(1)Identication of SISO nonlinear Wiener systems Corinne Ledoux

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(1)Identication of SISO nonlinear Wiener systems Corinne Ledoux. ERTEKNIK REGL. AU. OL TOM ATIC CONTR. LINKÖPING. Department of Electrical Engineering Linkoping University S-581 Linkoping, Sweden Linkoping 1996.

(2) Acknowledgments There are many people I would like to thank. First of all, I want to thank Professor Lennart Ljung for giving me the opportunity to work with him and supervising this research. Special thanks to you, Lennart ! I would like also to thank Ulla Salaneck for her kindness and help. Thanks to Magnus Andersson for being a nice oce mate and from whom I have learned to say uently in swedish \Kan jag oppna fonstret ?" and \Kan jag stanga fonstret ?" Thanks to all the sta from Reglerteknik for its welcome and the nice atmosphere to work in. Special and a ective thoughts to my parents who took care of all my business involved in Paris, while I was away. Thanks also to Volker Rezek and Tobias Kuhme for their friendships. I would like to thank my friends Philippe and Stefan for those memorable Christmas holidays in Stockholm. Finally, special thanks to Magnus Ericksson for teaching me the philosophy of life and the nice time spent with him. Sayonara ! Thanks to all of them and those I forgot.. Corinne.

(3) Contents. I Context of the research. 5. 1 Introduction. 7. 2 Concepts of system identication. 8. 1.1 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.2 Outline : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :. 2.1 2.2 2.3 2.4 2.5. Introduction : : : : : : : : : : : : : : Dynamical systems : : : : : : : : : : Models : : : : : : : : : : : : : : : : : The system identication procedure Use of a prior knowledge : : : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. 7 7. 8 8 9 9 10. II Dening the problem. 13. 3 Describing the Wiener system identication problem. 15. 3.1 Describing a Wiener nonlinear system : : : : : : : : : : : : : : : : : : : : : : : : : 3.2 Identifying a Wiener nonlinear system : : : : : : : : : : : : : : : : : : : : : : : : : 3.3 Some restrictions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :. 4 A novel identication method 4.1 4.2 4.3 4.4 4.5. Notations : : : : : : : : : : : : : : : : : : Grey models versus black box models : : Theoretical identication scheme : : : : : Major diculties : : : : : : : : : : : : : : Approach implemented : : : : : : : : : : : 4.5.1 General approach : : : : : : : : : : 4.5.2 Estimating the nonlinearity : : : : 4.5.3 Estimating the ARX model : : : : 4.5.4 Estimating the state space model :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. : : : : : : : : :. 15 16 16. 17. 17 17 18 21 21 21 22 23 24.

(4) 4. III Simulations. 27. 5 Identifying a Wiener system. 5.1 Objective : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.2 Presenting the system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.2.1 The ARX model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.2.2 The polynomial model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.3 Describing the simulations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.3.1 Estimating the models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.3.2 Pre-treating the data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4 First simulation set : Starting with \true" linear output using an ARX model : : : 5.4.1 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4.2 Initial conditions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4.3 Intermediate results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4.4 Final results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4.5 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.5 Second simulation set : Starting with \true" linear output using a state space model 5.5.1 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.5.2 Initial conditions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.5.3 Intermediate results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.5.4 Final results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.5.5 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6 Third simulation set : Realistic initialization : : : : : : : : : : : : : : : : : : : : : 5.6.1 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6.2 Initial conditions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6.3 Intermediate results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6.4 Final results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6.5 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.7 Black box model approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.7.1 Pre-treating the data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.7.2 Architecture of the neural network : : : : : : : : : : : : : : : : : : : : : : : 5.7.3 Estimation algorithm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.7.4 Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.7.5 Comparing the grey and black box approach : : : : : : : : : : : : : : : : :. IV Analyses. 6 The complexity of the system identication task 6.1 6.2 6.3 6.4 6.5. Objectives : : : : : : : : : : : : : : : : : : : : : : Maximum Likelihood criterion : : : : : : : : : : Derivatives of the Maximum Likelihood criterion Analysis of the criterion minimized : : : : : : : : Derivatives of the criterion minimized : : : : : :. V Conclusions. 7 Conclusions. 29. 29 29 29 30 31 31 31 31 31 32 32 34 36 37 37 37 37 38 38 41 41 41 41 43 46 46 46 46 46 46 47. 49 : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. : : : : :. 51 51 51 53 57 57. 61. 63.

(5) Part I. Context of the research.

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(7) 1 Introduction 1.1 Objectives A novel approach is presented for the design of an identication algorithm for nonlinear Wiener models. A nonlinear Wiener system consists of a Linear Time Invariant system followed by a nonlinearity. Two di erent philosophies can be applied to solve this problem : 1. Black box modelling. The key idea is to use a set of exible models that can accomodate the system, without looking into its internal structure. 2. Grey modelling. The key concept is to incorporate physical insight into the model set so as to estimate both the LTI system and the nonlinearity. The identication viewpoint of interest is the grey modelling, nevertheless the black box modelling has been looked into for comparison purposes. The identication algorithm consists of : 1. parameterizing individually both the LTI system and the nonlinearity, 2. identifying the parameters by minimizing the prediction error at the output of the LTI system, through an iterative sequence of standard least square problems. The major diculty lies in the estimation of the prediction error at the output of the LTI system, since the only available data consist of the input to the LTI system and the output from the nonlinearity. We have examined two identication problems. In the rst problem, the LTI system is an ARX model, whereas in the second one, it is a state space model.. 1.2 Outline This report consists of seven chapters, where the rst one is this introduction. The second chapter is dedicated to a brief introduction of the main concepts involved in system identication. Then the problem dealt with is presented in the third chapter. The fourth chapter describes in further details the identication algorithm proposed. The system to be identify and the results from the simulations are the topics of chapters 4 and 5. The sixth chapter aims at analyzing the complexity of the identication task. Finally, the seventh chapter concludes on the results reached so far..

(8) 2 Concepts of system identication 2.1 Introduction The current research lies within the context of system identication. System identication deals with the problem of building mathematical models of dynamical systems based on observed data from the systems. Briey, system identication is performed essentially by adjusting parameters within a given model until its outputs incide as well as possible with the measured outputs. Following sections present in further details the various concepts involved in system identication such as dynamical systems, models ... The materials used here can be found in 1] and 2].. 2.2 Dynamical systems Roughly speaking, a system is an object in wich di erent variables interact and produce observable signals. The observable signals we are interested in are called outputs. The system is also a ected by external stimuli. External signals that can be manipulated by the observer are called inputs. Other stimuli are called disturbances and can be divided in : those that are directly measured, denoted by w, those that are only observed through their inuence on the outputs, denoted by v. If we denote the inputs and outputs of the system by u and y respectively, the relationship can be depicted as in Figure 2.1. v. u w. System. Figure 2.1: A system. y.

(9) 2.3 Models Dynamic systems imply that the current output value depends not only on the current external stimuli but also on their past values.. 2.3 Models The purpose of a model is to describe how the various variables of a system relate to each other. When these relationships are expressed in terms of mathematical equations, the resulting model is a mathematical model. So far, a question may rise : why building models of a system ? There are many reasons for using models of a real system : the system may be not exist properly, running the system may be too expensive, advanced control systems require a model of the system, models are instruments for simulation and forecasting. Building of models is achieved through observed data. As far as mathematical models are concerned, one can distinguish :. modelling, that consists in splitting up the system into subsystems, whose properties are well known from past experience. Then these subsystems are joined mathematically and a model of the whole system is reached.. system identication. This approach is based on experimentations input and output signals from the system are collected and analyzed to infer a model.. 2.4 The system identication procedure The system identication problem is to estimate a model of a system based on observed input output data. Several ways to describe a system and to estimate such descriptions exist. The procedure to determine a model of a dynamical system from observed input output data involves three basic ingredients : 1. the input - output data, which depend both on the true system and the experimental conditions. 2. a set of candidate models : the model structure. Selecting the right model structure is among the toughest steps in system identication. It involves : specifying the type of model set to use .i.e. choosing between linear or nonlinear models, between black boxes, semi-physical and physically parameterized models ... specifying the size of the model set. It consists in choosing the possible variables, and combinaison of variables to use in the model description, setting up orders and degrees of the model type. parameterizing the model so that estimation procedures can lead to \good" parameters values. 3. a criterion to select a particular model in the set, based on the information in the data : the identication method.. 9.

