Model Reduction by

Moment Matching Alessandro Astolfi

Imperial College London and

University of Rome Tor Vergata

Inspired by and

dedicated to Alberto

The model reduction problem

Given a dynamical system determine a simpler model with essentially the same properties

•  Simpler: number of equations, complexity of functions, simulation time

•  Properties: frequency response, steady-state response for selected signals, oscillations, instabilities, I/O properties

Wiring in IC Performance

Ground Plane Logic Gate

Logic Gate

Wire and ground plane form a capacitor Wire has resistance

M’s of gates and km’s of wires: finite element approximation of wires

Reduced order model to reduce simulation time, development time

Reduced order model needs to preserve impedance

(from J. White, MIT)

Flow Around Sails

Discretized physical model with 15M states (dependent of wind speed and angle)

24 hours CPU time

30 Gbyte of RAM

32 parallel processors

Not for real-time applications!

(from A. Quarteroni, EPFL and Polimi)

Exploited by Alinghi in the 33rd America’s

Cup

System Model Approximation Properties

Jet engine Navier-Stokes eqs Finite-element Instabilities

MEMS resonator Euler-Lagrange eqs Maxwell eqs Finite-element Modes

Image processing RGB values Rank reduction Shapes

Russian service module (ISS) Euler-Lagrange eqs Finite-element Modes

Pulmonary circle Blood flow/vessel eqs Interface eqs Electrical equivalent Pressure

Wide hydraulic basins

Free surface eqs Transport eqs

Finite element/volume discretization Waves

Alloy solidification Stochastic transport eqs Finite element Concentration

Further examples

Model Reduction – The Big Picture

Data

Physical models

Artificial models

PDEs Discretization

ODEs Linearization

Model reduction

Simulation Prediction

Control

Model Reduction – The Big Picture

Linear systems

Nonlinear systems

Hankel norm approximation X X

Balanced truncation X X

Empirical Gramians X

H∞ model reduction X X

Moment matching X X

Petrov-Galerkin projections X

Proper orthogonal decomposition X

Error bounds/Stability Structural properties/Complexity

Goal of this presentation and tools

To develop a model reduction theory by moment matching for nonlinear systems

To re-visit the linear theory

Nonlinear regulator theory

Centre manifold

Interpolation theory

Frequency response of nonlinear systems

Projections Structural properties

Invariance

The key ingredient

Ideally:

The system (n dimensional)

The model (ν dimensional)

+

-

The key ingredient – Moment matching

is the order of the interpolation is the interpolation point

+

-

The notion of moment – Linear systems

0-moment at :

k-moment at :

The notion of moment – Linear systems

The interpolation point The system

Asymptotic stability

Observability

Steady state response

Moments

The notion of moment – Linear systems

The interpolation point The system

Steady state response

Moments

Alternatively

The notion of moment – Linear systems – Swapped

The system The interpolation point

Asymptotic stability

Controllability

Steady state response

Moments

The notion of moment – Linear systems – Summary

The system The interpolation point

Steady state response

Moments

Steady state response

Moments

Krylov projectors

The notion of moment – Nonlinear systems

The interpolation point The system

Asymptotic stability

Observability

Poisson stability

Moments

Steady state response

The notion of moment – Nonlinear systems

The interpolation point The system

Moments

Steady state response

The notion of moment – Nonlinear systems

The signal generator captures the requirement that one is interested in studying the behaviour of the system only in specific circumstances

The interconnected system possesses an invariant manifold and the dynamics restricted to the manifold are a copy of the

dynamics of the signal generator

The interpolation point The system

is by definition the moment of the nonlinear system at

Reduced order model

The system The model

Reduced order model – 1

Reduced order model – 2

Matching!

The reduced order model

Reduced order model – Construction – 1

The interpolation point The system Interpolation

points System to be

reduced

Reduced order model – Construction – 2

The interpolation point The system

The moment Compute the

moment

Select a mapping

Reduced order model – Construction – 3

The interpolation point The system

The moment

Reduced order model – Construction – 4

The interpolation point The system

The moment

Solve the matching equations

Reduced order model – Construction – 5

The interpolation point The system

The moment

The model Compute the

model

Reduced order model – Construction – 6

The interpolation point The system

The moment

The model

Reduced order model – Construction – 7

Select a mapping

A change of coordinates

A family of parameterized models achieving moment matching

Reduced order model – Construction – 8

Select a mapping

Reduced order model – Construction – 9

Select a mapping

A free mapping

The interpolation points

The moment

Reduced order model – Construction – 10 (end)

Select a mapping

A free mapping

The free mapping assigns

•  stability properties •  relative degree •  zero dynamics •  passivity properties •  gain properties •  error bounds •  monotonicy •  physical properties

A simple example – The Cuk converter

A simple example – The Cuk converter

The moment at 0

The reduced model

Some take-away messages

Accurate physical models are far too complex even for nowadays computer power

Reduced order models are essential for simulation, prediction, control, re-design

A truly nonlinear theory for nonlinear model reduction can be developed

... and to conclude

Thank you Alberto!