IMPROVED TPWL BASEDNONLINEAR MOR FOR FAST

SIMULATION OF LARGE CIRCUITSAmmu ChathukulamDebashree Sarkar

Shifali KalraM.Nabi

Department of Electrical EngineeringIndian Institute of Technology Delhi

New Delhi, IndiaEmail: achathukulam@gmail.com

KEYWORDS

Model Order Reduction(MOR); Trajectory Piece-wise Linearisation(TPWL); Principal Angles; Lineari-sation Points(LPs)

ABSTRACT

Trajectory Piecewise Linearisation(TPWL) is oneof the nonlinear Model Order Reduction(MOR) tech-niques that involves multiple linearisations around suit-ably selected linearisation points(LPs). Selection of lin-earisation points play a major role in the accuracy ofthe reduced order model in TPWL technique. Standardlinearisation point selection methods based on uniformdistance criteria are found to be less efficient. This pa-per proposes a new adaptive scheme for the selectionof LPs by taking into account of the ‘sensitivity’ ofthe nonlinear system towards the state. The proposedmethod has been implemented on two nonlinear bench-mark problems, nonlinear transmission line circuit andnonlinear chain of inverter circuit and the simulationresults are compared with the conventional schemes.

INTRODUCTION

Mathematical modelling of physical systems resultsin systems of very large order. The storage and sim-ulation of these higher order systems is a challengedue to the huge requirement of memory and time[1].This points towards the necessity of model order re-duction(MOR) where the reduced order models withsimiliar input-output mappings are generated.

Common MOR methods for nonlinear systems areProper Orthogonal Decomposition(POD)[2], DEIM(Discrete Empirical Interpolation)[3], Linear or Polynomialexpansion of the system nonlinearity[4], bi-linearisation[5], or Volterra series expansion [6] and TrajectoryPiecewise Linearisation(TPWL).

The idea of TPWL was first introduced by MichalRewienski and Jacob White[7] in which it wasimplemented on a nonlinear transmission line cir-cuit model and micromachined switch[8][9]. Laterthis method was applied to diverse areas including

fluid dynamics[15],nonlinear electronic circuit simula-tion[17][13][14],electromagnetics[16], etc.Since this method involves considering multiple lineari-sations about suitably selected states of the system,judicious location of LPs play an important role in theaccuracy of the generated reduced models. The LPselection methods proposed by Rewienski[7] takes dis-tances from the previously selected LPs as the criete-ria for location of LPs on either the exact or the ap-proximate trajectory. This method was improved byS.A. Nahvi[18] giving the idea of FAS which includes a‘coarse run’ over the whole trajectory, followed by selec-tive refinement in areas where more LPs are required.

A new method has been proposed in this paper thatadaptively decide the placement of the LPs along thetrajectory. This is inspired by[11], where the authorshave used adaptive sampling technique in parametriclinear MOR with the example of a linear beam. In thispaper, this adaptive sampling idea is incoorporated intoTPWL method to locate the LPs. The regions wherethe nonlinear system is more sensitive to the state areidentified based on the concept of distance between thesubspaces or principal angles.The proposed method hasbeen implemented on two nonlinear benchmark prob-lems,nonlinear transmission line circuit and nonlinearchain of inverter circuit and the simulation results arecompared with the conventional schemes.

The next two sections describe the basic TPWL pro-cess and the conventional methods for LP selection andtheir drawbacks respectively. The last section intro-duces the proposed adaptive scheme followed by its im-plementation on benchmark problems and error com-parison.

TRAJECTORY PIECEWISELINEARISATION

The TPWL process includes the following steps:

• The higher order system is simulated for a traininginput to obtain the nonlinear system trajectory.• Location of suitable Linearisation Points along theexact or approximate trajectory[7] followed by the sub-

Proceedings 32nd European Conference on Modelling and Simulation ©ECMS Lars Nolle, Alexandra Burger, Christoph Tholen, Jens Werner, Jens Wellhausen (Editors) ISBN: 978-0-9932440-6-3/ ISBN: 978-0-9932440-7-0 (CD)

sequent linearisations around these LPs.• Finding out the dominant subspace and projectingthe local-submodels into this dominant subspace.• Combine the local reduced models using properweight assignment stategies to form the final reducedmodel.

