model reduction for nonlinear daesintroduction trajectory piecewise linear (tpwl) proper ortogonal...

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Introduction Trajectory Piecewise Linear (TPWL) Proper Ortogonal Decomposition for DAEs

Model reduction fornonlinear DAEs

A. Verhoeven1,2 P. Astrid1 T. Voss2

[email protected]

1Eindhoven University of Technology (CASA)

2Philips Electronic Design & Tools / Analogue Simulation

MOR course, April 12, 2006



Outline

1 IntroductionModel reduction for nonlinear DAEsSurvey of existing nonlinear MOR algorithmsFormulation as optimization problem

2 Trajectory Piecewise Linear (TPWL)Introduction to TPWLError control of TPWL

3 Proper Ortogonal Decomposition for DAEsDescription of method for ODEsApplication to DAEsMissing Point Estimation



Introduction

Consider the following nonlinear DAE system

ddt q(t , x) + j(t , x) = Bu, x(0) = x0

y = Dx

We are interested in the input-output behaviour of this nonlinearsystem in the time-domain.We want to replace this model by the low-order model:

ddt q(t , x) + j(t , x) = Bu, x(0) = x0

y = Dx



Applications

If the nonlinear behaviour of the original model is essential.Optimization loop needs fast but accurate models.Verification of IC designs only considers the input-outputbehaviour.Design of controllers need models of low dimension.Models with different time-scalesApplications in electrical engineering, aeronauticengineering, mechanical engineering, etc.



Nonlinear balanced truncation IObservability and controllability functions

Lc(x0) = min{12

Z 0

−∞‖u(t)‖2dt : u ∈ L2(−∞, 0), x(−∞) = 0},

Lo(x0) =12

Z ∞

0‖y(t)‖2dt ,∀τ∈[0,∞)u(τ) = 0.

These functions can be solved from PDEs of Lyapunovand Hamilton-Jacobi type.The transformation x = φ(z) balances the system if

Lo(φ(z)) =12

zT z

Lc(φ(z)) =12

zT diag (τ1(z), . . . , τn(z)) z

∂Lc

∂zi(φ(z) = 0 ⇔ ∂Lo

∂zi(φ(z) = 0

where τ1(z) ≥ . . . ≥ τn(z) are the singular value functions.



Nonlinear balanced truncation II

If τn(. . . , zn) ≥ 0 the axis singular value functions ρi can bedefined by

τi(. . . , zi , . . .) = ρ2i (zi).

The Hankel norm for nonlinear systems equals

‖Σ‖2H = max

u∈L2[0,∞)

‖H(u)‖2

‖u‖2 = maxx

Lo(x)

Lc(x)= max

s{ρ1(s)}.

This balanced form can be used for nonlinear modelreduction.Stability is preserved.Generalization of balanced truncation



Trajectory Piecewise Linear (TPWL)

Nonlinear model is approximated by piecewise linearmodel along a trajectory;In each ball the linear model is reduced by linear MORtechniques, which results in a basis Pl ;Calculate the SVD of P = [P1, . . . , PN ] and define theglobal subspace as P = U(:, 1 : r).Create a weighted representation of the reduced model.Evaluation costs are also reduced!



POD

We need to compute snapsnots from the nonlinear system.A SVD is made from the correlation matrix;Then a global basis Φ is derived;Project the problem:

ddt Φ

T q(t ,Φx) + ΦT j(t ,Φx) = ΦT Bu, x(0) = x0y = DΦx

Notice that the evaluation costs for POD are not reduced!



Empirical balanced truncation (EBT)

The empirical observability and controllability Gramiansare derived from several snapshot series withrepresentative inputs and initial values.The global basis is derived by balancing these empiricalGramians.Same projection as POD.Input-output behaviour is taken into account;Very expensive model extraction



Volterra seriesThe method is only applicable if the system dependslinearly on the inputs.

x = f(x) + Bu(t).

A bilinear system for

x⊗ =

xx⊗ x

...

is constructed which approximates the first moments of thenonlinear system.

ddt

x⊗ = A⊗x⊗ +∑

i

N⊗i ui(t)x⊗ + B⊗u(t).

This bilinear system can be reduced by nested Krylovmethods, e.g. Arnoldi.The method is not applicable for DAEs because the pencilof the bilinear system is not regular.



Comparison of nonlinear MOR methods

For nonlinear DAEs only TPWL, POD and EBT arepossible.TPWL and EBT really match input-output behaviour, incontradiction to POD.POD and EBT need snapsnots of the nonlinear system,which can be expensive.Only TPWL also reduces the model evaluation costs!



Some literature of nonlinear MOR

Antoulas, Sorensen: General overview of MORScherpen, Fujimoto: Nonlinear balanced truncationWhite, Rewienski, Voss: TPWLAstrid, Wilcox, Petzold: PODLall, Marsden, Glavaski: EBTPhillips, Bai, Schetzen: Volterra series



Optimization problem

For all MOR techniques we are looking for a transformationx = φ(z), which balances the system or the input-outputbehaviour (Hankel operator).This means that for all r the first r elements of thetransformed system is automatically the optimal solution,which minimizes the reduction error! This can be the errorof the state or the input-output map.Such a balanced system can easily be reduced byselecting only the first r equations and variables.For linear systems one get always the linear transformationx = Pz, where P solves a system of Lyapunov equationsand can be approximated by Krylov methods.



Classification of MOR methods

Global transformations do not depend on t , in contradictionto local transformations. They balance the system on thecomplete interval. For nonlinear systems they always needsnapshots combined with a SVD in order to get a globalbasis.Local transformations balance the system along a shorttime interval on which the system matrices can beassumed to be constant! They do not need empirical databut apply linear methods to the locally linearized system.



