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

Authors: Abdelkader Mbarek; Kais Bouzrara
Journal: Mathematical Problems in Engineering
 (2019)

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## 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.


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## Body

## 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 on0,∞ 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 Section2, 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 by(1)yk=∑s=1Lμsγk−1ysk,where yk is the multiple model output, L is the submodel number, μsγk is the weighting function associated to the sth submodel, γk is the decision variable in general is selected as the input or the output of the system, and ysk is the output of the sth submodel. The weighting functions μsγk 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:(2)∑s=1Lμsγk=1,0≤μsγk≤1∀s=1,…,L.The weighting functions can be constructed from continuous functions derivatives such as Gaussian functions as follows:(3)wsγk=exp−γk−cs2σs2,μsγk=wsγk∑j=1Lwjγk,where σs and cs are, respectively, the dispersion and the center of the indexed variable γk.


### 2.2. MIMO ARX Multiple Models

A strictly causal discrete time MIMO nonlinear system withp inputs and m outputs can be represented by a MIMO ARX multiple models where each nonlinear multiple-input single-output (MISO) system yik for i=1,…,m can be represented by a MISO ARX multiple models written as(4)yik=∑s=1Lμsγ¯k−1yisk,where yisk is the output of the sth submodel for the MISO model, μsγ¯k−1 is the weighting function associated with the sth submodel for the MISO model, and γ¯k−1 is the decision variable selected as the inputs or the outputs of the system:(5)γ¯k−1=γ1k−1,γ2k−1,…,γhk−1,where γik−1=uik−1 for i=1,…,p, h=p, if the decision variable is selected as the inputs and γik−1=yik−1 for i=1,…,m, h=m, if the decision is variable selected as the outputs. By defining the following matrices,(6)C=c11c12⋯c1Lc21c22⋯c2L⋮⋮⋮⋮ch1ch2⋯chL∈ℝL×h,σ=σ1σ2⋯σh,where cij and σi are, respectively, the jth center, j=1,…,L, and the dispersion of the ith decision variable, i=1,…,h, where σi is defined as(7)σi=1M∑k=1Mγik−γi¯.Then, the weighting functionμsγ¯k is defined as(8)wsγ¯k=∏j=1hexp−γjk−Cjs2σj2,μsγ¯k=wsγ¯k∑j=1Lwjγ¯k.Thesth MISO linear submodel can be described by its output equation yisk given by a MISO autoregressive with exogenous input (MISO ARX) model as follows:(9)yisk=∑t=1p∑j=1nahaitsjutk−j+∑i=1m∑j=1nbhbirsjyik−j,where haitsj and hbirsj are the model parameters of the sth MISO ARX submodel, utk and yik for t=1,…,p and i=1,…,m are, respectively, the tth system inputs and the ith system outputs, and na and nb are the model orders associated, respectively, with the inputs and the outputs of every sth MISO ARX submodel. The sth MISO ARX submodel can be written in the matrix form as(10)yisk=φTkθis,with(11)φk=u1k−1,…,u1k−na,…,upk−1,…,upk−na,y1k−1,…,y1k−nb,…,ymk−1,…,ymk−nbT,θis=hai1s1,…,hai1sna,…,haips1,…,haipsna,hbi1s1,…,hbi1snb,…,hbips1,…,hbipsnbT,then, the MISO ARX multiple models given by relation (4) can be rewritten as(12)yik=∑s=1Lμsγk−1φTkθis.The output of the MISO ARX multiple modelsyik can be rewritten as(13)yik=ψTkΘi,with(14)ψk=μ1γk−1φk⋮μLγk−1φkT,Θi=θi1⋮θiL,where each MISO ARX multiple models is characterized by a Ni parameters number determined as follows:(15)Ni=Lp.na+m.nb.Then, the MIMO ARX multiple models are characterized by am.Ni parameter number. From relation (15), we can conclude that the complexity of the MIMO ARX multiple models increases according to the submodel orders na and nb. In order to reduce the number of parameters, we will proceed with the decomposition of these coefficients haitsj and hbirsj for i,r=1,…,mandt=1,…,p of the MIMO ARX multiple models given by (9), for i=1,…,m, on Laguerre orthonormal bases.


## 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 by(1)yk=∑s=1Lμsγk−1ysk,where yk is the multiple model output, L is the submodel number, μsγk is the weighting function associated to the sth submodel, γk is the decision variable in general is selected as the input or the output of the system, and ysk is the output of the sth submodel. The weighting functions μsγk 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:(2)∑s=1Lμsγk=1,0≤μsγk≤1∀s=1,…,L.The weighting functions can be constructed from continuous functions derivatives such as Gaussian functions as follows:(3)wsγk=exp−γk−cs2σs2,μsγk=wsγk∑j=1Lwjγk,where σs and cs are, respectively, the dispersion and the center of the indexed variable γk.