Mathematical Formulation

For the nonlinear dynamical system in the followingstate space form:

dx

dt= f(x) +Bu (1)

y = Cx (2)

Where x ∈ Rn is a vector of system states, f : Rn →Rn is the nonlinear vector field, B ∈ Rn×p is the inputmatrix, C ∈ Rq×n is the output matrix, u ∈ Rk is theinput and y ∈ Rq is the output. The Taylor series first-order approximation of f(x) about an initial state x0is given by

f(x) = f(x0) +A0(x− x0) (3)

Where A0 is the Jacobian of f(x) evaluated at x0.Clearly, the error in this approximation is:

e(x) = f(x)− f(x) (4)

The linearised system at x0 is hence

dx

dt= f(x0) +A0(x− x0) +Bu (5)

y = Cx (6)

The dynamics are restricted to the dominant subspaceby the state transformation x = V z. V ∈ Rn×r isthe projection matrix spanning the dominant subspace,z ∈ Rr, V TV = I and r n. The reduced model atx0 is:

dz

dt= A0rz + V T (f(x0)−A0x0)) +Bru (7)

y = Crz (8)

where A0r = V TA0V,Br = V TB, and Cr = CV .Assuming m similar linearised models generated alongthe training trajectory about the states x0, ..xi, ..xm−1,the final TPWL model is expressed as the weightedsum:

dz

dt=

∑m−1i=0 wi(z)(Airz + V T (f(xi)−Aixi)) +Bru(9)

y = Crz (10)

where Ai is the Jacobian of f(x) evaluated at xi andAir = V TAiV .

CONVENTIONAL METHODS FOR LPSELECTION

Uniform Division of Trajectory

[10]The first proposed and most widely used meth-ods for LP selection in TPWL based MOR are the ones

which divide the system trajectory uniformly to createlinearised systems. Two such methods, very similar toeach other, but differing in whether the nonlinear sys-tem trajectory is exact or approximate, are given in[7]. The first method can be found in [7, p.45]. Init, given a training input u(t) and an initial state x0,the nonlinear system is simulated and a finite numberof linear systems are generated on its trajectory. Thelinear systems are created by a uniform division of thetrajectory at a constant, pre-selected euclidean distanceδ. The second method given in [7, p.53], is similar tofirst, but it circumvents simulating the full-order non-linear system. It is called Fast Approximate Simulation(FAS), and in it successively selected sub-models of thenonlinear system are simulated to get an approximatetrajectory.

The uniform division of the trajectory suffers fromtwo major drawbacks as pointed out in[12]. The firstone is the heuristic nature of the δ being chosen. Se-lection of δ is mainly a game of ‘hit and trial’ and isunrelated with the nonlinearity involved. Moreover,these algorithms rely on the fact that the linearisationof a nonlinear function f at the state xi is an accurateenough approximation at some other state x, providedthat x is close enough to xi, i.e., ||x − xi|| < δ, or xlies within a ball of radius δ centered at xi. The pro-cedure contains an implicit assumption that dividingthe trajectory uniformly would lead to acceptable sam-pling. In other words, it is assumed that δ is a constantthroughout the trajectory, which is not necessarily true.There can be situations in which the trust regions don’toverlap resulting in inadequate sampling and the trustregion around one point covering the other trust regionresulting in over-sampling as shown in Fig[1],[2].

Fig. 1. The trajectory is inadequately sampled between the twotrust regions

PROPOSED METHOD FOR ADAPTIVESELECTION OF LPS

Uniform Division of the trajectory has proven to bea less efficient method for LP selection because of theabove mentioned reasons. This section presents an al-gorithm to adaptively decide the number and place-ment of the linearisation points along the trajectory by

Fig. 2. The trust region around x covers x1

placing more LPs in highly sensitive zones and lightlygridding the less sensitive zones. The high and low‘sensitive’ regions are identified based on the conceptof distance between the subspaces or principal angles.

