Time dependent weight functions for the Trajectory Piecewise-Linear approach?

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1 Tme epenent weght functons for the Trajectory Pecewse-Lnear approach? Juan Pablo Amorocho an Heke Faßbener Abstract Moel orer reucton (MOR) has become an ubqutous technque n the smulaton of large-scale ynamcal systems (.e an more equatons). One technque for non-lnear MOR s the trajectory pecewse-lnear approach (TPWL). TPWL approxmates a non-lnear fferental system by a weghte sum of lnear systems whch have a sgnfcantly reuce number of equatons. One open queston s whch weghtng schemes are approprate. We scuss whether a tme epenent weghtng scheme whch s computatonally faster as well as computatonally cheaper than the orgnally propose one s approprate. 1 The Trajectory Pecewse-Lnear approach (TPWL) Moel orer reucton (MOR) spees up the smulaton of hgh mensonal ynamcal systems by reucng ts menson whle keepng the output error small. MOR for lnear tme-nvarant systems s well unerstoo an use n nustry, however for non-lnear systems there s stll much research to o. The trajectory pecewse-lnear approach [4] s a numercal metho for the moel orer reucton of non-lnear systems escrbe by frst-orer, non-lnear ornary fferental equatons an non-lnear fferental-algebrac equatons. Here we conser the ornary fferental equaton: Juan Pablo Amorocho D. Insttut Computatonal Mathematcs, Technsche Unverstät Braunschweg, D Braunschweg, e-mal: jamoroch@tu-braunschweg.e Heke Faßbener Insttut Computatonal Mathematcs, Technsche Unverstät Braunschweg, D Braunschweg, e-mal: h.fassbener@tu-braunschweg.e 1

2 2 Juan Pablo Amorocho an Heke Faßbener t x(t) = f(x(t))+dv(t), 0 t T, x(0) = x 0, (1a) y(x(t)) = L T x(t), (1b) where x(t) R N s the state vector, f : R N R N s a non-lnear functon, D R N M s the nput matrx, v(t) R M s the nput, L T R P N s the matrx mappng the state vector nto the output, an y(x(t)) R P s the output of the system. The TPWL reuces the hgh computatonal cost of the evaluaton of the non-lnearty by approxmatng equaton (1a) through a non-lnear convex combnaton of s fferent lnear moels aroun (t,x ), = 0,...,s 1 where x = x(t ). Let ˆf(x(t)) be the frst orer Taylor approxmaton of f(x(t)) aroun x : ˆf(x(t)) = f + F (x(t) x ), where f = f(x ) an F R N N s the Jacoban matrx of f(x(t)) evaluate at x. Wth ths, the pecewse approxmaton of system (1) s gven by s 1 t ˆx(t) = ŵ ( ˆx(t))(F ˆx(t)+B u(t)) =0 (2a) ŷ( ˆx(t)) = L T ˆx(t) (2b) wth the weghtng functons ŵ : R N [0,1], an s 1 =0 ŵ(x(t)) = 1 for all t [0,T], B = [D, f F x ] R N (M+1), an u(t) = [v(t),1] T R M+1. The tuples (t,x ) are foun va a smulaton of system (1a), see [4] for more etals. The resultng trajectory s calle the tranng trajectory. Next, lnear moel orer reucton s apple to the system (2) as follows: Bul the projector π q = V q W T q, such that q N an V q,w q R N q,w T q V q = I q (I q : entty matrx of menson q). Next, use π q to apply a Petrov-Galerkn projecton to system (2). The resultng ynamcal system s calle the TPWL moel of system (1) gven by: s 1 t x r(t) = =0 y r (x r (t)) = (L r ) T x r (t), w (x r (t))(f r x r(t)+b r u(t)), (3a) (3b) where x r (t) R q, an w : R q [0,1], s 1 =0 w (x r (t)) = 1 for all t [0,T]. Matrces F r,br an L r are gven by F r = Wq T F V q,b r = W q T B,(L r ) T = L T V q. Note that x r (t) = Wq T ˆx(t). The use of the weghts w (x r (t)) n (3a) nstea of ŵ ( ˆx(t)) ams to reuce the cost of the computaton of the weght functon from O(Ns) to O(qs). In case the non-lnearty of (1a) s not too pronounce, the TPWL moel (3a) can prect the evoluton of the soluton trajectory even for ntal contons an/or nput sgnal fferent from the ones use to set up the TPWL moel. In [4] two algorthms to bul the projector π q are propose: One that nclues all lnear moels an another that only uses the lnear moel at the ntal conton. The latter s faster than the former, but the former yels a better approxmaton than the latter. Furthermore, [4] uses Krylov subspace methos to bul π q. However ths s not the only opton, [7] employs balance truncaton methos, an [1] combnes the proper orthogonal ecomposton metho (POD) together wth the TPWL approach.

