CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ

CONTROL with LIMITED INFORMATION ;

SWITCHING ADAPTIVE CONTROL

Daniel Liberzon

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

IAAC Workshop, Herzliya, Israel, June 1, 2009

SWITCHING CONTROL

Classical continuous feedback paradigm:

u yP

C

u yPPlant:

But logical decisions are often necessary:

u y

C1

C2

l o g i c

P

REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above






Plant

Controller

INFORMATION FLOW in CONTROL SYSTEMS

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

[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

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

QUANTIZATION

Encoder Decoder

QUANTIZER

finite subset

of

is partitioned into quantization regions

QUANTIZATION and ISS

– assume glob. asymp. stable (GAS)



no longer GAS


quantization error

Assume

class

Solutions that start in

enter and remain there

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

In time domain:


quantization error

Assume

LINEAR SYSTEMS

Quantized control law:

9 feedback gain & Lyapunov function

Closed-loop:

(automatically ISS w.r.t. )

DYNAMIC QUANTIZATION


– zooming variable

Hybrid quantized control: is discrete state




Zoom out to overcome saturation




After ultimate bound is achieved,recompute partition for smaller region




ISS from to ISS from to small-gain conditionProof:

Can recover global asymptotic stability

QUANTIZATION and DELAY

Architecture-independent approach

Based on the work of Teel

Delays possibly large

QUANTIZER DELAY

QUANTIZATION and DELAY

where

hence

Can write

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:

FINAL RESULT

Need:

small gain true

FINAL RESULT

Need:

small gain true

FINAL RESULT

solutions starting in

enter and remain there

Can use “zooming” to improve convergence

Need:

small gain true

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance



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:

HYBRID SYSTEMS as FEEDBACK CONNECTIONS

continuous

discrete

[Nešić–L, ’05, ’06]

• Other decompositions possible

• Can also have external signals

SMALL – GAIN THEOREM

Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:

• Input-to-state stability (ISS) from to :

• ISS from to :

• (small-gain condition)

SUFFICIENT CONDITIONS for ISS

[Hespanha-L-Teel ’08]

# of discrete events on is

• ISS from to if:

and

• ISS from to if ISS-Lyapunov function [Sontag ’89]:

LYAPUNOV – BASED SMALL – GAIN THEOREM

Hybrid system is GAS if:

•

and # of discrete events on is

•

•

SKETCH of PROOF

is nonstrictly decreasing along trajectories

Trajectories along which is constant? None!

GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ’07]

quantization error

Zoom in:

where

ISS from to with gain

small-gain condition!

ISS from to with some linear gain

APPLICATION to DYNAMIC QUANTIZATION

RESEARCH DIRECTIONS

• Modeling uncertainty (with L. Vu)

• Disturbances and coarse quantizers (with Y. Sharon)

• Avoiding state estimation (with S. LaValle and J. Yu)

• Quantized output feedback

• Performance-based design

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

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






MODELING UNCERTAINTY

Adaptive control (continuous tuning)

vs. supervisory control (switching)

unmodeleddynamics

0parametricuncertainty

Also, noise and disturbance

EXAMPLE

Could also take

controller index set

Scalar system:

, otherwise unknown

(purely parametric uncertainty)

Controller family:

stable

not implementable

SUPERVISORY CONTROL ARCHITECTURE

Plant

Supervisor

Controller

Controller

Controller u1

u2

um

yu...

candidate controllers

...

– switching controller

– switching signal, takes values in

TYPES of SUPERVISION

• Prescheduled (prerouted)

• Performance-based (direct)

• Estimator-based (indirect)

TYPES of SUPERVISION

• Prescheduled (prerouted)

• Performance-based (direct)

• Estimator-based (indirect)

OUTLINE

• Basic components of supervisor

• Design objectives and general analysis

• Achieving the design objectives (highlights)

OUTLINE




SUPERVISOR

Multi-Estimator

y1

y2

u

y

...

...yp

estimation errors:

ep

e2

e1

...