Principal Angle Concept

Let V1 and V2 be the orthonormal projection matri-ces at the linearization points x1 and x2. The largestsubspace angle across these two points is computed as

θ12 = arcsin(√

(1− σ2r)) = arccos(σr)

σr is the smallest singular value of V T1 V2.[11]

Fig. 3. Principal angles between V1, V2, V3

Higher the angle between the subspaces, higher thesensitivity of the ROM and hence more LPs should beconsidered. Hence we have exploited the concept ofsubspace angles as a crieteria for deciding the selec-tion the linearization points.This method hence assuresmore judicious placement of the LPs when comparedwith the uniform trajectory division.The new schemefor adaptive selection of LPs first perform a rough uni-form sampling by selectiong a very high value of δ fol-lowed by the refinement in those regions where the sys-tem is more sensitive to state. In other words, themethod adds more LPs to those regions where the sub-space angle is more than θmax, the maximum tolerable

subspace angles. The proposed adaptive sampling al-gorithm is given in the following section.

Automatic Adaptive Sampling Algorithm

• Input θmax, maximum error between the sub-space angles that can be tolerated.• Select linearization points x1, x2, .....xk by ini-tial rough sampling by selecting a very highvalue for δ.• While all li,i+1 > 1a) Calculate the projection matrices V1, V2...Vkcorresponding to each of these valuesx1, x2, .....xk.b) Compute subspace anglesV12, V23, .......Vk−1,k between these Vis, eachtaken pairwise.c) Calculate l12 = θ12

θmax, l23 = θ23

θmax,.......

d) Divide the interval between x1 and x2 intol12 further intervals. Likewise do the same forall the other intervals.

IMPLEMENTATION ON NONLINEARBENCHMARK PROBLEMS

The adaptive method for LP selection is implementedon the two nonlinear bench mark problems ie, Nonlin-ear Transmission Line and Inverter Chain Circuit. Theresults are compared with the LP selection method us-ing uniform division of trajectory.

Example-1 : Nonlinear Transmission Line

Fig. 4. Nonlinear transmission line circuit

The first example considered is a nonlinear trans-mission line circuit model shown in Figure 4. The cir-cuit consists of resistors, capacitors, and diodes witha constitutive equation id(v) = exp(40v) − 1, wherev is the voltage between diodes terminals. For sim-plicity we assume that all the resistors and capaci-tors have unit resistance and capacitance, respectively(R1 = R2 = ..... = Rn = 1, C1 = C2 = .... = Cn = 1).The input is the current source entering at node 1:u(t) = i(t), and the output is chosen to be the voltageat node 1: y(t) = v1(t). Using constitutive relationsfor capacitors, resistors, and diodes, as well as kirch-hoffs current law, we obtain an input affine nonlineardynamical system given by equations 1 and 2. This

higher order nonlinear circuit is reduced using TPWLmethod with adaptive selection of LPs and the resultsare compared with the uniform trajectory division case.In both the cases the reduced order and the number ofLPs are kept same. In this case, the value of δ is 1 foruniform trajectory division and while using adaptivecase the trajectory is first roughly sampled with δ=3and then used the adaptive algorithm keeping maxi-mum error tolerance angle, θmax as 5. As a result, thenumber of LPs came out to be 5 in both the cases. Theresults show that the proposed method gives only 4.2%error at the output voltage compared to 6.74% error inthe conventional case.

Error Analysis

The error in the output voltage between Full OrderModel(FOM) and Reduced Order Model(ROM)

e(t) = (yr(t)− y(t)) (11)

where yr and y are the output voltages of ROM andFOM respectively. The absolute error for the entiretime span can be calculated as

E =

∫ T

0

|e(t)|dt (12)

Using the above equations, the absolute error for uni-form trajectory division, Eu = 0.6233 V while in theadaptive case, the error is Ea = 0.3129 V, clearly show-ing Ea < Eu.The output voltages for both the cases are plotted inFig 5 and 6 along with the FOM response. It is clearlyvisible from the graphs that the error peak during time3.5 sec to 4.5 sec is reduced drastically in adaptive sam-pling case.