## 2.2. MIMO ARX Multiple Models

A strictly causal discrete time MIMO nonlinear system withp inputs and m outputs can be represented by a MIMO ARX multiple models where each nonlinear multiple-input single-output (MISO) system yik for i=1,…,m can be represented by a MISO ARX multiple models written as(4)yik=∑s=1Lμsγ¯k−1yisk,where yisk is the output of the sth submodel for the MISO model, μsγ¯k−1 is the weighting function associated with the sth submodel for the MISO model, and γ¯k−1 is the decision variable selected as the inputs or the outputs of the system:(5)γ¯k−1=γ1k−1,γ2k−1,…,γhk−1,where γik−1=uik−1 for i=1,…,p, h=p, if the decision variable is selected as the inputs and γik−1=yik−1 for i=1,…,m, h=m, if the decision is variable selected as the outputs. By defining the following matrices,(6)C=c11c12⋯c1Lc21c22⋯c2L⋮⋮⋮⋮ch1ch2⋯chL∈ℝL×h,σ=σ1σ2⋯σh,where cij and σi are, respectively, the jth center, j=1,…,L, and the dispersion of the ith decision variable, i=1,…,h, where σi is defined as(7)σi=1M∑k=1Mγik−γi¯.Then, the weighting functionμsγ¯k is defined as(8)wsγ¯k=∏j=1hexp−γjk−Cjs2σj2,μsγ¯k=wsγ¯k∑j=1Lwjγ¯k.Thesth MISO linear submodel can be described by its output equation yisk given by a MISO autoregressive with exogenous input (MISO ARX) model as follows:(9)yisk=∑t=1p∑j=1nahaitsjutk−j+∑i=1m∑j=1nbhbirsjyik−j,where haitsj and hbirsj are the model parameters of the sth MISO ARX submodel, utk and yik for t=1,…,p and i=1,…,m are, respectively, the tth system inputs and the ith system outputs, and na and nb are the model orders associated, respectively, with the inputs and the outputs of every sth MISO ARX submodel. The sth MISO ARX submodel can be written in the matrix form as(10)yisk=φTkθis,with(11)φk=u1k−1,…,u1k−na,…,upk−1,…,upk−na,y1k−1,…,y1k−nb,…,ymk−1,…,ymk−nbT,θis=hai1s1,…,hai1sna,…,haips1,…,haipsna,hbi1s1,…,hbi1snb,…,hbips1,…,hbipsnbT,then, the MISO ARX multiple models given by relation (4) can be rewritten as(12)yik=∑s=1Lμsγk−1φTkθis.The output of the MISO ARX multiple modelsyik can be rewritten as(13)yik=ψTkΘi,with(14)ψk=μ1γk−1φk⋮μLγk−1φkT,Θi=θi1⋮θiL,where each MISO ARX multiple models is characterized by a Ni parameters number determined as follows:(15)Ni=Lp.na+m.nb.Then, the MIMO ARX multiple models are characterized by am.Ni parameter number. From relation (15), we can conclude that the complexity of the MIMO ARX multiple models increases according to the submodel orders na and nb. In order to reduce the number of parameters, we will proceed with the decomposition of these coefficients haitsj and hbirsj for i,r=1,…,mandt=1,…,p of the MIMO ARX multiple models given by (9), for i=1,…,m, 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 haitsj and hbirsj are absolutely summable and they satisfy(16)∑j=1∞haitsj<∞,t=1,…,p,∑j=1∞hbirsj<∞,r=1,…,m,s=1,…,L,i=1,…,m.Therefore, these coefficients belong to the Lebesgue spaceℓ20,+∞. Noting that the orthogonal Laguerre functions form an orthogonal base belong also to the Lebesgue space, the coefficients haitsj and hbirsj can be, respectively, developed on the Laguerre bases ℑaits=ℓaitsjj=0∞ and ℑbirs=ℓbirsjj=0∞ as follows [20, 21]:(17)haitsj=∑n=0∞gait,nsℓait,nsξaits,j,t=1,…,p,hbirsj=∑n=0∞gbir,nsℓbir,nsξbirs,j,r=1,…,m,i=1,…,m,s=1,…,L,where for s=1,…,L,t=1,…,p,andi,r=1,…,m, gait,nas and gbir,nbs are the Fourier coefficients, ℓait,nsξaits,j and iℓbir,nsξbirs,j are the Laguerre functions, and ξaits and ξbirs are the poles defining, respectively, the orthogonal bases ℑaits and ℑbirs. Taking into account the stability condition (16), the sth MISO ARX multiple models given by (9) can be written as(18)yisk=∑t=1p∑j=1∞haitsjutk−j+∑r=1m∑j=1∞hbirsjyrk−j,where haitsj=0 if j>na and hbirsj=0 if j>nb.By substitutinghaitsj and hbirsj given by (17) in the sth MISO ARX submodel defined by (18), the resulting submodel can be written as(19)yisk=∑t=1p∑j=1∞∑n=0∞gait,nsℓait,nsξaits,jutk−j+∑r=1m∑j=1∞∑n=0∞gbir,nsℓbir,nsξbirs,jyrk−j,for s=1,…,Landi=1,…,m.The relation (19) can be written, for s=1,…,Landi=1,…,m, as(20)yisk=∑t=1p∑n=0∞gait,ns∑j=1∞ℓait,nsξaits,jutk−j+∑r=1m∑n=0∞gbir,ns∑j=1∞ℓbir,nsξbirs,jyrk−j.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:(21)Xisk+1=AisXisk+Baisu¯k+Bbisy¯k,yisk=CisTXisk,yik=∑s=1Lμsγ¯k−1yisk,where Cis, for i=1,…,m and s=1,…,L, are the parameter vectors regrouping the Fourier coefficients gait,ns, for n=0,…,Nas−1 and t=1,…,p, and gbir,ns, for n=0,…,Nbs−1 and r=1,…,m.(22)Cis=gai1,0s,…,gai1,Nas−1s,gai2,0s,…,gai2,Nas−1s,…,gaip,0s,…,gaip,Nas−1s,gbi1,0s,…,gbi1,Nbs−1s,gbi2,0s,…,gbi2,Nbs−1s,…,gbim,0s,…,gbim,Nbs−1sT,and the matrices Ais are defined by(23)Ais=Aai1s⋯0¯Nas×Nas0¯Nas×Nbs⋯0¯Nas×Nbs⋮⋱⋮⋮⋮⋮0¯Nis×Nis⋯Aaips0¯Nas×Nbs⋯0¯Nas×Nbs0¯Nbs×Nas⋯0¯Nbs×NasAbi1s⋯0¯Nbs×Nbs⋮⋮⋮⋮⋱⋮0¯Nbs×Nas⋯0¯Nbs×Nas0¯Nbs×Nbs⋯Abims,where the matrices Aaits, t=1,…,p and Abirs, r=1,…,m are defined according to the poles ξaits and ξbirs as given, respectively, by relations (A.13) and (A.14).The matricesBais and Bbis are defined by(24)Bais=Bais′0¯p.Na×p,Bbis=0¯m.Nb×mBbis′,where Bais′ and Bbis′ are defined as follows:(25)Bais′=bai1s0¯Nas×1⋯0¯Nas×10¯Nas×1bai2s…0¯Nas×1⋮⋮⋱⋮0¯Nas×10¯Nas×1⋯baips,Bbis′=bbi1s0¯Nbs×1⋯0¯Nbs×10¯Nbs×1bbi2s⋯0¯Nbs×1⋮⋮⋱⋮0¯Nbs×10¯Nbs×1⋯bbims,where the vectors Baits, t=1,…,p and Bbirs, r=1,…,m are defined according to the poles ξaits and ξbirs as given, respectively, by relations (A.15) and (A.16).The output of theith MISO ARX-Laguerre multiple models yik given by (21) can be written as(26)yik=C¯iTXik,where C¯i and Xik are defined as follows:(27)C¯i=Ci1TCi2T⋯CiLTT,(28)Xik=μ1γ¯k−1Xi1kTμ2γ¯k−1Xi2kT⋯μLγ¯k−1XiLkTT.The MISO ARX-Laguerre multiple models given by relation (21) is characterized by L.p.Na+m.Nb parameters as given by relation (27) and m+p.L poles defined as follows:(29)ξ¯i=ξai11,…,ξaip1,ξbi11,…,ξbim1,ξai12,…,ξaip2,ξbi12,…,ξbim2,…,ξai1L,…,ξaipL,ξbi1L,…,ξbimLT.The proposed MIMO ARX-Laguerre multiple models given, fori=1,…,m, 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.Figure 1
Parallel structure diagram of MIMO ARX-Laguerre multiple models in the case of 2 inputs/2 outputs.By defining the output vectorsy¯ and y¯s as(30)y_k=y1k,y2k,…,ymkT,y_sk=y1sk,y2sk,…,ymskT,the MIMO ARX-Laguerre multiple models given by relation (21) can be written as(31)Xsk+1=AsXsk+Ba,sU¯k+Bb,sy¯k,y¯sk=CsXsk,y¯k=∑s=1Lμsγ¯k−1y¯sk,with(32)Xsk=X1skT,X2skT,…,XmskTT,Cs=C1sT0⋯00C2sT⋯0⋮⋮⋱000⋯CmsT,As=A1s0⋯00A2s⋯0⋮⋮⋱000⋯Ams,Ba,s=Ba1sBa2s⋮Bams,Bb,s=Bb1sBb2s⋮Bbms.Then, the MIMO ARX-Laguerre multiple models can be written in the following recursive representation:(33)Xk+1=AXk+Bau¯k+Bby¯k,y¯k=CXk,where the state vector Xk and the matrices C, A, Ba, and Bb are defined, respectively, as(34)Xk=X1kT,X2kT,…,XLkTT,(35)C=μ1γ¯k−1C1μ2γ¯k−1C2…μLγ¯k−1CL,(36)A=A10¯…0¯0¯A2…0¯⋮⋮⋱0¯0¯0¯…AL,Ba=Ba,1Ba,2⋮Ba,L,Bb=Bb,1Bb,2⋮Bb,L.


## 4. Identification of the MIMO ARX-Laguerre Multiple Models

A strictly causal discrete time nonlinear MIMO system withp inputs and m outputs can be described by a MIMO ARX-Laguerre multiple models where each MISO system, for i=1,…,m, is represented by the recursive representation given by relation (21) and the ith output yik is rewritten by the vectorial representation as given in relation (26). Each MISO ARX-Laguerre multiple model is characterized by the parameter vector C¯i defined by relation (27) and the poles vector ξ¯i given by relation (29). To ensure a significant parametric reduction, it is necessary to optimize the MIMO ARX-Laguerre multiple model poles. The identification of the parameter vectors C¯i and the optimization of the poles ξ¯i are presented in the next sections.


### 4.1. Recursive Identification of the Fourier Coefficients

This method proposed by Abdelwahed et al. [21] is based on the minimization of a regularized square error given as(37)Ji=∑k=1hy¯ik−yik2+αi∑s=1L∑t=1p∑n=1Nasgait,nsh−1−gait,nsh2+∑r=1m∑n=1Nbsgbir,nsh−1−gbir,nsh2,where(i)
y¯ik and yih are, respectively, the ith output system and the ith output of the MIMO ARX-Laguerre multiple models(ii)
gait,nsh−1, gbir,nsh−1, and gait,nsh, gbir,nsh are, respectively, the Fourier coefficients at time instants h−1 and h(iii)
αi, αi>0, is a regularization constant which regulates the importance that we give either to the quadratic error between y¯ik and yih or to that between the Fourier coefficients.From relation (26) and at time instant h, the square error Ji can be written in matrix form as(38)Jih=Y¯ih−ΨihC¯ihTY¯ih−ΨihC¯ih+C¯ih−1−C¯ihTαiC¯ih−1−C¯ih,where C¯ih−1 and C¯ih are the parameter vectors regrouping the Fourier coefficients associated with the ith output at time instant h−1 and h, respectively, and the vector Y¯ih and the matrix Ψih are defined as(39)Y¯ih=y¯i1y¯i2⋮y¯ih,Ψih=Xi1TXi2T⋮XihT=Ψih−1XihT.The optimal parameter vectorC¯ih of each ith MISO system is obtained by minimizing the quadratic regularised square error Jih given by relation (38). The gradient of the criterion Jih with respect to C¯ih is written as(40)∂Jih∂C¯ih=−2ΨihY¯ih−ΨihTΨihC¯ih−2αiC¯ih−1−C¯ih.From relation (40), we can calculate the estimated parameter vector C¯ih of each ith MISO system as(41)C¯ih=ΨihTΨih+αiI−1ΨihY¯ih+αiC¯ih−1.The recursive parametric identification of the MIMO ARX-Laguerre multiple models representing a nonlinear MIMO system ofp inputs and m outputs where each MISO system is decomposed into L submodel is described by the following algorithm.


