«NEW PARADIGMS FOR CONTROL THEORY»

«NEW PARADIGMS«NEW PARADIGMS FOR CONTROL FOR CONTROL THEORY»THEORY»

«NEW PARADIGMS«NEW PARADIGMS FOR CONTROL FOR CONTROL THEORY»THEORY»

Romeo OrtegaLSS-CNRS-SUPELEC

Gif-sur-Yvette, France

ContentContentContentContent

BackgroundProposalExamples

FactsFactsFactsFactsModern (model-based) control theory is not providing solutions to new practical control problems

Prevailing trend in applications: data-based « solutions »

Neural networks, fuzzy controllers, etc They might work but we will not understand why/when

New applications are truly multidomain

There is some structure hidden in «complex systems »

Revealed through physical laws Pattern of interconnection is more important than detail

Why?Why?Why?Why?Signal processing viewpoint is not adequate:

= Input-Output-Reference-Disturbance.

Classical assumptions not valid: linear + «small » nonlinearities interconnections with large impedances time-scale separations lumped effects

Methods focus on stability (of a set of given ODEs)

no consideration of the physical nature of the model.

ProposalProposalProposalProposalReconcile modelling with, and incorporate energy information into, control design.

How?How?Propose models that capture main physical ingredients:

energy, dissipation, interconnection

Attain classical control objectives (stability, performance) as by-products of:

Energy-shaping, interconnection and damping assignment.

Confront, via experimentation, the proposal with current practice.

Prevailing paradigmPrevailing paradigmPrevailing paradigmPrevailing paradigm

• Models

• Control objectives

• Controller design

Signal procesing viewpoint

Models

C

Pd

u

z

yzd

r s

P

r

du

s

zy

:

Uncertainty

Known structure,

• •

RHd D L

Control objectives z-zd « small »

effect of d on z « small »

Controller

y

zu:C

d

Class of admissible systems TOO LARGE !!

Conservativeness (min max designs)

High gain (sliding modes, backstepping…)

Complexity

Practically useless

Intrinsic to signal-processing viewpoint

Drawbacks!!!Drawbacks!!!

C I

Unmodelled

environment

i i i

x

c

ec

e

v v v

(Energy-based) Control by interconnection

Proposed alternativeProposed alternativeProposed alternativeProposed alternative

Models

PLANT:

• H(x) energy function, x state,

• (v,i) conjugated port variables,

• Geometric (Dirac) structure capturing energy exchange

• Dissipation

ENVIRONMENT:

• Passive port

• Flexibility and dissipation effects

• Parasitic dynamics

Control objectives

• Focus on energy and dissipation

• Shape and exchange pattern

Controller

C

I

• controller, Hc(z) energy

• power preserving

IDA-PBC of mechanical IDA-PBC of mechanical systemssystemsIDA-PBC of mechanical IDA-PBC of mechanical systemssystems

To stabilize some underactuated mechanical devices it is necessary to modify the total energy function. In open loop

Where qRn, pRn are the generalized position and momenta, respectively, M(q)=MT(q)>0 is the inertia matrix, and V(q) is the potential energy

MODELMODEL

Control uRm, and assume rank(G)=m < nConvenient to decompose u=ues(q,p)+udi(q,p)

TARGET TARGET DYNAMICSDYNAMICS

Desired (closed loop) energy function

where Md=MdT>0 and Vd(q)

with port controlled Hamiltonian dynamics

where

All assignable energy All assignable energy functions are characterized by a functions are characterized by a PDE!!PDE!!

All assignable energy All assignable energy functions are characterized by a functions are characterized by a PDE!!PDE!!

The PDE is parameterized by The PDE is parameterized by two free matrices (related to two free matrices (related to physics)physics)

ExamplesExamplesExamplesExamples

BALL AND BEAMBALL AND BEAM

Ball and BeamBall and BeamBall and BeamBall and Beam

Ball and BeamBall and BeamBall and BeamBall and Beam

Vertical take-off and landing Vertical take-off and landing aircraftaircraftVertical take-off and landing Vertical take-off and landing aircraftaircraft

Cart with inverted pendulumCart with inverted pendulumCart with inverted pendulumCart with inverted pendulum

ExamplesExamplesExamplesExamples(PASSIVE) (PASSIVE) WALKING WALKING

• Plant: double pendulum

• Environement:

elastic (stiff)

Model

(Passive) walking(Passive) walking(Passive) walking(Passive) walking

Control objetive:

Shape energy

(Passive) walking(Passive) walking(Passive) walking(Passive) walking

(Passive) walking(Passive) walking(Passive) walking(Passive) walking

other other mechatronic mechatronic systems: systems:

teleoperators, robots in interaction (with environement)

• Plant: (controlled) wave eq.

• Environment: passive mech. contact

• model

• control objective: shape energy

Piezoelectric actuatorsPiezoelectric actuatorsPiezoelectric actuatorsPiezoelectric actuators

Control through long cablesControl through long cablesControl through long cablesControl through long cablesE.g., overvoltage in drivesE.g., overvoltage in drives

• model

• control objective: change interconnection to suppress waves

Dual to teleoperators

Many examples in power electronics and power systems

Thank Thank you!!you!!Thank Thank you!!you!!