(10) 10. Concepts of system identication. The two rst steps lead to a model set over which the search for a model can be conducted. The model set can be very large but the use of a priori knowledge often reduces its size. The identication process amounts to repeatedly selecting a model structure, computing the best model in the structure, and evaluating the properties of this model to see if they are satisfactory. The cycle is depicted in Figure 2.2 and can be run as follows : 1. Design an experiment and collect input output data from the process to be identied, 2. Examine the data. Some pre treatements may have to be applied, 3. Select and dene a model structure within wich a model is to be found, 4. Compute the best model in the model structure according to the input output and a given criterion of t, 5. Examine the properties of the model obtained, 6. If the model is good enough then stop, otherwise go back to the third step and try another model set. Other estimation methods can also be tried (fourth step), further pre treatements can be applied to the data (rst and second steps). Experimentations. Prior knowledge. Data. Choose model set Choose criterion of fit. Choose the estimation algorithm. Estimate the parameters Calculate Model. Validate the model. OK. OK Exploitation. Figure 2.2: The system identication procedure. 2.5 Use of a prior knowledge When designing models, a prior knowledge about the system can be exploited di erent ways, leading either to white, gray or black box models. When all the physical insight about the behaviour of the system is available and put into the model structure, one performs a physically parameterized modelling. The resulting model, also referred as to a white box model, leads to a full and exact characterization of the system. This kind of modelling is seldom applied since such amount of prior knowledge is never available. If no prior knowledge is used when building a model .i.e. one relies only on the information contained in the input output data set, one reaches a black box model..

(11) 2.5 Use of a prior knowledge. 11. In the middle zone, somewhere in between white modelling and black modelling, lies gray modelling. This is the case when some physical insight is available and used, but several parameters remain to be determined from the observed data. This is the viewpoint adopted in this report. A good reference on this topic is 3]..

(12) 12. Concepts of system identication.

(13) Part II. Dening the problem.

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(15) 3 Describing the Wiener system identication problem 3.1 Describing a Wiener nonlinear system A Wiener nonlinear system consists of a series connection of a Linear Time Invariant system followed by a nonlinearity (see Figure 3.1). v(k). u(k). z(k). Linear Time Invariant. Nonlinearity. +. y(k). system. Figure 3.1: A nonlinear Wiener system The system we are interested in identifying is assumed to be made of : 1. a Linear Time Invariant model. Depending on the simulations performed, it can be : a state space model described as follows :. (. x(k + 1) = Ax(k) + Bu(k) z (k ) = Cx(k) + Du(k). (3.1). wherein x(k) Rn denotes the state vector, u(k) Rm is the input vector and z (k) Rl is the output vector. an ARX model 2. a nonlinear function f . w (k ) = f ( z ( k ) ) (3.2) 3. The output from the whole system is then y(k) = f ( z (k) ) + v(k) wherein v(k) is a zero mean stochastic process of arbitrary color.. (3.3).

(16) 16. Describing the Wiener system identication problem. 3.2 Identifying a Wiener nonlinear system The Wiener system identication problem that is studied in this report is the same as analyzed in 4] and is stated as follows. Let the following data sequences of input and output data. u(k) u(k + 1) . y(k) y(k + 1). . . u(k + N ) y (k + N ). be made available. Assume that the input sequence fu(k)g is suciently persistently exciting and statistically independent from the perturbation fv(k)g, the task is to nd a good estimate of the LTI system as well as an estimate of the nonlinearity.. 3.3 Some restrictions At rst, for sake of simplicity, we restrict to single input single output systems this implies that l = m = n = 1. Secondly, the nonlinearity is a static nonlinear function. Thirdly, the data are noise free..

(17) 4 A novel identication method 4.1 Notations u(t). Linear Time Invariant. z(t). Nonlinearity. model g. y(t). f. Figure 4.1: Notations used The output from the Linear Time Invariant system can be expressed as. z ( t) = g(u(t)). (4.1) wherein g(:) refers to the transfer function of the Linear Time Invariant system, and u the input signal applied (see Figure 4.1). Similarly, we can dene the output from the nonlinear system as. y( t) = f (z (t)). (4.2). The most general description of the nonlinear Wiener system is then. y( t) = f ( g(u(t)) ). (4.3). 4.2 Grey models versus black box models The problem is to determine a predictor of y based on (4.3). Two di erent philosophies can guide the choice of parameterized model sets : Black box model structures. The key idea is to use a set of exible models that can accomodate the system, without looking into its internal structure. The resulting predictor takes the form y^(t ) = h(!N ) wherein !N refers to the input - output data set available for identication .i.e. !N = yt ut ] = y(1) u(1) y(N ) u(N )], h is a nonlinear function of past data, and is a parameter vector .i.e. Rp.

(18) 18. A novel identication method Grey models structures. The key idea is to incorporate physical insight into the model set so as to estimate what is actually unknown about the system. As far as a nonlinear Wiener system is concerned, we mean by physical insight :. { the series connection of a Linear Time Invariant model and a nonlinear function, { the structure of the Linear Time Invariant model. The resulting predictor has the form y^(t ) = f (g(u(t) ) ), wherein f () represents a particular nonlinear model, parameterized by a nite dimension parameter vector Rm , g() is a particular Linear Time Invariant model parameterized by Rn . Both parameter vectors and have to be estimated from the data set !N . One should always take advantages of the physical insight about the system to identify in order not to estimate what is already known. Several applications have demonstrated that the more a priori knowldege is put into the model, the better the modelling results. Let us mention for instance the solar heated problem described in 5], 6] or the DC motor problem studied in 3]. Grey modelling is the way we have decided to investigate. Nevertheless, we have looked into the black box modelling for comparison purposes.. 4.3 Theoretical identication scheme This section presents in further details the scheme we propose to estimate both the LTI model and the static nonlinearity. For sake of clarity, let us introduce some more notations. G denotes the model set over which the search for a LTI model has to be conducted. Using the a priori knowledge about the system to identify, this set is reduced signicantly to the model set of linear state space models or ARX models depending on the simulations. F is the model set over which the search for a static nonlinear function has to be performed. Let us assume now that the models in the sets G and F can be parameterized using respectively the nite dimension parameter vectors Rn and Rm . For instance, g() denotes a particular model within G , parameterized by .. z^( t ) denotes the output obtained from a particular parameterization of G at time t. It can be written as. z^( t ) = g(u(t) ). (4.4). whereas z (t) refers to the measurement provided at time t. Similarly, y^( t ) denes the output from a particular parameterization of F at time t. It can be expressed as. y^( t ) = f (z (t) ). (4.5). Actually, y( t ) is computed based on the output estimate from the LTI model. A more accurate description of the predictor of y is. y^( t ) = f (^z (t ) ) y^( t ) = f (g(u(t) ) ). (4.6) (4.7). Thus, equation (4.7) leads to the most general description of the predictor of the system. Now, let us remind about our objective : nd among the model sets G and F the best estimate models that t the data. In other words, it means nd the parameter vectors ^ and ^ that minimize the Maximum Likelihood criterion :.

