an introduction to model order reduction
TRANSCRIPT
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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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
Outline
1 Motivation & Objectives
2 Why MOR Works?
3 Moment Matching
4 Krylov Subspace Methods
5 Issues
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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 & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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 & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
Applications
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
Why MOR Works?
x ≈ Vz
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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.
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Classification of MOR Methods
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Classification of MOR Methods
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’
Issue: GS orthogonalization is also unstableRemedy: Use Arnoldi/Lanczos methods.
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’Issue: GS orthogonalization is also unstable
Remedy: Use Arnoldi/Lanczos methods.
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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-conditioningRemedy: To orthogonalize the vectors using‘Gram-Schmidt Orthogonalization’Issue: GS orthogonalization is also unstableRemedy: Use Arnoldi/Lanczos methods.
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
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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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
Advanced Issues in MOR
Parametric Model Order Reduction (pMOR)Structure Preserving MORMOR of Nonlinear SystemsCombinations
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Motivation & Objectives Why MOR Works? Moment Matching Krylov Subspace Methods Issues
Thanks!
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