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

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)

QUANTIZATION and ISS

QUANTIZATION and ISS

no longer GAS

QUANTIZATION and ISS

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 and ISS

quantization error

Assume

LINEAR SYSTEMS

Quantized control law:

9 feedback gain & Lyapunov function

Closed-loop:

(automatically ISS w.r.t. )

DYNAMIC QUANTIZATION

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

Zoom out to overcome saturation

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

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

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

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])

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

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

REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above

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

• Basic components of supervisor

• Design objectives and general analysis

• Achieving the design objectives (highlights)

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

• Basic components of supervisor

• Design objectives and general analysis

• Achieving the design objectives (highlights)

Controller

Plant

Multi-estimator

qe

y

uqy

y

fixed

CANDIDATE CONTROLLERS

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

=>

CANDIDATE CONTROLLERS

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

CANDIDATE CONTROLLERS

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=>