A MULTI-OBJECTIVE APPROACH WITH A RESTART META-HEURISTIC FOR THE LINEAR DYNAMICAL SYSTEMS INVERSE MATHEMATICAL PROBLEM

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1 Internatonal Journal on Informaton Technologes & Securty, 1 (vol.10), A MULTI-OBJECTIVE APPROACH WITH A RESTART META-HEURISTIC FOR THE LINEAR DYNAMICAL SYSTEMS INVERSE MATHEMATICAL PROBLEM 1 Ivan Ryzhkov, 2 Chrstna Brester, 3 Eugene Semenkn Insttute of Computer Scence and Telecommuncatons, Reshetnev Sberan State Unversty of Scence and Technologes, e-mals: 1 van.s.ryzhkov@gmal.com, 2 chrstna.brester@gmal.com, 3 eugenesemenkn@yandex.ru Russa Abstract: Ths study s focused on the automatc modellng of lnear dynamcal systems based on system output observatons. Identfcaton of a dfferental equaton degree and ts parameters requres the ntal pont because the output of the dynamcal system for the gven nput depends on the ntal pont value. Generally, t s mpossble to estmate the system state and ts dervatves usng the observaton values, so the nverse mathematcal modellng problem for dynamcal systems can be reduced to a smultaneous approxmaton of the dfferental equaton and the ntal pont. Solvng the reduced two-crteron extremum problem requres a searchng algorthm wth specfc problem-orented modfcatons. Key words: dynamcal system modellng, lnear tme-nvarant systems, ntal pont estmaton, mult-objectve optmzaton, restart meta-heurstc 1. INTRODUCTION The mathematcal modellng of dynamcal systems, as wth mathematcal modellng n general, s a complex and mportant problem, snce the model s the bass for the problems of system control, analyss and prognoses. At the same tme, the tools for solvng these knds of problems requres a model n a specfc form, such as dfferental equatons or dfference equatons, so t s preferable for the modellng approach to receve an dentty of ths class of models. The basc way of solvng the system dentfcaton problem s ts reducton to an extremum problem wth some crteron, and that s why the goal of ths work s to propose the reducton scheme and mprove the extremum-seekng approach appled to the reduced dentfcaton problem. Ths study comes from an ntersecton of the system dentfcaton and mult-crteron optmzaton felds. Ths ntersecton s well-known n some partcular cases, such as statc system parameter dentfcaton or statstcal modellng, and nverse mathematcal modellng for statc systems, but here we consder dynamcal systems n a more generalsed way than s common n other approaches. Moreover, we need to pont out that dynamcal system

2 94 Internatonal Journal on Informaton Technologes & Securty, 1 (vol. 10), 2018 modellng s related to mult-crteron optmzaton problem solvng because n the case of the dynamcs, we need to dentfy not just some set of parameters, whch determnes the transent process, but also an ntal pont, whch s used for the numercal estmaton of the dynamcal system output. Actually, many dfferent approaches are based on the hypothess that ths ntal value s known or can be estmated, but from a practcal pont of vew, ths assumpton can lead to a sgnfcant error rate arsng, snce the numercal schemes used n the approxmaton of the model s parameters are senstve to the ntal value. Statstcal methods are not a general tool to approxmate the ntal value, because there s a need to estmate the functon wth, for example, some of ts dervatves, and n practce, the sample sze can be small or the sample data can be flat, so the estmaton of the ntal value s a dstnct problem. At the same tme, we could decde f the ntal pont s good or bad on the bass of how the model fts the sample data for some partcular ntal pont. Here we can see that ntal value estmaton s not an solated problem. As can be seen, the nverse mathematcal modellng problem s related to the ntal value estmaton and the ntal value estmaton s related to nverse mathematcal modellng, so each of these problems should be solved smultaneously. For solvng the proposed two-objectve optmzaton problem, a specfc extremum-seekng tool s requred. Frstly, prevous studes have shown that the effcency of ths tool s provded by ts problem-orentaton - [7] and [6], snce the dynamcal system dentfcaton problem s a hard optmzaton problem. Secondly, mplementng meta-heurstcs, such as a restart metaheurstc, provdes an mprovement n the effcency of the mult-objectve extremumseekng algorthm [4]. In ths study, we used Preference Inspred Co-Evolutonary Algorthms-goals (PICEAg) wth a restart meta-heurstc. The proposed approach was appled for a set of dentfcaton problems for the systems of varous dfferental equaton degrees, ther nput functons and ntal values. The proposed approach performance was estmated and analysed, and the expermental results are presented wth fgures and tables. Ths paper conssts of an ntroducton, three chapters: Inverse Lnear Tme Invarant System Modellng, Mult-Crtera Optmzaton and Restart Meta-Heurstc, and Expermental Results, and conclusons. In the frst secton, the man prncples of the proposed meta-heurstc are stated, n the second, the problem defnton s gven and n the thrd, the numercal results are presented. Attempts at solvng the consdered problem were based on solvng an addtve form of the consdered two-crteron extremum problem, but ths approach allows only one pont on the non-domnated feld to be estmated, dependng on the coeffcents of crtera lnear combnaton [6] and [7]. There are many related works on nverse mathematcal modellng for dynamcal systems, but wth a dfferent problem defnton. Lnear tme-nvarant system dentfcaton problems, where an evoluton-based or a nature-nspred optmzaton tool s appled, are stated n [4] references. A deeper look at dynamcal system dentfcaton shows that these areas have a lot of applcaton felds and nclude many dfferent problem defntons, whch dffer also n model class, but each problem needs ts own hgh performance processng and optmzaton algorthm. The PICEA-g wth the restart meta-heurstc was successfully mplemented for solvng the hexadecane dsntegraton reacton modellng problem [5], and proved ts hgh effectveness. The studes on restart meta-heurstc development are stated n [1], [2] and