3 Tme epenent weght functons for the Trajectory Pecewse-Lnear approach? 3 In our mplementaton we use the algorthm gven n [3] usng only the lnear moel at the ntal conton an settng W q = V q whch yels an orthogonal projector. In bulng the pecewse approxmaton (2) two questons arse: () how to select the lnear moels along the tranng trajectory? () how to choose sutable weght functons? In [4] the extracton starts wth the lnear moel foun at the ntal conton x 0 an contnues to smulate an select lnear moels keepng the tranng trajectory covere by balls B α x, = 0,...,s 1 of constant raus α each centere at x. Startng at the lnear moel at x 0, a new lnear moel s selecte at (t,x ) when x(t ) oes not longer belong to B α x 1. Ths process s contnue untl x(t) s reache. Here we aopt the proceure from [4]. Concernng the secon queston, t s far to say that sutable weght functons shoul aequately approxmate the system s non-lnearty by the resultng convex combnaton of the lnearze moels. In [4] the weghts are gven by w ( x r (t)) = e β /m s 1 =0 e β /m, = 0,...,s 1 (4) where = x r (t) x 2, an m = mn an β s a postve parameter usually =0,...,s 1 set to 25. Ths ensures that the weghts change raply as x r (t) evolves n the state space. Here, x are gven by [ x 0, x 1,..., x s 1 ] = [W T x 0,W T x 1,...,W T x s 1 ], where ether W = W q as n (3a) or W results from the orthonormalzaton of the columns of W = {W q,x 0,..., x s 1 } an x r (t) s gven by x r (t) = W T ˆx(t). In the latter case, n (3a) W s use nstea of W q. However, ths ncreases the menson of the reuce space whch s unesrable. An alternatve to compute wthout havng to bul W s to project x r (t) back to the full state space,.e. V q x r (t) an compute the as V q x r (t) x 2 [5]. We use ths approach n our experments. We refer to the reaer to [2, 6] for other weght functons. 2 Tme epenent weght functons State-epenent weght functon are qute expensve to evaluate. Therefore, one mght want to conser weght functons whch are cheaper to evaluate, e.g., weght functons that are only a functon of tme. The ea behn these weght functons s smple: At each t we efne a pecewse weght functon w (t) as w (l) (t) f t [t 1,t ] w (t) = w (r) (t) f t [t,t +1 ] for = 0,...,s 1, 0 otherwse where w (l) 0 (t) = w(r) s 1 (t) an w (r) (t) are cubc poly- (t 1 ) = t w(l) (t ) = 0 nomals such that w (l) (t) = 0. Here, the functons w(l) (t ) = w (r) (t ) = 1, w (l) (5) (t 1 ) = t w(l)

4 4 Juan Pablo Amorocho an Heke Faßbener Fg. 1 Top: State-epenent weghts (4) from the tranng nput. Mle: Tme-epenent weghts (5) from tranng an test nput. Bottom: state-epenent weghts (4) from test nput. an w (r) (t +1 ) = t w(r) (t ) = t w(r) (t +1 ) = 0. If t s 1 < T, then we nee an atonal conton for w s 1 : w s 1 (t) = 1,t t s 1. Wth these cubc splnes, (3a) approxmates (1a) for each t [t,t +1 ] by at most two lnear fferental equatons, t x r(t) = w (r) (x r (t))(f r x r (t)+b r u(t))+w (l) +1 (x r(t))(f+1x r r (t)+b r +1u(t)). (6) Fgure 1 shows an example of tme-epenent an state-epenent weghts for the example consere n Secton 3. The TPWL moel constructe s smulate once for a tranng nput (whch was also use for smulatng the orgnal problem when settng up the TPWL moel) an once for a test nput. The tme-epenent weght functon oes not change for fferent nput functons, whle the stateepenent one oes change. As can be seen, consere as functons of tme, n case of the tranng nput, both weghtng functons o have a smlar behavor, the state-epenent weghts are almost pecewse constant (ether 0 or 1), whle the tme-epenent weght functons are smoother an allow (n the sense of the above equaton) for a better overlap of the two lnear moels whch lve on the nterval [t,t +1 ]. But for the test nput, the state-epenent weghts allow for a much stronger swtch between the fferent lnear moels than the tme-epenent weghts o. The tme-epenent weghts ffer from those n [2, 4, 6] n several ways. Frst, the computaton of the tme-epenent weghts s O(s) nstea of O(qs), as the computaton of the tme epenent weghts requres an evaluaton of a scalar polynomal of egree 3 per tme step. In contrast, n orer to compute the weght functons (4) sums of vectors of menson q have to be etermne n every weght w per tme step. Tme-epenent weght functons not only spee up the TPWL, but also reuce the storage, as there s no nee to bul the bass W as escrbe at the en of the prevous secton. Atonally, the use of these new weghts makes the pecewse system a lnear tme epenent one as the weghts are no longer state epene. Ths woul enable the use of the theory of lnear tme epenent systems to analyze the TPWL moel. But (6) has a serous problem when the TPWL moel s smulate wth ntal contons an/or nput sgnal fferent from the ones use to set up the TPWL

