NONLINEAR CONTROL with

LIMITED INFORMATION

Daniel Liberzon

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

Plenary talk, 2nd Indian Control Conference, Hyderabad, Jan 5, 2015 1 of 21

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 actuators

• Coarse sensing

• Theoretical interest3 of 21

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

• Deterministic & stochastic models

• Tools from information theory

• Mostly for linear plant dynamics

BACKGROUNDPrevious 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 Decoder

QUANTIZER

finite subset of

is partitioned into quantization regions

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QUANTIZATION and ISS

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– assume glob. asymp. stable (GAS)

QUANTIZATION and ISS

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QUANTIZATION and ISS

no longer GAS

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QUANTIZATION and ISS

quantization error

Assume

class

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Solutions that start inenter and remain there

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

In time domain:

QUANTIZATION and ISS

quantization error

Assume

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LINEAR SYSTEMS

Quantized control law:

∃ 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

ISS from to ISS from to small-gain conditionProof:

Can recover global asymptotic stability

[L–Nešić, ’05, ’06, L–Nešić–Teel ’14]9 of 21

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

Can write

hence

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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 inenter 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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NETWORKED CONTROL SYSTEMS [Nešić–L]

NCS: Transmit only some variables according to time scheduling protocol

Examples: round-robin, TOD (try-once-discard)

QCS: Transmit quantized versions of all variables

NQCS: Unified framework combining time scheduling and quantization

Basic design/analysis steps:• Design controller ignoring network effects

• Apply small-gain theorem to compute upper bound onmaximal allowed transmission interval (MATI)

• Prove discrete protocol stability via Lyapunov function

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ACTIVE PROBING for INFORMATION

dynamic

dynamic

(changes at sampling times)

(time-varying)

PLANT

QUANTIZER

CONTROLLER

Encoder Decoder

very small16 of 21

NONLINEAR SYSTEMS

sampling times

Example:

Zoom out to get initial bound

Between samplings

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NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

Let

Example:

Between samplings

• grows at most by the factor in one period

The norm

on a suitable compact region(dependent on )

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Pick small enough s.t.

NONLINEAR SYSTEMS (continued)

• grows at most by the factor in one period

• is divided by 3 at each sampling time

The norm

If this is ISS w.r.t. as before, then 18 of 21

ROBUSTNESS of the CONTROLLER

ISS w.r.t.

Same condition as before (restrictive, hard to check)

checkable sufficient conditions ([Hespanha-L-Teel])

ISS w.r.t.

Option 1.

Option 2. Look at the evolution of

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LINEAR SYSTEMS

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LINEAR SYSTEMS

[Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda]

Between sampling times,

where is Hurwitz0

• divided by 3 at each sampling time

• grows at most by in one period

amount of static infoprovided by quantizer

sampling frequency vs. open-loop instability

global quantity:

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OTHER RESEARCH DIRECTIONS

• Modeling uncertainty (with L. Vu)

• Disturbances and coarse quantizers (with Y. Sharon)

• Multi-agent coordination (with S. LaValle and J. Yu)

• Quantized output feedback and observers (with H. Shim)

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

• Performance-based design (with F. Bullo)

• Quantized control of switched systems

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