3 Internatonal Journal on Informaton Technologes & Securty, 1 (vol.10), [3]. In these studes, the authors demonstrate the benefts of applyng a restart metaheurstc, but t s desgned and mplemented for sngle-objectve optmzaton problems. 2. INVERSE LINEAR TIME INVARIANT SYSTEM MODELLING In ths chapter, we gve a problem defnton for lnear dynamcal system modellng. Let the dynamcal system observaton be represented wth sample data: dynamcal system output observatons Y y, 1, s and tme ponts T t, 1, s, where s s the sample sze. It s also known that the system nputs are determned wth a vector-functon U t u1 t... um t, where m s the number of nputs. Due to the assumpton of system lnearty, ts model can be determned as follows: d m a x t bk U t, (1) k 0 k1 where d s the hghest degree of the equaton. After dvdng (1) by ad 0, we receve the equaton n a more sutable form: d d1 m k, (2) k x t a x t b U t 0 k1 The output of the real system s the soluton of the Cauchy problem for equaton (2) wth the ntal value x t0 p, 1, d 1, (3) d 1 where p R s the vector of ntal values. Now we assume that the observatons are random values y x t, ~ N 0,, (4) where N 0, s a normal dstrbuton wth a zero expected value and standard devaton. Accordng to (2) and (3), we need to dentfy the dfferental equaton maxmum degree dˆ N, dˆ dˆ D, whch s lmted by a postve value D, vectors aˆ R and ˆ m b R, and the ntal value p ˆ the followng crtera where 1 dˆ R 1. All of these parameters can be dentfed wth a mnmzaton of ˆ ˆ ˆ ˆ s ˆ С d, a, b, p y x t mn, 1 arg mnt 2 0 ˆ ˆ ˆ ˆ d d1 d D, aˆ, br, pˆ R С pˆ y pˆ mn, ˆx t s the soluton of the Cauchy problem d dˆ 1 dˆ 1 pˆ R ˆ k, (6) k xˆ t aˆ xˆ t b U t m 0 k1 (5)

