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SMALL - GAIN THEOREMS of

LASALLE TYPE for

HYBRID SYSTEMS

Daniel Liberzon (Urbana-Champaign)

Dragan Nešić (Melbourne)

Andy Teel (Santa Barbara)

CDC, Maui, Dec 2012

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MODELS of HYBRID SYSTEMS

…

[Goebel-Sanfelice-Teel]

Flow:

Jumps:

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HYBRID SYSTEMS as FEEDBACK CONNECTIONS

continuous

discrete

Every hybrid system can be thought of in this way

But this special decomposition is not always the best one

E.g., NCS:

network protocol

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HYBRID SYSTEMS as FEEDBACK CONNECTIONS

HS1

HS2

Can also consider external signals

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SMALL – GAIN THEOREM

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

• (small-gain condition)

• Input-to-state stability (ISS) from to [Sontag ’89]:

• ISS from to :

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SUFFICIENT CONDITIONS for ISS

This gives “strong” ISS property [Cai-Teel ’09]

• For :

• For :

where

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LYAPUNOV – BASED SMALL – GAIN THEOREM

on•

(small-gain condition)•

is a Lyapunov function for the overall hybrid system

Then

Pick s.t.

on•

Assume:

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LYAPUNOV – BASED SMALL – GAIN THEOREM

Generalizes Lyapunov small-gain constructions for continuous [Jiang-Mareels-Wang ’96] and discrete [Laila-Nešić ’02] systems

decreases along solutions of the hybrid systemOn the boundary, use Clarke derivative

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LIMITATION

on•

on•

The strict decrease conditions

are often not satisfied off-the-shelf

E.g.:

Since and we would typically have

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LASALLE THEOREM

• all nonzero solutions have both flow and jumps

•

Assume:

•

•

As before, pick and let

Then is non-increasing along both flow and jumps

and it’s not constant along any nonzero traj. GAS

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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 ‘05]

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QUANTIZED STATE FEEDBACK

QUANTIZER

CONTROLLER

PLANT

Hybrid quantized control: is discrete state

– zooming variable

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QUANTIZED STATE FEEDBACK

QUANTIZER

CONTROLLER

PLANT

Hybrid quantized control: is discrete state

Zoom out to overcome saturation

– zooming variable

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QUANTIZED STATE FEEDBACK

QUANTIZER

CONTROLLER

PLANT

Hybrid quantized control: is discrete state

After the ultimate bound is achieved,recompute partition for smaller region

Can recover global asymptotic stability

– zooming variable

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SMALL – GAIN ANALYSIS

quantization error

Zoom in:

where

ISS from to with gain small-gaincondition!

ISS from to with some linear gain

Can use quadratic Lyapunov functions to compute the gains

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CONCLUSIONS

• Basic idea: small-gain analysis tools are naturally

applicable to hybrid systems

• Main technical results: (weak) Lyapunov function

constructions for hybrid system interconnections

• Applications:

• Quantized feedback control

• Networked control systems

• Event-triggered control [Tabuada]

• Other ???