There has een uc curren in terest in dev eloping suc tec hniques fo cus on bilinearization metho d whic extends Krylo subspace tec hniques for linear systems In this approac h the nonlinear system is 64257rst appro ximated bilinear system through Ca ID: 30294 Download Pdf

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There has een uc curren in terest in dev eloping suc tec hniques fo cus on bilinearization metho d whic extends Krylo subspace tec hniques for linear systems In this approac h the nonlinear system is 64257rst appro ximated bilinear system through Ca

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Krylo Subspace ec hniques for Reduced-Order Mo deling of Nonlinear Dynamical Systems Zhao jun Bai Daniel Sk ogh Departmen of Computer Science Sw edish Defense Researc Agency and Departmen of Mathematics Division of System ec hnology Univ ersit of California Departmen of Autonomous Systems Da vis, CA 95616, USA 172 90 Sto kholm, Sw eden Abstract Means of applying Krylo subspace tec hniques for adaptiv ely extracting accurate reduced- order mo dels of large-scale nonlinear dynamical systems is relativ ely op en problem. There has een uc curren in terest in dev eloping suc tec

hniques. fo cus on bi-linearization metho d, whic extends Krylo subspace tec hniques for linear systems. In this approac h, the nonlinear system is ﬁrst appro ximated bilinear system through Carleman bilinearization. Then reduced-order bilinear system is constructed in suc that it matc hes certain um er of ultimomen ts corresp onding to the ﬁrst few ernels of the olterra-Wiener represen tation of the bilinear system. It is sho wn that the o-sided Krylo subspace tec hnique matc hes signiﬁcan more um er of ultimomen ts than the corresp onding one-side tec hnique. In tro

duction Sev eral mo del reduction tec hniques for nonlinear dynamical systems ha een studied re- searc hers in arious ﬁelds. Tw of the most ell-kno wn metho ds are the Karh unen-Lo ev decom- osition based metho ds and metho ds of balanced truncation. Karh unen-Lo ev decomp osition based metho ds are also kno wn as prop er orthogonal decomp osition (POD) metho ds. Metho ds of balanced truncation extend the success of balanced truncation of linear systems to nonlinear systems. The in terested reader is referred to [6] and [14 ]. The latest ork includes [7 and [11 ]. Means of applying

Krylo subspace tec hniques for adaptiv ely extracting accurate reduced-order mo dels of large-scale nonlinear dynamical systems is relativ ely op en problem. There has een uc curren in terest in dev eloping suc tec hniques. will brieﬂy discuss metho ds, whic extend Krylo subspace tec hniques for linear dynamical systems. consider single-input single-output nonlinear dynamical systems of the form: u, (1.1) with initial condition (0) where is the state ariables, is the dimension of the state space. and are inputs and outputs, resp ectiv ely is the input distribution arra is the output

measuremen arra assume that the nonlinear state ev olution function is smo oth, i.e., and has an equilibrium. Without loss of generalit tak this equilibrium at i.e., Examples of the origins of nonlinear dynamical systems of the form (1.1) include the sim ulation of time-v arying nonlinear circuit elemen ts indep enden excitation source [4 3], and MEMS, suc as

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micro-pressure sensor [8 ]. The mo deling of the dynamical eha vior of oltage-con trolled parallel- plate electrostatic actuator also deriv es set of state equations of the form (1.1) [15 p.138]. Suc an electrostatic

actuator in ok es ulti-domain parameters, suc as mass, stiﬀness and damping in the mec hanical domain, and an excitation force net ork in the electrical domain. Linearization metho will discuss metho ds for the reduced-order mo deling of the nonlinear system (1.1). The ﬁrst metho is called the line arization metho It linearizes the system around the equilibrium oin t, and then extracts Krylo subspace for reduced-order mo deling. Sp eciﬁcally supp ose that the er series expansion of ab out the equilibrium oin is written as (2.2) where is the Jacobian or the ﬁrst

deriv ativ of and is the second deriv ativ matrix of and so on. is the Kronec er pro duct. linearize the original nonlinear system (1.1) only using the ﬁrst term in the expansion (2.2) of and obtain linear system: u, (2.3) can then apply reduced-order mo deling metho for the linearized system (2.3), and obtain line ar reduced-order mo del. The output is an appro ximation of the output of the original system (1.1). If are in terested in small region of the state space near the equilibrium oin t, or so-called small-signal analysis, then as demonstrated in [4], this approac pro vides an

