A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems

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1 International Journal of Automation and Computing 14(6), December 2017, DOI: /s A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems Nesrine Bahri 1 Asma Atig 1 Ridha Ben Abdennour 1 Fabrice Druaux 2 Dimitri Lefebvre 2 1 University of Gabes, National School of Engineers of Gabes-Tunisia, Street Omar Ibn Khattab 6029, Gabes 2 University of Havre, 25 Rue Philippe Lebon, Le Havre, France Abstract: In this paper, multimodel and neural emulators are proposed for uncoupled multivariable nonlinear plants with unknown dynamics. The contributions of this paper are to extend the emulators to multivariable non square systems and to propose a systematic method to compute the multimodel synthesis parameters. The effectiveness of the proposed emulators is shown through two simulation examples. The obtained results are very satisfactory, they illustrate the performance of both emulators and show the advantages of the multimodel emulator relatively to the neural one. Keywords: Uncoupled multimodel, neural networks, emulation, multivariable nonlinear systems, parameters estimation. 1 Introduction The study of industrial processes requires generally a precise knowledge of the process dynamics. The presence of strong nonlinearities in complex system dynamics makes the linearized models inefficient. Thus, the use of nonlinear models becomes necessary. Promising alternatives to the usual linearization methods are obtained with the use of neural networks and multimodel approaches. Thanks to their approximation properties, neural networks have been successfully investigated and applied for the emulation of nonlinear plants [1 3]. In our recent works, we have developed emulators based on recurrent neural networks and a real time recurrent algorithm [4, 5]. These emulators are composed of a small number of nodes with a set of parameters that evolve autonomously from zero initial conditions. They were proved to be able to track the system dynamics within a short time window. Consequently, they provide useful information for the control design of nonlinear systems and have been successfully applied in the domain of chemical reactors [5, 6]. Unfortunately, these emulators suffer from two strong limitations: they were developed only for square multi-input multi-output (MIMO) systems and their performance depends strongly on a specific initialization parameter which is difficult to compute. The present paper aims to overcome these limitations. The first contribution is to extend the neural emulator (NE) for non square MIMO nonlinear systems. The second contribution is to replace NE by an MIMO multimodel emulator in order to avoid the problem related to the selection of the initialization parameter. For this purpose, we continue our investigation with multimodel emulators that have been originally Research Article Manuscript received April 2, 2014; accepted March 4, 2015; published online June 20, 2016 Recommended by Associate Editor Sheng Chen c Institute of Automation, Chinese Academy of Sciences and Springer-Verlag GmbH Germany 2016 developed for single input single output (SISO) nonlinear systems [7, 8]. The basic idea of this approach is the decomposition of the full operation range of the process into several operating regimes. In each operating regime, a simple local model is considered. The parameters of the local models result from an offline identification procedure of the considered nonlinear system. The global multimodel output can be obtained by fusion of different outputs of models library [9 14]. In this paper, an uncoupled multimodel approach is preferred [7 13]. The main advantages of this structure are to have completely independent sub-models and to result in an easy application of the linear system techniques. The fusion requires the determination of weighting function parameters. Another contribution of this work is to extend the classification methods introduced for this determination in the case of SISO systems [9, 15, 16] to MIMO nonlinear systems. The performance of the obtained emulators are finally compared. This paper is organized as follows. Section 2 is about neural emulator algorithm. The multimodel emulator is presented in Section 3. The systematic determination of the synthesis parameters is also detailed in Section 3. Section 4 concludes the paper. 2 Neural emulator (NE) Let us define N IN and N OUT, respectively as the number of plant inputs and outputs where IN and OUT represent the set of inputs and outputs. The NE developed in this section extends our previous works that have been originally proposed for SISO systems [8] and square MIMO systems [6]. In this work, we investigate non square MIMO systems. The proposed NE is developed with fully connected recurrent neural networks (Fig. 1). Inputs and outputs signals are normalized in the range [ 1, 1]. In the proposed struc-

