MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng.,

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





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)


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no longer GAS

quantization error

Assume

class


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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


7 of 36class , e.g.

Solutions that start in


This is input-to-state stability (ISS) w.r.t. measurement errors

In time domain: [Sontag ’89]

quantization error

Assume

class


7 of 36class , e.g.

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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– zooming variable

Hybrid quantized control: is discrete state

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Zoom out to overcome saturation




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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

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


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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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])



After zoom-in:

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TALK OVERVIEW

ContextConcept


Input-to-state stability

Adaptive controlMinimum-phase property


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


Input-to-state stability

Adaptive controlMinimum-phase property


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


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


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





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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Documents

MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng.,