TOWARDS a UNIFIED FRAMEWORK for

NONLINEAR CONTROL with

LIMITED INFORMATION

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

Workshop dedicated to Roger Brockett’s 70th birthday, Cancun, 12/8/081 of 14

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

• Theoretical interest2 of 14

[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

no longer GAS

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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

class , e.g. 6 of 14

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

ISS from to ISS from to small-gain conditionProof:8 of 14

SMALL-GAIN ANALYSIS of HYBRID SYSTEMS

continuous

discrete

Hybrid system is GAS if we have:

• (small-gain condition)

Can use Lyapunov techniques to check this [Nešić–L ’06]

• ISS from to :

• ISS from to :

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QUANTIZATION and DELAY

Architecture-independent approach

Based on result of [Teel ’98]

Delays possibly large

QUANTIZER DELAY

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SMALL – GAIN ARGUMENT

ISS w.r.t. actuator errors gives

where

if then we have ISS w.r.t. Small gain:11 of 14

hence

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 gain is nonlinear although plant is linear; cf. [Martins])

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

After zoom-in:

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ONGOING RESEARCH

• Modeling uncertainty (with L. Vu, ThB08.2 )

• Disturbances and coarse quantizers (with Y. Sharon)

• Coordination with coarse sensing (with S. LaValle and J. Yu, TuC17.2 )

• Quantized output feedback and ISS observer design

• Vision-based control (with Y. Ma and Y. Sharon)

http://decision.csl.uiuc.edu/~liberzon

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