2 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 743 tures, any node is either an input node or an output one but not both at the same time. There is a total number N e of neurons that satisfies N e = N IN +N OUT. These neurons are separated into input neurons and output neurons to decouple the output and the input signals. The N IN inputs of the NE are the system inputs and the N OUT outputs are the estimated system outputs. The size of such networks depends only on the number of inputs and outputs. Moreover, input signals do not perturb the outputs. It is also important to notice that the aim of the NE is to emulate the instantaneous outputs but not to memorize the dynamics of the system. For this reason, the size of the resulting NE remains small even for complex systems. 1 τ e(k) where η e(k) is the emulator learning rate. The variations of y nl with respect to w ij are estimated with the introduction of sensitivity functions whose dynamics have been studied in our previous work [8] and that are trivially extended to the non square MIMO case in the present work. An advantage of the proposed method is that the adaptive time parameter and learning rate ηe(k) arealso updated with (4) and (5). As a consequence, the NE was proved to emulate efficiently nonlinear dynamics [4, 6]. Δη e(k) = ΔT OUT l=1 Δτ e(k) = η e(k) ΔT (ỹ l (k 1)) yn l (k 1) (4) η e OUT l=1 (ỹ l (k 1)) yn l (k 1). (5) τ e Fig. 1 Neural emulator with fully connected structure The variations of y nl with respect to η e and τ e are estimated with the introduction of another set of sensitivity functions v l (k) whose variations are given by (6) v l (k +1)=e τe(k) ΔT v l (k)+ N e (1 e τe(k) ΔT )(α l (k) w lh (k)v h (k)+ εe τ ). h=1 e(k) (6) 2.1 Autonomous adaptation algorithm In discrete time, k stands for the time variable with sampling period ΔT.Fori =1,,N e, the dynamics of the N e neurons of the neural emulator, that generalizes our previous work [8], are defined by the following equation s i(k +1)=e τe(k) ΔT s i(k)+ ( ) ) (1 e τe(k) ΔT e tanh w ij(k)s j(k)+x i(k) j=1 where s i(k) is the state of the i-th neuron whose input is 1 x i(k), w ij(k) and τ e(k) represent respectively the weights from j-th neuron to i-th neuron and the adaptive time parameter, x i(k) = u i(k) if i {1,,N IN}, x i(k) = 0 if i {N IN +1,,N e}, and y ni NIN (k) = s i(k) if i {N IN+1,,N e} represents the estimation of the plant outputs. As in [8], real time recurring learning (RTRL) algorithm is used in discrete time to update the parameters of NE. For each time k, the global estimation error e e(k) is introduced as e e(k) = 1 2 OUT l=1 (ỹ l (k)) 2 = OUT l=1 (1) e el (k) 2 (2) where ỹ l (k) =y nl (k) y l (k) is the scalar instantaneous emulation error for output y l (k). A gradient method is used to update the parameters w ij(k) with(3) Δw ij(k) = η e(k) ΔT OUT l=1 (ỹ l (k 1)) yn l (k 1) (3) w ij In (6), let us focus on the parameter ε e. This parameter [4, 6, 8] has been introduced and studied in our previous works in order to start with zero initial conditions for all other parameters: w ij, η e and τ e. Consequently, another characteristic of the proposed NE is that their performance depends on the value of a single parameter. The selection of this parameter is also the main difficulty with this method. 2.2 Numerical examples In this subsection, simulations are proposed to illustrate the performance of NE for square and non square MIMO nonlinear systems. The structure and the dynamics of the considered systems (subsequently defined by (9) and (10) are assumed completely unknown to the adaptation process. For each output y l of the considered systems, the mean square error (MSE l ) and the variance-accounted-for (VAF l ) are respectively computed with (7) and (8) N H MSE l = 1 (y nl (k) y l (k)) 2 (7) N H k=1 VAF l = { max 1 var {yn (k) y } l l(k) : k =1 N H}, 0 100% var {y nl (k) : k =1 N H} (8) where y nl (k) andy l (k) are respectively the emulator and the system outputs and N H is the simulation horizon. MSE is a mean indicator for the estimation error over a complete trajectory. On the contrary, VAF gives an estimation of the dispersion of the instantaneous errors.

3 744 International Journal of Automation and Computing 14(6), December NE for square MIMO nonlinear systems To evaluate the effectiveness of the neural emulator, for square MIMO nonlinear system, we consider the following two-inputs, two-outputs nonlinear system: y 1(k) =0.2y 1(k 1) + u 2(k 2) 3 + u 1(k 2) 2 0.2y 1(k 1)u 1(k 2) 0.1u 2(k 1) y 2(k) =0.05y 2(k 1) + u 1(k 2) 3 + u 2(k 1) y 2(k 2)u 2(k 1). Then a four-neuron neural network is capable to emulate the considered nonlinear system. This number of neurons depends only on the input and the output numbers (N e = N IN + N OUT =4,N IN =2andN OUT =2). Accordingtothesystem s dynamics, the sampling period is retained equal to 0.1 s. The term ε e is chosen to ensure the starting of the system emulation with zero initial conditions for all other parameters. With a good choice of ε e, we obtain good performance. For example, results of neural emulation using ε e = 10 are given