(19) 4.3 Theoretical identication scheme. VN ( . 19 N 1X N (y(t). !N ) =. t=1. ;. y^(t ))2. (4.8). With (4.7), VN ( !N ) can be rewritten as N X. VN ( !N ) = N1 (y(t) ; f (g(u(t) ) ))2 t=1 So the parameter vectors ^ and ^ are dened as follows N ^ ^ = arg min 1 X(y(t) N t=1. ;. f (g(u(t) ) ))2. (4.9). (4.10). Since the parameters and are independent, minimizing the Maximum Likelihood criterion (4.9) can be achieved by : at rst, minimizing a loss function with respect to leading to ^, secondly, based on ^, minimizing the loss function (4.8) with respect to leading to ^ At rst, let us deal with the problem of estimating ^. The parameter vector ^ we are looking for is the one that minimizes the following loss function N X. VN ( Z N ) = N1 (z (t) t=1 VN ( Z N ) =. ;. z^(t ))2. N 1X (t )2 N. (4.11) (4.12). t=1. wherein (t ) usually refers to the prediction error and Z N the input output data set, associated to the LTI model .i.e. Z N = z t ut ] = z (1) u(1) z (N ) u(N )]. The parameter vector ^ is thus dened as. ^ = arg min V ( Z N ) N. (4.13). Similarly, the parameter vector ^ we are looking for is the one that minimizes the loss function given by : N X. VN ( Y N ) = N1 (y(t) t=1 N X. ;. VN ( Y N ) = N1 (t )2 t=1. y^(t ))2. (4.14) (4.15). wherein Y N refers to the input output data set associated to the nonlinear model .i.e. Y N = yt z t] = y(1) z (1) y(N ) z (N )]. The criterion (4.9) is the classical Maximum Likelihood criterion minimized in system identication the parameter vectors ^ and ^ are estimated so that the prediction error at the output of the.

(20) 20. A novel identication method. system be the smallest. What precedes shows that identifying a nonlinear Wiener system based on the minimizing of (4.9) is not so trivial since the data sets Z N and Y N are not directly available, only !N is provided. Therefore, we propose to apply a criterion which is slightly di erent from (4.9) and leads to the estimation of Z N and Y N . Let us assume now that the function f be invertible so that an estimate of z can be obtained through:. z^(t ) = f ;1 ( (y(t) ). (4.16). Since z is the output from the LTI model, it can also be reached through. z^(t ) = g( u(t) ). (4.17). A condition for which z^(t ) is close to g( u(t) ) is that good estimates of the parameter vectors and have been reached. The criterion minimized then writes. VN ( . N 1X N (^z(t ). !N ) =. t=1. ;. g( u(t) ))2. (4.18). The parameter vectors ^ and ^ are then dened as follows :. ^ ^ = min min . X. (^z(t ) ; g(u(t) )))2. (4.19). To sum up, the key idea of our approach lies in : 1. the estimation of the LTI model, based on its input output data set being made available. Then z^( t ) = g(u(t) ) is computed. 2. the estimation of the nonlinear model parameterized by , based on the input output data set yt z^t()]. 3. the estimation of z^( t ) based on(4.16). This basic loop is repeated until the t between y(t) and y^(t ) or z^( t ) and z^( t ) is fulllled. The proposed iterative identication scheme then consists of : 1. Find initial values of the variable z namely z^(t î î ), with i = 0 P 2. Compute î+1 so that î+1 = arg min (^z( t î î ) ; g( u(t) î ))2 . 3. Compute ^z(t î+1 î ) = g( u(t) î+1 ). P(y(t) ; f (^z(t ^ ^ ))2 4. Compute î+1 so that î+1 = arg min i+1 i 5. Compute z^(t î+1 î+1 ) = f ;1 (y(t) î+1 î+1 ) 6. Return to 2 with i = i + 1 until the desired accuracy has been reached.

(21) 4.4 Major diculties. 21. 4.4 Major di culties The identication scheme described in section 4.3 rises so far two major diculties : 1. Can one start the iterative process with good initial conditions without any a priori knowledge? 2. Once initial values of z have been found, can the identication scheme proposed lead to a good modelling? It is quite obvious that the quality of the modelling can su er from bad initial conditions. The major problem faced here is that no insight can be retrieved to provide such good initial values. So, as a rst approximation and for initialization purposes only, we consider the nonlinear function f () as the identity function. Hence, the initialization step consists in stating z (t) = y(t). When estimating the LTI model, one hopes that the linear features of the output signal would be captured, leading to an estimate z^(t ) of suitable quality. Then taking advantages of the ability of a exible black box model to approximate any nonlinear continuous function, the building of the nonlinear mapping that transforms z^(t ) into y(t) is straightforward. Applying then y(t) to the input of the inverted black box model leads to a more accurate estimation of z . By doing this several times, one hopes to get closer and closer to the true system description. Another diculty can rise from the use of an iterative procedure the problem of global convergence. Little is currently known about the global behaviour of the algorithm. So far, no proof of convergence to global minimum is provided.. 4.5 Approach implemented 4.5.1 General approach One who is fully familiar with black box model based approximation function should know that inverting a black box model is far from easy. Therefore, the approach implemented is slightly di erent from the theoretical one, described in section 4.3. Rather than estimating the nonlinear function f that maps z^(t ) into y(t) and then inverting it to get z^(t ) , the fourth step now consists in estimating the nonlinear function that maps y(t) into z^(t ). A single loop of the identication process consists of steps 2, 3, 4, 5 and 6. Once a desired accuracy in the modelling of the LTI model and the nonlinear inverse function has been reached, the direct function that maps z^(t ) into y(t) is approximated. At that point, the identication of the nonlinear Wiener system is completed. So, the iterative identication scheme implemented is the following : 1. Initialize the variable z with z^(t î î ) = y(t), and i = 0 P 2. î+1 = arg min (^z ( t î î ) ; g( u(t) î ))2 . 3. Compute z^(t î+1 î ) = g( u(t) î+1 ) P 4. î+1 = arg min (^z(t î+1 î ) ; f ;1 (y(t) î+1 î ))2 . 5. Compute z^(t î+1 î+1 ) = f ;1 (y(t) î+1 î+1 ) 6. Return to 2 with i = i + 1 until the desired accuracy has been reached P(y(t) ; f (^z(t ^ ^ ))2 7. ^ = arg min i+1 i 8. Compute y^(t î+1 ^) = f (^z(t î+1 ^).

