MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS

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MEETING THE NEED FOR ROBUSTIFIED NONLINEAR
SYSTEM THEORY CONCEPTS
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
Plenary lecture, ACC, Seattle, 6/13/08
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TALK OVERVIEW
Context
Concept
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TALK OVERVIEW
Context
Concept
Stability of switched systems
Observability
Minimum-phase property
Adaptive control
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INFORMATION FLOW in CONTROL SYSTEMS
Plant
Controller
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INFORMATION FLOW in CONTROL SYSTEMS

• Coarse sensing

• Limited communication capacity

• Need to minimize information transmission

• Event-driven actuators

• Theoretical interest

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BACKGROUND
Previous work
Brockett, Delchamps, Elia, Mitter, Nair, Savkin,
Tatikonda, Wong,

• Deterministic stochastic models
• Tools from information theory

• Mostly for linear plant dynamics

• Unified framework for
• quantization
• time delays
• disturbances

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OUR APPROACH
(Goal treat nonlinear systems handle
quantization, delays, etc.)
Caveat This doesnt work in general, need
robustness from controller
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QUANTIZATION
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QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
assume glob. asymp. stable (GAS)
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QUANTIZATION and INPUT-to-STATE STABILITY
no longer GAS
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QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
quantization error
Assume
Solutions that start in enter and remain
there
This is input-to-state stability (ISS) w.r.t.
measurement errors
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LINEAR SYSTEMS
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
Hybrid quantized control is discrete state
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DYNAMIC QUANTIZATION
Hybrid quantized control is discrete state
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DYNAMIC QUANTIZATION
zooming variable
Zoom out to overcome saturation
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DYNAMIC QUANTIZATION
After the ultimate bound is achieved, recompute
partition for smaller region
Can recover global asymptotic stability
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QUANTIZATION and DELAY
Architecture-independent approach
Delays possibly large
Based on the work of Teel
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QUANTIZATION and DELAY
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SMALL GAIN ARGUMENT
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FINAL RESULT
Need
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FINAL RESULT
Need
small gain true
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FINAL RESULT
Need
small gain true
Can use zooming to improve convergence
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State quantization and completely unknown
disturbance
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State quantization and completely unknown
disturbance
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State quantization and completely unknown
disturbance
After zoom-in
Issue disturbance forces the state outside
quantizer range
Must switch repeatedly between zooming-in and
zooming-out
Result for linear plant, can achieve ISS w.r.t.
disturbance
(ISS gains are nonlinear although plant is
linear cf. Martins)
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TALK OVERVIEW
Context
Concept
Input-to-state stability
Stability of switched systems
Observability
Minimum-phase property
Adaptive control
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OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem
is glob. asymp. stable (GAS) if s.t.
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SWITCHED SYSTEMS
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Theorem (common weak Lyapunov function)
Switched linear system is GAS if
Want to handle nonlinear switched systems and
nonquadratic weak Lyapunov functions
Need a suitable nonlinear observability notion
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OBSERVABILITY MOTIVATING REMARKS
Several ways to define observability (equivalent
for linear systems)
Benchmarks

• observer design or state norm estimation

• detectability vs. observability

• LaSalles stability theorem for switched systems

Joint work with Hespanha, Sontag, and Angeli
No inputs here, but can extend to systems with
inputs
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STATE NORM ESTIMATION
This is a robustified version of
0-distinguishability
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OBSERVABILITY DEFINITION 1 A CLOSER LOOK
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DETECTABILITY vs. OBSERVABILITY
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OBSERVABILITY DEFINITION 2 A CLOSER LOOK
Definition
and
Theorem This is equivalent to definition 1
(small-time obs.)
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STABILITY of SWITCHED SYSTEMS
Theorem (common weak Lyapunov function)
Switched system is GAS if
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MULTIPLE WEAK LYAPUNOV FUNCTIONS
Example Popovs criterion for feedback systems
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TALK OVERVIEW
Context
Concept
Control with limited information
Input-to-state stability
Stability of switched systems
Observability
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MINIMUM-PHASE SYSTEMS
Robustified version LSontagMorse
output-input stability

• implies minimum-phase for nonlinear systems
  (when applicable)

• reduces to minimum-phase for linear systems
  (SISO and MIMO)

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UNDERSTANDING OUTPUT-INPUT STABILITY
OSS (detectability)
w.r.t. extended output,
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CHECKING OUTPUT-INPUT STABILITY
For systems affine in controls, can use structure
algorithm for left-inversion to check the
input-bounding property
Input bounding
OUTPUT-INPUT STABLE
Detectability
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CHECKING OUTPUT-INPUT STABILITY
For systems affine in controls, can use structure
algorithm for left-inversion to check the
input-bounding property
Example 2
Output-input stability allows to distinguish
between the two examples
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FEEDBACK DESIGN
Example global normal form
Output-input stability guarantees closed-loop GAS
It is stronger than minimum-phase ISS internal
dynamics
In general, global normal form is not needed
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CASCADE SYSTEMS
For linear systems, recover usual detectability
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ADAPTIVE CONTROL
Linear systems
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ADAPTIVE CONTROL
Plant
Design
Controller
model
Nonlinear systems

• plant is minimum-phase
• system inside the box is output-stabilized
• controller and design model are detectable

If
then the closed-loop system is detectable through
e (tunable Morse 92)
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ADAPTIVE CONTROL
Plant
Design
Controller
model
Nonlinear systems

• plant is output-input stable
• system inside the box is output-stabilized
• controller and design model are detectable

If
then the closed-loop system is detectable through
e (tunable Morse 92)
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ADAPTIVE CONTROL
Plant
Design
Controller
model
Nonlinear systems

• plant is output-input stable
• system in the box is output-stabilized
• controller and design model are detectable

If
then the closed-loop system is detectable through
e (tunable Morse 92)
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ADAPTIVE CONTROL
Plant
Design
Controller
model
Nonlinear systems
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TALK SUMMARY
Context
Concept
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ACKNOWLEDGMENTS
Special thanks go to

• Roger Brockett

• Steve Morse

• Eduardo Sontag

• Colleagues, students and staff at UIUC

• Financial support from NSF and AFOSR

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