by Fig. 2 (the emulated outputs are represented with solid lines and the system outputs are represented with dashed lines). The neural emulator provides, in this case, a satisfactory estimation of the process outputs. The low values of the outputs estimation errors plotted in logarithmic scale (Fig. 3) confirm this constatation NE for non square MIMO nonlinear systems To evaluate the effectiveness of the neural emulator, for non square MIMO process, we consider the following oneinput, two-output nonlinear system: y 1(k) = 1 (( z11(k 1))y1(k 1)+ 2 z 11(k 1)u 1(k 1)) (9) y 2(k) = 1 ((1.5 z12(k 1) 0.2z22(k 1))y2(k 1)+ 2 z 12(k 1)u 1(k 1)) (10) with z 11(k 1) = 0.6u1(k 1) 0.06y1(k 1) 1+0.2y 1(k 1) 0.5u1(k 1) z 12(k 1)= 1+0.5y 2(k 1) 0.07y2(k 1) z 22(k 1)= 1+0.1y 2(k 1). In this case, a three-neuron neural network is capable to emulate the considered nonlinear system. This number of neurons still depends only on the input and the output numbers (N e = N IN + N OUT =3,N IN =1andN OUT =2). The sampling period is retained equal to 0.1 s. With a suitable choice of ε e = 20, we obtain the results in Figs. 4 and 5 shows the variations of the outputs estimation errors plotted in logarithmic scale. These figures show that the neural emulator provides, in this case, a relatively satisfactory estimation of the process outputs. It should be noted that the selection of an appropriate value of ε e is not obtained in a systematic way. This choice is made by trial and error. Indeed, an arbitrary choice of ε e can affect the performance. For example, the choice of ε e equals to 0.4 for the square MIMO system and ε e equals to 0.8 for the non square MIMO system lead to an emulation with relatively important outputs estimation errors. To confirm that obtained performance depends on the emulator term, the mean square error (MSE l ) and the variance-accounted-for (VAF l ) have been calculated for an arbitrary and a good choice of ε e. They are summarized in Tables 1 and 2 respectively for the square and non square MIMO nonlinear systems. We also note that the online neural emulation needs an important computing load. To overcome these problems, a multimodel emulator for multivariable nonlinear systems will be proposed in the next section. Fig. 2 Variations of the square MIMO nonlinear system outputs and the corresponding NE outputs (ε e = 10)

4 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 745 Fig. 3 Variations of the neural emulation squared errors in logarithmic scale (square MIMO nonlinear system, ε e = 10) Fig. 4 Variations of the non square MIMO nonlinear system outputs and the corresponding NE outputs (ε e = 20) Fig. 5 Variations of the neural emulation squared errors in logarithmic scale (non square MIMO nonlinear system, ε e = 20) Table 1 MSE l and VAF l (l =1, 2) for both outputs, in open-loop emulation case with an arbitrary and a good choice of ε e (square MIMO nonlinear system) Neural emulator ε e =0.4 Neural emulator ε e =10 MSE MSE VAF % 95.49% VAF % 96.99% Table 2 MSE l and VAF l (l =1, 2) for both outputs, in open-loop emulation case with an arbitrary and a good choice of ε e (non square MIMO nonlinear system) Neural emulator ε e =0.8 Neural emulator ε e =20 MSE MSE VAF % 91.33% VAF % 93.51%

5 746 International Journal of Automation and Computing 14(6), December Multimodel emulator (ME) The multimodel formalism is based on the decomposition of the full operating range of the process into a finite number of operating regimes. In each operating regime, a linear local model is considered. A nonlinear interpolation between these linear submodels is, then, used to yield the global model (Fig. 6). Thus the multimodel structure gives the possibility to reduce the complexity of nonlinear systems. Different techniques exist to obtain the multimodel. In all cases, three major problems should be resolved. At first, the premise variables ξ(k) of weighting functions should be defined. In the present work, these variables are defined as the input signals. Secondly, the operating space should be decomposed and the corresponding weighting functions be characterized. Initially, the static characteristic of the considered system has been used for the operating space decomposition [7, 8, 17, 18]. Then, methods based on classification using Kohonen map and the Chiu classification were proposed for SISO nonlinear systems [9 11, 16, 19, 20]. In the present work, this method will be extended to the MIMO case. Finally, the structure of the multimodel should be determined and the parameters of each submodel should be identified. In the present work, a Levenberg-Marquardt optimization type algorithm is used for that purpose. 