Kais Bouzrara

Decomposition of an ARX model on Laguerre orthonormal bases.

In this paper, we propose a new reduced complexity model by expanding a discrete-time ARX model on Laguerre orthonormal bases. To ensure an efficient complexity reduction, the coefficients associated to the input and the output of the ARX model are expanded on independent Laguerre bases, to develop a new black-box linear ARX-Laguerre model with filters on model input and output. The parametric complexity reduction with respect to the classical ARX model is proved theoretically. The structure and parameter identification of the ARX-Laguerre model is achieved by a new proposed approach which consists in solving an optimization problem built from the ARX model without using system input/output observations. The performances of the resulting ARX-Laguerre model and the proposed identification approach are illustrated by numerical simulations and validated on benchmark manufactured by Feedback known as Process Trainer PT326. A possible extension of the proposed model to a multivariable process is formulated.

Online identification of the ARX model expansion on Laguerre orthonormal bases with filters on model input and output

This article proposes a new representation of the ARX models on independent and orthonormal Laguerre bases by filtering the process input and output using Laguerre orthonormal functions. The resulting model, entitled ARX–Laguerre model, ensures the parameter number reduction with a recursive and easy representation. However, this reduction is still subject to an optimal choice of the Laguerre poles defining both Laguerre bases. Therefore, we propose an analytical solution to optimise the Laguerre poles which depend on Fourier coefficients defining the ARX–Laguerre model, and that are identified using the regularised square error. The identification procedures of the Laguerre poles and Fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the ARX–Laguerre model. The proposed algorithm is tested on numerical simulation and validated on a benchmark system manufactured by Feedback known as Process Trainer PT326.

Non-linear predictive controller for uncertain process modelled by GOBF-Volterra models

This paper proposes a new approach for synthesising a predictive control for non-linear uncertain process based on a proposed reduced complexity discrete-time Volterra model known as GOBF-Volterra model. This model, provided by expanding each Volterra kernel on independent generalised orthonormal basis functions (GOBF), is efficient for the synthesis of non-linear model-based predictive control (NMBPC) which copes with physical constraints and geometrical constraints due to parameter uncertainties. A quadratic criterion is optimised and a new optimisation algorithm, formulated as a quadratic programming (QP) under linear and non-linear constraints, is proposed. Simulation results on a chemical reactor are presented to illustrate the performance of the proposed NMBPC strategy for uncertain process. This reveals that the stability performance of the resulting closed-loop system depends on the choice of the tuning parameters.

Optimal expansions of discrete-time bilinear models using Laguerre functions

In this paper, we propose a new reduced complexity model by expanding discrete-time bilinear model on Laguerre orthonormal bases. Thus the coefficients associated to the input, to the output and to the crossed product of the bilinear model are expanded on three independent Laguerre bases. The resulting model is entitled bilinear-Laguerre model with filters on model input and output. The parametric complexity reduction of the proposed model with respect to the classical bilinear model is proved theoretically. The structure and the parameter identification of the bilinear-Laguerre model is achieved by a new proposed approach which consists in solving an optimization problem built from the bilinear model without using system input/output observations. The performances of the proposed bilinear-Laguerre model and the proposed identification approach are illustrated on a numerical simulation and validated on a benchmark as the continuous stirred tank reactor system.

Nonlinear system modeling based on bilinear Laguerre orthonormal bases.

This paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input, to the output and to the crossed product on three independent Laguerre orthonormal bases. Compared to classical bilinear model, the resulting model entitled bilinear-Laguerre model ensures a significant parameter number reduction as well as simple recursive representation. However, such reduction still constrained by an optimal choice of Laguerre pole characterizing each basis. To do so, we develop a pole optimization algorithm which constitutes an extension of that proposed by Tanguy et al.. The bilinear-Laguerre model as well as the proposed pole optimization algorithm are illustrated and tested on a numerical simulations and validated on the Continuous Stirred Tank Reactor (CSTR) System.

Online identification of the bilinear model expansion on Laguerre orthonormal bases

This paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input, to the output and to the crossed product on three independent Laguerre orthonormal bases. Compared to classical bilinear model, the resulting model entitled bilinear-Laguerre model ensures a significant parameter number reduction as well as simple recursive representation. However, this reduction is still subject to an optimal choice of the Laguerre poles defining the three Laguerre bases. Therefore, we propose an analytical solution to optimise the Laguerre poles which depend on Fourier coefficients defining the bilinear-Laguerre model, and that are identified using the regularised square error. The identification procedures of the Laguerre poles and Fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the bilinear-Laguerre model. The bilinear-Laguerre model as well as the proposed algorithm are illustrated and tested on a numerical simulation and validated on the continuous stirred tank reactor system.

Expansion of MIMO ARX model on Laguerre orthonormal bases

In this paper, we propose a new dynamic linear MIMO system representation by using discrete-time MIMO AutoRegressive with eXogenous input (ARX) model. To provide a reduced complexity model, each polynomial function of the MIMO ARX model associated to the inputs and to the outputs is expanded on independent Laguerre orthonormal basis to develop a new black-box linear MIMO ARX-Laguerre model. This reduction is ensured once the poles characterising each Laguerre orthonormal basis are set to their optimal values. Simulation results show the effectiveness of the proposed modeling method.

Reduced complexity Volterra model of non-linear MISO system

In this paper, we propose a new dynamic non-linear MISO system model using discrete-time Volterra series. To provide a reduced complexity model, each Volterra kernel is expanded on independent generalised orthonormal bases (GOBs) associated to the inputs to develop a new black-box non-linear MISO-GOB-Volterra model. However, this reduction is ensured once the poles characterising each independent generalised orthonormal basis (GOB) are set to their optimal values. For the selection of optimal GOBs poles, we develop two new general approaches based on Gauss-Newton and exhaustive algorithms, the performances of which are illustrated and compared in simulation.

System approximations based on Meixner-like models

In this study, the authors investigate the parametric complexity reduction of the Meixner-like model for linear discrete-time system representation. The use of the Meixner-like functions is more suitable than the use of Laguerre functions and Kautz functions especially when the system have a slow initial onset or delay. The coefficients of the Meixner-like model can be estimated recursively from input–output data by the new representation. Noting that the selection of an arbitrary pole for the Meixner-like functions can raise the parameter number of the Meixner-like model. However, when the pole is set to its optimal value, an optimal expansion of transfer functions is produced. Therefore an optimisation technique is developed to generate the optimal Meixner-like pole, which is achieved by an iterative method, that consists in minimising the mean square error between the system output and the model output. Theoretical analysis and a numerical simulation show the efficiency of the approach.

Laguerre-based modelling and predictive control of multi-input multi-output systems applied to a communicating two-tank system (CTTS):

In this paper, a novel method is constructed for model predictive control (MPC) of multi-input multi-output (MIMO) systems. The latter are represented by a discrete-time MIMO ARX model expansion on Laguerre orthonormal bases. The resulting model, entitled the MIMO ARX-Laguerre model, provides a recursive representation with parameter number reduction. This reduction is strongly linked to the choice of Laguerre poles, and therefore we propose a new algorithm to optimize the Laguerre poles of the resulting model. The recursive formulation of the MIMO ARX-Laguerre model is used to obtain the MPC strategy. An ℓ 2 -norm finite moving horizon cost function is used to obtain a control law which is implemented as a quadratic programming (QP) problem. The effectiveness of the proposed controller that takes into account physical constraints is illustrated by a numerical simulation example and by a practical validation on an experimental communicating two-tank system (CTTS).