Data-based Continuous-time Modelling of Environmental Systems

05/09/2017
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Data-based Continuous-time Modelling of Environmental Systems

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1 Data-based Continuous-time Modelling of Environmental Systems Hugues Garnier, P.C. Young, S. Thil, M. Gilson R´esum´e—Les probl`emes li´es `a l’environnement sont tr`es pr´esents actuellement. Les ´evolutions technologiques ont ´et´e tr`es importantes r´ecemment, notamment au niveau de l’acquisition d’informations (nouveaux capteurs et r´eseaux de capteurs). Les applications li´ees `a l’environnement permettent ainsi d’envisager l’utilisation de techniques d´evelopp´ees dans d’autres disciplines. Cet article rentre dans ce cadre puisqu’il a pour objectif de d´evelopper et d’appliquer des m´ethodes d’identification de syst`emes de l’Automatique `a la mod´elisation exp´erimentale de syst`emes environnementaux. Pendant tr`es longtemps, c’est l’approche de mod´elisation d´eterministe qui a domin´e dans les diff´erents domaines des sciences de l’environnement. Cette approche s’appuie sur les lois de la Physique pour ´etablir des mod`eles dits de connaissance. Ces mod`eles de connaissance sont souvent non lin´eaires et refl`etent la perception du syst`eme environnemental par le scientifique comme un syst`eme dynamique complexe poss´edant un mod`ele parfait. Une fois le mod`ele de connaissance ´etabli, il reste cependant encore une phase d´elicate de d´etermination des coefficients physiques intervenant au niveau du mod`ele, coefficients qui s’av`erent souvent tr`es difficiles `a estimer. Dans cet article, nous pr´esentons une approche alternative de mod´elisation `a partir de donn´ees exp´erimentales. Cette approche stochastique prend en compte les incertitudes pr´esentes sur la plupart des syst`emes environnementaux et s’appuie sur l’utilisation de mod`eles `a temps continu et de m´ethodes statistiques de traitement de donn´ees. L’int´erˆet des approches d’identification directe de mod`eles `a temps continu, longtemps rest´ees dans l’ombre des m´ethodes d’identification de mod`eles `a temps discret, est `a pr´esent reconnu et des outils logiciels sont disponibles. Nous d´ecrivons les principales caract´eristiques des m´ethodes directes d’identification de mod`eles lin´eaires param´etriques `a temps continu `a partir de donn´ees exp´erimentales. Les boˆıtes `a outils logicielles CONTSID et CAPTAIN pour MATLAB qui comprennent la plupart des m´ethodes d´evelopp´ees au cours des trente derni`eres ann´ees sont bri`evement pr´esent´ees. Les principaux avan- tages de ces approches directes de mod´elisation `a temps continu sont ´egalement discut´es. L’approche de mod´elisation propos´ee est enfin illustr´ee pour la d´etermination `a partir de donn´ees exp´erimentales d’un mod`ele pluie-d´ebit dans une vall´ee. H. Garnier, S. Thil and M. Gilson are with Centre de Recherche en Automatique de Nancy, Nancy-Universit´e, CNRS, BP 239, F-54506 Vandœuvre-l`es-Nancy Cedex, France, firstname.lastname@cran.uhp-nancy.fr. P.C. Young is with Centre for Research on Environmental Systems and Statistics, Lancaster University, Lancaster LA1 4YQ, U.K. and Centre for Resource and Environmental Studies, Australian National University, Canberra, Australia, p.young@lancaster.ac.uk. R´esum´e—In this paper we argue that many environmen- tal processes are more naturally defined in the context of continuous-time differential equation models, normally derived by the application of natural laws, such as mass and energy conservation. As a result, there are advantages if such models are estimated directly in this continuous-time differential equation form, rather than being formulated and estimated as discrete-time models. This paper discusses these advantages and briefly describes the key features of the two toolboxes for use with MATLAB which supports the advanced time-domain methods for directly identifying linear continuous-time models from discrete-time data. The arguments presented in the paper are finally illustrated by a practical example in which the proposed data-based approach to direct CT modelling is applied to a set of rainfall-flow data. Index Terms—continuous-time systems ; environmental sys- tems ; linear differential equation ; rainfall-flow modelling, sampled data, system identification. INTRODUCTION Most mathematical models of environmental systems are formulated on the basis of natural laws, such as dynamic conservation equations, often expressed in terms of continuous-time (CT), linear or nonlinear differential equations. Paradoxically, Transfer Function (TF) models, which have been growing in popularity over the last decade because of their ability to characterise environ- mental data in efficient parametric (or parsimonious) terms, are almost always presented in the