3.1 The multimodel emulator structure Our goal is to represent a multivariable nonlinear system with a decoupled multimodel. For simplicity reasons, MIMO model is decomposed into several multi input single output (MISO) multimodels: each MISO multimodel is attached to one of the system outputs. Thereafter identification tools available in [8, 12, 17], although developed in the context of SISO systems, are directly used to identify the MIMO system. The uncoupled structure, used for the l-th base of model B l representing every MISO multimodel l {1,,N OUT }, is given by a state space representation in discrete time (Fig. 6) X l,i (k +1)=A l,i (θ l,i )X l,i (k)+b l,i (θ l,i )U(k) (11) y l,i (k) =C l,i (θ l,i )X l,i (k) (12) where, X l,i R n l,i and U(k) =[u 1(k),,u e(k),,u NIN (k)] T R N IN are the state vector and the input vector, y l,i (k) and n l,i are, respectively, the output and the dimension of the i-th local model. A l,i (θ l,i ), B l,i (θ l,i )and C l,i (θ l,i ) are, respectively, the state matrix, the input matrix and the output matrix of dimension 1 n l,i. The multimodel representing the l-th system output y ml (k) is defined by y ml (k) = ml i=1 μ l,i (ξ(k))y l,i (k) (13) where N ml is the number of local models for the l-th system output. ξ(k) =[ξ 1(k),,ξ N(k)] T is the decision variable vector (N = N IN or N = N OUT depending on the choice of the decision variables). In this work, the input vector is set as decision variables (ξ(k) = U(k)). The weighting function μ l,i (ξ(k)) quantifies the relative weight of the i-th submodel (A l,i,b l,i,c l,i )withrespectto the other ones in the global model. Such functions are generated by the decision variables vector to model the non- Fig. 6 The ME based structure for multivariable nonlinear systems

6 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 747 linearities of the system. These functions are either sigmoidal, triangular or Gaussian ones. In the present work, the Gaussian functions are used to generate the weighting functions: w l,i (ξ(k)) = N IN e=1 (ξe(k) c l,ie) 2 e σl,ie 2. (14) The weighting functions μ l,i (ξ(k)) are obtained by normalizing the Gaussian functions w l,i (ξ(k)) (see Fig. 6) μ l,i (ξ(k)) = w l,i(ξ(k)). (15) w l,p (ξ(k)) N ml p=1 These functions must satisfy the following convex sums properties [13, 14] : ml i=1 μ l,i (ξ(k)) = 1, 0 μ l,i (ξ(k)) 1 i =1,,N ml where c l,ie and σ l,ie are respectively the center and the dispersion of the i-th weighting function [7, 8, 11]. The choice of N ml, c l,ie and σ l,ie has a great influence on the modeling precision. In our previous work, the choice of these parameters was made according to an a priori information about the static characteristics of the considered system [18]. In this work, they will be selected systematically using clustering techniques. We present in the next section, a systematic method to generate the weighting functions centers basing on Chiu s classification method [15]. 3.2 Systematic generation of weighting function centers and dispersions Consider a set of numerical data, the classification procedure consists of selecting the cluster centers between these data using Chiu s method for the numerical data classification. In this work, we aim to extend this method to a systematic determination of the weighting function parameters for multimodel representation of MIMO nonlinear systems. Considering the set of persistent identification data sufficiently rich to cover the whole operating range of the considered MIMO nonlinear system, we define, for each MISO multimodel, a set of regression vector ϑ l,j = [y l (j),y l (j 1),u 1(j 1),,u NIN (j 1)] T,j = 1,,N H. Using the Chiu s algorithm to classify these vectors we associate, firstly, a potential Pot l,j (1) to each regression vector ϑ l,j as follows: Pot l,j (1) = N H h=1 e 2 4 ϑ l,j ϑ l,h r 2 l,a (16) where r l,a is a positive parameter controlling the decreasing rate of the potential. Each potential is a function of the distance of the corresponding regression vector with respect to all other ones. Thus, the potential increases exponentially as ϑ l,j has a few neighbouring regression vectors. After calculating all potentials, the first regression vector cluster center ϑ l,1 is selected as the vector whose potential Pot l,1 given by (15) is the maximum. In the next steps, after the (i 1)-th cluster center ϑ l,i 1 with the maximum potential Pot l,i 1 has been chosen, we revise the potentiel of each regression vector as follows: Pot l,j (i) =Pot l,j (i 1) Pot l,i 1e 4 ϑ l,j ϑ 2 l,i 1 r 2 l,b (17) where Pot l,j (i 1) are the potentials calculated previously. Chiu [15] introduced two positive parameters ε 1 and ε 2 (with the restriction ε 1 >ε 2) to determine the cluster selection. He proposed the following algorithm as stopping criterion for the selection of the clusters centers. Algorithm 1. if Pot l,j >ε 1Pot l,1 Selection sanctioned: accept ϑ l,j as a cluster center and continue else if Pot l,j <ε 2Pot l,1 Selection stopped: reject ϑ l,j and end the clustering process else Let d min be the shortest distance within the distances between ϑ l,j and all previously found cluster centers if ( dmin 1 Pot l,j ) r l,a Pot l,1 Accept ϑ l,j as a cluster center and continue. else Reject ϑ l,j and set the corresponding potential to 0. Select the regression vector with the next highest potential as the new ϑ l,j and re-test end if end if ϑ l,j is the current