4 96 Internatonal Journal on Informaton Technologes & Securty, 1 (vol. 10), 2018 ˆ ˆ xˆ mn t p, 1, d 1. As can be seen, the lnear dynamcal system can be fully determned wth a set of the consdered parameters, whch brngs the extremum to a two-objectve optmzaton problem (5). The frst objectve functon of ths problem s the nadequacy degree estmaton, whch ndcates how well the model output fts the observaton data. The second objectve functon s the dstance between the frst observatons and the estmated ntal value frst coordnate, whch s the approxmaton of the system output. Any other crtera could be used here nstead of the proposed one. However, ths one was chosen because of ts applcablty n a general case when the sample sze s small. In studes [6] and [7] the degree s the varable of the extremum problem (5), but n the current study, we propose enumeratve searchng algorthm, whch enumerates the degree. There are two reasons for ths. The frst s an nvestgaton of the approach performance n approxmatng the whole set of parameters, and the second s based on fact that dfferent degrees combned wth dfferent parameters could gve us the same outputs, whch decreases our confdence n t. The searchng s organzed n the followng way: dˆ 1, D, С ˆ ˆ ˆ ˆ 1 a, b, p mn, С2 p mn. (7) ˆ ˆ ˆ d d1 dˆ 1 aˆ, br, pˆr pˆ R Ths problem s complex, t s multmodal and these crtera are evaluated va numercal ntegraton schemes solvng (7) and requre a mult-objectve optmzaton algorthm of hgh performance. In the next chapter, we descrbe the algorthm and proposed meta-heurstc, whch ncreased the effectveness of the algorthm and made t more problem-orented. 3. MULTI-CRITERIA OPTIMIZATION AND RESTART META-HEURISTIC Let us start wth the problem defnton. Let X be the space of alternatves and there s a need to fnd the soluton for the followng two-crteron extremum problem С x extrem, С x extrem, 1 2 (8) xx1 xx2 where С : X R, 1,2 are the objectve functons and X1, X2 X are subspaces of the whole search space X. The functons (8) are calculable, but, generally, perform a Black-Box Optmzaton Problem (BBOP), whch s mult-modal, complex and there s no addtonal nformaton we have about ths problem or the alternatve space. What we actually have s (5) or (7). Evoluton-based and nature-nspred optmzaton algorthms tend to be the most powerful tools to solve problems of ths knd, but as wth any other 0-th order stochastc algorthms, there s a rsk of t becomng stagnated n some basn of a local optmum. Of course, ths rsk s not as hgh as for determnstc algorthms, but t s stll crucal n partcular cases. The reducng of ths rsk leads to a contradcton between the search n breadth and the search n depth and the restart operator s an easy way out. In ths study, we consder a partcular restart meta-heurstc whch s used to mprove the ntal optmzaton algorthm performance. Ths meta-heurstc works on a smple dea: the seekng algorthm restarts f some condton s met, and can be appled to any teratve

5 Internatonal Journal on Informaton Technologes & Securty, 1 (vol.10), and stochastc extremum-seekng algorthm. It has varous mplementatons for one-crteron extremum problems, but for mult-crteron problem solvng, we suggested a new one n [4]. Let the set of non-domnated solutons, whch are the Pareto set estmaton, be denoted as S and the Pareto front estmaton be denoted as F. The front and the set can change p after each teraton of algorthm, and the next front estmaton domnates the prevous one. In a smlar way to the restart operator for a sngle-objectve optmzaton problem, we need the characterstcs of the searchng process, and that s why we desgned a specfc metrc between the front estmatons c c u, v R R R, where card u j jcard v 1 1 u, v mn u v, card u card s the functon that returns the cardnalty of an argument, and uv, are the next and the prevous front approxmatons, respectvely. To estmate f there s stagnaton, whch can be seen by the slow changng of the metrc (8), we need to hold a set contanng these metrcs for the several prevous algorthm teratons to the -th one p p 1 o M F, F : l j, (10) where lo 0, the controllng parameter, s the number of metrc observatons. Usng ths set, we can predct f stagnaton wll take place by checkng the followng condton max M mn M, (11) where s another controllng parameter. After each restart, the algorthm s non-domnated set s stored n ts memory and used for performng the ntal populaton on the next step. The next two parameters and p s control the probablty of generatng each ndvdual n the ntal populaton randomly aganst makng t a modfed copy of a randomly chosen soluton from the memory. Each gene of the soluton s mutated wth probablty p s. The so-called ftness functon n ths work s the mappng, based on crtera (7) and s calculated for each crteron as follows 1 f x 1 C x, (12) where C x s the crteron n terms of (8) and x s a pont on the subspace X 1 or X 2, dependng on whch crteron s beng evaluated. The restart meta-heurstc s mplemented nto PICEA-g as proposed by Wang n 2013 [8]. Ths algorthm relates to a class of preference-nspred co-evolutonary algorthms (PICEAs) whch are based on the concept of co-evolvng the populaton wth decsonmaker preferences. It was used because there are a varety of successfully solved problems wth ths algorthm and ts speed. To estmate the algorthm effcency, t was appled to a set of dentfcaton problems for systems of dfferent orders and wth dfferent numbers of control nputs. p (9)

