Willems - Dissipative Dynamical Structures

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DISSIPATIVE DYNAMICAL SYSTEMS

Jan C. Willems

ESAT-SCD (SISTA)

SISTA seminar, January 9, 2002

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THEME

Adissipative systemabsorbs supply (e.g., energy).

How do we formalize this?

Involves thestorage function.

How is it constructed? Is it unique?

KYP,LMIs,AREs.

Where is this notion applied in systems and control?

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OUTLINE

Lyapunov theory

Dissipative systems

Physical examples

Construction of the storage function

LQ theory LMIs, etc.

Applications in systems and control

Extensions

Polynomial spectral factorization

Recapitulation

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LYAPUNOV THEORY

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LYAPUNOV FUNCTIONS

Consider the classical dynamical system, the flow

with , thestate space, . Denote the set of

solutions by , thebehavior.The function

is said to be a Lyapunov function for if along

Equivalent to

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Typical Lyapunov theorem:

and for

there holds for global stability

Refinements:LaSalles invariance principle.

Converse: Kurzweils thm.

LQ theory Lyapunov (matrix)

equation. A linear system is stable iff it has a quadratic positive

definite Lyapunov function.

Basis for most stability results in control, physics, adaptation,

even numerical analysis, system identification.

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V

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V

X

Aleksandr Mikhailovich Lyapunov (1857-1918)

Studied mechanics, differential equations.

Introduced Lyapunovssecond methodin his Ph.D. thesis (1899).

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DISSIPATIVE SYSTEMS

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A much more appropriate starting point for the study of dynamics

areopen systems.

SYSTEM outputinput

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INPUT/STATE/OUTPUT SYSTEMS

Consider the dynamical system

: the input, output, state.

Behavior all solns

Let

be a function, called thesupply rate.

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models something like thepowerdelivered to the system

when the input value is and output value is .

supply

input

output

SYSTEM

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DISSIPATIVITY

is said to be dissipative w.r.t. the supply rate if

called the storage function, such that

along input/output/state trajectories ( .

This inequality is called the dissipation inequality.

Equivalent to

for all .

If equality holds:conservative system.

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Dissipativeness Increase in storage Supply.

SUPPLY

DISSIPATION

STORAGE

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Special case: closed system:

then dissipativeness is a Lyapunov function.

Dissipativity is a natural generalization of LF to open systems.

Stabilityfor closed systems Dissipativityfor open systems.

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PHYSICAL EXAMPLES

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System Supply Storage

Electrical

circuit voltage

current

energy in

capacitors and

inductors

Mechanical

system force, velocity: angle, torque

potential +

kinetic energy

Thermodynamic

system heat, work

internal

energy

Thermodynamic

system heat, temp.

entropy

etc. etc. etc.

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Electrical circuit:

(potential, current)

Dissipative w.r.t. (electrical power).

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Mechanical device:

(position, force, angle, torque)

Dissipative w.r.t. (mech. power).

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Thermodynamic system:

(work)

(heatflow, temperature)

Conservative w.r.t. ,

Dissipative w.r.t.

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THE CONSTRUCTION OF STORAGE FUNCTIONS

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Central question:

Given (a representation of ) ,the dynamics,and

given ,the supply rate,

is the system dissipative w.r.t. , i.e.,

does there exista storage function such that the

dissipation inequality holds?

Assume henceforth that a number of (reasonable) conditions hold:

;

Maps and functions (including ) smooth;State space of connected:

every state reachable from every other state;

Observability.

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Thm:Let and be given.

Then is dissipative w.r.t. iff

for allperiodic .

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Two universal storage functions:

1. The available storage

2. The required supply

Storage fns form convex set, every storage function satisfies

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LINEAR SYSTEMS with QUADRATIC SUPPLY RATES

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Assume linear, time-invariant,finite-dimensional:

and quadratic: e.g.,

E.g., for circuits , etc.

Assume controllable, observable., the transfer function of .