eﬃcien to ol for analyzing the nonlinear system (1.1). Alternativ ely one ma also use the linearized mo del (2.3) to extract Krylo pro jection subspace spanned Then, substituting in to the original nonlinear system (1.1), nonline ar reduced-order mo del is obtained: u, where ), and assume that is an orthonormal basis of the pro jection subspace. One of the issues asso ciated with this approac is that one ust ha represen tation of that can eﬃcien tly stored and ev aluated. The hallenge of this issue is highligh ted in [8 ]. If has certain structure, then one ma exploit suc

structure to deriv an eﬃcien represen tation of or example, in [3, 2], is considered as quadratic function Ax ), and in [5 ], is represen ted as gradien of scalar function ). It is often the case that in order to obtain some pre-kno wledge ab out the dynamical eha vior of the full-order nonlinear system, in ten tionally linearize system ev en if it is not near the equilibrium and accept some degree of error rather than confron the full-order nonlinear system. understand the limitation of the linearization approac h, namely when reduced-order mo del strictly based on the Jacobian of the

nonlinear state ev aluation function is accurate enough for particular application, ma in ok the to ol of ariational analysis to analyze the con tribution of the higher order nonlinear term [12 p.113]. As y-pro duct, ma also use the resulting sequence of linearized systems to dev elop tec hnique for the reduced-order mo del of the nonlinear dynamical system, as rep orted in [9 ].

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Bilinearization metho The second approac is in tended to explicitly incorp orate the higher order nonlinear terms in the er series expansion (2.2) of in to the construction of Krylo pro jection

subspace. The approac is based on the Carleman bilinearization of nonlinear system. The follo wing one-sided Krylo metho is similar to the metho presen ted in [10 ]. By Carleman bilinearization (see, for example, [12 13 ]), the nonlinear system (1.1) can ap- pro ximated bilinear system giv en in the follo wing form u, (3.4) The olterra-Wiener represen tation of the bilinear systems (3.4) with the ernel in regular form is giv en =1 ), where the degree- subsystem is giv en eg dt dt with the asso ciated -th degree regular ernel eg The ulti-dimensional Laplace transform of eg yields the transfer

function (3.5) rom the er series expansion of it is natural to deﬁne the corresp onding multi- moments as 1) (3.6) where are nonnegativ in tegers. The expressions of the transfer function (3.5) and the asso ciated ulti-momen ts (3.6) suggest that in order to matc the momen ts for the degree- ernel, can ﬁrst generate the subspace of nested Krylo subspaces with depth deﬁned span (3.7) for and then tak union of the subspaces span =1 span (3.8) Once the basis of the pro jection subspace is extracted, can appro ximate the state ector another state ector constrained to the

subspace span i.e., let ). This yields reduced-order mo del of the bilinear system (3.4): u, (3.9) This approac can explicitly incorp orate higher order nonlinear terms of the state ev olution function

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matc desired um er of ultimomen ts, the dimension of the one-sided Krylo subspace span can ecome quite large ev en for lo er degree ernels, whic results in high dimension of the reduced-order bilinear mo del. or example, to matc the ultimomen ts of the ﬁrst, second and third order ernels up to order where it requires subspace of dimension oid suc rapid gro wth of the

dimension, prop ose to build pair of biorthogonal bases for oth left and righ Krylo subspaces to matc the ultimomen ts in more eﬃcien The result is more dramatic than the diﬀerence et een one-sided and o-sided Krylo metho ds for linear systems. In the o-sided metho d, the righ subspace is the same as (3.8). or the left subspace, ﬁrst construct the nested Krylo subspaces with depth span where The left pro jection subspace is then tak en as the union of these subspaces span =1 span urthermore, the bases and are constructed to biorthogonal. The system matrices in the

reduced-order bilinear mo del (3.9) are then deﬁned as NV It can sho wn that the reduced-order mo del matc hes all ultimomen ts that can represen ted through the scalar pro duct 1) where span and span It matc hes more um er of ultimomen ts than the total um er of basis ectors, whereas using only one-sided basis, it generally only matc hes the same um er of ultimomen ts as the um er of basis ectors. Example 1. Let the righ and left subspaces span and span By using the com bined information in the basis and it can sho wn that the reduced-order bilinear mo del matc hes 13 momen ts of the

degree-1 ernel 13 40 ultimomen ts of the degree-2 ernel

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10 0.005 0.01 0.015 0.02 0.025 10 −0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 Figure 1: ransien resp onses of nonlinear circuit system with full-order system (solid line), linearized system (dash line), and the reduced-order bilinear system (dot-dot line). The plots of full-order bilinear system erlap the plots of the reduced-order one and is omitted. Left: The input signal is the ramp function t/ 5). Righ t: the input signal is the sin function sin( ). where 4; 4; or 7, and 16 ultimomen ts of the