Fig. 5. TPWL Using Uniform Division of Trajectory for Non-linear Transmission Line Model

Example-2 : Inverter Chain Circuit

The inverter circuit shown in Fig.7 is a nonlinearcircuit consisting of an input voltage source, capacitors,

Fig. 6. TPWL Using Adaptive Selection of LP for NonlinearTransmission Line Model

Fig. 7. Inverter Chain Circuit

resistors, and inverters whose input-output relation is

Vout = f(Vin) = Vddtanh(AVin) (13)

where Vdd is the supply voltage and A is a parameter.We choose Vdd = 1 and A = 5. Input is the voltagesource u(t) and output is the voltage at node 1, v1(t).The states are the corresponding voltage at each node.The parameter values are selected as R1 = R2 = ..... =Rn = 1 Ω, C1 = C2 = .... = Cn = 1 F. With theabove given values the circuit has input affine form asgiven by equations 1 and 2. This higher order nonlinearcircuit is reduced using TPWL method with adaptiveselection of LPs and the results are compared with theuniform trajectory division case. In both the cases thereduced order and the number of LPs are kept same.In this case, the value of δ is 6 for uniform trajectorydivision and while using adaptive case the trajectoryis first roughly sampled with δ=9 and then used theadaptive algorithm keeping maximum error toleranceangle, θmax as 10. As a result, the number of LPs cameout to be 4 in both the cases. By using the proposedmethod we are able to reduce the percentage error atthe output from 11.5% to 9% without increasing thenumber of LPs. Hence we can conclude that, in thiscase also the adaptive sampling with Ea = 36.2522 Vgives a less erroneous ROM as compared to uniformdivision with Eu = 54.8893 V.

Fig. 8. TPWL using Adaptive Selection of LPs

Fig. 9. Absolute error in the output voltage between the FOMand ROM for adaptive LP selection method and uniform trajec-tory division method

NL transmission line Chain of invertersAdaptive Uniform Adaptive Uniform

Full Order 100 100 100 100Reduced Order 10 10 20 20

% Error in Output 4.2 6.74 9 11.5No of LPs 5 5 4 4

Online Simulation Time 0.1s 0.1s 0.1s 0.1s

Table 1: Comparison of proposed method with uni-form selection of LPs for nonlinear transmission lineand chain of inverter circuit.

CONCLUSION

This paper presents an improved TPWL method fornonlinear MOR in which the LPs are placed more accu-rately. The adaptive LP selection algorithm takes intoaccount of the ‘sensitivity’ of the nonlinear system to-wards state removing the uncertainities associated withuniform division methods. Hence it is able to reduce thepercentage error in the output between FOM and ROMcompared to unform trajectory division by judiciouslylocating the LPs on the trajectory.The new method isthus able to develop more efficient reduced order models

without increasing the number of linearisation pointsas seen from the simulation results of the two nonlin-ear benchmark problems. The proposed method canbe applied to more nonlinear higher order systems andhence fast simulation of such systems can be achievedmore efficiently.

References

[1] P. Benner, “Solving large-scale control problems,” IEEEControl Systems Magazine, vol.24, no.1, pp. 44-59, Feb. 2004.

[2] P. Astrid, Reduction of Process Simulation Models: A ProperOrthogonal Decomposition Approach, PhD thesis, Depart-ment of Electrical Engineering, Eindhoven University ofTechnology, Eindhoven, The Netherlands, 2004.

[3] S. Chaturantabut, D.C. Sorensen, Discrete empirical interpo-lation for nonlinear model reduction, Report TR09-05, RiceUniversity (2009)

[4] Y. Chen, Model Order Reduction for Nonlinear Systems,M.S. dissertation, MIT, Cambridge, MA, 1999.

[5] Z. Bai, “Krylov subspace techniques for reduced-order mod-eling of largescale dynamical systems”, Applied NumericalMathematics, vol. 43, no. 1-2, pp. 9-44, 2002.

[6] J.R. Phillips, “Projection frameworks for model reduction ofweakly nonlinear systems”, Proc. Design Automation Con-ference, vol. 37, pp.184-189, 2000.

[7] M. Rewienski, A trajectory piecewise-linear approach tomodel order reduction of nonlinear dynamical systems, PhDThesis, MIT, Cambridge, MA, June 2003.