regression vector center and ϑ l,1, ϑ l,2,,ϑ l,j 1 are the last selected ones. Once the regression vector cluster centers are selected, referring to the decision variable to be considered, the weighting functions centers are chosen. Since in this work the input vector will serve as decision variables (ξ(k) = U(k)), so the weighting functions centers are the N IN last elements of the regression vectors. Finally, it remains to determine the corresponding dispersion. To do this, we can use one of the following two expressions: Expression 1. The dispersion σ l,ie can be adjusted accordingtothemeandistancetothen nearest neighbors [20] : 1 n σ l,ie = α l c l,ie c l,ke (18) n k=1 where c l,ie is the current center and c l,ke are the n nearest neighbor centers to c l,ie. α l is a scaling factor defining the degree of overlap between the weighting functions. Expression 2. The dispersion σ l,ie is considered to be in proportion to the distance separating the center c l,ie and the n nearest centers [20] σ l,ie = 1 α l min ( c l,ie c l,ke ) (19) k=1,,n

7 748 International Journal of Automation and Computing 14(6), December 2017 where α l is a positive factor defining the degree of overlap between the weighting functions. 3.3 Parametric estimation procedure For each MISO multimodel, let us define the vector of unknown parameters θ l as follows: θ l =[θ T l,1 θ T l,iθ T l,n ml ] T. (20) This vector is partitioned into N ml column blocks where each column block θ l,i is formed by the q l,i unknown parameters (A l,i,b l,i,c l,i ) of the particular submodel. The identification of the set of all parameters are then based on the optimization of a quadratic global criterion expressed as a function of the difference between the system outputs and the multimodel ones over a simulation horizon N H J ml = 1 2 N H k=1 (y ml (k) y l (k)) 2 = N H k=1 e ml (k). (21) Different algorithms can be used to ensure the minimisation procedure. In this work, Levenberg-Marquardt s algorithm is used [21] θ l (it +1)=θ l (it) Δ l (it)(h l (θ l )+λ l (it)i) 1 G l (θ l ) (22) where H l (θ l )andg l (θ l ) are respectively the Hessian matrix and the gradient vector. They are computed from the calculation of sensitivity functions of output multimodel with respect to local models parameters ( y l,i(k) ) as follows: θ l with G l (θ l )= J H ml = θ l H l (θ l )= 2 J ml θ l θ T l (y ml (k) y l (k)) y ml(k) θ k=1 l N H y = ml (k) y ml (k) θ k=1 l θl T N y ml (k) ml = μ l,i (ξ(k)) y l,i(k). θ l θ l i=1 (23) I is the identity matrix of appropriate dimension. At each iteration it, θ l (it) is the vector of the l-th multimodel parameters. Δ l (it) is a relaxation coefficient introduced to minimise the criterion in the direction of vector H 1 l G l and λ l (it) is the regularization parameter. The values of λ l (it) and Δ l (it) are adjusted, usually by means of a heuristic based on the evolution of the criterion [21]. To improve the understanding an algorithm explaining, the parametric estimation procedure of the multimodel is provided in the Appendix. 3.4 Numerical examples In order to illustrate the performance of the multimodel emulator, we consider the square and non square MIMO nonlinear systems described respectively by (9) and (10) ME for square MIMO nonlinear systems The identification of the two MISO multimodels is realized with a global criterion (defined by (20). The inputs u e(k), e=1, 2 of the system, that are set as decision variables for the weighting functions (ξ(k) = U(k)), consist of a signal with variable amplitude (u e(k) [ 1, 1], e=1, 2). The identification data set is used to generate the set of regression vectors ϑ l,j = [y l (j),y l (j 1),u 1(j 1),u 2(j 1)] T for every MISO multimodel. Using the extended classification procedure we treat these vectors to generate the regression vector cluster centers. For each MISO multimodel, the algorithm generates three cluster centers. By exploiting these selected regression vectors and the expressions (13) and (14), every MISO multimodel comprises (N m = 9) sub-models. The corresponding dispersions are deduced using the relation (17). Fig. 7 gives the resulting weighting functions for each MISO multimodel. Tables 3 and 4 detail the centers and dispersions obtained with the systematic approach (Table 3) and with the static characteristics (Table 4). One can notice that the systematic method generates additional models in comparison with the other method. Fig. 7 The weighting functions using the systematic generation of synthesis parameters (square MIMO nonlinear system)