### 4.2. Pole Optimization of the MIMO ARX-Laguerre Multiple Models Using Metaheuristic Algorithms

To guarantee a significant parametric reduction of the proposed MIMO ARX-Laguerre multiple models, it is necessary to optimize theL.m+p Laguerre bases poles. Several optimization methods are proposed in the literature in order to put the Laguerre poles into their optimal values, such as the gradient algorithm, the Newton–Raphson algorithm, and the Bouzrara el al.’s method. In this paper, since we have many poles to be optimized, L.m+p poles, we propose the use of genetic algorithms (GAs) as a metaheuristic optimization method. The values of the MIMO ARX-Laguerre multiple models poles are optimized by minimizing the normalized mean squared error NMSEi of each MISO ARX-Laguerre multiple models given by(42)NMSEi=∑k=1My¯ik−yik2∑k=1My¯ik2,where y¯ik and yih are the ith output system and the ith output of the MIMO ARX-Laguerre multiple models, respectively.The pole optimization algorithm of the proposed MIMO ARX-Laguerre multiple models using the genetic algorithm is summarized in the following algorithm.


## 4.1. Recursive Identification of the Fourier Coefficients

This method proposed by Abdelwahed et al. [21] is based on the minimization of a regularized square error given as(37)Ji=∑k=1hy¯ik−yik2+αi∑s=1L∑t=1p∑n=1Nasgait,nsh−1−gait,nsh2+∑r=1m∑n=1Nbsgbir,nsh−1−gbir,nsh2,where(i)
y¯ik and yih are, respectively, the ith output system and the ith output of the MIMO ARX-Laguerre multiple models(ii)
gait,nsh−1, gbir,nsh−1, and gait,nsh, gbir,nsh are, respectively, the Fourier coefficients at time instants h−1 and h(iii)
αi, αi>0, is a regularization constant which regulates the importance that we give either to the quadratic error between y¯ik and yih or to that between the Fourier coefficients.From relation (26) and at time instant h, the square error Ji can be written in matrix form as(38)Jih=Y¯ih−ΨihC¯ihTY¯ih−ΨihC¯ih+C¯ih−1−C¯ihTαiC¯ih−1−C¯ih,where C¯ih−1 and C¯ih are the parameter vectors regrouping the Fourier coefficients associated with the ith output at time instant h−1 and h, respectively, and the vector Y¯ih and the matrix Ψih are defined as(39)Y¯ih=y¯i1y¯i2⋮y¯ih,Ψih=Xi1TXi2T⋮XihT=Ψih−1XihT.The optimal parameter vectorC¯ih of each ith MISO system is obtained by minimizing the quadratic regularised square error Jih given by relation (38). The gradient of the criterion Jih with respect to C¯ih is written as(40)∂Jih∂C¯ih=−2ΨihY¯ih−ΨihTΨihC¯ih−2αiC¯ih−1−C¯ih.From relation (40), we can calculate the estimated parameter vector C¯ih of each ith MISO system as(41)C¯ih=ΨihTΨih+αiI−1ΨihY¯ih+αiC¯ih−1.The recursive parametric identification of the MIMO ARX-Laguerre multiple models representing a nonlinear MIMO system ofp inputs and m outputs where each MISO system is decomposed into L submodel is described by the following algorithm.


## 4.2. Pole Optimization of the MIMO ARX-Laguerre Multiple Models Using Metaheuristic Algorithms

To guarantee a significant parametric reduction of the proposed MIMO ARX-Laguerre multiple models, it is necessary to optimize theL.m+p Laguerre bases poles. Several optimization methods are proposed in the literature in order to put the Laguerre poles into their optimal values, such as the gradient algorithm, the Newton–Raphson algorithm, and the Bouzrara el al.’s method. In this paper, since we have many poles to be optimized, L.m+p poles, we propose the use of genetic algorithms (GAs) as a metaheuristic optimization method. The values of the MIMO ARX-Laguerre multiple models poles are optimized by minimizing the normalized mean squared error NMSEi of each MISO ARX-Laguerre multiple models given by(42)NMSEi=∑k=1My¯ik−yik2∑k=1My¯ik2,where y¯ik and yih are the ith output system and the ith output of the MIMO ARX-Laguerre multiple models, respectively.The pole optimization algorithm of the proposed MIMO ARX-Laguerre multiple models using the genetic algorithm is summarized in the following algorithm.


## 5. Moving Horizon Fault Estimation Based on MIMO ARX-Laguerre Multiple Models

In the following, we propose to use the moving horizon fault estimation (MHE) based on the MIMO ARX-Laguerre multiple models to solve the fault estimation of the MIMO nonlinear system. The MHE is used to estimate the actuator faults from the error between the estimated outputy^¯k and the system output y¯k. The MIMO ARX-Laguerre multiple models taking account of actuator faults can be written from (33) as(43)X^k+1=AX^k+Bau¯k+f¯k+Bby¯^k,y¯^k=CX^k,where X^k and y¯^k are the estimated state vector and the estimation outputs model, respectively, when the actuator faults f¯k are defined as(44)f¯k=f1k,f2k,…,fpkT.The estimate actuators faultsf¯^k are obtained by an online minimization at every sample time of the quadratic cost function Jfk defined as follows:(45)Jfk=∑j=k−Nf+1ky¯^j−y¯j2,where Nf is the estimation fault horizon, y¯^j is the estimation outputs model when the actuator faults are considered as given by (43), and y¯j is the outputs system defined as(46)y¯j=y¯11,y¯2j,…,y¯mjT.The quadratic criterionJfk given by (45) can be rewritten as(47)Jfk=∑j=k−Nf+1kCAX^j−1+Bau¯j−1+f¯^j+Bby¯^j−1−y¯j2,where f¯^ is the fault actuators to be optimized using the MHE method:(48)f¯^j=f^1jf^2j⋯f^pjT.The MHE method is formulated by minimizing the criterionJfk given by (47) at every sample time:(49)F^k−1=minFk−1∈SfJfk,where F^k−1 is defined as follows:(50)F^k−1=f¯^k−Nf+1f¯^k−Nf+2⋯f¯^k−1T,and Sf is the admissible set of fault actuators constraints defined as(51)Sf=F^k/Fmin≤F^k≤Fmax,where Fmin and Fmax are p.Nf-dimensional vectors defined as follows:(52)Fmin=f¯mf¯m⋮f¯m,Fmax=f¯Mf¯M⋮f¯M.where f¯m and f¯M are the bounds of actuator faults that can be chosen from the physical constraints defined as(53)f¯m=f1,mf2,m⋮fp,m,f¯M=f1,Mf2,M⋮fp,M.


## 6. MIMO Nonlinear Fault Tolerant Control via Model Predictive Control

In this paper, we propose a MIMO nonlinear fault tolerant control via model predictive control using MIMO ARX-Laguerre multiple models (MIMO NFTC-MPC). The proposed MIMO NFTC-MPC is applied to a nonlinear system and used to compensate online the actuator faults by including the effects of the faults in the model predictive control optimization problem. Then, the MIMO NFTC-MPC strategy requires a fault estimation unit to estimate the actuator faults and a MIMO ARX-Laguerre multiple models observer to estimate the system state. The MIMO NFTC-MPC strategy is described by a block diagram as represented in Figure2.Figure 2
Block diagram of MIMO NFTC-MPC strategy.The MIMO NFTC-MPC is characterized by three essential steps. The first step is devoted to the calculation of the system output predictiony¯k+j/k to determine the process behavior on the prediction horizon k+1,k+Np+1 depending on the inputs applied to the process until the instant k−1T. The second step is dedicated to the actuator fault optimization until the instant k−1, the outputs measured until the instant kT, and the equations describing the model, T, is the sample time. The third step is dedicated to the optimization of a criterion to determine a future control sequence over the control horizon Nu.