alternative, discrete-time (DT) terms. One reason for this paradox is that environmental data are normally sampled at regular intervals over time, so forming discrete-time series that are in a most appropriate form for DT modelling [10], [4]. Another is that most of the technical literature on the statistical identification and estimation (calibration) of TF models deals with these DT models. Closer review of the literature, however, reveals apparently less well-known publications dealing with estimation methods that allow for the direct identification of CT (differential equation) models from discrete-time, sampled data [9], [2], [3]. This paper first discusses the formulation, identification and estimation1 of linear CT models. It then briefly des- 1The statistical meaning of these terms will be used here, where ‘identification’ is taken to mean the specification of an identifiable model structure and ‘estimation’ relates to the estimation of the parameters that characterize this identified model structure. e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6 2 cribes the main features of the user-friendly CAPTAIN and CONTSID toolboxes for MATLAB which supports the computationally efficient CT model identification algorithms. In addition, we present a practical example, where the proposed data-based approach to direct CT modelling is applied to a typical set of effective rainfall- flow data. Our main aim is to demonstrate the many ad- vantages of direct continuous-time model estimation, in relation to its discrete-time alternative, and so encourage its practical application. CONTINUOUS-TIME MODELS For simplicity of presentation, the theoretical basis for the statistical identification and estimation of linear, continuous-time models from discrete-time, sampled data can be outlined by considering the case of a linear, single input, single output system. It should be noted, however, that the analysis extends straightforwardly to multiple input systems and, in a more complex manner, to full multivariable systems x(t) = B(s) A(s) u(t − τ) (1) Here A(s) and B(s) are the following polynomials in the derivative operator s = d/dt : A(s) = sn + a1sn−1 + ... + an−1s + an (2) B(s) = b0sm + b1sm−1 + ... + bm−1s + bm (3) in which n and m can take on any positive integer values ; τ is any pure time delay in time units ; u(t) and x(t) denote, respectively, the deterministic input and output signals of the system. Of course, the model (1) can also be written in the following differential equation form, which is often more familiar to physical scientists dn x(t) dtn + a1 dn−1 x(t) dtn−1 + . . . + anx(t) = b0 dm u(t − τ) dtm + . . . + bmu(t − τ) (4) The structure of the CT model is denoted by the triad [n m τ]. In most practical situations, the observed input and output signals u(t) and y(t) will be sampled in discrete-time. In the case of uniform sampling, at a constant sampling interval Ts, these sampled signals will be denoted by u(tk) and y(tk) and the output observation equation then takes the form, y(tk) = x(tk) + ξ(tk) k = 1, · · · , N (5) where x(tk) is the sampled value of the unobserved, noise-free output x(t). Given the discrete-time, sampled nature of the data, the measured output is assumed to be contaminated by a noise or stochastic discrete-time disturbance signal ξ(tk). This noise is assumed to be independent of the input signal u(tk) and it represents the aggregate effect of all the stochastic inputs to the system, including distributed unmeasured inputs, measu- rement errors and modelling error. If ξ(tk) has rational spectral density, then it can be modelled as an Auto- Regressive (AR) or Auto-Regressive, Moving-Average (ARMA) process but this restriction is not essential. The objective is then to identify a suitable model structure defined by the triad [n m τ] for (1) and estimate the parameters that characterize this struc- ture, based on the sampled input and output data ZN = {u(tk) ; y(tk)} N k=1. IDENTIFICATION METHODS FOR DATA-BASED CT MODELLING The problem of identifying continuous-time models, ba- sed on discrete-time sampled data can be approached in two main ways : – The direct approach : here, it is necessary to identify the most appropriate CT model structure ; and then directly estimate the TF parameters ai, bi, and τ that characterise this structure. Of course, some approxi- mation will be incurred in this estimation procedure because the inter-sample behaviour of input signal u(t) is not known and it must be interpolated over the sampling interval in some manner. – The indirect approach : here, a DT model for the original CT system is first obtained by applying conventional stochastic DT model identification and estimation procedure. This estimated DT model is then converted to a CT model, again using some assumption about the nature of the input signal u(t) over the