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A TUTORIAL INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS, PART 11: MODELlNG AND CONTROL Luis Antonio Aguirre Centro de Pesquisa e Desenvolvimento em Engenharia Elétrica Universidade Federal de Minas Gerais

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A TUTORIAL INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS, PART 11: MODELlNG AND CONTROL Luis Antonio Aguirre Centro de Pesquisa e Desenvolvimento em Engenharia Elétrica Universidade Federal de Minas Gerais Av. Antônio Carlos 6627 : Belo Horizonte, M.G., Brazil Fax: El , Phone: 55 : :3 aguirre(q.icpc1ee.ufmg.br ABSTRACT - This is the second and final pari. of a series of two papers on nonlinear dynamics and chaos. In the first pari some tools. developed for analysing nonlinear systems, were c!esniijec! in conjunction with a sei. of moc!els commonly llsed as benchmarks in the literature. This papel' investigates a llllillber of isslles concerning the modeling, signal processillg anel control of nonlinear e1ynamics. This is carrieel ou! llsillg tbe tools and 1D0elels described in the first papel'. This inwstigation has th1'own some new light on relevant p1'oblems such as modei parametrization, modei validation. data smoothing anel control of nonliear systems. These issues are inwstigated using NARMAX polynomial modeis but it is believed that the conclusions are relevant to nonlinear represelltajiolls in general. Some numerical examples are includecl. 1 INTRODUCTION Chaotic systems have attractec! a great deal of attention in the last three decades. Systems and modeis which undergo chaotic regimes for a rather wide range of operating conditions have been found in virtually every branch of science and engineering. In the evolution of the study of chaotic systems, several distinct but sometimes co-existent phases can be distinguished. In the first phase, chaos was recognised as a deterministic dynamical regime which could be responsible for f!uctuations that hitherto had been regarded as noise and therefore modeled as stochastic processes (Lorenz, 1963). oartigo submetido em 04/07/95; Revisão em 27/11/95 Aceito por recomendação do Ed. Consultor Prof.Dr. Liu Hsu In a subsequent phase, it was necessary both to develop criteria to detect chaotic dynamics and to establish C!ynamical invariants to quantify chaos (Guckenheimer, 1982; Eckmann and Ruelle, 1985; Denton and Diamond, 1991). Having succeeded in diagnosing chaos, the next step was to build modeis which would learn the dynamics f1'om data on the strange attractor. In this respect a number of model structures have been investigated such as local linear mappings (Farmer and Sidorowich, 1987; Crutchfield and McNamara, 1987), radial basis functions (B1'Oomhead and Lowe, 1988; Casdagli, 1989), neural networks (EIsner, 1992) and global nonlinear polynomials (Aguirre and Billings, 199.5c; Kadtke et alii, 1993). This phase is being currently investigated with other equa.lly important issues concerning nonlinear dynamics such as noise reduction and control. It is the objective of this papel' to p1'ovide a brief int1'oduction to these issues. The outline of the papel' is as follows. Section 2 discusses a numbel' of issues concerning the modeling of nonlinea dynamics. Embedding techniques in general and NARMAX modeis in particular are brief!y presented in that section. Section 3 is concerned with the noise reduction problem, which is a major limitation in the modeling of nonlinear dynamics. Section 4 provides a very superficial int1'oduction to the subject of control and synchronization of chaos, nonetheless several references are p1'ovided for further reading. Some final remarks are made in section 5. 2 MODELlNG NONLlNEAR DYNAMICS This section gives a quick view at nonlinear dynamics 1D0deling. This vital issue is discussed under several related headings. A helpful account of some of the main points on 50 SBA Controle & Automação jvol.7 no. IjJan., Fev., Mar. e Abril 1996 y'l\/\fçj1 to Y3 y(t) E IR n Figure 1 - The n time series defined by the state variables of an ntn-order dynamical system can be used to compose the trajeetory in state space. Yl y(to) this subject can be found in the literature (Casdagli et alii, 1992). 2.1 Embedding Techniques An n th-order dynamical system ij = f(y) can be represented as a set of n first-order ordinary differential equations each one governed by a state variable. The global system would therefore have n time variahles {Yl, Y2,..., y,,} and the solution of such a system could be thought of as 11 time series. In a sense, the n time series mentioned above are obtained from the original n th-order system by decomposition. AIso, given th 11 times series it is possible to recover the original n-dimensional solution by taking each state variable to be a coordinate of a 'reconstruction space' and to represent each time series in such a space. Thus n time series can be used to compose ar reconstruct the system solution OI' trajectory. This is iiiustrated in figure 1. A di:fficulty encountered in