### 6.1. Thej-Step Ahead Predictor including the Actuator Faults

The MIMO ARX-Laguerre multiple models taking into account actuator faults can be written from relation (31) as(54)X˜sk+1=AsX˜sk+Ba,su¯k+f¯^k+Bb,sy¯˜k,y¯˜sk=CsX˜sk,y¯˜k=∑s=1Lμsγ¯ky¯˜sk,where f¯^k is the vector of actuator faults estimated using the MHE method. To calculate the j-step ahead predictor on the MIMO ARX-Laguerre multiple models output taking into account actuator faults, y˜k+j/k, we use the incremental form given from (54) as(55a)δX˜sk=AsδX˜sk−1+Ba,sδu¯k−1+Ba,sδf¯^k−1+Bb,sδy˜¯k−1,(55b)y¯˜k=y¯˜k−1+∑s=1Lμsγ¯k−1CiTδX˜sk+∑s=1Lδμsγ¯k−1CsX˜sk,where δu¯k, δy˜¯k, δf¯^k, δX˜sk, and δμsγ¯k are the control increments, the output increments, the actuator faults increments, the state vector increments, and the weighting function increments, respectively, defined as(56)δu¯k=u¯k−u¯k−1;δy˜¯k=y˜¯k−y˜¯k−1;δf¯^k=f¯^k−f¯^k−1;δX˜sk=X˜sk−X˜sk−1;δμsγ¯k=μsγ¯k−μsγ¯k−1.By successive substitutions and after some reorganization, we give the following proposition.Proposition 1.
The j-step ahead predictor including the actuator faults using the MIMO ARX-Laguerre multiple models,y¯˜k+j/k, is given as(57)y¯˜k+j/k=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−INsδX˜sk+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδU¯k+l−1+Ba,sδf¯^k+l−1+Bb,sδy˜¯k+l−1,where INs is the Ns-dimensional identity matrix, Ns is the truncation order of the sth L submodel for each MISO submodel, and Ks,n is defined as(58)Ks,n=∑r=0nAsr,forn≥0,Ks,n=0,forn<0.
See proof in AppendixB.Thej-step ahead predictor given by (57) is split into two components, the free and the forced components:(59)y¯˜k+j/k=y¯˜frk+j/k+y¯˜fok+j,where y¯˜frk+j/k is the free components determined using the measured outputs until kT and the controls up to k−1T:(60)y¯˜frk+j/k=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−INsδX˜sk+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδf¯^k+l−1+Bb,sδy˜¯k+l−1.The free componenty¯˜frk+j/k given by relation (60) can be rewritten as(61)y¯˜frk+j/k=y¯˜fr1k+j/k+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδf¯^k+l−1+Bb,sδy˜¯k+l−1,where y¯˜fr1k+j/k is defined as(62)y¯˜fr1k+j=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−l−INsδX˜sk,and from relation (A.25), the forced component y¯˜fok+j/k resulting from the action of future control can be written as(63)y¯˜fok+j/k=∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδu¯k+l−1.Forj=1,…,Np, relation (59) becomes(64)Y˜k=Y˜fok+Y˜frk,where(65)Y˜k=y¯˜k+1/k⋯y¯˜k+Np/kT,Y˜frk=y¯˜frk+1/k⋯y¯˜frk+Np/kT,Y˜fok=y¯˜fok+1/k⋯y¯˜fok+Np/kT.From relations (71) and (73) and for j=1,…,Np, the vectors Y˜frk and Y˜fok can be written as(66)Y˜frk=Y˜fr1k+GaδF^k+GbδY˜k,Y˜fok=GaδUk,where(i)
Y˜fr1k is a m.Np-dimensional vector computed from relation (72) and defined as(67)Y˜fr1k=y¯˜fr1k+1/k⋯y¯˜fr1k+Np/kT.(ii)
δY˜k is a m.Np-dimensional vector defined as(68)δY˜k=δy¯˜k⋯δy¯˜k+Np−1T.(iii)
δUk is a p.Nu-dimensional future control increments vector where Nu is the control horizon, defined as(69)δUk=δu¯k,δu¯k+1,…,δu¯k+Nu−1T.(iv)
δF^k is a p.Nu-dimensional actuator faults increments vector given by(70)δF^k=δf¯^k,δf¯^k+1,…,δf¯^k+Nu−1T.(v)
Ga and Gb are, respectively, p.Nu-dimensional matrix and m.Np-dimensional matrix given as follows:(71)Ga=g¯a,100¯m×p⋯0¯m×pg¯a,21g¯a,20⋯0¯m×p⋮⋮⋱⋮g¯a,NuNu−1g¯a,NuNu−2⋯g¯a,Nu0⋮⋮⋱⋮g¯a,NpNpg¯a,NpNp−1⋯g¯a,NpNp−Nu+1,where 0¯m×p is a m×p-dimensional vector of zeros.(72)Gb=g¯b,100¯m×p⋯0¯m×pg¯b,21g¯b,20⋯0¯m×p⋮⋮⋱⋮g¯b,NpNp−1g¯b,NpNp−2⋯g¯b,Np0,where 0¯m×m is a m×m-dimensional vector of zeros.According to relations (71) and (73), the components g¯a,jn and g¯b,jn n=0,…,Np−1 and j=1,…,Np can be written as(73)g¯d,jn=∑s=1Lμsγ¯k+jCsKs,nBd,s,d=a,b.Then, from relation (66), the prediction vector Y˜k given by (54) can be written in matrix form as(74)Y˜k=Y˜fr1k+GaδF^k+GaδUk+GbδY˜k.


### 6.2. Control Calculation

The control calculation of a MIMO system withp inputs and m outputs is based on the minimization of the following performance quadratic criterion:(75)J2δUk=∑i=1mρi∑j=1Npy˜ik+j/k−wik+j2+∑t=1pϱt∑j=0Nu−1δutk+j2,where wik+j, i=1,…,m are the ith reference trajectories at time instant k+j, ρi are the weights on the tracking error considered constant and equal to ρ, and ϱt are positive weights generally considered constant and equal to ϱ.The quadratic criterionJ2δUk given by (75) can be rewritten on the following compact matrix:(76)J2δUk=Y˜k−WkTΛyY˜k−Wk+δUkTΛuδUk,where Y˜k is the mNp-dimensional vector of predicted outputs defined in relation (65) and the vector Wk is defined from the reference trajectories wi as follows:(77)Wk=w1k+1,…,wmk+1,…,w1k+Np,…,wmk+NpT.δ
U
k is the vector of future control increments given by (69), Λy and Λu are positive definite matrices of dimension, respectively, m.Np and p.Nu, defined as follows:(78)Λy=bloc_diagρ¯,…,ρ¯︸Nptimes,Λu=bloc_diagϱ,…,ϱ︸Nutimes,where(79)ρ¯=ρ1⋯0⋮⋱⋮0⋯ρm,ϱ¯=ϱ1⋯0⋮⋱⋮0⋯ϱp.By replacingY˜k by its expression given by (74), the matrix form of criterion J2δU¯k given by (76) becomes(80)J2δUk=Y˜fr1k+GaδF^k+GaδUk+GbδY˜k−WkTΛyY˜fr1k+GaδF^k+GaδUk+GbδY˜k−Wk+δUkTΛuδUk.The model predictive control methods allow to take into account some constraints optimization problem subject to physical constraints on the inputs and the outputs due to the actuator technology, the control system security, or the quality desired for the output of the controlled process. In the case of MIMO system ofm outputs and p inputs, the constraints are given as(81)δut,m≤δutk+j≤δut,M,∀j∈0,…,Nu−1,t=1,…,p,(82)ut,m≤utk+j≤ut,M,∀j∈0,…,Nu−1,t=1,…,p,(83)yi,m≤yik+j≤yi,M,∀j∈1,…,Np,i=1,…,m,by analogy to the matrix form of constraints given by Mbarek et al. [22], these constraints can be expressed in terms of future control increments and written in matrix form as(84)ΓδUk≤V¯k,with(85)Γ=IpNu−IpNuD1−D1Ga−Ga,V¯k=δU¯M−δU¯mU¯M−V1k−1−U¯m+V1k−1Y¯M−Y˜fr1k−GaδF^k−GbδY˜k−Y¯m+Y˜fr1k+GaδF^k+GbδY˜k,where the future control increment vector is defined by relation (69), and the square matrix D1 and the vector V¯1k−1 of dimensions p×Np, are defined as follows:(86)D1=Ip0¯p×p⋯0¯p×pIpIp⋱0¯p×p⋮⋮⋱⋮IpIp⋯Ip,V¯1k−1=u¯k−1u¯k−1⋮u¯k−1,and the vectors U¯M, U¯m, δU¯M, and δU¯m of dimension p.Nu are defined as follows:(87)U¯M=u¯MT,…,u¯MTT,U¯M=u¯mT,…,u¯mTT,δU¯M=δu¯MT,…,δu¯MTT,δU¯M=δu¯mT,…,δu¯mTT,and the vectors y¯M and y¯m of dimension m.Np are given by(88)y¯M=y¯MT,…,y¯MTT,y¯m=y¯mT,…,y¯mTT.These constraints define a future control increments admissible set as follows:(89)Ψ=δUk/ΓδUk≤V¯k.Then, the optimization problem subject to the physical constraints on the inputs and the outputs is written as follows:(90)minδU¯k∈ΨJ2δUk,where Ψ is the future control increments admissible set given by (89) and J2δUk is the quadratic criterion that can be written, from (90), as(91)J2δUk=δUTkGaTΛyGa+ΛuδUk+2Y˜fr1k+GaδF^k+GbδY˜k−W¯kTΛyGaδUk+Y˜fr1k+GaδF^k+GbδY˜k−W¯kTΛyY˜fr1k+GaδF^k+GbδY˜k−W¯k.The optimization problem (90) subject to (84) is a convex quadratic programming problem (QP), for which the global solution is unique. As with all predictive control methods, where a moving horizon strategy is applied, at instant kT, only the first optimized control U¯k is applied to the system, with(92)u¯k=u¯k−1+δu¯k.