5 Tme epenent weght functons for the Trajectory Pecewse-Lnear approach? 5 1 N Fg. 2 Non-lnear transmsson lne. [4] moel. In that case, there s no guarantee that, for nstance, n the frst tme nterval [0,t 1 ] the lnear moels F 0 an F 1 are the best lnear moels to approxmate the ynamcs. It mght be better to use, say, F j an F k. State-epenent weght functons mght be able to allow for ths choce, a tme-epenent weght functon can not acheve ths, as t wll always be local wth respect to tme. Therefore, the use of tme-epenent weght functons s very lmte, unless the TPWL moel s bul from several smulaton runs wth fferent parameter settng for the ntal conton an nput. In that case one mght be able to mofy the tme-epenent weghts approach above to be comparatve to state-epenent weghts. 3 Numercal Experments We have teste our weght representatons on a non-lnear transmsson lne [4] shown n Fg. 2. Applyng the mofe noal analyss to ths crcut yels the ornary fferental equaton x 1 (t) = 2x 1(t)+x 2 (t) 1,0 1,2 + v(t) x j(t) = x j 1 (t) 2x j (t)+x j+1 (t)+ j 1, j j, j+1, j = 2,...,N 1 x N (t) = x N 1 x N + N 1,N, where x k s the kth noe voltage an k,l (t) = exp(40(x k (t) x l (t))) 1 s the oe s current between noes k an l. l = 0 means the groun noe. We have smulate the above crcut usng the TPWL wth two fferent nputs, one for the tranng nput 1 f 0 t < 0.5 v(t) = 1 t otherwse an one for the test nput v(t) = cos( π 2 t), an wth two fferent weght functons: the state epenent weght functon (4) an the tme epenent weght functon (5). In the fgures an tables below we refer to each case as Rewensk an splnes, respectvely. The smulatons run from 0 to 10 tme unts, the ntal conton s zero, N = 100, an the output y(t) s the voltage at noe 1. The lnear moels of the TPWL moel are extracte usng the algorthm escrbe n [4], the number of extracte lnear moels s 10 an the matrx V q s bul usng the algorthm of [3] an has rank 10. The smulaton was one usng MATLAB R2007b mplct ODE solver

6 6 Juan Pablo Amorocho an Heke Faßbener TPWL smulaton of non lnear transmsson lne usng fferent weghts 0.04 Voltage noe 1 (V) Splnes tme Exact soluton Rewensk state Tme Fg. 3 Smulaton of a non-lnear transmsson lne usng TPWL. oe15s on a Intel Core 2 DUO 1.80Ghz wth 2.0Gb of RAM memory runnng a Lnux kernel Fgure 3 shows the output of the smulaton, where the exact soluton s obtane by a smulaton of the full system. Ths s a worse case scenaro where the shape of the tranng nput s fferent from the test nput. The ba performance of the tmeepenent weghts can be unerstoo by lookng at the weght functons n Fgure 1. The test nput requres the TPWL to strongly swtch between the lnear moels, somethng moel (6) oes not o. One mght be tempte to reverse the orer of the nputs so that the TPWL s set up wth a more oscllatng nput. However our experments show that moel (6) wll stll perform poorly, but more nterestngly moel (1) usng weghts (4) ntally follows the exact soluton, but eventually verges. Acknowlegements Ths work, part of the SyreNe network, s supporte by the German Feeral Mnstry of Eucaton an Research (BMBF) grant no. 03FAPAE2. References 1. Bechtol, T., Strebel, M., Mohaghegh, K., ter Maten, E.J.W.: Nonlnear moel orer reucton n nanoelectroncs: Combnaton of POD an TPWL. In: PAMM, vol. 8, pp. 10,057 10,060. WILEY-VCH Verlag GmbH & Co. (2008). 79th Annual Meetng of the Internatonal Assocaton of Apple Mathematcs an Mechancs (GAMM) 2. Dong, N., Roychowhury, J.: Automate nonlnear macromoelng of output buffers for hghspee gtal applcatons. In: ACM/IEEE Desgn Automaton Conference, pp (2005). 3. Oabasoglu, A., Celk, M., Plegg, L.T.: PRIMA: passve reuce-orer nterconnect macromoelng algorthm. IEEE Transactons on Computer-Ae Desgn of Integrate Crcuts an Systems 17(8), (1998). DOI / Rewensky, M.: A trajectory pecewse-lnear approach to moel orer reucton of nonlnear ynamcal systems. Ph.D. thess, Massachusetts Insttute of Technology (2003) 5. Strebel, M., Rommes, J.: Moel orer reucton of non-lnear systems: status, open ssues, an applcatons. Tech. Rep. CSC/08-07, Technsche Unverstät Chemntz (2008) 6. Twary, S., Rutenbar, R.A.: Scalable trajectory methos for on-eman analog macromoel extracton. In: ACM/IEEE Desgn Automaton Conference, pp (2005). 7. Vaslev, D., Rewensky, M., Whte, J.: A TBR-base trajectory pecewse-lnear algorthm for generatng accurate low-orer moels for analog crcuts an mems. DAC pp (2003)