(22) 22. A novel identication method. 4.5.2 Estimating the nonlinearity. Estimating the nonlinear model consists in approximating the nonlinear function that maps z^(t ) into y(t). Similarly, the approximation of the function that maps y(t) into z^(t ) results in the estimation of the inverse nonlinear model. There are several techniques that can be used to approximate functions. For instance, it is possible to decompose the objective function on a basis of elementary functions these elementary functions must generate a base of dense functions in the space of the functions to be approximated. In the context of this study, the "dense" model used is a feedforward neural network. The following paragraphs only present briey feedforward neural networks. For deeper details, see 5], 7].. Feed Forward Neural Networks Articial neural networks are computing systems consisting of many simple nonlinear nodes interconnected by links. Connections between nodes are called "synaptic weights". In "feed forward" networks, the nodes are organized in layers, with links from each node in the kth layer being directed to each node in the (k +1)th layer. Inputs from the environment enter the rst layer, outputs from the network are read from the output layer. Intermediate layers, between the input and the output layers, are called "hidden" layers. A three layer neural network is depicted in Figure 4.2.. Wi1 Wi2 Neuron j. Wij Oj Win. Input layer. Hidden layer. Output layer. Figure 4.2: The neuron network used. Transfer. xi function Oi φ. Neuron i. Figure 4.3: An articial neural. Let us denote i the index of a neuron, located in the (k + 1)th layer, j refers to the neurons belonging to the kth layer and therefore connected to the neuron i. Wij represents the weight of the connection between the neuron i and a particular neuron j . Each neuron i is characterized by an input state xi and an output state Oi . The input state xi is reached through the computation of a dot product between its vector Oj ] and its weight vector Wij ] (see Figure 4.3). The weighting sum xi (4.21) is then passed through a squashing function to produce the state of the neuron i denoted by Oi (4.21). The most common function is the hyperbolic tangent.. xi =. n X j =1. Wij Oj. Oi = (xi ). Neural networks as universal function approximators. (4.20) (4.21). Let us consider a function F : Rm ! Rn that is to be approximated. This function F is not analytically known, but rather known through a set of numerical samples SP = fS1 S2 SP g.

(23) 4.5 Approach implemented. 23. with Si = (xi yi ) that is generated by a process that is governed by F , so that F (xi ) = yi . From the available samples, the objective is to build an approximation of the function F . The nonlinear mapping performed by the neural network is NSW : Rm ! Rn . NSW consists of basis functions i : R ! R that are represented by the hidden and output nodes and the synaptic weights. For a three layer feedforward neural network, containing m input nodes, h hidden nodes and n output nodes, its ith output is NSW (x)i. =. h X j =1. m X. Wij j (. k=1. Wjk xk ). (4.22). It has been demonstrated that the feedforward network NSW is able to approximate any continuous function F up to a desired accuracy 8]. The approximation depends on the estimation samples available, the architecture of the network, the estimation algorithm that determines the parameters Wij from SP .. Estimation process. Estimating a neural network involves the modication of the synaptic weights, until the function approximated by the neural network is fully satisfactory. The estimation process is divided into several steps. At rst, samples are presented to the network. Then, the quadratic error between the values predicted by the network and the desired values provided by the data is calculated over the entire estimation set as follows :. E (W ) =. h m n X 1 X X X y ; W ( Wjk xk )]2 i ij j 2 i=1 (xy)Sp. j =1. k=1. (4.23). Afterwards, the weights are modied so that the mean square error (4.23) be minimal. The estimation of the synaptic weights is thus reduced to a complex multivariate minimization procedure of the global error function given by equation (4.23). The whole process is then repeated until the mean square error falls below a determined threshold. Most of the optimization methods used to minimize the error function are based on the same strategy. The minimization is a local iterative process in which an approximation of (4.23) in the neighbourhood of the current point in the weight space is minimized. The approximation is based on the rst or second order Taylor expansion of the error function. The estimation algorithm used here is the Gauss Newton second order method, implemented in an o -line way and available in the Neural Network ToolBox, developed by Jonas Sjoberg in the matlab language. Once the neural network has been estimated, its ability to approximate the desired function can be evaluated on further samples the estimation process did not use.. 4.5.3 Estimating the ARX model. The most general description of an ARX model is given by the following linear di erence equation:. y(t) + a1 y(t ; 1) + + ana y(t ; na ) = b1 u(t ; nk ) + + bnb u(t ; nk ; nb + 1). (4.24). which relates the current output y(t) to a nite number of past outputs y(t ; k) and inputs u(t ; k). The numbers na and nb are the orders of the ARX model. Two methods are available in the System Identication ToolBox to estimate the coecients a and b in the ARX model structure, once the orders and time delay have been set up :. the least squares method which minimizes the sum of squares of the right hand side minus the left hand side of (4.24), with respect to a and b..

(24) 24. A novel identication method the instrumental variables method, that determines a and b so that the error between. the right and left hand sides becomes uncorrelated with a particular combination of the inputs (see section 7.6 of 1]). The estimation algorithm applied is the least square method.. 4.5.4 Estimating the state space model. Estimating a state space model can be handled through the use of : a predition error minimization method, a subspace method.. Prediction error minimization method. Let us briey remind about the basic concepts of the prediction error minimization method, already introduced in section 4.3. The standard setting can thus be described as follows. Given the estimation data set consisting of an input sequence f u(t) g and an output sequence f y(t) g and dened by !N = yt ut ]t=1N and a model structure M dening a mapping from the parameter space DM to the predictor y^(t ), the objective is to nd the value ^ which minimizes the loss function N X. VN ( !N ) = N1 j (t ) j2 t=1. (4.25). (t ) is the prediction error, which writes (t ) = y(t). ;. y^(t ). (4.26). The parameter vector ^ is dened by. ^ = arg min VN ( !N ) DM. (4.27). Further details on this topic, refer to 1]. The Prediction Error Method relies on a parametrization of the model. Finding a suitable parameterization among all the possible parameterizations is not so trivial. This is especially true for higher order Multi Input Multi Output systems. Usually a priori knowledge about the system to identify is required for some guidance purposes, but it is not always available. An extra parameterization of the initial state is also needed when estimating a state space model from data measured on a plant with nonzero initial conditions. The objective criterion is usually expressed in terms of a nonlinear function of the parameters. This implies that a nonlinear optimization process has to be performed. Widely used nonlinear optimization algorithms present the disadvantages of being iterative and sensitive to initial conditions. There is also no garantee of convergence. Once has often to deal with the problem of local minima.. Subspace method. SubSpace State Space IDentication methods, also refered to N4SID, are alternatives techniques to estimate state space models. The key idea lies in the estimation of the state space matrices through projections of input and output data. They do not rely on any canonical form, involving a minimal number of parameters, which is sometimes hard to nd. Furthermore, the order of the.

(25) 4.5 Approach implemented. 25. system can be determined through inspection of the dominant singular values of a matrix that is calculated during the estimation. They also present the attracting advantage of not requiring any nonlinear iterative optimization algorithm. Therefore, they do not su er from local minimum, sensitivity to initial conditions ... For these reasons, it is the method adopted to estimate the state space model. For some more details about N4SID techniques, refer to 9], 10]..

(26) 26. A novel identication method.

(27) Part III. Simulations.

(28)

(29) 5 Identifying a Wiener system 5.1 Objective Results from previous simulations have stressed the diculty in identifying both the LTI system consisting in a state space model and the static nonlinearity. So far, two main explanations are provided : 1. the identication scheme relies on the approximation of the inverse nonlinear function. This may lead to an additional diculty one does not suspect. 2. Neural networks are too exible, .i.e., the more hidden units are added, the more the neural network adapts to errors resulting from the LTI system modelling. The trade o between a too exible model and a not exible enough model is here particularly hard to nd. In order to nd out whether the approximation of the inverse nonlinear function makes the identication task even more dicult, one has decided to study a simpler problem.. 5.2 Presenting the system The system consists of a series connection of an ARX model followed by a polynomial model (see Figure 5.1). u(t). z(t) ARX model. Polynomial model. y(t). Figure 5.1: System to identify. 5.2.1 The ARX model. The most general description of an ARX model is given by the following linear di erence equation:. y(t) + a1 y(t ; 1) + + ana y(t ; na ) = b1 u(t ; nk ) + + bnb u(t ; nk ; nb + 1). (5.1).