## 6.1. Thej-Step Ahead Predictor including the Actuator Faults

The MIMO ARX-Laguerre multiple models taking into account actuator faults can be written from relation (31) as(54)X˜sk+1=AsX˜sk+Ba,su¯k+f¯^k+Bb,sy¯˜k,y¯˜sk=CsX˜sk,y¯˜k=∑s=1Lμsγ¯ky¯˜sk,where f¯^k is the vector of actuator faults estimated using the MHE method. To calculate the j-step ahead predictor on the MIMO ARX-Laguerre multiple models output taking into account actuator faults, y˜k+j/k, we use the incremental form given from (54) as(55a)δX˜sk=AsδX˜sk−1+Ba,sδu¯k−1+Ba,sδf¯^k−1+Bb,sδy˜¯k−1,(55b)y¯˜k=y¯˜k−1+∑s=1Lμsγ¯k−1CiTδX˜sk+∑s=1Lδμsγ¯k−1CsX˜sk,where δu¯k, δy˜¯k, δf¯^k, δX˜sk, and δμsγ¯k are the control increments, the output increments, the actuator faults increments, the state vector increments, and the weighting function increments, respectively, defined as(56)δu¯k=u¯k−u¯k−1;δy˜¯k=y˜¯k−y˜¯k−1;δf¯^k=f¯^k−f¯^k−1;δX˜sk=X˜sk−X˜sk−1;δμsγ¯k=μsγ¯k−μsγ¯k−1.By successive substitutions and after some reorganization, we give the following proposition.Proposition 1.
The j-step ahead predictor including the actuator faults using the MIMO ARX-Laguerre multiple models,y¯˜k+j/k, is given as(57)y¯˜k+j/k=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−INsδX˜sk+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδU¯k+l−1+Ba,sδf¯^k+l−1+Bb,sδy˜¯k+l−1,where INs is the Ns-dimensional identity matrix, Ns is the truncation order of the sth L submodel for each MISO submodel, and Ks,n is defined as(58)Ks,n=∑r=0nAsr,forn≥0,Ks,n=0,forn<0.
See proof in AppendixB.Thej-step ahead predictor given by (57) is split into two components, the free and the forced components:(59)y¯˜k+j/k=y¯˜frk+j/k+y¯˜fok+j,where y¯˜frk+j/k is the free components determined using the measured outputs until kT and the controls up to k−1T:(60)y¯˜frk+j/k=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−INsδX˜sk+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδf¯^k+l−1+Bb,sδy˜¯k+l−1.The free componenty¯˜frk+j/k given by relation (60) can be rewritten as(61)y¯˜frk+j/k=y¯˜fr1k+j/k+∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδf¯^k+l−1+Bb,sδy˜¯k+l−1,where y¯˜fr1k+j/k is defined as(62)y¯˜fr1k+j=y¯˜k+∑s=1L∑l=1jδμsγ¯k+lCsX˜sk+∑s=1Lμsγ¯k+jCsKs,j−l−INsδX˜sk,and from relation (A.25), the forced component y¯˜fok+j/k resulting from the action of future control can be written as(63)y¯˜fok+j/k=∑s=1Lμsγ¯k+jCs∑l=1jKs,j−lBa,sδu¯k+l−1.Forj=1,…,Np, relation (59) becomes(64)Y˜k=Y˜fok+Y˜frk,where(65)Y˜k=y¯˜k+1/k⋯y¯˜k+Np/kT,Y˜frk=y¯˜frk+1/k⋯y¯˜frk+Np/kT,Y˜fok=y¯˜fok+1/k⋯y¯˜fok+Np/kT.From relations (71) and (73) and for j=1,…,Np, the vectors Y˜frk and Y˜fok can be written as(66)Y˜frk=Y˜fr1k+GaδF^k+GbδY˜k,Y˜fok=GaδUk,where(i)
Y˜fr1k is a m.Np-dimensional vector computed from relation (72) and defined as(67)Y˜fr1k=y¯˜fr1k+1/k⋯y¯˜fr1k+Np/kT.(ii)
δY˜k is a m.Np-dimensional vector defined as(68)δY˜k=δy¯˜k⋯δy¯˜k+Np−1T.(iii)
δUk is a p.Nu-dimensional future control increments vector where Nu is the control horizon, defined as(69)δUk=δu¯k,δu¯k+1,…,δu¯k+Nu−1T.(iv)
δF^k is a p.Nu-dimensional actuator faults increments vector given by(70)δF^k=δf¯^k,δf¯^k+1,…,δf¯^k+Nu−1T.(v)
Ga and Gb are, respectively, p.Nu-dimensional matrix and m.Np-dimensional matrix given as follows:(71)Ga=g¯a,100¯m×p⋯0¯m×pg¯a,21g¯a,20⋯0¯m×p⋮⋮⋱⋮g¯a,NuNu−1g¯a,NuNu−2⋯g¯a,Nu0⋮⋮⋱⋮g¯a,NpNpg¯a,NpNp−1⋯g¯a,NpNp−Nu+1,where 0¯m×p is a m×p-dimensional vector of zeros.(72)Gb=g¯b,100¯m×p⋯0¯m×pg¯b,21g¯b,20⋯0¯m×p⋮⋮⋱⋮g¯b,NpNp−1g¯b,NpNp−2⋯g¯b,Np0,where 0¯m×m is a m×m-dimensional vector of zeros.According to relations (71) and (73), the components g¯a,jn and g¯b,jn n=0,…,Np−1 and j=1,…,Np can be written as(73)g¯d,jn=∑s=1Lμsγ¯k+jCsKs,nBd,s,d=a,b.Then, from relation (66), the prediction vector Y˜k given by (54) can be written in matrix form as(74)Y˜k=Y˜fr1k+GaδF^k+GaδUk+GbδY˜k.