8 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 749 Table 3 The weighting functions parameters for both MISO multimodels using the systematic generation method (square MIMO nonlinear system) Table 4 The weighting functions parameters for both MISO multimodels using the static characteristics (square MIMO nonlinear system) Multimodel synthesis parameters (systematic method) c 1,11 = , σ 1,11 = c 1,21 = , σ 1,21 = c 1,31 = , σ 1,31 = c 1,ie and σ 1,ie c 1,12 = , σ 1,12 = c 1,22 = , σ 1,22 = c 1,32 = , σ 1,32 = c 2,11 = , σ 2,11 = c 2,21 = , σ 2,21 = c 2,31 = , σ 2,31 = c 2,ie and σ 2,ie c 2,12 = , σ 2,12 = Multimodel synthesis parameters (classical choice [18] ) c 1,11 = 0.6, σ 1,11 =0.4 c 1,21 =0.6, σ 1,21 =0.4 c 1,ie and σ 1,ie c 1,12 = 0.7, σ 1,12 =0.2 c 1,22 =0.4, σ 1,22 =0.2 c 1,32 =0.7, σ 1,32 =0.2 c 2,11 = 0.6, σ 2,11 =0.4 c 2,21 =0,σ 2,21 =0.4 c 2,ie and σ 2,ie c 2,31 =0.6, σ 2,31 =0.4 c 2,12 = 0.5, σ 2,12 =0.6 c 2,22 =0.6, σ 2,22 =0.6 c 2,22 = , σ 2,22 = c 2,32 = , σ 2,32 = Fig. 8 Variation of square MIMO nonlinear system and multimodel emulator outputs using the systematic generation of synthesis parameters Fig. 9 Variations of the multimodel emulation squared errors in logarithmic scale (square MIMO nonlinear system)

9 750 International Journal of Automation and Computing 14(6), December 2017 We note that, in general, the local models must have a structure as simple as possible. Only the offline validation phase of the base of models may increase the structure complexity. In this case, a base of first order models provides sufficient precision. Indeed, Fig. 8 (the emulated outputs are represented with solid lines and the system outputs are represented with dashed lines) illustrates the validation of the multimodel emulation results obtained using the systematic generation of the weighting functions parameters. This figure shows the variations of the real nonlinear system and the multimodel outputs (y l (k) andy ml (k), l =1, 2). Fig. 9 shows that the multimodel emulation squared errors obtained in this case and plotted in the logarithmic scale are smaller than those obtained using the neural emulator (Fig. 3). The simulation results confirm that the uncoupled multimodel emulator offers a satisfactory modeling precision compared to the neural one. Table 5 summarizes the mean square error and the variance-accounted-for calculated in open-loop emulation case for neural emulator and both multimodel emulators (using the usual and the proposed method to select the weighting functions parameters). In this table, we note that neural emulation depends on the choice of the starting parameter Results obtained using a multimodel emulator remain better than the ones obtained for neural emulator. The performance of both multimodel emulators are quite similar. However, it is important to notice that the systematic method proposed in the present paper avoids any tedious effort to find a good value of the synthesis parameters. In comparison, the method based on the static characteristics is efficient only if numerous candidates are tested and checked for the centers and dispersions. To conclude, the present determination is a nice way to quickly obtain an accurate multimodel. Table 5 MSE l and VAF l (l =1, 2) for both outputs, in open-loop emulation case for both neural and multimodel emulators (square MIMO nonlinear system) Classical Systematic Neural multimodel multimodel emulator emulator [18] emulator MSE (ε e =0.4) (ε e = 10) MSE (ε e =0.4) (ε e = 10) 72.93% (ε e =0.4) 99.26% 99.15% VAF % (ε e = 10) 82.9% (ε e =0.4) 99.93% 99.65% VAF % (ε e = 10) ME for non square MIMO nonlinear systems For the non square MIMO system, two SISO multimodels are identified with a global criterion. The inputs u(k) of the system, that will serve as decision variables for the weighting functions (ξ(k) = u(k)), consist of a signal with variable amplitude (u(k) [ 1, 1]). The identification data set is used to generate the set of regression vectors ϑ l,j =[y l (j),y l (j 1),u(j 1)] T for every MISO multimodel. Using the extended classification procedure we treat these vectors to generate the regression vector cluster centers. For each MISO multimodel, the algorithm generates three clusters centers. By exploiting these selected regression vectors and the expressions (13) and (14), every MISO multimodel comprises (N m = 3) submodels. The corresponding dispersions are deduced using the relation (18). Fig. 10 gives the resulting weighting functions evolution for each MISO multimodel. The centers and dispersions obtained with the systematic approach and the static characteristics are respectively presented in Tables 6 and 7. In this case, we can also notice that the systematic method generates additional models in comparison with the classical method. Figs. 11 and 12 illustrate the multimodel validation results. Fig. 11 gives the variations of the non square MIMO nonlinear system and the multimodel emulator outputs. The multimodel can describe properly the nonlinear system behaviour. Fig. 12 gives the variations of the multimodel emulation squared errors in logarithmic scale. Compared to Fig. 5, this figure confirms that performance recorded using a multimodel emulator is far better than that using neural one. Table 6 The weighting functions parameters for both MISO multimodels using the systematic generation method (non square MIMO nonlinear system) Multimodel synthesis parameters (systematic method) c 1,1 = , σ 1,1 = c 1,i and σ 1,i c 1,2 = , σ 1,2 = c 1,3 = , σ 1,3 = c 2,1 = , σ 2,1 = c 2,i and σ 2,i c 2,2 = , σ 2,2 = c 2,3 = , σ 2,3 = Table 7 The weighting functions parameters for both MISO multimodels using the static characteristics (non square MIMO nonlinear system) Multimodel synthesis parameters (classical choice [18] ) c 1,i and σ 1,i c 1,1 = 0.9, σ 1,1 =0.8 c 1,2 =0.9, σ 1,2 =0.8 c 2,i and σ 2,i c 2,1 = 0.7, σ 2,1 =0.8 c 2,2 =0.7, σ 2,2 =0.8

