Slide 1

SWITCHING ADAPTIVE CONTROLDaniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign1REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above2MODELING UNCERTAINTYAdaptive control (continuous tuning)vs. supervisory control (switching)unmodeleddynamics

parametricuncertainty

Also, noise and disturbance

3Similarities with adaptive control for those who know, but the knowledge of adaptive control is not neededEXAMPLECould also take

controller index setScalar system:

, otherwise unknown

(purely parametric uncertainty)

Controller family:

stable

not implementable4The origin is excluded from $\P$ to preserve controllabilityEvery possible plant is stabilized by some controller from the familyWhat really matters is the sign of p^*, finite family is easier to work withSUPERVISORY CONTROL ARCHITECTUREPlantSupervisor

Controller ControllerController u1u2umyu...candidate controllers...

switching controller

switching signal, takes values in

5dont actually run all controllers at all times explain state sharing, switched systemTYPES of SUPERVISION Prescheduled (prerouted) Performance-based (direct) Estimator-based (indirect)6Discuss each briefly

TYPES of SUPERVISION Prescheduled (prerouted) Performance-based (direct) Estimator-based (indirect)7OUTLINE Basic components of supervisor Design objectives and general analysis Achieving the design objectives (highlights) 8Last bullet: will not really state results formally, but will illustrate by examples may not even get to itIn fact, this is going to be a relatively informal talk, the material is rather difficult, also end of the day want to give a flavor

OUTLINE Basic components of supervisor Design objectives and general analysis Achieving the design objectives (highlights) 9Last one: will not really state results formally, but will illustrate by examplesIn fact, this is going to be a relatively informal talk, the material is rather difficult, also last day of school want to give a flavor

SUPERVISORMulti-Estimatory1y2uy......ypestimation errors:

epe2e1...

Want to be small

Then small indicates likely

10we now zoom in on supervisornext: what does small mean exactly?EXAMPLE

Multi-estimator:

exp fast

=>11EXAMPLEMulti-estimator:

disturbance

exp fast

=>

12STATE SHARING

Bad! Not implementable if is infinite

The systemproduces the same signals

13The family of signals is still infinite but we can look them up, take min, etc.SUPERVISOR...Multi-Estimator

y1y2ypepe2e1uy......MonitoringSignalsGenerator

...

Examples:

14Explain why we need monitoring signals (history of estimation errors)L_2 normscaling, forgetting factorEXAMPLE

Multi-estimator: can use state sharing

15The family of signals is still infinite but we can look them up, take min, etc.SUPERVISORSwitchingLogic

...Multi-Estimator

y1y2ypepe2e1uy......MonitoringSignalsGenerator

...Basic idea:

Justification?

small=>

small=>plant likely in

=>

gives stable closed-loop system(certainty equivalence)Plant , controllers:

16Assume for now that Q=PExplain CE verballyIve been lying! Prompt students to explain whats wrongSUPERVISORSwitchingLogic

...Multi-Estimator

y1y2ypepe2e1uy......MonitoringSignalsGenerator

...Basic idea:

Justification?

small=>

small=>plant likely in

=>

gives stable closed-loop systemonly know converse!Need:

small=>

gives stable closed-loop systemThis is detectability w.r.t.

Plant , controllers:

17DETECTABILITYWant this system to be detectableoutput injection matrixasympt. stableview as output

is Hurwitz

Linear case:plant in closedloop with

18let me remind you what detectability isoutput injection relates detectability to stabilitySUPERVISORSwitchingLogic

...Multi-Estimator

y1y2ypepe2e1uy......MonitoringSignalsGenerator

...Switching logic (roughly):

This (hopefully) guarantees that is small

Need:small=>stable closed-loop switched system

We know: is small

This is switched detectability19But lets take a more careful look at what we have so farDETECTABILITY under SWITCHINGneed this to be asympt. stableplant in closedloop withview as output

Assumed detectable for each frozen value of

Output injection:

slow switching (on the average) switching stops in finite timeThus needs to be non-destabilizing

:Switched system:Want this system to be detectable:

20SUMMARY 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

21SUMMARY of BASIC PROPERTIESAnalysis:1 + 3 => is small

2 + 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 &

22advantage of modular architectureOUTLINE Basic components of supervisor Design objectives and general analysis Achieving the design objectives (highlights) 23Last one: will not really state results formally, but will illustrate by examplesIn fact, this is going to be a relatively informal talk, the material is rather difficult, also last day of school want to give a flavor

ControllerPlant Multi-estimator

fixedCANDIDATE CONTROLLERS

24Walk through the diagramCANDIDATE CONTROLLERSLinear: overall system is detectable w.r.t. if system inside the box is stableplant is detectable

fixed

Plant Multi-estimator

Controller

PC

E

Need to show:=>

PC

E

, ,

=>E

C

,i

=>

=>

Pii

=>

25CANDIDATE CONTROLLERSLinear: overall system is detectable w.r.t. if system inside the box is stableplant is detectable

fixed

Plant Multi-estimator

Controller

PC

E

Nonlinear: same result holds if stability and detectability are interpreted in the ISS / OSS sense:

external signal26ISS-stabilization is a problem of current interest in nonlinear controlModular architecture: reduces to this problemCANDIDATE CONTROLLERSLinear: overall system is detectable w.r.t. if system inside the box is stableplant is detectable

fixed

Plant Multi-estimator

Controller

PC

E

Nonlinear: same result holds if stability and detectability are interpreted in the integral-ISS / OSS sense:

27Especially suitable because monitoring signals measure integral energy of errorsSWITCHING 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

28Initialization: either smart (argmin) or arbitrarySWITCHING LOGIC: HYSTERESISInitialize

Find

noyes

monitoring signals

hysteresis constant

?

or (scale-independent)

29Avoid problems of dwell time, but slow switching is not directly enforcedSWITCHING LOGIC: HYSTERESISThis applies to exp fast,

finite, bounded switching stops in finite time=>

Initialize

Find

noyes

?

Linear, bounded average dwell time=>

30Explain verbally why switching stops (variation gets small)Covers exact matching case both linear and nonlinearCan regulate average dwell time via hThis connects with results presented earlier (ISS under ADT)More info: Chapter 6 of my bookREASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above31PARKING PROBLEMUnknown parameters correspond to theradius of rear wheels and distance between them

p1p2p1

32