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

1 of 15Workshop on Control and Optimization, UIUC, Dec 5, 2007

Plant

Controller

INFORMATION FLOW in CONTROL SYSTEMS

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• many control loops share network cable or wireless medium

• microsystems with many sensors/actuators on one chip

• Need to minimize information transmission (security)

• Event-driven actuators2 of 15

[Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…]

• Deterministic & stochastic models

• Tools from information theory

• Mostly for linear plant dynamics

BACKGROUND

Earlier 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 via deterministic 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 DecoderQUANTIZER

finite subset

of

is the range, is the quantization error bound

For , the quantizer saturates

Assume such that

is partitioned into quantization regions

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QUANTIZATION and ISS

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QUANTIZATION and ISS

quantization error

Assume

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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 and ISS

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quantization error

Assume

cf. linear:

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 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 conditionSwitching strategy based on dwell time or quantized state

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EXTERNAL DISTURBANCES [Nešić-L]

State quantization and completely unknown disturbance

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

EXTERNAL DISTURBANCES [Nešić-L]

State quantization and completely unknown disturbance

Developed two different strategies:

• continuous-time vs. sampled-data implementation

• Lyapunov-based vs. trajectory-based analysis

Result: for linear plant, can achieve ISS w.r.t. disturbance

(ISS gains are nonlinear although plant is linear [cf. Martins])9 of 15

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QUANTIZATION and DELAY

QUANTIZER DELAY

Architecture-independent approach

Based on the work of Teel

QUANTIZATION and DELAY

Assuming ISS w.r.t. actuator errors:

In time domain:

where

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SMALL – GAIN ARGUMENT

hence

ISS property becomes

if

then we recover ISS w.r.t. [Teel ’98]

Small gain:

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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

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Can use “zooming” to recover global asymptotic stability

Need:

small gain true

RESEARCH DIRECTIONS

• Modeling uncertainty (robust, adaptive control)

• Disturbances and coarse quantizers [Sharon-L]

• Moving away from estimation-based approach

• Quantized output feedback

• Performance-based design

• Vision-based control and other applications

• how many variables to transmit?

• nonlinear ISS observer design

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