6 98 Internatonal Journal on Informaton Technologes & Securty, 1 (vol. 10), EXPERIEMENTS AND RESULTS To provde the nvestgaton of the algorthm wth effcency and the benefts from solvng the consdered dentfcaton problem as a mult-objectve one, we performed two sets: a set of dynamcal systems wth ts parameters and ntal values and a set of control nputs wth control functons and control parameters. The frst one s gven n Table 1. Table 1. Lnear dynamcal system. x 0 Dfferental equaton Intal value, 1 (4) x t x t 4 x t 2 xt xt ut T 2 x t 4 x t 3 xt xt ut T 3 x t 2 xt xt ut 0 0 T Table 2, smlar to Table 1, s gven below and here the rght-sde functons are presented. Number of nputs Control functon 1 2 u t 2 sn t 2 3 ut 1sn tsn t sn 2t sn t 3 4 ut 1 0.5t 0.25 t cost Table 2. Control nputs. In ths nvestgaton, we compare the values of ftness functons (12). Accordng to the gven determnaton of the ftness functon, the closer ts value to 1, the better soluton s. We nvestgate the algorthms wth dfferent settngs for the problems below, where for each partcular dentfcaton problem we take 200 ponts randomly from the tme nterval 0, 20, on whch the systems were ntegrated numercally. The standard devaton of error was set to 0. For all restarts, the border value for changng the metrcs was set at 10 5, accordng to the ftness functon. At frst, we would lke to show the benefts provded by applyng the mult-objectve approach nstead of the sngle-crteron approach. In Fgure 1, the Pareto front estmaton s gven, whch s the unon of all estmatons from 20 algorthm runs wth restart and settngs: l 100, 1 and p 1. We receve a set of solutons whch dffer n ther fttng o crtera and n ther parameter values. s

7 Internatonal Journal on Informaton Technologes & Securty, 1 (vol.10), f 2 Fg. 1. Pareto front approxmaton dstrbuton for the problem: dynamcal system 1 (Table 1) and nputs 1 (Table 2). Solvng the dentfcaton problem as a two-objectve optmzaton problem gves us the possblty to estmate the set of solutons, whch could be closer to fttng the observaton data, but dstant from the ntal value measurement and vce-versa. The result of a sngle algorthm wth a restart run also proves ths assumpton and t s shown n Fgure 2. f 1 f 2 Fg. 2. Pareto front approxmaton. Sngle algorthm run. However, Fgure 1 shows that all the solutons gve very good results for the second crteron and ths s proven by all the experments n whch the values of the second crteron vary slghtly. Because of ths, and snce the frst crteron s actually more f 1

8 100 Internatonal Journal on Informaton Technologes & Securty, 1 (vol. 10), 2018 mportant, the performances of the algorthms wll be approxmated and compared by ther frst crteron values. Next, we consdered the combnatons of systems gven n Table 1 and control nputs gven n Table 2 to estmate algorthm performance. Now we can compare the algorthm wth a restart meta-heurstc and one wthout t by varyng dfferent parameters. A summary demonstratng the characterstcs of all the algorthms s gven n Fgure 3, where the ftness functon medum values, 10 and 90 percent quntles and ther mnmum and maxmum values are gven. The results were acheved on the bass of 20 runs for each algorthm for every partcular dentfcaton problem; the number of objectve functon evaluatons s To ft the pcture, each algorthm s named n the followng way: populaton sze, l o, and optonally and p s f these do not equal 1. Wth dashed and dotted lnes, we marked the best and medum values of the best consdered algorthm wthout a restart. Fg. 3. Comparng the algorthm performance. In ths fgure, the followng notaton s used: algorthms 1-2 are PECIA-g wthout a restart and have 4000 generatons aganst 800 ndvduals and 8000 generatons aganst 400 ndvduals, respectvely; the other algorthms are wth restart meta-heurstcs. The other algorthms wth the restart mplementaton have the settngs gven n Table 3, where the rows are: 1 generatons number (p.s.), 2 populaton sze (.n.), 3 restart set cardnalty (r.s.c.), 4 probablty of ndvdual random generaton (p.r.g.) and 5 super-mutaton parameter (s-m.p.). The numercal results let us make a concluson that l 0 s the parameter, the adjustng of whch sgnfcantly mproves the algorthm performance and makes ths algorthm more effcent than the algorthm wthout a restart. Increasng the number of generatons and