(30) 30. Identifying a Wiener system. which relates the current output y(t) to a nite number of past outputs y(t ; k) and inputs u(t ; k). The numbers na and nb are the orders of the ARX model. Regarding the notations previously introduced, the ARX model to identify is dened by. z (t) + a z (t ; 1) = b u(t) + c u(t ; 1) (5.2) wherein z is the output signal, u is the input signal and the parameters a b c being respectively equal to 0:6 1:5 0:7. Let denote the parameter vector a b c]. The input and output signals are depicted in Figures 5.2 and 5.3. The 300 rst values are used for estimation purposes and the remaining values are kept for validation. Input applied to the ARX model. Output from the ARX model. 1. 1.6. 0.9. 1.4. 0.8. 1.2. 0.7 1 0.6 0.8 0.5 0.6 0.4 0.4 0.3 0.2. 0.2. 0. 0.1 0 0. 50. 100. 150. 200. 250. 300. −0.2 0. 50. 100. 150. 200. 250. 300. Figure 5.2: Data used for the estimation of the ARX model Input applied to the ARX model. Output from the ARX model. 1. 1.6. 0.9. 1.4. 0.8. 1.2. 0.7 1 0.6 0.8 0.5 0.6 0.4 0.4 0.3 0.2. 0.2. 0. 0.1 0 0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. −0.2 0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. Figure 5.3: Data used for the validation of the ARX model. 5.2.2 The polynomial model. The nonlinearity consists of a second order polynomial model. f (x) = x2 + x + . (5.3). wherein the parameters being respectively equal to 2 2 1. Let denote the parameter vector ]. The output from the whole system is expressed as.

(31) 5.3 Describing the simulations. 31. y(t) = f (z (t)) y(t) = z (t)2 + z (t) + . (5.4) (5.5) In Figures 5.4 and 5.5 are presented the data used for the estimation and validation purposes. Output from the polynomial model. Output from the polynomial model. 9. 9. 8. 8. 7. 7. 6. 6. 5. 5. 4. 4. 3. 3. 2. 2. 1. 1. 0 0. 50. 100. 150. 200. 250. 300. Figure 5.4: Data used for the estimation. 0 0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. Figure 5.5: Data used for the validation. 5.3 Describing the simulations 5.3.1 Estimating the models. As far as the 'grey' identication approach is concerned, both models are estimated as follows : the ARX model is estimated from input output data, using the least square procedure available in the matlab System Identication ToolBox. the inverse nonlinear model is estimated using a neural network. The network is a Multi Layer Perceptron. The estimation algorithm is the Gauss Newton second order optimization technique, available in the Neural Networks ToolBox. The best architectures are reached through successive trial errors. Several initial congurations for the weights are tested. The approximation of the direct nonlinear model can be performed either by a neural network or by a polynomial model. The results provided by the previous approach are also compared with those provided by the black box approach.. 5.3.2 Pre-treating the data. Before estimating both models from the available data, some pre-treatements are performed. As far as the ARX modelling is concerned, detrending - removing of the means - is or is not applied, depending on the simulations. As far as the approximation by neural networks is concerned, data are normalized so as to lie between ;1 and +1.. 5.4 First simulation set : Starting with \true" linear output using an ARX model 5.4.1 Objectives. The purpose of this rst simulation set is to demonstrate that the identication scheme proposed leads to accurate estimates of both models, when starting with good initial conditions .i.e. the \true" output from the LTI system ..

(32) 32. Identifying a Wiener system. 5.4.2 Initial conditions. The output from the true ARX model description is chosen as initial values for the variable z . Data used to estimate the ARX models are not detrended.. 5.4.3 Intermediate results. When iteratively estimating the ARX model and the inverse nonlinear function, two comparisons are proposed : the output from the estimated ARX model is compared with the signal z^ reached at the end of the previous step .i.e. after estimating the inverse nonlinear model. This is the only information available when dealing with real problem identication. the output from the estimated ARX model is compared with the output from the true ARX model description. This is of course not feasible in the case of real problem identication. Nevertheless, to fully characterize the algorithm proposed, it is quite obvious this comparison is relevant. Furthermore, at the end of every single iteration step, the direct polynomial model is approximated, using a polynomial model. It allows us to evaluate the quality of the currently estimated ARX model. Table 5.1 provides, for every single iteration of the identication algorithm : the description of the best estimate models, the ts on validation and estimation sets. The ts are computed based on the values of z^ reached at the end of the previous iteration step .i.e. after estimating the inverse nonlinear function. For instance, for the 1ft step, the ts associated to the ARX model are computed based on the initial values given to z . The ts associated to the inverse nonlinear model are computed based on the output from the estimated ARX model. For the 2sd step, output from the inverse nonlinear model is used to estimate a new ARX model. It is the reference signal for the computation of the ts associated to the new ARX model. The same way as previously, output from the newly estimate ARX model is used to estimate the inverse nonlinear model. It is the reference signal for the computation of the ts associated to the inverse nonlinear function. Every iteration step, the true description of the ARX model is retrieved, and the inverse nonlinear function is approximated with a good accuracy. Regarding the rst ARX model estimated, the accuracy is pretty high it is not surprising since the model is estimated based on the data generated by its true description. Therefore, one would have expected the 2sd , 3td and 4th steps to lead to better ts when identifying the ARX models. Nevertheless, these good results do not provide any insight of how far from the true model descriptions the estimated models are. One can notice that only four iterations of the identication scheme have been performed. Since the quality of the identication is, on one hand, very similar from one iteration to another, and on the other hand, satisfactory, it is not worth performing some more. Table 5.2 aims at stressing the inuence of the exibility of the neural network on the quality of the identication. It shows, for every iteration step, the ts on the estimation and validation sets resulting from di erent number of hidden neurons. Since the estimate of the ARX model is of good quality, the inuence of the exibility of the neural network can not be seen here. As another consequence, the neural network approximates with a good accuracy the inverse nonlinear function. Indeed, when plotting in Figures 5.6 and 5.7 : the inverse function to be approximated (dashdot line), the function approximated by the neural network (solid line),.

(33) 5.4 First simulation set : Starting with \true" linear output using an ARX model 33. Step 1 Step 2 Step 3 Step 4. ARX modelling Inverse nonlinear function na nb nk Fit on est. Fit on val. Hid. units Fit on est. Fit on val. 1 2 0 1:48 10;8 1:60 10;8 3 4:71 10;4 1:50 10;3 ; 4 ; 3 1 2 0 4:70 10 2:19 10 3 4:87 10;4 1:40 10;3 1 2 0 4:80 10;4 1:45 10;3 3 5:68 10;4 1:40 10;3 ; 4 ; 3 1 2 0 5:68 10 1:41 10 3 4:45 10;4 1:50 10;3 Table 5.1: Intermediate results, for the rst simulation set. the measurements from which a function has to be retrieved (dotted line), no di erence can be seen between the curves. Step Number Nb of hidden units 1 2 3 4 2 2 3 4 3 2 3 4 4 2 3 4. Fit on est. 1:80 10;3 4:71 10;4 7:98 10;4 1:80 10;3 4:87 10;4 1:10 10;3 1:80 10;3 5:68 10;4 9:39 10;4 1:80 10;3 4:45 10;4 1:00 10;3. Fit on val. 2:20 10;3 1:50 10;3 1:30 10;3 2:20 10;3 1:40 10;3 1:40 10;3 2:20 10;3 1:40 10;3 1:20 10;3 2:20 10;3 1:50 10;3 1:30 10;3. Table 5.2: Inuence of the exibility of the neural network on the approximation of the inverse nonlinear function, for the rst simulation set In order to nd out about the real distance from the ARX models estimated to the true model description, the output from the various ARX models previously estimated is plotted in Figures 5.8, 5.9, 5.10, 5.11 and compared with the output from the true ARX model description. The output from the ARX models is depicted in dashed line. The solid line represents the output signal from the true ARX model description. A good t is obvious. In order to check that the algorithm converges to the desired values, the approximation of the direct nonlinear function is performed by a polynomial model every iteration step. The data set used consists of the output from the currently estimated ARX model and the measured output from the nonlinear system. Table 5.3 provides the results from these approximations. What can be concluded is that a second order polynomial model, whose parameter estimates are equal to those expected, is retrieved every iteration step. Step Number ^ ^ ^ 1 2 2 1 2 2.0001 1.9999 1 3 1.9992 2.0005 1.0002 4 1.9979 2.0016 1.0004. Fit on est. 1:05 10;7 1:70 10;3 1:90 10;3 2:20 10;3. Fit on val. 1:13 10;7 1:04 10;2 5:02 10;3 5:02 10;3. Table 5.3: Approximation of the polynomial model, after every iteration step, for the rst simulation set.

