results---- Example 2
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Simulation results---- Example 2 switching
properties
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Summary of simulation results

• Simulation results have demonstrated
• SCM can stabilize the considered systems
• SCM can achieve satisfactory performance
• The final controller may or may not be the
  correct one, but always a satisfactory one
• The initial choice of the candidate controller is
  important

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Conclusions

• Proposed a dwell-time-switching based MMAC scheme
  for general nonlinear systems with nonlinear
  parameterization
• Provided sufficient conditions for closed-loop
  stability
• Characterized the maximum number of switchings
• i.e. Strictly less than the number of candidate
  controllers
• Designed an SCM for a special class of nonlinear
  systems and solved the adaptive control problem
  successfully.
• Obtained satisfactory simulation results no
  matter the final controller is the correct one
  or not

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Future research topics

• Model selection issues
• Model type
• The number of models
• The choice of representative parameter points
• Output feedback controller design
• Not addressed here
• However, results have been obtained and will be
  presented in another paper
• Reduce the complexity of the existing design
• Minimum dynamic order MMAC problem
• More complex problems
• Systems with unmodelled dynamics
• Removal of the assumption of a known compact set

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• Thank you!
• Questions?