[8] M. Rewienski and J. K. White, “A trajectory piecewise-linear approach to model order reduction and fast simulationof nonlinear circuits and micromachined devices”, Proc. ofIEEE/ACM International Conference on Computer Aided-Design, pp. 252-7, San Jose, CA, USA, November 2001.

[9] M. Rewienski and J. White, “A trajectory piecewise-linearapproach to model order reduction and fast simulation ofnonlinear circuits and micromachined devices”, IEEE Trans.Computer-Aided Design, vol. 22, no. 2, pp. 155-70, Feb 2003.

[10] Shahkar Ahmad Nahvi, Mashuq-un-Nabi and S. Janard-hanan, “Trajectory Based Methods for Nonlinear MOR:Review and Perspectives”, Proc. of the 2012 IEEE Inter-national Conference on Signal Processing, Computing andControl, Shimla, India, March 2012, pp. 1-6.

[11] Maria Cruz Varona, Mashuq-un-Nabi and Boris Lohmann,“Automatic Adaptive Sampling in Parametric Model Or-der Reduction by Matrix Interpolation”, 2017 IEEE Inter-national Conference on Advanced Intelligent Mechatronics(AIM), Sheraton Arabella Park Hotel, Munich, Germany,July 3-7, 2017

[12] Shahkar Ahmad Nahvi, Reduced Order Modelling and fastsimulation Strategies for nonlinear Dynamical Systems, PhdThesis, IITD

[13] T. Voss, R. Pulch, E. Maten, and A. Guennouni, “Tra-jectory piecewise linear approach for nonlinear differential-algebraic equations in circuit simulation”, Scientific Comput-ing in Electrical Engineering, Gabriela Ciuprina and DanielIoan, Eds., vol. 11 of Mathematics in Industry, pp. 167173.Springer Berlin Heidelberg, 2007.

[14] Michael Striebel and Joost Rommes, Model Order Reduc-tion of Nonlinear Systems in Circuit Simulation: Status andApplications, Chapter in Lecture Notes in Electrical Engi-neering January 2011

[15] David Gratton, Reduced-order, trajectory piecewise-linearmodels for nonlinear computational fluid dynamics, MS The-sis, MIT

[16] M. Nassar Albunni,Volker Rischmuller,ThomasFritzsche,and Boris Lohmann, “Model-Order Reductionof Moving Nonlinear Electromagnetic Devices ” IEEETransactions on Magnetics, Vol. 44, No. 7, July 2008

[17] Xiaoda Pan, Fan Yang, Xuan Zeng and Yangfeng Su, “AnEfficient Transistor-Level Piecewise-Linear MacromodelingApproach for Model Order Reduction of Nonlinear Circuits”,Design, Automation & Test in Europe Conference & Exhibi-tion (DATE), 2010

[18] S.A. Nahvi, M. Nabi and S. Janardhanan, “Adaptive sam-pling of nonlinear system trajectory for Model Order Reduc-tion”, Proceedings of 2012 International Conference on Mod-elling, Identification and Control, Wuhan, China, June 24-26,2012

AMMU CHATHUKULAMAmmu Chathukulam is currently pursuingM.Tech in Control and Automation fromIndian Institute of Technology Delhi. Herresearch field is nonlinear model order re-duction using trajectory piecewise lineari-sation.

DEBASHREE SARKARDebashree Sarkar got her M.Tech in Con-trol and Automation at Indian Instituteof Technology Delhi. Her research inter-ests includes reduced order modelling andits applications in circuit simulation. Sheis currently working in Airport Authorityof India.

SHIFALI KALRAShifali Kalra is working as research scholarin Department of Electrical Engineering atIndian Institute of Technology Delhi. Herresearch interests includes linear and non-linear model order reduction.

M.NABIM.Nabi is currently working as Asso-ciate Professor in IIT Delhi. His re-search includes Control Systems, Guid-ance and Control, Computational meth-ods for Modeling, Simulation and Con-trol, Finite Element Method, DistributedParametered Systems, Flexible Struc-tures,Electromagnetic and Coupled Sys-tems, Electromagnetic NDT.