sampling interval Ts. Various statistical methods of identification and esti- mation have been proposed to implement the two ap- proaches outlined above and these have been formulated in both the time and frequency domains. However, only estimation in the time domain will be considered here, and only one direct efficient estimation method will be briefly discussed (and later used in the practical example section) : the Refined Instrumental Variable method for Continuous-time Systems (RIVC) [14]. The RIVC method and its simplified version SRIVC (when the additive measurement noise is assumed to be white) are the only time-domain methods that can be interpreted in optimal statistical terms, so providing an estimate of the parametric error covariance matrix and, therefore, estimates of the confidence bounds on the parameter estimates. The SRIVC/RIVC algorithms are available in both the CAPTAIN and CONTSID toolboxes which are briefly described in the next section. This paper also promotes the Data-Based Mechanistic (DBM) modelling philosophy (see [11] and the prior references therein) based on the direct identification and e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6 3 estimation of a CT model. This involves the application of an advanced method of direct continuous-time TF model estimation and identification and the interpretation of this estimated model in physically meaningful terms. This DBM modelling approach can be contrasted with ‘black-box’ modelling, since DBM models are only dee- med credible if, in addition to explaining the time series data in a statistically efficient, parsimonious manner, they also provide an acceptable physical interpretation of the system under study. They can also be contrasted with ‘grey-box’ models, because the model structure is infer- red inductively from the data, rather than being assumed a priori before model identification and estimation in a hypothetico-deductive manner (see the discussion in [12]). SOFTWARE SUPPORT FOR DATA-BASED CT MODELLING The CONTSID toolbox The CONtinuous-Time System IDentification (CONT- SID) toolbox2 for MATLAB supports continuous-time transfer function or state-space model identification di- rectly from regularly or irregularly time-domain sampled data, without requiring the determination of a discrete- time model. The motivation for developing the CONT- SID toolbox was first to fill in a gap since no software support was available to serve the cause of direct time- domain identification of continuous-time linear models but also to provide the potential user a platform for testing and evaluating these data-based CT modelling techniques. It includes most of the direct deterministic methods, the stochastic SRIVC/RIVC techniques as well as output error and subspace-based methods. The toolbox is designed as an add-on to the Mathwork’s System IDentification (SID) toolbox and has been given a similar setup [1]. The CAPTAIN toolbox The Computer Aided Program for Time series Analysis and Identification of Noisy systems (CAPTAIN)3 for MATLAB is a more general toolbox intended not only for the identification of CT (and DT) transfer function models but also for the extrapolation, interpolation and smoothing of nonstationary and nonlinear time series[8]. The CT (and DT) identification algorithms are all based on Refined Instrumental Variable (RIV) estimation [10]. In particular, CT model identification is provided by the SRIVC/RIVC algorithms discussed above. 2http ://www.cran.uhp-nancy.fr/contsid. 3http ://www.es.lancs.ac.uk/cres/captain. ADVANTAGES OF DIRECT DATA-BASED CT MODELLING As recalled in the previous section, there are two fun- damentally different time-domain approaches to the pro- blem of obtaining a CT model of a naturally CT system from its sampled input-output data. The indirect approach has the advantage that it uses well-established DT model identification methods [4]. Examples of such methods are the maximum likelihood and prediction error methods, which are known to give consistent and statistically efficient estimates under very general conditions. On the surface, the choice between the two approaches may seem trivial. However, some recent studies have brought some serious shortcomings of the indirect route through DT models to the fore. An extensive analysis aimed at comparing direct and indirect approaches has been indeed discussed recently. The simulation model used in this analysis provides a very good test for CT and DT model estimation methods : it was first suggested by Rao and Garnier [6] (see also [2], [7]) ; further investigations confirmed the results [5]. This simulation example illustrates some of the well-known difficulties that may appear in DT modelling under less standard conditions such as rapidly sampled data or relatively wide-band systems : – relatively high sensitivity to the