(34) 34. Identifying a Wiener system Estimation set. Estimation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Validation set. Validation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.6: Approximation of the inverse function obtained at the 1ft step, for the rst simulation set. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.7: Approximation of the inverse function obtained at the 4th step, for the rst simulation set. 5.4.4 Final results. Both the ARX model and the direct nonlinearity are now estimated. At rst, an ARX model is identied using the output signal from the inverse nonlinearity approximated at the 4th step. Secondly, the direct nonlinearity is approximated based on the output from the new ARX model as input. At this point, the identication scheme is fully completed. Tables 5.4 and 5.5 provide the description of the best estimate models, the ts on validation and estimation sets. The output ts from the ARX model are computed based on the signal resulting from the 4th step. The ts from the nonlinear model are computed based on the output signal generated by the true polynomial model description. One can conclude that both the ARX and the polynomial models have been identied with a good accuracy. Once again here, stating that a good model estimate of the LTI system has been reached is not so obvious. Results from Table 5.4 simply tell that the ts between the output from the model estimated at the 5th step and the output provided by the models estimated at the 4th step is good. One can emphasize that adding more neurons to the hidden layer does not lead to a more accurate modelling, it only implies more parameters in the model. na nb nk Fit on est. Fit on val. 1 2 0 4:38 10;4 1:43 10;3 Table 5.4: Identication of the ARX model, for the rst simulation set Let us remind that the data used so far are simulated .i.e. the output from the true ARX model description is available. Therefore, one can compare the ARX model estimated at the 5th step with its true model description (see Figure 5.12). The dashed line represents the output from the ARX model estimated whereas the solid line represents the output from its true model description. The ts between both curves is very good. A good estimate has been reached. As a consequence, one can state that a good estimate of the nonlinear model has also been obtained..

(35) 5.4 First simulation set : Starting with \true" linear output using an ARX model 35. Estimation set − Fit 0.0004716 2. 1.5. 1.5. ARX model output. ARX model output. Estimation set − Fit 1.481e−08 2. 1 0.5 0 −0.5 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 1 0.5 0 −0.5 0. 300. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output Validation set − Fit 0.002206. 2. 2. 1.5. 1.5. ARX model output. ARX model output. Validation set − Fit 1.604e−08. 1 0.5 0 −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 1 0.5 0 −0.5 0. 250. Figure 5.8: Outputs from the ARX model estimated at the rst step, for the rst simulation set 1.5. 1.5. 1 0.5 0. 1 0.5 0 −0.5 0. 300. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 1.5. 1.5. 1 0.5 0 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 300. Validation set − Fit 0.001375 2 ARX model output. ARX model output. Validation set − Fit 0.001451 2. −0.5 0. 250. Estimation set − Fit 0.0005745. 2 ARX model output. ARX model output. Estimation set − Fit 0.0004837. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. Figure 5.9: Outputs from the ARX model estimated at the second step, for the rst simulation set. 2. −0.5 0. 300. 250. 1 0.5 0 −0.5 0. Figure 5.10: Outputs from the ARX model estimated at the third step, for the rst simulation set. Nb of hidden units Nb of parameters 2 4 3 6 4 8 5 10. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.11: Outputs from the ARX model estimated at the fourth step, for the rst simulation set. Fit on est. 3:13 10;3 9:24 10;4 3:59 10;3 2:09 10;3. Fit on val. 3:73 10;3 4:76 10;3 4:06 10;3 4:43 10;3. Table 5.5: Approximation of the direct nonlinear function, for the rst simulation set.

(36) 36. Identifying a Wiener system Estimation set − Fit 0.0004409. ARX model output. 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. Validation set − Fit 0.001429. ARX model output. 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.12: Comparison of the ARX model identied with its true model description Reading the structure of a nonlinear system that has been approximated by a black box model is far from easy. Therefore, a polynomial approximation is performed based on the same data set as the one used by the neural network. Thus, one will reach more condence in the quality of the ARX model and a better understanding of the structure of the direct nonlinear model estimated. Table 5.6 provides the results obtained from the best polynomial model estimate. Let us remind that the direct nonlinear model initially used to generate the data is a second order polynomial model whose parameters are = 2 = 2 = 1. The polynomial model estimated is the one expected. The conclusion that can be driven is that good estimates of both models have been reached. When comparing Table 5.6 with Table 5.5, it is obvious that the second order polynomial model leads to a better approximation. By using a second order polynomial model, some a priori knowledge is introduced into the model structure, leading to a better modelling. Parameter estimates Fit ^ ^ ^ est. val.. 1.9999 2.0000 1.0002 1:50 10;3 4:90 10;3 Table 5.6: Approximation of the direct nonlinear function by a polynomial model, for the rst simulation set. 5.4.5 Conclusions The purpose of this set of simulations is to prove that a good modelling of both the LTI system and the nonlinearity can be reached, when starting with good intial conditions. Here the output generated by the true description of the ARX model is used as initial conditions. The results obtained have shown that the iterative identication scheme proposed in section 4.3 leads to accurate estimates..

(37) 5.5 Second simulation set : Starting with \true" linear output using a state space model 37. 5.5 Second simulation set : Starting with \true" linear output using a state space model 5.5.1 Objectives. The very rst objective of this research is to identify a Wiener nonlinear system, wherein the LTI system is a state space model whose structure is presented in section 3.1. So, the same simulations as the previous ones are performed on the same data set but involving a state space model. Since no noise is introduced, a state space model is equivalent to an ARX model, so no major di erence in the results should be observed. The estimation algorithm is the N4SID method.. 5.5.2 Initial conditions. The output from the true state space model description is chosen as initial values for the variable z.. 5.5.3 Intermediate results. Table 5.7 provides, for every single iteration of the identication algorithm : the description of the best estimate models, the ts on validation and estimation sets. As in the previous simulation set, they are computed based on the values of z^ reached at the end of the previous iteration step .i.e. after estimating the inverse nonlinear function. Since no noise is introduced, a state space model is equivalent to an ARX model, comparing the results provided by Table 5.7 with those of Table 5.1 is then meaningfull. It is not surprising the performances are pretty similar. Figures 5.13 and 5.14 agree in the same sense. Step 1 Step 2 Step 3 Step 4. State Space modelling Inverse nonlinear function order Fit on est. Fit on val. Hid. units Fit on est. Fit on val. 1 1:48 10;8 1:60 10;8 4 6:82 10;4 1:20 10;3 ; 4 ; 3 1 6:32 10 1:36 10 3 9:41 10;4 1:20 10;3 ; 4 ; 3 1 8:57 10 1:30 10 3 9:40 10;4 1:30 10;3 1 9:31 10;4 1:36 10;3 4 4:27 10;4 1:30 10;3 Table 5.7: Intermediate results, for the second simulation set. Table 5.8 provides the results from the approximation of the direct nonlinearity by a polynomial model, for every iteration step. Performing those approximations aims at checking that the algorithm proposed converges to the desired values. The conclusion driven is that a second order polynomial model, whose parameter estimates are equal to those expected, is retrieved every iteration step. Step number 2 3 4. Parameter estimates Fit ^ ^ est. val. 1:9971 2:0042 1:0003 1:50 10;3 5:00 10;3 1:9974 2:0059 1:00010 2:20 10;3 5:10 10;3 2:0001 2:0041 1:0009 1:5 10;3 5:10 10;3. ^. Table 5.8: Approximation of the nonlinear function by a second order polynomial model, for the second simulation set In Figures 5.15, 5.16, 5.17, 5.18 are plotted the output signals from the various state space models estimated, and compared with the output signal from its true model description. The solid line.