## 6.2. Control Calculation

The control calculation of a MIMO system withp inputs and m outputs is based on the minimization of the following performance quadratic criterion:(75)J2δUk=∑i=1mρi∑j=1Npy˜ik+j/k−wik+j2+∑t=1pϱt∑j=0Nu−1δutk+j2,where wik+j, i=1,…,m are the ith reference trajectories at time instant k+j, ρi are the weights on the tracking error considered constant and equal to ρ, and ϱt are positive weights generally considered constant and equal to ϱ.The quadratic criterionJ2δUk given by (75) can be rewritten on the following compact matrix:(76)J2δUk=Y˜k−WkTΛyY˜k−Wk+δUkTΛuδUk,where Y˜k is the mNp-dimensional vector of predicted outputs defined in relation (65) and the vector Wk is defined from the reference trajectories wi as follows:(77)Wk=w1k+1,…,wmk+1,…,w1k+Np,…,wmk+NpT.δ
U
k is the vector of future control increments given by (69), Λy and Λu are positive definite matrices of dimension, respectively, m.Np and p.Nu, defined as follows:(78)Λy=bloc_diagρ¯,…,ρ¯︸Nptimes,Λu=bloc_diagϱ,…,ϱ︸Nutimes,where(79)ρ¯=ρ1⋯0⋮⋱⋮0⋯ρm,ϱ¯=ϱ1⋯0⋮⋱⋮0⋯ϱp.By replacingY˜k by its expression given by (74), the matrix form of criterion J2δU¯k given by (76) becomes(80)J2δUk=Y˜fr1k+GaδF^k+GaδUk+GbδY˜k−WkTΛyY˜fr1k+GaδF^k+GaδUk+GbδY˜k−Wk+δUkTΛuδUk.The model predictive control methods allow to take into account some constraints optimization problem subject to physical constraints on the inputs and the outputs due to the actuator technology, the control system security, or the quality desired for the output of the controlled process. In the case of MIMO system ofm outputs and p inputs, the constraints are given as(81)δut,m≤δutk+j≤δut,M,∀j∈0,…,Nu−1,t=1,…,p,(82)ut,m≤utk+j≤ut,M,∀j∈0,…,Nu−1,t=1,…,p,(83)yi,m≤yik+j≤yi,M,∀j∈1,…,Np,i=1,…,m,by analogy to the matrix form of constraints given by Mbarek et al. [22], these constraints can be expressed in terms of future control increments and written in matrix form as(84)ΓδUk≤V¯k,with(85)Γ=IpNu−IpNuD1−D1Ga−Ga,V¯k=δU¯M−δU¯mU¯M−V1k−1−U¯m+V1k−1Y¯M−Y˜fr1k−GaδF^k−GbδY˜k−Y¯m+Y˜fr1k+GaδF^k+GbδY˜k,where the future control increment vector is defined by relation (69), and the square matrix D1 and the vector V¯1k−1 of dimensions p×Np, are defined as follows:(86)D1=Ip0¯p×p⋯0¯p×pIpIp⋱0¯p×p⋮⋮⋱⋮IpIp⋯Ip,V¯1k−1=u¯k−1u¯k−1⋮u¯k−1,and the vectors U¯M, U¯m, δU¯M, and δU¯m of dimension p.Nu are defined as follows:(87)U¯M=u¯MT,…,u¯MTT,U¯M=u¯mT,…,u¯mTT,δU¯M=δu¯MT,…,δu¯MTT,δU¯M=δu¯mT,…,δu¯mTT,and the vectors y¯M and y¯m of dimension m.Np are given by(88)y¯M=y¯MT,…,y¯MTT,y¯m=y¯mT,…,y¯mTT.These constraints define a future control increments admissible set as follows:(89)Ψ=δUk/ΓδUk≤V¯k.Then, the optimization problem subject to the physical constraints on the inputs and the outputs is written as follows:(90)minδU¯k∈ΨJ2δUk,where Ψ is the future control increments admissible set given by (89) and J2δUk is the quadratic criterion that can be written, from (90), as(91)J2δUk=δUTkGaTΛyGa+ΛuδUk+2Y˜fr1k+GaδF^k+GbδY˜k−W¯kTΛyGaδUk+Y˜fr1k+GaδF^k+GbδY˜k−W¯kTΛyY˜fr1k+GaδF^k+GbδY˜k−W¯k.The optimization problem (90) subject to (84) is a convex quadratic programming problem (QP), for which the global solution is unique. As with all predictive control methods, where a moving horizon strategy is applied, at instant kT, only the first optimized control U¯k is applied to the system, with(92)u¯k=u¯k−1+δu¯k.


## 7. Numerical Example

The proposed MIMO ARX-Laguerre multiple models, the fault detection algorithm, and the fault tolerant control via MPC are validated on numerical simulation using the following nonlinear MIMO discrete-time system characterized by two inputsp=2 and two outputs m=2 [14, 15]:(93)ys1k=0.4ys1k−1+u1k−11+u12k−1+0.2u13k−1+0.5u2k−1,ys2k=0.2ys2k−1+u2k−11+u22k−1+0.4u23k−1+0.2u1k−1.The MIMO nonlinear system given by (93) was simulated, and a set of training input-output data were obtained. These inputs and outputs systems are represented, respectively, by Figures 3 and 4.Figure 3
Input signals.Figure 4
Output signals.
(a)
(b)To describe the MIMO nonlinear system by the proposed MIMO ARX-Laguerre multiple models, it is necessary to determine the weighting functionsμsγk defined by relation (3) for L=3 submodels where we use the output systems as the decision variables. The weighting functions μsγk for s=1,2,3 are represented in Figure 5.Figure 5
Weighting functionsμsγk, s = 1, 2, 3.From the input/output couples, we use 800 couples to identify the proposed nonlinear MIMO ARX-Laguerre multiple models where we applied the Algorithm1 to identify the Fourier coefficients and the Algorithm 2 to optimize the poles. After several simulations for different truncation orders Nas and Nbs for s=1,2,3, the optimal choice of the latter is Nas=Nbs=N=2. Then, in our case, the nonlinear MIMO ARX-Laguerre multiple models is obtained by optimizing the m.L.m+p=24 poles ξ¯i, i=1,…,m, given by relation (29) and identifying the m.L.N.m+p=48 Fourier coefficients C¯i, i=1,…,m, defined by relation (27). In the following, we give the optimized values of the poles and the Fourier coefficients (Algorithm 3):(94)C1=0.7643,−0.0075,0.6586,0.0119,0.3399,−0.0028,−0.3532,−0.0014,0.8595,−0.019,0.5649,−0.0813,−0.0198,−0.0039,−0.0406,0.0197,0.7640,−0.0268,0.5633,−0.1051,0.1968,−0.0255,−0.1096,0.0314,C2=0.2262,0.0067,0.8839,−0.0045,−0.1075,−0.0033,0.1704,0.0016,0.2018,−0.0021,0.9055,0.0004,0.0146,−0.0002,−0.0070,−0.0000,0.2082,0.0094,0.9045,0.0102,−0.0187,−0.0093,−0.0087,0.0032,(95)ξ¯1=0.0777,0.5907,0.2871,0.6100,0.4433,0.5557,0.4709,0.5924,0.2050,0.5529,0.4413,0.6274,ξ¯2=0.0777,0.5907,0.2871,0.6100,0.4433,0.5557,0.4709,0.5924,0.2050,0.5529,0.4413,0.6274.Algorithm 1:Recursive parametric identification.
(1)
Assume we haveM inputs/outputs couples u1,…,up and y¯1,…,y¯m
(2)
Choose the number of submodels and fix the truncating ordersNis=N
(3)
Assume that the Laguerre poles vectorsξ¯i, i=1,…,m are optimized
(4)
Calculate the matricesAis and Bis for i=1,…,m and s=1,…,L given by (23) and (25), respectively
(5)
Initialize the parameter vectorsC¯i=0¯N.Lp+m×1, yi1=0 and the state vectors Xis1=0¯N.p+m×1 for i=1,…,m and s=1,…,L
(6)
Forh=1,…,M(a)
CalculateXih for i=1,…,m from relations (28) and (A.17)(b)
Deduce the matrixΨih and form the vector Y¯ih as given in relation (39)(c)
Compute the estimated Fourier coefficients at time instanth, C¯ih, using (41)(d)
h=h+1, go to step (a)Algorithm 2:Pole optimization algorithm.
(1)
Assume we haveM measured input/output system u1,…,up and y¯1,…,y¯m
(2)
Choose the size of populationN
(3)
Fix a stopping criterion
(4)
Fix an initial set ofN pole vectors ξ¯i, i=1,…,m, given by relation (29)
(5)
For each poles vector, identify the parameters vectorC¯i for i=1,…,m, using Algorithm 1
(6)
Compute the outputsyi for i=1,…,m from relation (21)
(7)
Evaluate the fitness values of the initial populationNMSEi for each MISO model, i=1,…,m
(8)
do(a)
Select parents(b)
Crossover(c)
Mutation(d)
Next population(e)
Identify parameter vectors and compute the outputs for each poles vectors(f)
Evaluate the fitness values of the populationNMSEi for every MISO model, i=1,…,m
(9)
Until stopping criteria reached
(10)
The last population is the optimal value of the MIMO ARX-Laguerre multiple models polesAlgorithm 3:MIMO nonlinear fault tolerant control via MPC (MIMO NFTC-MPC).
(1)
The nonlinear system is modeled by the MIMO ARX-Laguerre multiple models forL fixed linear submodels for each MISO system, where the parameters vector C¯i, i=1,…,m, is identified by Algorithm 1 and the pole vector ξ¯i, i=1,…,m, is optimized by Algorithm 2
(2)
Choose the prediction horizonNp, the control horizon Nu, the weights on the tracking error ρ, and the positive weights ϱ
(3)
Determine the positive definite matricesΛy and Λu given by relation (78)
(4)
Choose the physical constraints given by (81)–(83)
(5)
Fix the fault actuators constraintsfm and fM
(6)
Compute the vectorsU¯M, U¯m, ΔU¯M, and ΔU¯m given by (87) and the vectors y¯M and y¯m given by (88)
(7)
Compute these matricesA, Ba, and Bb given by (36) and the matrix C given by (35)
(8)
Calculate the matricesGa, Gb, and D1 given, respectively, by (71), (72), and (86)
(9)
Compute the matrixΓ given by (85)
(10)
Choose the reference signalswik and initialize the input ui1, for i=1,…,m
(11)
fork=2 until M−Np,(a)
Apply the inputsuik−1 to the system and read the output yik, for i=1,…,m(b)
Optimize the fault actuatorsf^k−1 using the MHE method from relation (49)(c)
ComputeδX˜sk and X˜sk given, respectively, by relations (54) and (55a)(d)
Forj=1,…,Np, compute y¯˜fr1k+j/k given by relation (72)(e)
Compute these vectorsY˜fr1k, δY˜k, and δF^k given, respectively, by relations (77), (78), and (80) and compute the vector Wk given by (77)(f)
Compute the vectorV¯k−1 given by (85)(g)
Solve the optimization problem given by (90)(h)
Calculate the control signalu¯k from relation (92)
(12)
ENDUsing the optimal polesξ¯i, i=1,…,m, given by (95), we can deduce the matrices A, Ba, and Bb given by relation (36), and using the identified parameters C¯i, i=1,…,m, given by (94) and the weighting function μsγk for s=1,2,3, we can determine the vector C defined by relation (32). Then, the state space representation (31) of the proposed MIMO ARX-Laguerre multiple models representing the nonlinear MIMO system given by (93) is determined. In Figures 6 and 7, we draw the validation of the identified model.Figure 6
Validation of first output.Figure 7
Validation of second output.The identified MIMO ARX-Laguerre multiple models are used to detect the fault actuator by applying the MHE technique developed in Section5. To validate the capability of the MHE using MIMO ARX-Laguerre multiple models, we applied to the nonlinear MIMO system given by (93), a constant input u1=0.5 and u2=0.6 and we added to the input the fault given in Figure 8. By considering constraint to the faults given by relation (96), the optimal faults are given in Figure 9.(96)0≤f1≤0.25,0≤f2≤0.25.Figure 8
Fault input.Figure 9
Fault optimization.To validate the proposed fault tolerant control algorithm for nonlinear MIMO systems via MPC based on ARX-Laguerre multiple models Algorithm3, we used constant reference trajectories w1k=0.4 and w2k=0.6 for k=1,…,200 and we applied to the numerical model given by relation (93), an input fault at each sample time as given in Figure 10. In Figures 11 and 12, we represent the trajectory references and the system outputs, respectively, for the first output and second output without FTC where we observe the influence of the input faults on the output systems. In Figures 13 and 14, we draw the system outputs and the reference trajectories in the case of FTC via MPC using nonlinear MIMO ARX-Laguerre multiple models, and in Figure 15, we plot the fault optimization.Figure 10
Input faults in the case of FTC.Figure 11
First output without FTC.Figure 12
Second output without FTC.Figure 13
First output in the case of FTC via MPC.Figure 14
Second output in the case of FTC via MPC.Figure 15
Input fault optimization.


