Mathematical Problems in Engineering

Volume 2019, Article ID 9012182, 26 pages

https://doi.org/10.1155/2019/9012182

## Fault Tolerant Control for MIMO Nonlinear Systems via MPC Based on MIMO ARX-Laguerre Multiple Models

Research Laboratory of Automatic Signal and Image Processing, National School of Engineers of Monastir, University of Monastir, Rue Ibn ELJazzar, 5019 Monastir, Tunisia

Correspondence should be addressed to Abdelkader Mbarek; nt.unr.mine@kerabm.redakledba

Received 9 June 2019; Revised 5 August 2019; Accepted 9 September 2019; Published 3 October 2019

Academic Editor: Jean Jacques Loiseau

Copyright © 2019 Abdelkader Mbarek and Kais Bouzrara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this article, we propose a fault tolerant control for multiple-input multiple-output (MIMO) nonlinear systems via model predictive control. The MIMO nonlinear systems are approximated by MIMO ARX-Laguerre multiple models. The latter is obtained by expanding a discrete-time MIMO ARX multiple model parameters on Laguerre orthonormal bases. The resulting model ensures an efficient complexity reduction with respect to the classical MIMO ARX multiple models. This parametric complexity reduction still subjects to an optimal choice of the Laguerre poles defining Laguerre bases. The parameter and structure identifications of the MIMO ARX-Laguerre multiple models are achieved by the recursive method and a metaheuristic algorithm, respectively. The proposed model is built from the system input/output observations and is used to synthesize a MIMO nonlinear fault tolerant control algorithm via MPC. So, we develop a fault detection and isolation (FDI) scheme based on the proposed model. The scheme of the fault detection is applied at every step of MPC control calculation, where we determine the actuator faults and we use it in the MPC optimization problem to determine the new control with respect to the actuator faults. The proposed strategy is tested on numerical simulation and validated on the real system.

#### 1. Introduction

The security of technological systems is a major concern in the last decade [1–4]. Ensuring the security and the environment of a process require the knowledge of its operating status as finely as possible at every moment. In particular, we have to be able to decide if the working system is normal or if a malfunction has occurred. In this case, it is interesting to know the nature of this dysfunction, which is the main objective of the diagnosis. So, in the context of increasing autonomy of the systems, once the dysfunction is detected and identified, this knowledge must be taken into account when calculating a closed-loop control law to counter the influence of defects. This strategy is entitled fault tolerant control (FTC) [5].

The goal of this work is to synthesize fault tolerant control for nonlinear multiple-input multiple-output (MIMO) systems via model predictive control (MPC) based on reduced complexity multiple models. Model-based predictive control is a well-established online control strategy which iteratively computes control signals by solving an optimization problem over a future time horizon under certain process constraints [6–9]. This optimization uses a prediction model of the future plant behavior. The closed-loop performance depends on the choice of an appropriate model for prediction and several tuning parameters. To model MIMO nonlinear systems, several models are used like neural network [10], MIMO nonlinear autoregressive with exogenous input model (NARX) [11], fuzzy logic model [12], and MIMO autoregressive with exogenous input (ARX) multiple models known as MIMO Takagi–Sugeno model [13]. However, these models are constrained by a high parameter number. Furthermore, the complexity of MIMO nonlinear models handicaps the synthesis of a control law by increasing the computation time of the control. To overcome this problem, several works are developed in the literature in the case of single-input single-output (SISO) linear and nonlinear models and MIMO linear model by expansion on Laguerre orthonormal bases [14–18].

In this context, we propose in this paper to reduce the parametric complexity of ARX MIMO multiple models by decomposing its parameter associated with the inputs and the outputs on independent Laguerre orthonormal bases. This decomposition can be realized since the coefficients of the ARX MIMO multiple models are absolutely summable on in the sense of the bounded-input bounded-output (BIBO) stability criterion of the system. The new model, entitled MIMO ARX-Laguerre multiple models, ensures the parameter number reduction with a recursive and easy representation. The proposed model is characterized by a set of poles where an optimal choice of these poles is compulsory to lessen considerably the number of MIMO ARX-Laguerre multiple model parameters. In this paper, we propose to use a recursive method to identify the Fourier coefficients and a metaheuristic algorithm to optimize the MIMO ARX-Laguerre multiple model poles.

The synthesis of a MIMO nonlinear fault tolerant control via MPC (MIMO NFTC-MPC) requires fault detection and estimation procedures. About that, we propose in this paper to use the moving horizon fault estimation (MHE) [19] based on the proposed MIMO ARX-Laguerre multiple models. The MHE is used to estimate the actuator faults from the error between the estimated outputs and the system outputs. The main contributions of this paper is basically threefold. (1) We present a new reduced complexity model for nonlinear MIMO systems by expanding the MIMO ARX multimodel on independent Laguerre bases. The resulting MIMO ARX-Laguerre multiple models ensures the parameter number reduction with a recursive and easy representation. (2) We develop a fault actuator detection and estimation based on the identified MIMO ARX-Laguerre multimodel and using the moving horizon fault estimation. (3) By combining the fault estimation procedure and the model predictive control for MIMO nonlinear system based on the MIMO ARX-Laguerre multiple models, we develop a MIMO nonlinear fault tolerant control via model predictive control.