initialization. DT mo- del identification often require computationally costly minimization algorithms without even guaranteeing convergence (to the global optimum). In fact, in many cases, the initialization procedure for the identification scheme is a key factor to obtain satisfactory estimation results compared to direct methods ; – numerical issues in the case of fast sampling because the eigenvalues lie close to the unit circle in the complex domain, so that the model parameters are more poorly defined in statistical terms ; – a priori knowledge of the relative degree is not easy to accommodate ; – non inherent data prefiltering. Further, the question of parameter transformation bet- ween a DT description and a CT representation is non- trivial. The zeros of the DT model are first not as easily translatable to CT equivalents as the poles. Secondly, due to the discrete nature of the measurements they do not contain all the information about the CT signals. To describe the signals between the sampling instants some additional assumptions have to be made, for example, assuming that the excitation signal is constant between the sampling intervals (zero-order hold assumption). Vio- lation of these assumptions may lead to severe estimation errors. Advantages of direct CT identification approaches over the alternative DT identification methods can be summa- rized as follows : e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6 4 – they directly provide differential equation models whose parameters can be interpreted immediately in physically meaningful terms. As a result, they are of direct use to scientists and engineers who most often derive models in differential equation terms based on natural laws and who are much less familiar with ‘black-box’ discrete-time models ; – they can estimate fractional time-delay system ; – the estimated model is defined by a unique set of pa- rameter values that are not dependent on the sampling interval Ts ; – there is no need for conversion from discrete to continuous-time, as required in the indirect route based on initial DT model estimation ; – the direct continuous-time methods can easily handle the case of irregularly sampled data ; – the a priori knowledge of the relative degree is easy to accommodate and therefore allows the identification of more parsimonious models than in discrete-time ; – they also offer advantages when applied to systems with widely separated modes ; – they include inherent data filtering ; – they are well suited in the case of over-sampling. This is particularly interesting since today’s data acquisition equipments can provide nearly continuous-time mea- surements and, therefore, make it possible to acquire data at very high sampling frequencies. Note that as mentioned in [5], the use of prefiltering and decimation strategies may lead to better results in the case of DT modelling but these can be not so obvious for a practitioner while the CONTSID toolbox methods are free of these difficulties. All these advantages will facilitate for the user the appli- cation of the proposed data-based modelling procedure. PRACTICAL EXAMPLE : RAINFALL-FLOW MODELLING This example concerns the modelling of the daily effec- tive rainfall-flow data from the ephemeral River Canning in Western Australia, as shown in Figure 1. Effec- tive rainfall is a nonlinear transformation of measured rainfall that is a function of the soil–water storage in the catchment and provides a measure of the rainfall that is effective in causing flow variations (rather than that retained by the soil). Further information on the modelling of rainfall-flow processes is given in [12] and the references therein. Another hydrological example is discussed in a recent, related tutorial paper [13] that reinforces the results reported here. The best identified TF model based on the data in Figure 1 takes the form : y(t) = bos2 + b1s + b2 s2 + a1s + a2 u(t) + ξ(t) (6) with SRIVC parameter estimates, ˆa1 = 0.457(0.032) ˆb0 = 0.0138(0.002) ˆa2 = 0.025(0.005) ˆb1 = 0.051(0.002) ˆb2 = 0.0046(0.0008) where the figures in parentheses are the estimated stan- dard errors. The coefficient of determination based on the simulated output of this model is R2 T = 0.980 (i.e. 98% of the flow variance explained by the model output). The model can be interpreted in a physically meaningful manner as a parallel pathway process with an instantaneous pathway (rainfall affecting flow within one day) ; a ‘quick’ flow pathway reflecting surface water processes, modelled as a first order process with a time constant (residence time) of 2.5 days ; and a ‘slow- flow’ pathway, again modelled as a first order process reflecting ground water effects, this time with a much longer time constant of 15.9 days. Indirect estimation produced mixed results. Two cases were considered : without (OE) and with noise