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Theorem:The following are equivalent:

1. is dissipative w.r.t. (i.e., there exists a storage function ),

2. ,

,

3. for all ,

4. a quadratic storage fn, ,

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5. there exists a solution to the

Linear Matrix Inequality (LMI)


Algebraic Riccati Inequality (ARIneq)


Algebraic Riccati Equation (ARE)

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Solution set (of LMI, ARineq) is convex, compact, and attains

itsinfimumand itssupremum:

These extreme solns and themselves satisfy the ARE.

Extensive theory, relation with other system representations,

many applications, well-understood (also algorithmically).

Connection with optimal LQ control, semi-definite programming.

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Important refinement:

Existence of a (i.e., bounded from below) (energy?)

In LQ case

,

Note def. of -norm !

soln to LMI, ARineq, ARE.

KYP-lemma.

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APPLICATIONS

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Synthesis of RLC-circuits

Robust stability and stabilization

Norm estimation

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Interconnection laws:

Interconnected system:

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Assume

dissipative, storage fn ,

dissipative, storage fn

.

For example,

,

;

or , .

Then is a Lyapunov functionfor the interconnected system .

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Proof:

Small gain theorem, Positive operator theorem,Robust stability.

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ROBUST STABILITY

: linear, time-invariant, transfer fn ,

:uncertain system,

e.g. memoryless: with

Then

stable

Quadratic LF: , from LMI, ARIneq, or ARE.

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Leads to:

SYSTEM

sensor output

control input

UNCERTAINPLANT

!! Stabilize robustly

Find a controller that stabilizes for a whole class of systems at once.

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sensor output

control input

output

CONTROLLER

input

PLANT

CLOSED LOOP SYSTEM

!! Givenplant , findcontroller such that

-control theory, synthesis of dissipative systems.

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Application:

SYSTEM NORM ESTIMATION

Model reduction:

Given a (linear, time-invariant) system , find a system ,

with a low dimensional state space, that approximates well.

Well small -norm of tf fn :

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+

_

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Some beautiful results have been obtained, in particular:

Balanced reductionof linear systems:

Let be a strictly proper, transfer function.

Then admits a representation

with controllable, observable, Hurwitz.

Moreover, can be made to bebalanced(controllability grammian = observability grammian):

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. . . . . .

. . . . . .

with theHankel SVsof the system.

Assume large, , small.

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Neglect (heuristic: these are the state components that

are bothmost difficult to reach and most difficult to observe).

With the obvious partitioning of as

we obtain the -orderreduced system

Call its transfer fn .

Question:

How close is to ?

!! Estimate = .

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Theorem (Glover):

with such that , where are

thedistinct Hankel SVsof the system.

bound sum of the neglected SVswithout repetition.

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Proof using dissipative systems:

Step 1: Neglect ONE (possibly repeated) SV:

. . .

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Consider the error system:

+

_

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Its dynamics:

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Relations between system parameters due to balancing:

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Now verify (straightforward, tedious):

Whence

Conclude,using LMI-theory,

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Step 2: Triangle inequality.In the obvious notation

where = the balanced representation truncated at :

neglect ; .

Whence

Combine step 1 and step 2:

Open problems: improve bound, find storage fn for .

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Generalizations

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Drawback 1: requires separation of interaction variables in

inputs and outputs

Behavioral systems.

Drawback 2: imposes storage function = state function.This is something one would like to prove!

Drawback 3: limited to dynamical (as opposed to distributed,

PDE) systems.

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Recap

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The notion of adissipative system:

Generalization of Lyapunov function toopensystems

Central concept in control theory: many applications to

feedback stability, robust ( -) control, adaptive control,

system identification, passivation control

Other applications: system norm estimates

passive electrical circuit synthesis procedures

Natural systems concept for the analysis of physical systems

Notable special case:second law of thermodynamics

Forms a tread through modern system theory

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More info, ms, copy sheets? Surf to

http://www.math.rug.nl/ willems

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THANK YOU !

Willems - Dissipative Dynamical Structures

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