praetice with this approach is that the order of the system n is seidom known and even when an accurate estimate of this variable exists the numbel' of measurements wiii not be as large as n. Take for instance the atmosphere which is usuaiiy thought of as a high-order system, nonetheless monitoring and weather forecasting stations only measure a very limited numbel' of variables of this system in order to make predictions. Y2 which operates on the entire state OI' phase space but which yields just a scala.r which is cailed the measured variable. The question which naturaiiy arises at this stage is the following: given.f : IR --;. IR and h(y) : IR ~ IR, is it possible to reconstruct a trajectory OI' sulution of.f from the scalar measurement h( y)? Fortunately, it turns out that this question has an a:ffirmative answer if certain requirements are met (Takens, 1980; Packard et alii, 1980; Sauer et alii, 1991). Thus embedology is concerned with how to reconstruet the phase space of a dynamical system of order n from a limited set of measurements q where q n, and more often than not q = 1. In other words, the objective is to reconstruct the phase space of a system from a single time series. The resulting phase space is usuaiiy referred to as embedded phase space, embedding space or just em.bedding. Another question which should be addressed is: why should we be concerned in reconstrueting the trajectories of a dynamical system? In the companion papel' it was shown that in state (01' phase) space the steady state dynamics of a system are represented by geometrical figures which are caiied attractors. A stable autonomous linear system only has one kind of attractor, a point attraetor. However, nonlinear systems may have more complicated a.ttractors such as limit cycles, tori and the so-caiied strange attraetors. Therefore if time series are used to reconstruct the phase space of dynamicai systems via embedding techniques, it is possible to use results from differentiai geometry and topology to analyse the resulting attractors which are geometrical objeets in the reconstructed space. Moreover, if the embedding is successful, both the reconstructed and the original attraetors are equivalent from a topological point of view, OI' in other words, they are said to be diffeomorphic. The practical consequences of this are obvious. No matter how complex a dynamical system might be, even if only one variahle ofsuch a system is measured, it is possible to reconstruct the originai phase space via embedding techniques. It is also possible to estimate qalitative and quantitative invariants of the original attraetor, such as Poincaré maps, fractal dimension and Lyapunov exponents, directly from the reconstructed attractor which is topologically equivalent to the originai one. These ideas are iiiustrated in figure 2. A convenient but by no means unique way of reconstructing phase spaces from scalar measurements is achieved by using delay coordinates (Packard et alii, 1980; Takens, 1980; Sauer et alii, 1991). Other coordinates include the singular value (Broomhead and King, 1986; Albano et alii, 1988) and derivatives (Baake et alii, 1992; Gouesbet and Maquet, 1992). A framework for the comparison of several reconstructions has been developed in (Casdagli et alii, 1991) and three of the most conu11on methods have been studied in (Gibson ei alii, 1992). A delay veetor has the foliowing form This can be described in a more mathematical way by considering the aetion of a measuring function h(y) : IR --;. IR SBA Controle &. Automação /Vol.7 no. I/Jan., Fev., Mar. e Abril window defined as (de - l)t (Albano et alú, 1988; Buzug and Pfister, 1992; Martinerie et alii, 1992). Some of these methods have recently been compareel in (Rosenstein et alii, 1994). There is some evielence that the 'optimum' vaiue of T in system ielentification problems is shorter than for phas-space reconstructions (Aguirre, 1994). Dynamical reconstructions frol11 nonuniformly sampleel data has been aelelressed in (Breeelon and Packarel, 1992) and phase space reconstruction of symmetric a.ttractors has been consielereel in (King anel Stewart, 1992). (De) Composition ) Figure 2 - In many practical situations the number of measured variables is limited. Embedding techniques enable the reconstruction of the state space even from a single measurement. The reconstructed (OI' embedded) and the original state spaces are equivaient. Taken's theorem gives sufficient conelitions for equation (2 to holel, that is, in order to be able to infer elynamical invariants of the original system from the time series of a single variable, however no indication is given as to how to estimate the map ft' A number of papers have been devoteel to this goal and snch methoels cano be separated into two major groups, namely local anel global a.pproximation techniqnes. The local approaches usually begin by partitioning the embedeling space into neighbourhoods {U;} ~\ within which the elynamics can be appropriately elescribed by a linear map gt : IR~ -+ IR such that y(k + T) ~ gt;(y(k)) for y(k) E Ui, i = 1,..., N n. (3) y(k) = [y(k) y(k - T)... y(k - (de - I)TW, (1) where de is the embedding dimension and T is the delay time. Clearly, y( k) can be represented as a point in the de-dimensionai embedding space. Takens (1980) has shown that embeddings with de 2n will be faithful generically so that there is a smooth map h : IRde -+ IR such that y(k + T) = h(y(h~)) for ali integers k, and where the forecasting time T and T are also assumed to be integers. A consequence of Taken's theorem is that the attractor reconstructed in IR~ is diffeomorphic to the original attractor in state space and therefore the fonner retains dynamical and topological characteristics of the latter. In the case of delay reconstructions, the choice of the reconstruction parameters, that is, the embedding dimension de and the delay time T is of the greatest importance since such parameters strongly affect the quality of the embedded space. The selection of de has been investigated in (Cenys and Pyragas, 1988; Aleksié, 1991; Cheng and Tang, 1992; Kennel et alii, 1992). The choice of the delay time has been discussed in (Albano et alii, 1991; Buzug et alii, 1990; Fraser, 1989; Kember and Fowler, 1993; Liebert and Schuster, 1989; Billings and Aguirre, 1995; Aguirre, 1995a). Many authors have suggested that in some applications it is more meaningful to estimate these parameters simultaneously, this is tantamount to estimating the embedding (2) Several choices for gt have been suggesteel in the literature such as linear polynomiais (Farmer anel Sielorowich, 1987; Casdagli, 1991) which can be interpolateel to obtain an approximation of the map h (Abarbanel et alii, 1990). Simpler choices incluele zeroth-order approximations, aiso known as local constant predictors (Farmer and Sielorowich, 19B7; Kennel anel Isabelle, 1992; Wayland et alii, 1993) anel a weighted predictor (Linsay, 1991). Global a.pproximators overcome some of the elifficulties faced by local maps. Although global moelels have problems of their own, some attention has been elevoted to the investigation of such models (Cremers anel Hübler, 1987; Crutchfield anel McNamara, 1987; Kaeltke et alii, 1993; Aguirre and Billings, 1995e). 2.2 Representation of Nonlinear Systems The Volterra series anel other related functional representations were among the first modeis to be useel in nonlinear approximation. A well known elifficulty with such representations is the enormous amount of parameters requireel in order to approximate simpie nonlinearities (Billings, 1980). Related techniques seem to suffer from the same problem and, in addition, tend to require very large data sets (Giona et alii, 1991). One of the most popular representations of dynamical modeis is the polynomial formo Apart from being easy to interpret, simulate and operate, algorithms for the estimation of the parameters of polynomial models are currently widely available. One of the disadvantages of global polynomials, however, is that even for polynomial models of moderate 52 SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996 oreler, the number of terms can become impractically large (Farmer anel Sielorowich, ; Caselagli, 1989). Using polynomials to forecast chaotic time series, Caselagli (1989) has reporteel that such preelictors blow up in the iterative proceelure anel suggests that this is because polynomial preelictors give bael approximants to the true elynamics except very elose to the attractor. On the other hanel, some of the problems relateel to global polynomials are believeel to be connecteel to the structure of the moelels (Aguirre anel Billings, 1995b) anel promising results have been reporteel for some systems using nonlinear global polynomials with simplifieel structure (Kaeltke ei aliz, 1993; Aguirre anel Billings, 1994c). Rational moelels share with polynomials the advantage of being linear in the parameters. This feature males it possible to use well known anel numerically robust algorithms to estimate the parameters of such models. Moreover, 1'30 tional moelels seem to extrapolate better than polynomials (Farmer and Sidorowich, ). The radial basis funetion (RBF) approach is a global interpolation technique with good localization properties anel it is easy to implement as the algorithm is essentially inelepenelent of the elimension (Broomheael and Lowe, 1988; Caselagli, 1989; Whaha, 1992). However performance of raelial basis functions depeneis critically upon the centres (Chen ei alii, 1990). For a few hunelreel elata points the choice of the centres is a elifficult task and the solution of the problem could become infeasible (Casdagli, 1989; Billings anel Chen, 1992). Local approximants are concerned with the mapping of a set of neighbouring points in a reconstructed state space into their future values. A major problem here is to select the neighbourhooels because such a choice is criticai anel there coulel be hundreels OI' even thousands of these (Farmer anel Sielorowich, 1988ab). The size of the neighbourhooels elepenels on the noise levei and the