(38) 38. Identifying a Wiener system Estimation set. Estimation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Validation set. Validation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.13: Approximation of the inverse function obtained at the 1ft step, for the second simulation set. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.14: Approximation of the inverse function obtained at the 4th step, for the second simulation set. represents the output signal from the true state space model description. The output from the state space model estimated is depicted in dashed line. Once again, the accurate ts are not surprising.. 5.5.4 Final results. Both the state space model and the direct nonlinearity are now estimated. At rst, a state space model is identied using the output signal from the inverse nonlinearity approximated at the 4th step. Secondly, the direct nonlinearity is approximated based on the output from the new state space model as input. At this point, the identication scheme is fully completed. Tables 5.9 and 5.10 provide the description of the best estimate models, the ts on validation and estimation sets. The output ts from the state space model are computed based on the signals resulting from the 4th step. The ts from the nonlinear model are computed based on the output signal generated by the true polynomial model description. Results are very similar to those that can be read in Tables 5.4 and 5.5. Accuracy in approximating the direct nonlinearity by a polynomial model, shown in Table 5.11, is close to the one reached in the previous simulation set. order Fit on est. Fit on val. 1 4:04 10;4 1:42 10;3 Table 5.9: Identication of the state space model, for the second simulation set In Figure 5.19 is depicted the output signal from the fth state space model estimated. It is compared with the output signal from its true model description.. 5.5.5 Conclusions. The purpose of this set of simulations is to prove that a good modelling of both the state space model and the nonlinearity can be reached, once good initial conditions have been found. Here.

(39) 5.5 Second simulation set : Starting with \true" linear output using a state space model 39 Estimation set − Fit 0.001066 State space model output. State space model output. Estimation set − Fit 0.0006825 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 2 1.5 1 0.5 0 −0.5 0. 300. 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Estimation set − Fit 0.001066. 1 0.5 0. 1.5 1 0.5 0. 1.5 1 0.5 0 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. Validation set − Fit 0.001177 State space model output. State space model output. Validation set − Fit 0.001104. 1.5 1 0.5 0 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. 2. −0.5 0. 300. 2. −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. Estimation set − Fit 0.001192. State space model output. State space model output. 1.5. Figure 5.16: Outputs from the state space model estimated at the third step, for the second simulation set. 2. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. 2. −0.5 0. Figure 5.15: Outputs from the state space model estimated at the second step, for the second simulation set. −0.5 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output Validation set − Fit 0.001104. State space model output. State space model output. Validation set − Fit 0.001233. 250. Figure 5.17: Outputs from the state space model estimated at the third step, for the second simulation set. Nb of hidden units 2 3 4 5. 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.18: Outputs from the state space model estimated at the fourth step, for the second simulation set. Fit on est. 3:53 10;3 6:19 10;4 1:77 10;3 1:23 10;3. Fit on val. 3:53 10;3 4:75 10;3 4:25 10;3 4:46 10;3. Table 5.10: Approximation of the direct nonlinear function performed by a neural network, for the second simulation set.

(40) 40. Identifying a Wiener system. State space model output. Estimation set − Fit 0.0009484 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. State space model output. Validation set − Fit 0.001287 2 1.5 1 0.5 0 −0.5 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.19: Comparison of the state space model identied with its true model description. Parameter estimates Fit ^ ^. ^ est. val. 2:0052 2:0005 1:0006 1:40 10;3 5 10;3 Table 5.11: Approximation of the nonlinear function by a second order polynomial model, for the second simulation set.

(41) 5.6 Third simulation set : Realistic initialization. 41. the output generated by the true description of the state space model is used as initial conditions. The results obtained have shown that the iterative identication scheme proposed in section 4.3 leads to accurate estimates.. 5.6 Third simulation set : Realistic initialization 5.6.1 Objectives. So far, the simulations have shown that the identication scheme proposed leads to accurate estimates of both the LTI system and the nonlinearity, when starting with good initial conditions. Next move aims at evaluating the behaviour of the full algorithm - initialization step + iterative identication procedure - on the same input output data set. The initialization step consists in stating, as a rst approximation, that the nonlinear function is the identity function. Let us mention that the LTI system consists of a state space model.. 5.6.2 Initial conditions. The output from the polynomial model is chosen as initial values for the variable z .. 5.6.3 Intermediate results. Table 5.12 provides, for every single iteration of the identication algorithm : the description of the best estimate models, the ts on validation and estimation sets. The ts are computed based on the values of z^ reached at the end of the previous iteration step .i.e. after estimating the inverse nonlinear function. The results reect the following behaviour of the algorithm : 1. Inuence of the initial values of z on the estimation of the state space model. The output signal from the nonlinear model is used as the output signal from the linear state space model. Hence, because of the nonlinearities introduced, the relationship between the input and output data is no longer linear, leading to a poor estimate of the state space model (see results from the rst step). 2. Improving of the quality of the t. The quality of the state space model estimate improves along with the number of identication steps performed. This is mainly due to the ability of the neural network to approximate accuratly the inverse nonlinear function, leading to a good estimate of the output signal from the state space model. The second step leads to a state space model whose order is the one expected. 3. Once good estimates of both the state space and the inverse nonlinear models have been retrieved, there is no need to carry on any longer the identication the e orts put into getting a better accuracy are not worth (see results from 5th 6th 7th 8th steps). The next step in the identication scheme is the modelling of the direct nonlinear model. In order to fully characterize the behaviour of the identication algorithm proposed, these results have to be compared with those of the second simulation set. What can be concluded from the comparison of Tables 5.12 and 5.7 is that : 1. the accuracy in estimating the models is far better in the second simulation set it is about 10;4 in the rst simulation set, whereas it is about 10;2 here. 2. the algorithm does not converge towards the same values..

(42) 42. Identifying a Wiener system. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8. State Space modelling Inverse nonlinear function order Fit on est. Fit on val. Hid. units Fit on est. Fit on val. 2 3:40 10;1 3:61 10;1 4 5:52 10;2 4:86 10;2 ; 2 ; 2 1 3:00 10 3:59 10 4 2:34 10;2 2:68 10;2 ; 2 ; 2 1 2:26 10 2:71 10 3 2:71 10;2 2:66 10;2 ; 2 ; 2 1 2:14 10 2:61 10 4 2:32 10;2 2:65 10;2 ; 2 ; 2 1 2:23 10 2:69 10 4 2:10 10;2 2:64 10;2 ; 2 ; 2 1 2:10 10 2:63 10 4 2:26 10;2 2:63 10;2 ; 2 ; 2 1 2:23 10 2:27 10 4 2:16 10;2 2:62 10;2 1 2:15 10;2 2:63 10;2 4 2:26 10;2 2:64 10;2 Table 5.12: Intermediate results, for the third simulation set. Estimation set. Estimation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. 0.5. 0 0. Validation set. Validation set 1 Function approximated. Function approximated. 1. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.20: Approximation of the inverse function obtained at the 1ft step, for the third simulation set. 0.5. 0 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Solid line : Model output, Dotted line : Measured output, Dashdot : Desired output. Figure 5.21: Approximation of the inverse function obtained at the 8th step, for the third simulation set.