### 7.1. Example 2: Communicating Two-Tank System

#### 7.1.1. Description of the Experimental System

The proposed faults tolerant control scheme presented in this paper is tested on a MIMO nonlinear MIMO communicating two-tank system (CTTS). The picture of the MIMO CTTS is given in Figure16, and its block diagram is represented in Figure 17. The CTTS contains two voltage-controlled motor pumps (P1 and P2) which deliver the flow rates Q1 and Q2 inside of the tank 1 and the tank 2, respectively. Each tank is equipped with a level sensor and a valve for draining. This experimental process allows the control of both water levels h1 and h2 in tanks 1 and 2, respectively, (outputs) by tuning water flow rates Q1 and Q2 provided by the motor-pumps P1 and P2 (inputs). The notations used in Figure 17 are given in Table 1. The experimental process shown in Figure 16 operates as follows: the motor-pump P1 fills the tank 1 with a flow rate Q1 and the motor-pump P2 fills the tank 2 with a flow rate Q2. The manual valves V1 and V2 ensure the draining of tanks 1 and 2, respectively, with flow rates Q10 and Q20. Furthermore, if the levels h1 and h2 are different, a draining of tank 1 or 2 with the flow rate Q12 is performed through the interconnection valve VC.Figure 16
Picture of the experimental CTTS.Figure 17
Block diagram of the CTTS.Table 1
Notations used for the CTTS system.
P
1
Motor-pump 1 with variable flow rate between 0 and7l/min at 12V DC
P
2
Motor-pump 2 with variable flow rate between 0 and3l/min at 12V DC
V
1
Manual drain valve of tank 1 of cross sectionS1
V
C
Manual interconnection valve between the two tanks of cross sectionS12
V
2
Manual drain valve of tank 2 of cross sectionS2
V
E
Electromagnetic valve of cross sectionSE
Q
1
Flow rate provided by motor-pump 1
Q
2
Flow rate provided by motor-pump 2
Q
12
Flow rate between tanks 1 and 2
Q
10
Flow rate drain of tank 1
Q
20
Flow rate drain of tank 2
V
3
V
Three-way valve
H
1
Level sensor in the tank 1
H
2
Level sensor in the tank 2The measurement platform and experimental process control represented in Figure18 are characterized by the interface card plugged in the computer slot generating two PWM (pulse width modulation), the DAS-1600 card for the acquisition of water levels h1 and h2 and the control card of the variation of the flow rates Q1 and Q2. This card transfers the PWM signals generated by the interface card to the motor-pumps P2 and P2.Figure 18
Block diagram of measurement and control of CTTS.The nonlinear CCTS system can be described by a nonlinear MIMO model. In this paper, we will model this system by the proposed nonlinear MIMO ARX-Laguerre multiple models. In the following, the identified model is used to validate the MIMO MHE method proposed in Section5 by the estimation of the actuator faults. Finally, we validate the proposed Algorithm 2 “MIMO nonlinear fault tolerant control via MPC” developed based on “nonlinear MIMO ARX-Laguerre multiple models.”


### 7.2. Identification of Nonlinear MIMO ARX-Laguerre Multiple Models

In this subsection, we validate the proposed nonlinear MIMO ARX-Laguerre multiple models to model the CTTS process. In this simulation, we fix the number of submodel toL=3. The process inputs Q1 and Q2 were chosen to be pseudo-random sequence with variable amplitude between 0% and 100% as represented in Figure 19.Figure 19
Process inputsQ1 and Q2.
(a)
(b)To describe the MIMO nonlinear CTTS by the proposed MIMO ARX-Laguerre multiple models, we use the output systems as decision variablesγk. From the input/output couples, we use 2000 couples to identify the proposed nonlinear MIMO ARX-Laguerre multiple models where we applied the Algorithm 1 to identify the Fourier coefficients and the Algorithm 2 to optimize the poles for truncation orders Nas=Nbs=N=2. And, we use last 1000 couples to validate the identified model as represented in Figures 20 and 21.Figure 20
Validation of first output of the CTTS process.Figure 21
Validation of second output of the CTTS process.


### 7.3. Fault Tolerant Control for CTTS via MPC

To control the CTTS using the proposed fault tolerant control algorithm via MPC based on MIMO ARX-Laguerre multiple models Algorithm3, we used the reference trajectories w1 and w2 and we applied to the CTTS process an input fault actuators f1 and f2 at each sample time as given in Figure 22. The weighting functions μsγk, for s=1,2,3, are determined in terms of the reference trajectories. In Figures 23 and 24, we draw the trajectory references and the system outputs, respectively, for the first outputs and second outputs without FTC where we observe the influence of the input faults on the output systems. By considering the constraint to the actuator faults given by relation (97), the optimal actuator faults f1opt and f2opt are given in Figure 22. In Figures 25 and 26, we draw the system outputs and the reference trajectories in the case of FTC via MPC using nonlinear MIMO ARX-Laguerre multiple models. We compare Figures 23–26, and observe the elimination of the effect of the actuator faults on the outputs by using the proposed fault tolerant control algorithm:(97)−10≤f1≤0,−5≤f2≤0.Figure 22
Fault actuatorsf1 and f2 and the optimal values F1opt and F2opt.
(a)
(b)Figure 23
Reference trajectory and first output in the case without FTC.Figure 24
Reference trajectory and second output in the case without FTC.Figure 25
Reference trajectory and first output in the case of FTC via MPC.Figure 26
Reference trajectory and second output in the case of FTC via MPC.


