MEETING THE NEED FOR ROBUSTIFIED
NONLINEAR SYSTEM THEORY CONCEPTS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Plenary lecture, ACC, Seattle, 6/13/08 1 of 36
TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
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TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
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Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
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INFORMATION FLOW in CONTROL SYSTEMS
• Limited communication capacity
• Need to minimize information transmission
• Event-driven actuators
• Coarse sensing
3 of 36• Theoretical interest
[Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…]
• Deterministic & stochastic models
• Tools from information theory
• Mostly for linear plant dynamics
BACKGROUND
Previous work:
• Unified framework for
• quantization
• time delays
• disturbances
Our goals:
• Handle nonlinear dynamics
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Caveat:
This doesn’t work in general, need robustness from controller
OUR APPROACH
(Goal: treat nonlinear systems; handle quantization, delays, etc.)
• Model these effects as deterministic additive error signals,
• Design a control law ignoring these errors,
• “Certainty equivalence”: apply control
combined with estimation to reduce to zero
Technical tools:
• Input-to-state stability (ISS)
• Lyapunov functions
• Small-gain theorems
• Hybrid systems
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QUANTIZATION
Encoder Decoder
QUANTIZER
finite subset
of
is partitioned into quantization regions
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QUANTIZATION and INPUT-to-STATE STABILITY
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– assume glob. asymp. stable (GAS)
QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
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no longer GAS
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
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Solutions that start in
enter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain: [Sontag ’89]
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36class , e.g.
Solutions that start in
enter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain: [Sontag ’89]
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
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LINEAR SYSTEMS
Quantized control law:
9 feedback gain & Lyapunov function
Closed-loop:
(automatically ISS w.r.t. )
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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After the ultimate bound is achieved,recompute partition for smaller region
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
Can recover global asymptotic stability
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QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
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QUANTIZATION and DELAY
where
hence
Can write
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SMALL – GAIN ARGUMENT
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
Assuming ISS w.r.t. actuator errors:
In time domain:
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
solutions starting in
enter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
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EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
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EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
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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])
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
After zoom-in:
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TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-state stability
Adaptive controlMinimum-phase property
Stability of switched systems
Observability
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OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem:
(observability with respect to )
observable=> GAS
Example:
is glob. asymp. stable (GAS) if s.t.
• is not identically zero along any nonzero solution
• (weak Lyapunov function)
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SWITCHED SYSTEMS
Want: stability of for classes of
Many results based on common and multiple
Lyapunov functions [Branicky, DeCarlo, …]
In this talk: weak Lyapunov functions
• is a collection of systems
• is a switching signal
• is a finite index set
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SWITCHED LINEAR SYSTEMS [Hespanha ’04]
Want to handle nonlinear switched systems
and nonquadratic weak Lyapunov functions
Theorem (common weak Lyapunov function):
• observable for each
Need a suitable nonlinear observability notion
Switched linear system is GAS if
• infinitely many switching intervals of length
• s.t. .
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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
• LaSalle’s stability theorem for switched systems
No inputs here, but can extend to systems with inputs
Joint work with Hespanha, Sontag, and Angeli
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STATE NORM ESTIMATION
This is a robustified version of 0-distinguishability
where
(observability Gramian)
Observability definition #1:
where
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OBSERVABILITY DEFINITION #1: A CLOSER LOOK
Initial-state observability:
Large-time observability:
Small-time observability:
Final-state observability:
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DETECTABILITY vs. OBSERVABILITY
Observability def’n #2: OSS, and can decay arbitrarily fast
Detectability is Hurwitz
Observability can have arbitrary eigenvalues
A natural detectability notion is output-to-state stability (OSS):
[Sontag-Wang]where
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and
Theorem: This is equivalent to definition #1 (small-time obs.)
Definition:
For observability, should have arbitrarily rapid growth
OSS admits equivalent Lyapunov characterization:
OBSERVABILITY DEFINITION #2: A CLOSER LOOK
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STABILITY of SWITCHED SYSTEMS
Theorem (common weak Lyapunov function):
• s.t.
Switched system is GAS if
• infinitely many switching intervals of length
• Each system
is observable:
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MULTIPLE WEAK LYAPUNOV FUNCTIONS
Example: Popov’s criterion for feedback systems
Instead of a single , can use a
family with additional
hypothesis on their joint evolution:
linear system observable
positive real
See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]
Weak Lyapunov functions:
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TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-state stability
Adaptive controlMinimum-phase property
Stability of switched systems
Observability
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MINIMUM-PHASE SYSTEMS
[Byrnes–Isidori]
Nonlinear (affine in controls): zero dynamics are (G)AS
Robustified version [L–Sontag–Morse]: output-input stability
Linear (SISO): stable zeros, stable inverse
• implies minimum-phase for nonlinear systems (when applicable)
• reduces to minimum-phase for linear systems (SISO and MIMO)27 of 36
where for some ;
UNDERSTANDING OUTPUT-INPUT STABILITY
uniform over :OSS (detectability) w.r.t. extended output,
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Output-input stability detectability + input-bounding property:
Sufficient Lyapunov condition for this detectability property:
CHECKING OUTPUT-INPUT STABILITY
For systems affine in controls,
can use structure algorithm for left-inversion
to check the input-bounding property
equation for is ISS w.r.t.
Example 1:
can solve for and get a bound on
can get a bound onOU
TPU
T-IN
PU
T S
TAB
LE
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Detectability:
Input bounding:
CHECKING OUTPUT-INPUT STABILITY
Example 2:
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For systems affine in controls,
can use structure algorithm for left-inversion
to check the input-bounding property
Does not have input-bounding property:
can be large near
while stays bounded
zero dynamics , minimum-phase
Output-input stability allows to distinguish
between the two examples
FEEDBACK DESIGN
Output stabilization state stabilization
( r – relative degree)
...
Output-input stability guarantees closed-loop GAS
In general, global normal form is not needed
Example: global normal form
GAS zero dynamics not enough for stabilization
Apply to have with Hurwitz
It is stronger than minimum-phase: ISS internal dynamics
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CASCADE SYSTEMS
For linear systems, recover usual detectability
If: • is detectable (OSS)
• is output-input stable
then the cascade system is detectable (OSS)
through extended output
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ADAPTIVE CONTROL
Linear systems
Controller
Plant
Designmodel
If: • plant is minimum-phase• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough (“tunable” [Morse ’92])
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ADAPTIVE CONTROL
If: • plant is minimum-phase• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Nonlinear systems
Controller
Plant
Designmodel
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ADAPTIVE CONTROL
If: • plant is output-input stable• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Nonlinear systems
Controller
Plant
Designmodel
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ADAPTIVE CONTROL
Nonlinear systems
If: • plant is output-input stable• system in the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Controller
Plant
Designmodel
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ADAPTIVE CONTROL
Nonlinear systems
If: • plant is output-input stable• system in the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough and its derivatives (“weakly tunable”)
Controller
Plant
Designmodel
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TALK SUMMARY
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
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ACKNOWLEDGMENTS
• Roger Brockett
• Steve Morse
• Eduardo Sontag
• Financial support from NSF and AFOSR
• Colleagues, students and staff at UIUC
http://decision.csl.uiuc.edu/~liberzon
• Everyone who listened to this talk
Special thanks go to:
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