9 Internatonal Journal on Informaton Technologes & Securty, 1 (vol.10), decreasng the populaton sze gves better results. It can also be seen that controllng the ntal populaton generaton parameters could mprove the performance too, whch suggests that adjustng the restart parameters, makng them adaptve or problem-orented, could provde a very promsng research drecton. Table 3. Algorthms. Alg p.s n r.s.c p.r.g s-m.p It s mportant to pont out that mplementng the restart operator makes t possble to receve a greater set of solutons, ncludng the approxmaton of the Pareto front, not only n ts cardnalty, but also n ts wdth. As can be seen, the algorthm wth the restart gves more varatons of the parameters, whch acheve relatvely the same results. 5. CONCLUSION In ths study, we proposed an approach based on the mult-crteron optmzaton problem reducton of the nverse mathematcal modellng problem for a lnear dynamcal system, and solvng the reduced problem wth a populaton-based optmzaton algorthm wth restart meta-heurstc mplementaton. The consdered problem s complex and t requres some problem-orented mprovements of a searchng algorthm. Expermental results show that wth adjusted settngs, the algorthm wth restart metaheurstcs works better n terms of medan and best value, and t gves the more explored Pareto set and front. However, at the same tme, meta-heurstcs parameter adaptaton, along wth optmzaton algorthm parameter adaptaton, becomes an mportant problem, whch wll be nvestgated n further studes. In addton, the further work wll be related not only to algorthm and meta-heurstcs mprovement, but also to a generalzaton of the proposed approach and ts applcaton to mult-output systems. ACKNOWLEDGEMENTS Ths research s performed wth the fnancal support of the Mnstry of Educaton and Scence of the Russan Federaton wthn state project assgnment /ПЧ. REFERENCES [1] Belganns G. N., Tsroganns G. A., Pntelas P. E. Restartngs: a technque to mprove classc genetc algorthms' performance. J. Glob. Optm., 1, 2014, pp [2] Fukunaga A. S. Restart Schedulng for Genetc Algorthm. Lecture Notes of Computer Scence, Vol. 1498, 1998, pp

10 102 Internatonal Journal on Informaton Technologes & Securty, 1 (vol. 10), 2018 [3] Loshchlov I., Schoenauer M., Sebag M. Alternatve Restart Strateges for CMA-ES. Lecture Notes n Computer Scence, vol. 7491, 2012, pp [4] Ryzhkov I., Brester C. and Semenkn E. Mult-objectve Order Reducton Problem Solvng wth Restart Meta-heurstc Implementaton. In Proceedngs of the 14th Internatonal Conference on Informatcs n Control, Automaton and Robotcs, Volume 1: ICINCO, 2017, p [5] Ryzhkov I., Brester C. and Semenkn E. Mult-objectve Dynamcal System Parameters and Intal Value Identfcaton Approach n Chemcal Dsntegraton Reacton Modellng. In Proceedngs of the 14th Internatonal Conference on Informatcs n Control, Automaton and Robotcs, Volume 1: ICINCO, [6] Ryzhkov I., Semenkn E. Restart operator meta-heurstcs for a problem-orented evolutonary strateges algorthm n nverse mathematcal MISO modellng problem solvng. IOP Conference Seres: Materals Scence and Engneerng, 2017, vol [7] Ryzhkov I., Semenkn E., Panflov I. Evolutonary optmzaton algorthms for dfferental equaton parameters, ntal value and order dentfcaton. Proceedngs of the 13th Internatonal Conference on Informatcs n Control, Automaton and Robotcs (ICINCO 2016), Volume 1, 2016, pp [8] Wang R. Preference-Inspred Co-evolutonary Algorthms. A thess submtted n partal fulfllment for the degree of the Doctor of Phlosophy, Unversty of Sheffeld, 2013, 231 p.. Informaton about the authors: Ivan Ryzhkov s a research fellow at the Reshetnev Sberan State Unversty of Scence and Technology (Krasnoyarsk, Russa). He receved hs PhD n Computer Scence n 2016 from the Reshetnev Sberan State Aerospace Unversty. Hs areas of research nclude global black-box optmzaton, meta-heurstcs for natural computng extremum-seekng algorthms, nverse mathematcal modellng and dynamcal system dentfcaton. Chrstna Brester s an assocate professor at the Department of Hgher Mathematcs, Reshetnev Sberan State Unversty of Scence and Technology (Krasnoyarsk, Russa). She receved her PhD n Computer Scence n 2016 from the Sberan Federal Unversty and the Insttute of Computatonal Modellng of the Sberan Branch of the Russan Academy of Scences. Her research nterests nclude evolutonary computaton, neuro-evolutonary algorthms, machne learnng and speech analyss. Eugene Semenkn s a professor of system analyss at the Reshetnev Sberan State Unversty of Scence and Technology (Krasnoyarsk, Russa). He receved hs PhD n Computer Scence from Lenngrad State Unversty (Lenngrad, USSR) n 1989 and hs DSc n Engneerng and Habltaton from the Reshetnev Sberan State Aerospace Unversty (Krasnoyarsk, Russa) n Hs areas of research nclude the modellng and optmzaton of complex systems, computatonal ntellgence and data mnng. Manuscrpt receved on 11 January 2018