## 7.1. Example 2: Communicating Two-Tank System

### 7.1.1. Description of the Experimental System

The proposed faults tolerant control scheme presented in this paper is tested on a MIMO nonlinear MIMO communicating two-tank system (CTTS). The picture of the MIMO CTTS is given in Figure16, and its block diagram is represented in Figure 17. The CTTS contains two voltage-controlled motor pumps (P1 and P2) which deliver the flow rates Q1 and Q2 inside of the tank 1 and the tank 2, respectively. Each tank is equipped with a level sensor and a valve for draining. This experimental process allows the control of both water levels h1 and h2 in tanks 1 and 2, respectively, (outputs) by tuning water flow rates Q1 and Q2 provided by the motor-pumps P1 and P2 (inputs). The notations used in Figure 17 are given in Table 1. The experimental process shown in Figure 16 operates as follows: the motor-pump P1 fills the tank 1 with a flow rate Q1 and the motor-pump P2 fills the tank 2 with a flow rate Q2. The manual valves V1 and V2 ensure the draining of tanks 1 and 2, respectively, with flow rates Q10 and Q20. Furthermore, if the levels h1 and h2 are different, a draining of tank 1 or 2 with the flow rate Q12 is performed through the interconnection valve VC.Figure 16
Picture of the experimental CTTS.Figure 17
Block diagram of the CTTS.Table 1
Notations used for the CTTS system.
P
1
Motor-pump 1 with variable flow rate between 0 and7l/min at 12V DC
P
2
Motor-pump 2 with variable flow rate between 0 and3l/min at 12V DC
V
1
Manual drain valve of tank 1 of cross sectionS1
V
C
Manual interconnection valve between the two tanks of cross sectionS12
V
2
Manual drain valve of tank 2 of cross sectionS2
V
E
Electromagnetic valve of cross sectionSE
Q
1
Flow rate provided by motor-pump 1
Q
2
Flow rate provided by motor-pump 2
Q
12
Flow rate between tanks 1 and 2
Q
10
Flow rate drain of tank 1
Q
20
Flow rate drain of tank 2
V
3
V
Three-way valve
H
1
Level sensor in the tank 1
H
2
Level sensor in the tank 2The measurement platform and experimental process control represented in Figure18 are characterized by the interface card plugged in the computer slot generating two PWM (pulse width modulation), the DAS-1600 card for the acquisition of water levels h1 and h2 and the control card of the variation of the flow rates Q1 and Q2. This card transfers the PWM signals generated by the interface card to the motor-pumps P2 and P2.Figure 18
Block diagram of measurement and control of CTTS.The nonlinear CCTS system can be described by a nonlinear MIMO model. In this paper, we will model this system by the proposed nonlinear MIMO ARX-Laguerre multiple models. In the following, the identified model is used to validate the MIMO MHE method proposed in Section5 by the estimation of the actuator faults. Finally, we validate the proposed Algorithm 2 “MIMO nonlinear fault tolerant control via MPC” developed based on “nonlinear MIMO ARX-Laguerre multiple models.”


## 7.1.1. Description of the Experimental System

The proposed faults tolerant control scheme presented in this paper is tested on a MIMO nonlinear MIMO communicating two-tank system (CTTS). The picture of the MIMO CTTS is given in Figure16, and its block diagram is represented in Figure 17. The CTTS contains two voltage-controlled motor pumps (P1 and P2) which deliver the flow rates Q1 and Q2 inside of the tank 1 and the tank 2, respectively. Each tank is equipped with a level sensor and a valve for draining. This experimental process allows the control of both water levels h1 and h2 in tanks 1 and 2, respectively, (outputs) by tuning water flow rates Q1 and Q2 provided by the motor-pumps P1 and P2 (inputs). The notations used in Figure 17 are given in Table 1. The experimental process shown in Figure 16 operates as follows: the motor-pump P1 fills the tank 1 with a flow rate Q1 and the motor-pump P2 fills the tank 2 with a flow rate Q2. The manual valves V1 and V2 ensure the draining of tanks 1 and 2, respectively, with flow rates Q10 and Q20. Furthermore, if the levels h1 and h2 are different, a draining of tank 1 or 2 with the flow rate Q12 is performed through the interconnection valve VC.Figure 16
Picture of the experimental CTTS.Figure 17
Block diagram of the CTTS.Table 1
Notations used for the CTTS system.
P
1
Motor-pump 1 with variable flow rate between 0 and7l/min at 12V DC
P
2
Motor-pump 2 with variable flow rate between 0 and3l/min at 12V DC
V
1
Manual drain valve of tank 1 of cross sectionS1
V
C
Manual interconnection valve between the two tanks of cross sectionS12
V
2
Manual drain valve of tank 2 of cross sectionS2
V
E
Electromagnetic valve of cross sectionSE
Q
1
Flow rate provided by motor-pump 1
Q
2
Flow rate provided by motor-pump 2
Q
12
Flow rate between tanks 1 and 2
Q
10
Flow rate drain of tank 1
Q
20
Flow rate drain of tank 2
V
3
V
Three-way valve
H
1
Level sensor in the tank 1
H
2
Level sensor in the tank 2The measurement platform and experimental process control represented in Figure18 are characterized by the interface card plugged in the computer slot generating two PWM (pulse width modulation), the DAS-1600 card for the acquisition of water levels h1 and h2 and the control card of the variation of the flow rates Q1 and Q2. This card transfers the PWM signals generated by the interface card to the motor-pumps P2 and P2.Figure 18
Block diagram of measurement and control of CTTS.The nonlinear CCTS system can be described by a nonlinear MIMO model. In this paper, we will model this system by the proposed nonlinear MIMO ARX-Laguerre multiple models. In the following, the identified model is used to validate the MIMO MHE method proposed in Section5 by the estimation of the actuator faults. Finally, we validate the proposed Algorithm 2 “MIMO nonlinear fault tolerant control via MPC” developed based on “nonlinear MIMO ARX-Laguerre multiple models.”


## 7.2. Identification of Nonlinear MIMO ARX-Laguerre Multiple Models

In this subsection, we validate the proposed nonlinear MIMO ARX-Laguerre multiple models to model the CTTS process. In this simulation, we fix the number of submodel toL=3. The process inputs Q1 and Q2 were chosen to be pseudo-random sequence with variable amplitude between 0% and 100% as represented in Figure 19.Figure 19
Process inputsQ1 and Q2.
(a)
(b)To describe the MIMO nonlinear CTTS by the proposed MIMO ARX-Laguerre multiple models, we use the output systems as decision variablesγk. From the input/output couples, we use 2000 couples to identify the proposed nonlinear MIMO ARX-Laguerre multiple models where we applied the Algorithm 1 to identify the Fourier coefficients and the Algorithm 2 to optimize the poles for truncation orders Nas=Nbs=N=2. And, we use last 1000 couples to validate the identified model as represented in Figures 20 and 21.Figure 20
Validation of first output of the CTTS process.Figure 21
Validation of second output of the CTTS process.


## 7.3. Fault Tolerant Control for CTTS via MPC

To control the CTTS using the proposed fault tolerant control algorithm via MPC based on MIMO ARX-Laguerre multiple models Algorithm3, we used the reference trajectories w1 and w2 and we applied to the CTTS process an input fault actuators f1 and f2 at each sample time as given in Figure 22. The weighting functions μsγk, for s=1,2,3, are determined in terms of the reference trajectories. In Figures 23 and 24, we draw the trajectory references and the system outputs, respectively, for the first outputs and second outputs without FTC where we observe the influence of the input faults on the output systems. By considering the constraint to the actuator faults given by relation (97), the optimal actuator faults f1opt and f2opt are given in Figure 22. In Figures 25 and 26, we draw the system outputs and the reference trajectories in the case of FTC via MPC using nonlinear MIMO ARX-Laguerre multiple models. We compare Figures 23–26, and observe the elimination of the effect of the actuator faults on the outputs by using the proposed fault tolerant control algorithm:(97)−10≤f1≤0,−5≤f2≤0.Figure 22
Fault actuatorsf1 and f2 and the optimal values F1opt and F2opt.
(a)
(b)Figure 23
Reference trajectory and first output in the case without FTC.Figure 24
Reference trajectory and second output in the case without FTC.Figure 25
Reference trajectory and first output in the case of FTC via MPC.Figure 26
Reference trajectory and second output in the case of FTC via MPC.


## 8. Conclusion

In this paper, we have proposed a new MIMO nonlinear fault tolerant via model predictive control to control MIMO nonlinear systems by taking into account fault actuators. The MIMO nonlinear systems are described by a MIMO ARX-Laguerre multiple models. One more important advantage of the proposed MIMO ARX-Laguerre multiple models is that its structure and its reduced parameter number can facilitate the controller design problem and reduce the computational effort of the MIMO NFTC-MPC. The fault actuators estimation is assured by the MHE method with respect to these constraints on these faults. The proposed MIMO NFTC-MPC is synthesized by taking into account the estimation fault actuators. The optimization problem of the MIMO NFTC-MPC is a minimization of a quadratic criterion with respect to constraints on input signals. The control algorithm is easily solved and yields efficient results. The stability of closed-loop system is guaranteed with respect to the choice of the tuning parameters.



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*Source: 9012182-2019-10-03.xml*