10 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 751 Fig. 10 The weighting functions using the systematic generation of synthesis parameters (non square MIMO nonlinear system) Fig. 11 Variation of non square MIMO nonlinear system and multimodel emulator outputs using the systematic generation of synthesis parameters Fig. 12 Variations of the multimodel emulation squared errors in logarithmic scale (non square MIMO nonlinear system)

11 752 International Journal of Automation and Computing 14(6), December 2017 Table 8 summarizes the mean square error and the variance-accounted-for calculated in open-loop emulation case for both neural and multimodel emulators. In this table, we note that neural emulation depends on the choice of the starting parameter ε e. Results obtained using a multimodel emulator with a systematic selection of the weighting functions parameters are far better than the ones obtained for both classical multimodel and neural emulator. Table 8 MSE l and VAF l (l =1, 2) for both outputs, in open-loop emulation case for both neural and multimodel emulators (non square MIMO nonlinear system) Neural Classical Systematic emulator multimodel multimodel emulator [18] emulator MSE (ε e =0.8) (ε e = 20) MSE (ε e =0.8) (ε e = 20) VAF % (ε e =0.8) 98.41% 99.67% 91.33% (ε e = 20) VAF % ( ε e =0.8) 97.10% 99.62% 93.51% (ε e = 20) 4 Conclusions In this paper, neural and multimodel emulators are proposed for multivariable non square and nonlinear system emulation. Particularly two contributions have been presented. First, the NE and ME approaches have been extended to non square MIMO systems. Secondly, for the ME approach, each system output is emulated by using a library of models. These models result from an offline identification procedure and a systematic generation of the synthesis parameters that are used to combine the submodels. The proposed method is based on a classification procedure over an identification data set. This systematic approach presents an important advantage i.e. avoiding any tedious effort to find an optimal value of the weighting function parameters. Another advantage of ME in comparison with NE is that the selection of the initialization parameter is no longer required. This scheme also reduces the computational complexity of the multivariable emulation [8, 18]. Several open loop simulations illustrate clearly that the proposed multimodel emulator leads to good performance relatively to the case where the neural emulator is applied. Appendix In this section we add an algorithm explaining the parametric estimation procedure. Algorithm 2. Input data: N H, U(k), y l (k) andn itersl Input data from Algorithm 1: N ml, c l,ie and σ l,ie Initialization: Δ l (0) and λ l (0) it 1 updatej ml 1(updateJ ml is a Boolean variable) while it n itersl if updatej ml =1 For each k [1,N H] Compute μ l,i (ξ(k)), X l,i R n l,i, y l,i (k), y ml (k) andthe sensitivity functions y l,i(k) θ l end for if it=1 Compute quadratic global criterion for the first iteration J end if Compute the gradient vector G l and the Hessian matrix H l end if Update the vector of parameters θ l Compute quadratic global criterion for the updated parameters J ml if (J ml J) Decrease λ l (it) andincreaseδ l (it) J J ml updatej ml 1 it it +1 else Increase λ l (it) and decrease Δ l (it) updatej ml 0 end if end while References [1] K. S. Narendra, K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, vol. 1, no. 1, pp. 4 27, [2] O. Nelles. Nonlinear System Identification, Berlin, Heidelberg, Germany: Springer, [3] R. J. Williams. Adaptive state representation and estimation using recurrent connectionist networks. Neural Networks for Control, Cambridge, USA: MIT Press, pp , [4] S. Zerkaoui, F. Druaux, E. Leclercq, D. Lefebvre. Stable adaptive control with recurrent neural networks for square MIMO non-linear systems. Engineering Applications of Artificial Intelligence, vol. 22, no. 4 5, pp , [5] A. Atig, F. Druaux, D. Lefebvre, K. Abderrahim, R. Ben Abdennour. Neural emulation applied to chemical reactors. In Proceedings of the th IEEE International Multi- Conference on System, Signals and Devices, IEEE, Amman, Jordan, pp. 1 6, [6] A. Atig, F. Druaux, D. Lefebvre, K. Abderrahim, R. Ben Abdennour. Adaptive control design using stability analysis and tracking errors dynamics for nonlinear square MIMO systems. Engineering Applications of Artificial Intelligence, vol. 25, no. 7, pp , [7] N. Bahri, A. Messaoud, R. Ben Abdennour. A multimodel emulator for nonlinear system controls. International Journal of Sciences and Techniques of Automatic Control & Computer Engineering (IJ-STA), vol. 5, no. 1, pp , 2011.

