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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!
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