model parameter estimation. In the first case, the indirectly identified CT model, based on SRIV estimation of the DT model has virtually the same parameter estimates and R2 T as the SRIVC estimated CT model. The indirectly identified model based on PEM(OE) estimation is almost as good, with R2 T = 0.979. However, as expected, the indirectly identified model based on ARX estimation is quite poor, with R2 T = 0.959. In the second case, the RIV-based estimation with AIC identified AR(4) model is only a little different from that obtained without noise model parameter estimation, with R2 T = 0.979. However, all reasonable PEM-based estimation results (MA(2), AR(4) and ARMA (2,4) noise models) are worse, with R2 T = 0.959, R2 T = 0.933 and R2 T = 0.932, respectively. More importantly in practical terms, none of these PEM- based models identified the long time constant, so would be rejected on physical grounds. In the case of the MA(2) noise model (the ARMAX model form), the eigenvalues have different signs which cannot be interpreted at all in physically meaningful terms. CONCLUSIONS This paper has outlined an approach to Data-Based Mechanistic (DBM) modelling of environmental systems based on the direct identification and estimation of continuous-time (transfer function or differential equa- tion) models. It has also argued that such models are more appropriate to modelling of environmental systems than their more widely known discrete-time eqLuiva- lents. The main advantage of these methods, over the alternative discrete-time methods, is that they provide differential equation models whose parameters can be interpreted immediately in physically meaningful terms, e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6 5 20 40 60 80 100 120 140 160 180 0 1 2 3 4 Flow(cumecs) 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 60 70 EffectiveRainfall(mm) Time (days) FIG. 1. A typical set of effective rainfall and flow data as our examples have shown. Consequently, they are of direct use to scientists and engineers who most often derive models in differential equation terms based on natural laws and who are much less familiar with ‘black- box’ discrete-time models. Moreover, the continuous- time methods can be adapted easily to handle the case of irregularly sampled data or non-integral time delays that are often encountered in the modelling of environmental systems. They are also much superior when applied to rapidly sampled data, where discrete-time methods often perform poorly because the eigenvalues lie close to the unit circle in the complex domain leading to estimation problems. These advantages, together with the parsimony that is a natural consequence of DBM CT transfer function modelling, should mean that any relationships between the CT model parameters and physical measures of the environmental process should be clearer and better defined statistically. This data-based DBM CT modelling approach has pro- ven successful in many practical applications and the most powerful statistical methods are available in the user-friendly CAPTAIN and CONTSID toolboxes. R´EF ´ERENCES [1] Garnier, H., Gilson, M., Bastogne, T., et Mensler, M. (2008). Identification of continuous-time models from sampled data, Cha- pitre The CONTSID toolbox : a software support for data-based continuous-time modelling. H. Garnier and L. Wang (Eds.), Springer-Verlag. [2] Garnier, H., Mensler, M., et Richard, A. (2003). Continuous- time model identification from sampled data. Implementation issues and performance evaluation. International Journal of Control, 76(13) :1337–1357. [3] H. Garnier and L. Wang (Eds.) (2008). Identification of continuous- time models from sampled data. Springer-Verlag. [4] Ljung, L. (1999). System identification. Theory for the user. Prentice Hall, Upper Saddle River, 2nd edition. [5] Ljung, L. (2003). Initialisation aspects for subspace and output-error identification methods. European Control Conference (ECC’2003), Cambridge (U.K.). [6] Rao, G. et Garnier, H. (2002). Numerical illustrations of the relevance of direct continuous-time model identification. 15th Triennial IFAC World Congress on Automatic Control, Barcelona (Spain). [7] Rao, G. et Garnier, H. (2004). Identification of continuous-time systems : direct or indirect ? Systems Science, 30(3) :25–50. [8] Taylor, C., Pedregal, D., Young, P., et Tych, W. (2007). Environ- mental time series analysis and forecasting with the captain toolbox. Environmental Modelling and Software, 22(6) :797–814. [9] Young, P. (1981). Parameter estimation for continuous-time models - a survey. Automatica, 17(1) :23–39. [10] Young, P. (1984). Recursive estimation and time-series analysis. Springer-Verlag, Berlin. [11] Young, P. (1998). Data-based mechanistic modeling of environ- mental, ecological, economic and engineering systems. Journal of Modelling & Software, 13 :105–122. [12] Young, P. (2002). Advances in real-time flood forecasting. Philo- sophical Trans. Royal Society, Physical and Engineering Sciences, 360(9) :1433–1450. [13] Young, P. et Garnier, H. (2006). Identification and estima- tion of continuous-time data-based mechanistic (DBM) models for environmental systems. Environmental Modelling and Software, 21(8) :1055–1072. [14] Young, P., Garnier, H., et Gilson, M. (2008). Identification of continuous-time models from sampled data, Chapitre Refined instrumental variable identification of continuous-time hybrid Box- Jenkins models. H. Garnier and L. Wang (Eds.), Springer-Verlag. [15] Young, P. et Jakeman, A. (1980). Refined instrumental variable methods of time-series analysis : Part III, extensions. International Journal of Control, 31 :741–764. e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6 6 Hugues Garnier received the Ph.D. degree in Automatic Control in 1995 from Universit´e Henri Poincar´e, Nancy 1, France. He has been with the Centre de Recherche en Automatique de Nancy (CRAN) at the Universit´e Henri Poincar´e, Nancy 1 since 1993, where he is currently a Professor. From Septembre 2003 to August 2004, he visited the Centre for Complex Dynamic Systems and Control, University of Newcastle, Australia. In 2006, he has held visiting positions at different universities in Australia including the University of Newcastle, the Royal Melbourne Institute of Technology and the Uni- versity of Technology in Sydney. Hugues Garnier is cur- rently the leader of the System Identification and Signal Processing research group at CRAN and was the co- leader of the working group on ”System identification” of the French GdR MACS from 2005 up to December 2007. He is member of the IFAC Technical Committee ”Modelling, Identification and Signal Processing” and the IEEE Technical Committee ”System Identification and Adaptive Control”. He was also member of the International Program Committee for the IFAC Sympo- sium on System Identification (SYSID’2006), hold in Newcastle, Australia in March 2006. Hugues Garnier’s main research interest is related to the analysis and modelling of stochastic dynamical systems. This includes signal processing, time series analysis and prediction, parameter estimation and system identification. He has written several recent contributions on new techniques for continuous-time model identification from sampled data and organised many invited sessions at international congresses on this research area in the past decade. He is also behind CONTSID (www.cran.uhp-nancy.fr/contsid), a MATLAB toolbox for continuous-time system iden- tification. Hugues Garnier is member of the Editorial Board of International Journal of Control and is the lead- editor of the book entitled ”Identification of continuous- time models from sampled data” to be published by Springer-Verlag in 2008. Peter Young is well known for his pioneering work on recursive estimation and time series analysis applied to both discrete and continuous-time system modelling. His research has involved numerous different areas of appli- cation, including hydrology, climate, macro-economics and business. His numerous publications include a mo- nograph on recursive estimation and time series ana- lysis, as well as the edited Concise Encyclopedia of Environmental Systems. His latest research has been in several areas : a continuation, with Hugues Garnier, of his early work on the identification and estimation of continuous-time systems from discrete-time data ; the modelling of nonlinear stochastic systems using state- dependent parameter estimation ; the emulation of large computer simulation models by reduced order, dominant mode models ; the use of recursive estimation in adaptive flood forecasting ; and non-minimal state space methods of control system design. St´ephane Thil was born in 1977 in Sarreguemines, France. He received the Ph.D. degree in Automatic Control in December 2007 from Universit´e Henri Poin- car´e, Nancy 1, France. He currently holds a post-doc po- sition within the Centre de Recherche en Automatique de Nancy (CRAN). His research interests include stochastic signal processing and system identification. Marion Gilson received her Ph.D. degree in Automatic Control in 2000 from Universit´e Henri Poincar´e, Nancy 1, France. She has been with the Centre de Recherche en Automatique de Nancy (CRAN) at the Universit´e Henri Poincar´e, Nancy 1 since 1997, where she is currently an Associate Professor. She pursues research in the areas of system identification and stochastic signal processing. From 1999, she has held several visiting positions at TU Delft University in the Netherlands. Marion Gilson is currently the co-leader of the working group on ”System identification” of the French GdR MACS. e-STA copyright © 2008 by see Volume 5 (2008), N°2 pp 1-6