Want to be small

Then small indicates likely

EXAMPLE

Multi-estimator:

exp fast=>

EXAMPLE

Multi-estimator:disturbance

exp fast=>

STATE SHARING

Bad! Not implementable if is infinite

The system

produces the same signals

SUPERVISOR

...Multi-

Estimator

y1

y2

yp ep

e2

e1

u

y

... ...

MonitoringSignals

Generator

1

2

p...

Examples:

EXAMPLE

Multi-estimator:

– can use state sharing

SUPERVISOR

SwitchingLogic

...Multi-

Estimator

y1

y2

yp ep

e2

e1

u

y

... ...

MonitoringSignals

Generator

1

2

p...

Basic idea:

Justification?

small => small => plant likely in => gives stable

closed-loop system

(“certainty equivalence”)

Plant , controllers:

SUPERVISOR

SwitchingLogic

...Multi-

Estimator

y1

y2

yp ep

e2

e1

u

y

... ...

MonitoringSignals

Generator

1

2

p...

Basic idea:

Justification?

small => small => plant likely in => gives stable

closed-loop systemonly know converse!

Need: small => gives stable closed-loop system

This is detectability w.r.t.

Plant , controllers:

DETECTABILITY

Want this system to be detectable

“output injection” matrix

asympt. stable

view as output

is Hurwitz

Linear case: plant in closedloop with

SUPERVISOR

SwitchingLogic

...Multi-

Estimator

y1

y2

yp ep

e2

e1

u

y

... ...

MonitoringSignals

Generator

1

2

p...

Switching logic (roughly):

This (hopefully) guarantees that is small

Need: small => stable closed-loop switched system

We know: is small

This is switched detectability

DETECTABILITY under SWITCHING

need this to be asympt. stable

plant in closedloop with

view as output

Assumed detectable for each frozen value of

Output injection:

• slow switching (on the average)

• switching stops in finite time

Thus needs to be “non-destabilizing”:

Switched system:

Want this system to be detectable:

SUMMARY of BASIC PROPERTIESMulti-estimator:

1. At least one estimation error ( ) is small

Candidate controllers:

3. is bounded in terms of the smallest

: for 3, want to switch to for 4, want to switch slowly or stop

conflicting

• when • is bounded for bounded &

Switching logic:

2. For each , closed-loop system is detectable w.r.t.

4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of

SUMMARY of BASIC PROPERTIES

Analysis: 1 + 3 => is small2 + 4 => detectability w.r.t.

=> state is small

Switching logic:

Multi-estimator:

Candidate controllers:

3. is bounded in terms of the smallest

4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of

2. For each , closed-loop system is detectable w.r.t.

1. At least one estimation error ( ) is small

• when • is bounded for bounded &

OUTLINE




Controller

Plant

Multi-estimator

qe

y

uqy

y

fixed

CANDIDATE CONTROLLERS


Linear: overall system is detectable w.r.t. if

i. system inside the box is stable

ii. plant is detectable

fixed

Plant

Multi-estimator

qe

y

uqy

y

Controller

P

C

E

Need to show: => P C E , ,

=> EC,i

=> => Pii

=>


Linear: overall system is detectable w.r.t. if i. system inside the box is stableii. plant is detectable

fixed

Plant

Multi-estimator

qe

y

uqy

y

Controller

P

C

E

Nonlinear: same result holds if stability and detectability are

interpreted in the ISS / OSS sense: external signal


Linear: overall system is detectable w.r.t. if i. system inside the box is stableii. plant is detectable

fixed

Plant

Multi-estimator

qe

y

uqy

y

Controller

P

C

E

Nonlinear: same result holds if stability and detectability are

interpreted in the integral-ISS / OSS sense:

SWITCHING LOGIC: DWELL-TIME

Obtaining a bound on in terms of is harder

Not suitable for nonlinear systems (finite escape)

Initialize

Find

no ? yes

– monitoring signals – dwell time

Wait time units

Detectability is preserved if is large enough

SWITCHING LOGIC: HYSTERESIS

Initialize

Find

no yes

– monitoring signals – hysteresis constant

?

or

(scale-independent)

SWITCHING LOGIC: HYSTERESIS

This applies to exp fast,

finite, bounded switching stops in finite time=>

Initialize

Find

no yes?

Linear, bounded average dwell time=>

Documents

CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