An Introduction to Model Order Reduction

Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues

An Introduction to Model Order Reduction

Mohammad Umar RehmanResearch Scholar

Control & Automation Group

Electrical Engineering DepartmentIIT Delhi

March 15, 2016

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Outline

1 Motivation & Objectives

2 Why MOR Works?

3 Moment Matching

4 Krylov Subspace Methods

5 Issues

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Motivation

Mathematical modeling is integral part of control systemsengineering wherein the aim is to study dynamicalsystems.Rising complexity in the systems gives rise to need ofmore accurate mathematical models.Lumped Parameter Systems–ODE, Distributed parametersystems–PDEPDE–FEM Discretization–Large no. of ODEsControl Design and simulation becomes computationallytedious.

Rescue path: Model Order Reduction

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Motivation

Mathematical modeling is integral part of control systemsengineering wherein the aim is to study dynamicalsystems.Rising complexity in the systems gives rise to need ofmore accurate mathematical models.Lumped Parameter Systems–ODE, Distributed parametersystems–PDEPDE–FEM Discretization–Large no. of ODEsControl Design and simulation becomes computationallytedious.Rescue path: Model Order Reduction

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MOR Objectives

High fidelity representation of the original large scalesystem.Considerable difference between the size of ROM andoriginal model.Small approximation error and/or global error bound.Preservation of system properties like stability/passivity.Numerically stable & efficient procedure.

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Applications

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Why MOR Works?

The fact that large scale dynamical systems are poorlycontrollable and observable becomes the workforce behindMOR.Dynamics is often restricted to a smaller subspace.MOR methods for such systems are essentially differentways to find this dominant subspace and Project the largesystem dynamics onto it.

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Why MOR Works?

x ≈ Vz

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Illustration

x = Vz, A = V−1AV, B = V−1B, C = CV

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Finding The Projection ‘Matrix’ V

How the matrix V is determined, gives rise to various methodsBalanced TruncationKrylov SubspaceProper Orthogonal Decomposition (for non-linear systems)etc.

Classification of MOR Methods


Moment Matching

Moments (Definition): The coefficients of the Taylor Series expansionof a transfer function are known as moments Given a TF H(s), it canbe expanded around a point s = s0 as:

H(s) = η0 + η1(s− s0) + η2(s− s0)2 + . . .+ ηk(s− s0)

k + . . .

where,

ηk = (−1)k dk

dsk H(s)∣∣∣s=s0

Moment matching aims to determine a lower order approximant H(s)of the original system H(s) such that ‘r’ moments of these match – rth

order approximation.

Explicit Moment Matching – numerically unstableIt is possible to achieve moment matching without explicitly computingthem, through the introduction of the Krylov subspaces.

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Moment Matching


H(s) = η0 + η1(s− s0) + η2(s− s0)2 + . . .+ ηk(s− s0)

k + . . .

where,

ηk = (−1)k dk



order approximation.Explicit Moment Matching

– numerically unstableIt is possible to achieve moment matching without explicitly computingthem, through the introduction of the Krylov subspaces.

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Moment Matching


H(s) = η0 + η1(s− s0) + η2(s− s0)2 + . . .+ ηk(s− s0)

k + . . .

where,

ηk = (−1)k dk



order approximation.Explicit Moment Matching – numerically unstableIt is possible to achieve moment matching without explicitly computingthem, through the introduction of the Krylov subspaces.

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Implicit Moment Matching:Krylov Subspace Method

Krylov Subspace is defined as:

Kn(A, b) = span[b,Ab,A2b, . . . ,An−1b

]Similar to Controllability Matrix in control.Krylov vectors readily align themselves along the dominanteigenvector – ill-conditioning

Remedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’Issue: GS orthogonalization is also unstableRemedy: Use Arnoldi/Lanczos methods.

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]Similar to Controllability Matrix in control.Krylov vectors readily align themselves along the dominanteigenvector – ill-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’

Issue: GS orthogonalization is also unstableRemedy: Use Arnoldi/Lanczos methods.

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]Similar to Controllability Matrix in control.Krylov vectors readily align themselves along the dominanteigenvector – ill-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’Issue: GS orthogonalization is also unstable

Remedy: Use Arnoldi/Lanczos methods.

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]Similar to Controllability Matrix in control.Krylov vectors readily align themselves along the dominanteigenvector – ill-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’Issue: GS orthogonalization is also unstableRemedy: Use Arnoldi/Lanczos methods.

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Arnoldi Method

Algorithm 1 Arnoldi1: Choose inital vector v, normalize it2: for j = 1 to m do3: hij = 〈Avj, vi〉 i = 1, 2, . . . , j4: wj = Avj −

∑ji=1 hijvj Gram Schmidt

5: hj+1,j = ‖w‖2, if hj+1,j = 0 stop6: vj+1 =

wjhj+1,j

7: end for

k-step Arnoldi applied to the pair (A,B) produces a matrix withorthonormal columns Vk. Using Vk as the projection matrix,moments up to order k can be matched.

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Remarks

Krylov based methods have lower number of operationsand memory requirements than conventional SVDmethods.Computational cost is also low as only matrix-vectormultiplications are required — no matrix factorizations orinversions are involved.Applicable to MIMO systems.

However,Suffer with loss of orthogonality,due to instability of theclassical Gram-Schmidt procedure.The remedy lies inre-orthogonalization.Stopping point of the iterative scheme is also an issue.

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Remarks

Krylov based methods have lower number of operationsand memory requirements than conventional SVDmethods.Computational cost is also low as only matrix-vectormultiplications are required — no matrix factorizations orinversions are involved.Applicable to MIMO systems.However,Suffer with loss of orthogonality,due to instability of theclassical Gram-Schmidt procedure.The remedy lies inre-orthogonalization.Stopping point of the iterative scheme is also an issue.

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Advanced Issues in MOR

Parametric Model Order Reduction (pMOR)Structure Preserving MORMOR of Nonlinear SystemsCombinations

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Thanks!

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An Introduction to Model Order Reduction

Engineering

Transcript of An Introduction to Model Order Reduction