This paper is organized as follows: in Section 2, we present the modeling of MIMO nonlinear systems where we recall the principle of MIMO multiple model approach and we give the definition of the MIMO ARX multiple models. In Section 3, we present the MIMO ARX-Laguerre multiple models obtained by the expansion of MIMO ARX multiple models on independent Laguerre bases. In Section 4, we propose the identification procedure of the MIMO ARX-Laguerre multiple models, where we develop a recursive method to identify the Fourier coefficients and we use a metaheuristic algorithm to optimize the poles. Section 5 is devoted to the development of a MHE to detect and estimate the actuator faults of the MIMO nonlinear system using the MIMO ARX-Laguerre multiple models. In Section 6, we synthesize the MIMO nonlinear fault tolerant control via MPC where we develop the *j*-step ahead predictor of the MIMO ARX-Laguerre multiple model outputs by taking into account the actuator faults and we present the control calculation by taking into account the constraint on the inputs and the outputs by resolving an optimization problem. Finally, Section 7 illustrates the proposed MIMO nonlinear fault tolerant control via MPC by a numerical example.

#### 2. Modeling of MIMO Nonlinear Systems

##### 2.1. Principle of Multiple Model Approach

A multiple model is a set of LTI (linear time invariant) and causal submodels aggregated by an interpolation mechanism to characterize the dynamic behavior of the overall nonlinear system. It is characterized by the number of submodels, their structure, and the choice of weighting functions. A multiple model structure is represented bywhere is the multiple model output, *L* is the submodel number, is the weighting function associated to the submodel, is the decision variable in general is selected as the input or the output of the system, and is the output of the submodel. The weighting functions allow to determine the relative contribution of each submodel according to the zone where the system operates, and they respect the convexity properties given as follow:

The weighting functions can be constructed from continuous functions derivatives such as Gaussian functions as follows:where and are, respectively, the dispersion and the center of the indexed variable .

##### 2.2. MIMO ARX Multiple Models

A strictly causal discrete time MIMO nonlinear system with *p* inputs and *m* outputs can be represented by a MIMO ARX multiple models where each nonlinear multiple-input single-output (MISO) system for can be represented by a MISO ARX multiple models written aswhere is the output of the submodel for the MISO model, is the weighting function associated with the *s*^{th} submodel for the MISO model, and is the decision variable selected as the inputs or the outputs of the system:where for , , if the decision variable is selected as the inputs and for , , if the decision is variable selected as the outputs. By defining the following matrices,where and are, respectively, the center, , and the dispersion of the decision variable, , where is defined as

Then, the weighting function is defined as

The MISO linear submodel can be described by its output equation given by a MISO autoregressive with exogenous input (MISO ARX) model as follows:where and are the model parameters of the MISO ARX submodel, and for and are, respectively, the system inputs and the system outputs, and and are the model orders associated, respectively, with the inputs and the outputs of every MISO ARX submodel. The MISO ARX submodel can be written in the matrix form aswiththen, the MISO ARX multiple models given by relation (4) can be rewritten as

The output of the MISO ARX multiple models can be rewritten aswithwhere each MISO ARX multiple models is characterized by a parameters number determined as follows:

Then, the MIMO ARX multiple models are characterized by a parameter number. From relation (15), we can conclude that the complexity of the MIMO ARX multiple models increases according to the submodel orders and . In order to reduce the number of parameters, we will proceed with the decomposition of these coefficients and for of the MIMO ARX multiple models given by (9), for , on Laguerre orthonormal bases.

#### 3. Expansion of MIMO ARX Multiple Models on Laguerre Orthonormal Bases

In this section, we use the Laguerre orthonormal bases to reduce the parametric complexity of the MIMO ARX multiple models defined by (9) [15, 18, 20]. This choice is due to the capability of Laguerre base on parametric reduction and for the classical recurrent representation. According to the stability condition of the system in the sense of bounded-input bounded-output criterion (BIBO), the coefficients and are absolutely summable and they satisfy

Therefore, these coefficients belong to the Lebesgue space . Noting that the orthogonal Laguerre functions form an orthogonal base belong also to the Lebesgue space, the coefficients and can be, respectively, developed on the Laguerre bases and as follows [20, 21]:where for and are the Fourier coefficients, and *i* are the Laguerre functions, and and are the poles defining, respectively, the orthogonal bases and . Taking into account the stability condition (16), the MISO ARX multiple models given by (9) can be written aswhere if and if .

By substituting and given by (17) in the MISO ARX submodel defined by (18), the resulting submodel can be written asfor .

The relation (19) can be written, for , as

By analogy to the development given by Mbarek et al. [22] for the MIMO case and from relation (A.3) and (A.4) given in Appendix A, the MISO ARX-Laguerre multiple model can be described by the following recursive representation:where , for and , are the parameter vectors regrouping the Fourier coefficients , for and , and , for and .and the matrices are defined bywhere the matrices , and , are defined according to the poles and as given, respectively, by relations (A.13) and (A.14).

The matrices and are defined bywhere and are defined as follows:where the vectors , and , are defined according to the poles and as given, respectively, by relations (A.15) and (A.16).

The output of the MISO ARX-Laguerre multiple models given by (21) can be written aswhere and are defined as follows:

The MISO ARX-Laguerre multiple models given by relation (21) is characterized by parameters as given by relation (27) and poles defined as follows:

The proposed MIMO ARX-Laguerre multiple models given, for , by the MISO ARX-Laguerre multiple modes (21) can be represented, as in Figure 1, by a parallel structure diagram in the case of 2 inputs/2 outputs where every MISO model is decomposed into *L* submodels.