(43) 5.6 Third simulation set : Realistic initialization. 43. 3. performing more iteration steps does not necessarily lead to better ts. The distance between the true inverse nonlinear model and the ones reached at the end of the rst and eigth steps has not decreased signicantly (Cf Figures 5.20 and 5.21). It is now relevant to plot the output signal from the various state space models estimated and compare it with the signal from the true state space model description. It provides good insight of how far from the true model description the estimated models are. The gures are 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29. The output from the estimated state space model is depicted in dashed line. The solid line represents the output signal from the true state space model description. By looking at the plots, it is quite obvious that stating from results provided by Table 5.12 that accurate models have been reached is risky : the signals estimated are far di erent from the expected one. The estimated models are rather poor.. 5.6.4 Final results. Tables 5.13 and 5.14 provide the description of the best estimate models, the ts on validation and estimation sets. The output ts from the state space model are computed based on the signal resulting from the 8th step. The ts from the nonlinear model are computed based on the output signal generated by the true polynomial model description. More relevant than the results from Tables 5.13 and 5.14 is the plot of the output signal from the state space model estimated at the 9th step and its objective values (cf Figure 5.30). The very poor t is obvious, and expected regarding the conclusions driven in section 5.6.3. No need to further comment on it ! Since one relies on the exibility of neural networks to approximate any continuous function, results from the identication of the nonlinear model are not worse (Cf Table 5.14). order Fit on est. Fit on val. 1 2:23 10;2 2:68 10;2 Table 5.13: Identication of the state space model, for the third simulation set Nb of hidden units 2 3 4 5. Fit on est. 3:30 10;2 3:29 10;2 3:29 10;2 3:29 10;2. Fit on val. 3:21 10;2 3:24 10;2 3:25 10;2 3:25 10;2. Table 5.14: Approximation of the direct nonlinear function performed by a neural network, for the third simulation set As a consequence of the poor state space model estimate reached, approximating the direct nonlinear function with a second order polynomial model reveals a model that is far di erent from the one expected (see Table 5.15). Parameter estimates Fit ^ ^ ^ est. val.. ;8:26 10;2 1:62 ;8:36 10;1 6:48 10;1 6:83 10;1 Table 5.15: Approximation of the nonlinear function by a second order polynomial model, for the third simulation set.

(44) 44. Identifying a Wiener system Estimation set − Fit 3.415 State space model output. State space model output. Estimation set − Fit 3.401 8 6 4 2 0 −2 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 8 6 4 2 0 −2 0. 300. 6 4 2 0 −2 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 8 6 4 2 0 −2 0. 250. Figure 5.22: Comparison of the output signal from the 1ft state space estimated with its true value 6 4 2 0. 8 6 4 2 0 −2 0. 300. 6 4 2 0 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 6 4 2 0 −2 0. 250. 6 4 2 0. 8 6 4 2 0 −2 0. 300. 6 4 2 0. Figure 5.26: Comparison of the output signal from the 5th state space estimated with its true value. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. Validation set − Fit 3.435 State space model output. State space model output. Validation set − Fit 3.436 8. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Estimation set − Fit 3.414. State space model output. State space model output. Estimation set − Fit 3.414. −2 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. Figure 5.25: Comparison of the output signal from the 4th state space estimated with its true value. 8. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. 8. Figure 5.24: Comparison of the output signal from the 3td state space estimated with its true value. −2 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output Validation set − Fit 3.437. State space model output. State space model output. Validation set − Fit 3.438 8. −2 0. 250. Estimation set − Fit 3.415. State space model output. State space model output. Estimation set − Fit 3.415. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. Figure 5.23: Comparison of the output signal from the 2sd state space estimated with its true value. 8. −2 0. 300. Validation set − Fit 3.44. 8. State space model output. State space model output. Validation set − Fit 3.445. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 250. 8 6 4 2 0 −2 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.27: Comparison of the output signal from the 6th state space estimated with its true value.

(45) 5.6 Third simulation set : Realistic initialization. Estimation set − Fit 3.413. 8. State space model output. State space model output. Estimation set − Fit 3.415. 6 4 2 0 −2 0. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 8 6 4 2 0 −2 0. 300. 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 6 4 2 0 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.28: Comparison of the output signal from the 7th state space estimated with its true value. 8 6 4 2 0 −2 0. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. State space model output. Estimation set − Fit 3.411. 6 4 2 0 50 100 150 200 250 Dashed line : Model output, Solid line : Measured output. 300. State space model output. Validation set − Fit 3.431 8 6 4 2 0 −2 0. 250. Figure 5.29: Comparison of the output signal from the 8th state space estimated with its true value. 8. −2 0. 300. Validation set − Fit 3.434 State space model output. State space model output. Validation set − Fit 3.436 8. −2 0. 45. 50 100 150 200 Dashed line : Model output, Solid line : Measured output. 250. Figure 5.30: Comparison of the state space model identied with its true model description.

(46) 46. Identifying a Wiener system. 5.6.5 Conclusions. Results from this last simulation set have stressed the diculty in identifying both the LTI system and the nonlinearity, without any insight about the output from the LTI system. At this point, four explanations are possible. The rst one involves the way the output signal from the LTI system is initialized. Stating as a rst approximation that the nonlinear function is the identity function leads to so poor initial conditions that accurate estimate models can not be reached. Moreover, too little is currently known about the structure of the nonlinearity and therefore a neural network o ers to much exibility. The third explanation is related to the problem of local minima resulting from the approximation of nonlinear functions by neural networks. The fourth explanation involves the criterion minimized, it may be too complex. Chapter 6 aims at analyzing the complexity of the identication task, through the analytic computations of the criterion minimized and its derivatives.. 5.7 Black box model approach 5.7.1 Pre-treating the data. The data are normalized so as to lie between ;1 and +1.. 5.7.2 Architecture of the neural network. The neural network used is a Multi Layer Perceptron. The best architectures - optimal number of hidden neurons and initial conguration of the synaptic weights - are reached through successive trial errors.. 5.7.3 Estimation algorithm. The estimation algorithm is the Gauss Newton second order optimization technique, available in the Neural Networks ToolBox.. 5.7.4 Results. Results read in Table 5.16 are provided by the NARX model whose orders are na = 1 nb = 2 and delay nk = 0. Since the system to be identied results in a rather simple nonlinear system of order 2, one would have expected to get better ts in both the estimation and the validation. It looks like the neural network is stuck in an obvious local minimum of rather poor quality. Architecture Nb of hidden neurons 1 2 3 4 5 6 7 8 9 10. Fit Est. Val. 1:55 10;1 1:83 10;1 1:37 10;1 1:56 10;1 1:38 10;1 1:52 10;1 1:31 10;1 1:52 10;1 1:89 10;1 2:17 10;1 1:90 10;1 2:17 10;1 1:56 10;1 1:81 10;1 1:53 10;1 1:75 10;1 1:73 10;1 2:01 10;1 1:82 10;1 2:07 10;1. Table 5.16: Black Box Modelling.

(47) 5.7 Black box model approach. 5.7.5 Comparing the grey and black box approach. 47. Concluding that a better identication of the Wiener nonlinear system has been reached through the grey box approach is somehow not so trivial. Comparing performances from Table 5.16 with those of Table 5.14 is rather convincing. Nevertheless, the plotting of the output from the LTI model identied and the comparison with the signal provided by the true model description leads to a certain loss of credibility..

(48) 48. Identifying a Wiener system.

(49) Part IV. Analyses.

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