complexity of the dynamics (Farmer anel Sielorowich, 1991). A simpie alternative to nonlinear modeling is the use of piecewise-linear representations (Billings anel Voon, 1987). The eliscontinuities among the several linear modeis which compose a piecewise-linear moelel, can provide effects similar to those observeel in nonlinear modeis such as elmos (Mahfouz and Badrakhan, ; MahfollZ and Baelrakhan, 1990b ). However, as other local representations, the final model is piecewise-linear and therefore eliscontinuous. Piecewise-linear moelels have been founel to be unreliahle indicators of the underlying dynamics in some cases (Billings and Voon, 1987), anel a possible explanation for this is that such modeis violate the physically motivated hypothesis of smooth elynamical systems (Crutchfield anel McNamara, 1987). Thus local predictors may not always be suitable for predicting invariant measures (Brown ei alii, 1991). Smooth interpolation functions have been suggested as a way of alleviating the problem causeel by eliscontinuities in piecewise-linear moelels (Johansen anel Foss, 1993). Such functions have localiseel properties which confer to the final moelels composeel in this way some similarities with raelial basis functions. As woulel be expecteel, the quality of the a.pproximation depeneis on the choice of several operating regimes where the system dynamics are approximately linear. This information has to be available a priori anel is somewhat criticai. The problem of selecting the operating points is similar to the choice of neighborhooels anel of centres in other approaches. Other representations for modelling nonlinear systems inelude Legenelre polynomials (Cremers anel Hübler, 1987), neural networks (EIsner, 1992; Principe ei alii, 1992) and weighteelmaps (Stokbro anel Umberger. 1992). At present no particular representation can be regareleel as the best for any application anel fineling a good representation is largely a matter of trial anel error (Farmer anel Sielorowich, ). On the other hanel, it seems that global polynomial models are in many respects simpler and therefore more convenient (Kadtke ei alii, 1993). The remainder of this section investigates the use of global polynomials with simplifieel structure to estimate elynamical invariants of strange attractors. 2.3 NARMAX Models Consider the nonlinear -ªutoregressive moving -ªverage model with e2:;ogenous inputs (NARMAX) (Leontaritis anel Billings, ; Leontaritis anel Billings, 1985b) y(k) = F( [y(k - 1), u(k - d), e(k), where n y, 71 anel n e are the maxílnull1 lags consielereel for the output, input and noise terms, respectively anel d is the elelay measured in sampling intervals, T s. Moreover, u( k) and y(k) are respectively input and output time series obtained by sampling the continuous data ll(k) anel y(ld at T s. Furthermore, e( k) accounts for uncertainties, possible noise, unmodelleel dynamics, etc. anel p([.] is some nonlinear function ofy(k), 1l.(k) anel e(k) with nonlinearity degree f.e Z+. In this papel', the map F([.] is taken to be a polynomial of elegreee. In oreler to estimate the parameters of this map, equation (4) has to be expresseel in preeliction errar form as where y(k - n y ), y(k) = wt(k - 1) 3 + ç(k) u(k-d-n,,+l),,e(k-n e )], (4) anel where w~ (k -1) is a matrix which contains linear anel nonlinear combinations ofoutput anel input terms up to and (5) (6) 5BA Controle & Automação IVo!.? no. 1/Jan., Fev., Mar. e Abril including time k-l. The matrices \Ii; ç(k:-l) and \Ii[(k-l) are defined similarly. The parameters corresponding to ~ach term in such matrices are the elements of the vectors 8 y , êy ç anel ê ç, respectively. Fina11y, é,( k:) are the residuais which are defined as the difference between the measureel data y(k) and the one-step-ahead prediction \lit(k - 1)8. The parameter vector 8 can be estimated by minimizing the fo11owing cost function (Chen ci alú, 1989). where II. II is the Euclidean norm. Moreover, least squares minimization is performed using orthogonal techniques in order to effectively overcome two major elifficulties in nonlinear model identification, namely i) numerical i11-conditioning and ii) structure selection. In order to circumvent such problems orthogonal techniques may be used (Bi11ings ei alá, 1988; Korenberg and Paarmann, 1991). 2.4 Structure Selection The number of terms in a polynomial grows very rapidly even for relativdy lo\\' values ofe, n y, n and n e. This is too difficult a problem to be solved by trial and error. However, effective anel elegant solutions to handle this problem are availahk. see (Bi11ings ci alá, 1988; Aguirre and Bi11ings, 199!)d) anel the survey paper by Haber anel Unbehauen (Haber anel Unbehauen, 1990). One solution is the crror ndl/clion ratio (ERR) test (Bi11ings ei alii, 1988; Bi11ings ei alii. H)S9: h:orenberg ei alii, 1988). Two advantages of this approach are i) it does not require the estimation of a complete model to determine the significance of a candidate term and its con

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