Outline






Introduction

The system is linearized at some linearization tuples (LT)(xli , tli ). In each region the linearized systems are reduced bylinear model reduction techniques. The LT are selected at atrajectory of typical initial values and inputs. Afterwards aweighted sum is constructed of all reduced linearized systems.



TPWL sequence I

Construct s linear reduced systems along a trajectory

Cli x + Gli x + Bli u(t) = 0.

Reduce each system with a linear MOR technique to getPli which represents the reduced subspace of the i-thsystem.Calculate a SVD of P =

[Pl1 , . . . , Pls

]= USVT and define

the global subspace as P := U(:, 1 : r).Reduce each linearized system with P

PT Cli Py + PT Gli Py + PT Bli u(t) = 0.



TPWL sequence II

Create a weighted representation

s−1∑i=0

wi(y, t) (Cir y + Gir y + Bir u(t)) = 0,

where Cir = PT Cli P, Gir = PT Gli P, Bir = PT Bli and wi(y, t)is a weighting function which describes how muchinfluence the i-th reduced linearized system has on theactual state.



Linear model order reduction

Krylov These methods match a certain number of moments ofthe transfer function. For this type it seems that PRIMAis the best one because of preserving stability andpassivity. All Krylov based methods are very fast butcreate too large subspaces.

TBR This method balances the system first and thentruncate the modes which are hardly to observe or tocontrol. In theory this method should perform very wellbut in practice there are a lot of problems in the case ofDAE models.

Poor Mans TBR This is a combination of multi point rationalinterpolation and TBR. Therefore the transfer functionis calculated at several frequencies and then a SVD isused to find an optimal subspace. This method isslower than Krylov but creates better approximations.



Linearization tuple controller

Error of TPWL model consists of linearization andreduction error.Locations of linearization tuples determine the linearizationerror.These tuples can be created during a rough transientsimulation, but with the restriction that this rough solutionlies in the neighbourhood of the exact solution.



Weighting

Weighting is needed to combine the local reduced systemsto a global linear system.Distance di = ‖y− yli‖ and minimum distance m.Weighting function:

wi(y, t) =1S

e−γdim .

Extended weighting using the Hessians of q and j.

di =12

Hj,i‖Py− xi‖2 + Hq,i‖Py− xi‖‖Py‖,

whereHj,i ≥ ‖ ∂2

∂x2 j(t , x)‖, Hq,i ≥ ‖ ∂2

∂x2 q(t , x)‖.



Inverter chain model

The state of the inverter chain model comprises of 100nodal voltages and some electrical currents. In total, thereare 104 state variables.The excitation signal Uop and the dynamical response ofthe inverter chain.



Numerical results of TPWL

Proper numerical results for r = 50.The higher accuracy of PMTBR can be used to get smallermodels.Reduced models rather alre still valid for different inputs.Linearization tuples are directly computed during a roughtransient simulation before.Weighting using the norms of the Hessians gives the bestresults.



Outline






Introduction

POD constructs a subspace which should consist the mostdominant part of the state of the nonlinear system.In contradiction to TPWL and EBT the state itself isapproximated.Snapshots are needed to create the reduced basis.In literature POD is only applied to ODEs.POD could also be applied in a trajectory piecewiseapproach like TPWL.



POD procedure

Collection of simulation (N snapshots)

Tsnap = [t(1), . . . , t(N)] .

Construction of correlation matrix M = 1N TsnapTT

snap.Eigenvalue decomposition of correlation matrix:

M = UΣUT .

Galerkin projection by Φ = U(:, 1 : r):

ddt Φ

T q(t ,Φx) + ΦT j(t ,Φx) = ΦT Bu, x(0) = x0y = DΦx



Problems of POD

Reduced DAEs are not always solvable in contradiction toODEs.Function evaluation costs are not reduced at all.



Least squares approach

Instead of solving ddt Φ

T q(t ,Φx) + ΦT j(t ,Φx) = ΦT Bu weconsider the problem:

ddt

q(t ,Φx) + j(t ,Φx) = 0. (1)

The (weak) solution x should minimize the residual of thisequation.Least squares system

ΦTn MT MΦny = ΦT

n MT b.

Now the reduced DAE for POD is always solvable.



Missing Point Estimation

Function evaluation ΦTn f(Φnx) is cheap if Φn is a

permutation matrix.Let P ∈ {0, 1}G×K be a permutation matrix where G << Kand define the restricted basis Φn as

Φn = PΦn. (2)

Combinatorial optimization problem

find Psuch that ‖

(Φ>n P>PΦn

)−1 − In‖ < TOLsubject to PP> = I

pij ∈ {0, 1}

(3)

Approximation: cond(Φ>n P>PΦn) < TOL.



Numerical results of POD

The POD basis Φ is found by solving the eigenvalue problemfor the correlation matrix.

The POD basis Φ comprises the eigenvectors corresponding to20 largest eigenvalues.



Comparison of TPWL and POD

For the same basis the TPWL model is much cheaper toevaluate than POD because all local Jacobians are alreadyreduced and stored. This is also the case if MPE is used.Therefore TPWL needs much more storage than POD.For very nonlinear models TPWL needs a lot oflinearization tuples, which implies a large reduced basis.In general the POD basis can be of smaller size, becausePOD directly projects the nonlinear model and only usesthe data of one simulation.TPWL will be more robust with respect to changes in theparameters and inputs.

model reduction for nonlinear daesintroduction trajectory piecewise linear (tpwl) proper ortogonal...

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