An INTRODUCTION to SWITCHING ADAPTIVE CONTROL Daniel Liberzon - - PowerPoint PPT Presentation

An INTRODUCTION to SWITCHING ADAPTIVE CONTROL Daniel Liberzon

Coordinated Science Laboratory and

• Dept. of Electrical & Computer Eng.,
• Univ. of Illinois at Urbana-Champaign

UTC-IASE online seminar, July 21-22, 2014

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Based on joint work with J.P. Hespanha (UCSB) and A.S. Morse (Yale)

SWITCHING CONTROL

Classical continuous feedback paradigm:

u y

P C

u y

P Plant: But logical decisions are often necessary:

u y C1 C2 l o g i c

P

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REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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MODELING UNCERTAINTY

Adaptive control (continuous tuning)

• vs. supervisory control (switching)

unmodeled dynamics

 

parametric uncertainty Also, noise and disturbance

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EXAMPLE

Could also take

controller index set

Scalar system: , otherwise unknown (purely parametric uncertainty) Controller family: stable 

not implementable

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SUPERVISORY CONTROL ARCHITECTURE

Plant Supervisor



Controller Controller Controller

u1 u2 um

y u . . .

candidate controllers

. . . – switching controller

 – switching signal, takes values in

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TYPES of SUPERVISION

• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)

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TYPES of SUPERVISION

• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)

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OUTLINE

• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)

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OUTLINE

• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)

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SUPERVISOR

Multi- Estimator

y1 y2

u y . . . . . .

yp

estimation errors:



  

  ep e2 e1

. . . Want to be small Then small indicates likely

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EXAMPLE

Multi-estimator: exp fast

=>

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EXAMPLE

Multi-estimator:

disturbance

exp fast

=>

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STATE SHARING

Bad! Not implementable if is infinite The system produces the same signals

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SUPERVISOR

. . . Multi- Estimator

  

   y1 y2 yp ep e2 e1

u y . . . . . . Monitoring Signals Generator

1



2



p

 . . . Examples:

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EXAMPLE

Multi-estimator: – can use state sharing

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SUPERVISOR

Switching Logic



. . . Multi- Estimator

  

   y1 y2 yp ep e2 e1

u y . . . . . . Monitoring Signals Generator

1



2



p

 . . . Basic idea: Justification? small => small => plant likely in

=>

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

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SUPERVISOR

Switching Logic



. . . Multi- Estimator

  

   y1 y2 yp ep e2 e1

u y . . . . . . Monitoring Signals Generator

1



2



p

 . . . Basic idea: Justification? small => small => plant likely in

=>

gives stable closed-loop system

• nly know converse!

Need: small => gives stable closed-loop system This is detectability w.r.t. Plant , controllers:

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DETECTABILITY

Want this system to be detectable

• asympt. stable

view as output

“output injection” matrix

is Hurwitz

9 L q: Aq ¡ L qCq _ x = (Aq ¡ L qCq)x + L qeq

Linear case:

plant in closed loop with

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SUPERVISOR

Switching Logic



. . . Multi- Estimator

  

   y1 y2 yp ep e2 e1

u y . . . . . . Monitoring Signals 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

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DETECTABILITY under SWITCHING

need this to be asympt. stable plant in closed loop 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:

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SUMMARY of BASIC PROPERTIES

Multi-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

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SUMMARY of BASIC PROPERTIES

Analysis: 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 &

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OUTLINE

• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)

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



Controller Plant Multi- estimator

q

e

y

u

q

y y

fixed

CANDIDATE CONTROLLERS

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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

q

e

y

u

q

y



 y

Controller

P C E

Need to show:

=>

P C E

, ,

=>

E C,

i

=> =>

P

ii

=>

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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

q

e

y

u

q

y



 y

Controller

P C E

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

external signal

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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

q

e

y

u

q

y



 y

Controller

P C E

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

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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

q

e

y

u

q

y



 y

Controller

P C E

For minimum-phase plants, it is enough to ask that the system inside the box be output-stabilized

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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 

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SWITCHING LOGIC: HYSTERESIS

Initialize Find

no yes

– monitoring signals

– hysteresis constant ?

• r

(scale-independent)

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SWITCHING LOGIC: HYSTERESIS

This applies to exp fast, finite, bounded switching stops in finite time

=>

Initialize Find

no yes

? Linear, bounded average dwell time

=>

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TOY EXAMPLE: PARKING PROBLEM

Unknown parameters correspond to the radius of rear wheels and distance between them

2 1 , p

p p1 p2 p1

 cos

1 1

w x    sin

1 2

w x  

2

w   

2

x

1

x



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