12 N. Bahri et al. / A Systematic Design of Emulators for Multivariable Non Square and Nonlinear Systems 753 [8] N. Bahri, A. Atig, R. Ben Abdennour, F. Druaux, D. Lefebvre. Multimodel and neural emulators for non-linear systems: Application to an indirect adaptive neural control. International Journal of Modelling, Identification and Control, vol. 17, no. 4, pp , [9] M. Ltaief, K. Abderrahim, R. Ben Abdennour, M. Ksouri. A fuzzy fusion strategy for the multimodel approach application. WSEAS Transactions on Circuits and Systems, vol. 2, no. 4, pp , [10] A. Messaoud, M. Ltaief, R. Ben Abdennour. Supervision based on partial predictors for a multimodel generalised predictive control: Experimental validation on a semi-batch reactor. International Journal of Modelling, Identification and Control, vol. 6, no. 4, pp , [11] A. Messaoud, M. Ltaief, R. Ben Abdennour. Supervision based on a multi-predictor for an uncoupled state multimodel predictive control. In Proceedings of the 6th International Conference on Electrical Systems and Automatic Control, Hammamet, Tunisia, [12] R. Orjuela, B. Marw, J. Ragot, D. Maquin. State estimation for non-linear systems using a decoupled multiple model. International Journal of Modelling, Identification and Control, vol. 4, no. 1, pp , [13] R. Orjuela. Contributionà l estimation d état et au diagnostic des systèmes représentés par des multimodèles, Ph. D. dissertation, National Polytechnic Institute of Lorraine, France, (in French) [14] D. Ichalal. Estimation et diagnostic de systèmes non linéaires décrits par un modèle de Takagi-Sugeno, Ph. D. dissertation, National Polytechnic Institute of Lorraine, France, (in French) [15] S. L. Chiu. Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems, vol.2, no. 3, pp , [16] M. Ltaief, R. Ben Abdennour, K. Abderrahim, P. Borne. A multimodel numerical control based on a new approach for the determination of a model s library for uncertain systems. In Proceedings of the 2002 IEEE International Conference on Systems, Man and Cybernetics, IEEE, Yasmine Hammamet, Tunisia, pp , [17] R. Orjuela, D. Maquin, J. Ragot. Nonlinear system identification using uncoupled state multiple-model approach. In Proceedings of Workshop on Advanced Control and Diagnosis, Nancy, France. [18] N. Bahri, A. Atig, R. Ben Abdennour, F. Druaux, D. Lefebvre. Emulation of multivariable non square and nonlinear systems. In Proceedings of 2013 the 14th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering, IEEE, Sousse, Tunisia, pp , [19] M. Ltaief, K. Abderrahim, R. Ben Abdennour, M. Ksouri. Contributions to the multimodel approach: Systematic determination of a models base and validities estimation. International Journal of Automation and Systems Engineering, vol. 2, no. 3, [20] M. Ltaief, A. Messaoud, R. Ben Abdennour. Optimal systematic determination of models base for multimodel representation: Real time application. International Journal of Automation and Computing, vol. 11, no. 6, pp , [21] D. W. Marquardt. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp , Nesrine Bahri received the engineering diploma in electric-automatic engineering, in 2009, and the master degree in automatic control and intelligent techniques, in 2010, from National School of Engineers of Gabes-Tunisia. She is currently a Ph. D. degree candidate at Research Unit of Numerical Control of Industrial Processes at ENIG (CONPRI) and at Electrical and Automatic Engineering Research Group at Le Havre University (GREAH). Her research interests include nonlinear process identification, multimodel and multicontrol approaches, neural and multimodel emulation and adaptive control. bahri.nesrine@gmail.com (Corresponding author) ORCID id: Asma Atig received the engineering diploma in electrical engineering, in 2007, the master degree in automatic control, in 2008, from the National School of Engineering of Sfax-Tunisia (ENIS), the Ph. D. degree in electrical engineering from the National School of Engineering of Gabes- Tunisia (ENIG) and from the University of Le Havre in automatic, signal processing and computing, in Actually she is a teaching assistant in Electrical Engineering Department at the High Institute of Industrial Systems of Gabes-Tunisia. She is a member of Research Unit of Numerical Control of Industrial Processes at ENIG (CONPRI). Her research interests include nonlinear process identification, neural emulation and adaptive control. asma.atig@issig.rnu.tn Ridha Ben Abdennour received the Speciality Ph. D. degree in automatic control from Higher School of Technique Education in 1987, and the Ph. D. degree in electrical engineering from the National School of Engineering of Tunis-Tunisia, in He is a professor in automatic control at the National School of Engineering of Gabes-Tunisia. He was chairman of the Electrical Engineering Department and the director of the High Institute of Technological Studies of Gabes. He is the head of the Research Unit of Numerical Control of Industrial Processes and the founder and the honorary President of the Tunisian Association of Automatic Control and Numerisation. He is the co-author of a book on Identification and Numerical Control of Industrial Processes and he is the author of more than 300 publications. He participated in the organization of many conferences and he was a member of some scientific committees of a number of congresses. His research interests include identification, multimodel & multicontrol approaches, numerical control and supervision of industrial processes. ridha.benabdennour@enig.rnu.tn

13 754 International Journal of Automation and Computing 14(6), December 2017 Fabrice Druaux received the B. Sc. degree in physics and mathematics in 1976, the M. Sc. degree in physic in 1981 and the Ph. D. degree in physic from University of Rouen, France in Since 1988, he has been an assistant professor at the Faculty of Sciences and Technology of Le Havre, France. Since 1999, he has been with the Electric and Automatic Engineering Research Group. His current research interests include modelling, control and fault detection using dynamical neural network. His principal applications under focus are electro-technical processes such as motors and wind generators. Dimitri Lefebvre is graduated from the Ecole Centrale of Lille, France in He received the Ph. D. degree in automatic control and computer science from University of Sciences and Technologies, Lille in 1994, and the HAB degree from University of Franche Comt Belfort, France in Since 2001, he has been a professor at Institute of Technology and Faculty of Sciences, University Le Havre, France. He is with the Electric and Automatic Engineering Research Group (GREAH). His research interests include Petri nets and electronic data systems (DESs), learning processes, adaptive control, fault detection and diagnosis and applications to electrical engineering. dimitri.lefebvre@univ-lehavre.fr