degree-3 ernel where 4. It matc hes total of 69 ultimomen ts with only 22 basis ectors. On the other hand, to able to matc the same um er of momen ts using only the left or righ subspace, will need 69 basis ectors. As result, the dimension of the reduced-order mo del using o-sided subspaces will of order 11, not the order of 69 using one-sided subspace. One critical issue asso ciated with the bilinearization metho is the gro wth of the dimension of the bilinear system (3.4) as result of Carleman linearization. or example, ev en if only use the ﬁrst terms in the er series expansion (2.2)

of the order of the resulting bilinear system is ab out ). Ho ev er, the matrices in the er series expansion (2.2) of are generally extremely sparse, and the matrices and in the bilinear system (3.4) are highly structured, so one can exploit these facts in Krylo pro cess, namely through the matrix-v ector ultiplications during the Lanczos or Arnoldi pro cess, to pro duce an eﬃcien reduced-order mo del. Example 2. In Fig. 1, sho the transien resp onses of circuit with nonlinear resistors as describ ed in [3 ]. The dimension of the original full-order nonlinear system is 100. Although the

order of the bilinear system using second-order appro ximation is of dimension 10100, the order of the reduced one is only 11. The pair of bases and is constructed as describ ed in Example 1. urther details of the bilinearization-based Krylo subspace tec hniques for reduced-order mo del- ing of large-scale nonlinear dynamical systems will rep orted in [1 ]. Ac kno wledgemen ts. Supp ort for this ork has een pro vided in part NSF under gran CI-9813362 and DOE under gran DE-F G03-94ER25219, and in part MICR pro ject (#00- 005) from Univ ersit of California and MSC.soft are Corp oration. DS as

also supp orted the oundation BLANCEFLOR Boncompagni-Ludo visi, ne Bildt and the Ro al Sw edish Academ of Sciences while visiting Univ ersit of California at Da vis during the academic ear of 2000-2001.

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References [1] Z. Bai and D. Sk ogh. Reduced-order mo deling of structured nonlinear systems. ork in progress. 2002. [2] J. Chen and S. M. Kang. An algorithm for automatic mo del-order reduction of nonlinear MEMS devices. In IEEE Inter. Symp. on Cir cuits and Systems Genev a, Switzerland, 2000. [3] Y. Chen and J. White. quadratic metho for nonlinear mo del order reduction. In

Inter. Conf. on Mo deling and Simulation of Micr osystems, Semic onductors, Sensors and ctuators San Diego, 2000. [4] R. W. reund and eldman. Small-signal circuit analysis and sensitivit computations with the PVL algorithm. IEEE ans. Cir cuits and Systems II 43:577–585, 1996. [5] L. D. Gabba J. E. Mehner, and S. D. Sen turia. Computer-aided generation of nonlinear reduced-order dynamic macromo dels I: Non-stress-stiﬀened case. J. of Micr ele ctr ome chan- ic al Systems 9:262–269, 2000. [6] Holmes, J. L. Lumley and G. Berk oz. urbulenc e, Coher ent Structur es, Dynamic al Systems and

Symmetry Cam bridge Univ. Press, 1996. [7] S. Lall, J. E. Marsden, and S. Gla aski. subspace approac to balanced truncation for mo del reduction of nonlinear con trol systems. Inter. J. on ubust and Nonline ar Contr ol 2000. submitted. [8] T. Mukherjee, G. edder, D. Ramasw am and J. White. Emerging sim ulation approac hes for micromac hined devices. IEEE ans. CAD 19:1572–1588, 2000. [9] J. Phillips. Automated extraction of nonlinear circuit macromo dels. In Pr dings of IEEE 2000 Custom Inte gr ate Cir cuits Confer enc pages 451–452, 2000. [10] J. Phillips. Pro jection framew orks for mo del

reduction of eakly nonlinear systems. In Pr o- dings of 2000 pages 184–189, 2000. [11] M. Rathinam and L. etzold. An iterativ metho for sim ulation of large scale mo dular systems using reduced order mo dels. In Pr c. IEEE Contr ol and De cision Confer enc Australia, 2000. [12] W. J. Rugh. Nonline ar system the ory The John Hopkins Univ ersit Press, Boltimore, 1981. [13] S. Sastry Nonline ar systems: nalysis, stability and ontr ol Springer, New ork, 1999. [14] J. M. A. Sc herp en. Balancing for nonlinear systems. Systems and Contr ol etters 21:143–153, 1993. [15] S. D. Sen turia. Micr osystem

design Klu er Academic Publishers, Boston, 2001.

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