Volume 16 Number 3
June 2019
Article Contents
Dalhoumi Latifa, Chtourou Mohamed and Djemel Mohamed. Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems. International Journal of Automation and Computing, vol. 16, no. 3, pp. 369-388, 2019. doi: 10.1007/s11633-016-1021-3
Cite as: Dalhoumi Latifa, Chtourou Mohamed and Djemel Mohamed. Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems. International Journal of Automation and Computing, vol. 16, no. 3, pp. 369-388, 2019. doi: 10.1007/s11633-016-1021-3

Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems

Author Biography:
  • Mohamed Chtourou received the B. Eng. degree in electrical engineering from National School of Engineers of Sfax, Tunisia in 1989, the M. Sc. degree in automatic control from National Institute of Applied Sciences of Toulouse, France in 1990, and the Ph. D. degree in process engineering from the National Polytechnic Institute Toulouse, France in 1993, and was a post-doctoral in automatic control in the National School of Engineers of Sfax, the Habilitation University, Tunisia till 2002. He is currently a professor in the Department of Electrical Engineering of National School of Engineers of Sfax, Tunisia. He is an author and co-author of more than thirty papers in international journals and of more than 50 papers published in national and international conferences.
    His research interests include learning algorithms, artiflcial neural networks and their engineering applications, fuzzy systems, and intelligent control.
    E-mail: Mohamed.chtourou@enis.rnu.tn

    Mohamed Djemel received the B. Sc., M. Sc., and Ph. D. degrees in electrical engineering from the High School of Technical Sciences of Tunis (ESSTT), Tunisia in 1987 and 1989 and 1996, respectively, and was a post-doctoral in the National School of Engineers of Sfax, the Habilitation University, Tunisia till 2006. He joined the Tunisian University since 1990, where he held difierent positions involved in research and education. Currently, he is a professor of automatic control at the Department of Electrical Engineering of the National School of Engineering of Sfax. He is a member of several national and international conferences.
    His research interests include order reduction and the stability, the control and the advanced control of the complex systems.
    E-mail: Mohamed.djemel@enis.rnu.tn

  • Corresponding author: Latifa Dalhoumi received the B. Eng. degree in electrical engineering from National School of Engineering of Sfax (ENIS), Tunisia in 2008, the M. Sc. degree in automatic and industrial computing from the same Institute, Tunisia in 2009. She is currently a Ph. D. candidate in Control and Energy Management Laboratory (CEM-Lab) in the Department of Electrical Engineering of National School of Engineers of Sfax, Tunisia.
    Her research interests include control of complex systems, interconnected linear and nonlinear systems, fuzzy model predictive control, and formulation of fuzzy model predictive control of nonlinear interconnected systems.
    E-mail: dalhoumilatifa@hotmail.com (Corresponding author)
    ORCID iD: 0000-0001-8359-5393
  • Received: 2015-06-27
  • Accepted: 2016-02-01
  • Published Online: 2016-12-05
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Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems

  • Corresponding author: Latifa Dalhoumi received the B. Eng. degree in electrical engineering from National School of Engineering of Sfax (ENIS), Tunisia in 2008, the M. Sc. degree in automatic and industrial computing from the same Institute, Tunisia in 2009. She is currently a Ph. D. candidate in Control and Energy Management Laboratory (CEM-Lab) in the Department of Electrical Engineering of National School of Engineers of Sfax, Tunisia.
    Her research interests include control of complex systems, interconnected linear and nonlinear systems, fuzzy model predictive control, and formulation of fuzzy model predictive control of nonlinear interconnected systems.
    E-mail: dalhoumilatifa@hotmail.com (Corresponding author)
    ORCID iD: 0000-0001-8359-5393

Abstract: This paper proposes fuzzy model predictive control (FMPC) strategies for nonlinear interconnected systems based mainly on a system decomposition approach. First, the Takagi-Sugeno (TS) fuzzy model is formulated in such a way to describe the behavior of the nonlinear system. Based on that description, a fuzzy model predictive control is determined. The system under consideration is decomposed into several subsystems. For each subsystem, the main idea consists of the decomposition of the control action into two parts: The decentralized part contains the parameters of the subsystem and the centralized part contains the elements of other subsystems. According to such decomposition, two strategies are deflned aiming to circumvent the problems caused by interconnection between subsystems. The feasibility and e–ciency of the proposed method are illustrated through numerical examples.

Dalhoumi Latifa, Chtourou Mohamed and Djemel Mohamed. Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems. International Journal of Automation and Computing, vol. 16, no. 3, pp. 369-388, 2019. doi: 10.1007/s11633-016-1021-3
Citation: Dalhoumi Latifa, Chtourou Mohamed and Djemel Mohamed. Decomposition Based Fuzzy Model Predictive Control Approaches for Interconnected Nonlinear Systems. International Journal of Automation and Computing, vol. 16, no. 3, pp. 369-388, 2019. doi: 10.1007/s11633-016-1021-3
  • Model based predictive control (MPC), also known as receding horizon control, is one of the few control techniques that enable to ensure stability and robustness of complex systems. Indeed, the main concept of the MPC strategy is the use of a model to predict the future evolution and to describe the behavior of the system. Therefore, a sequence of future control actions is computed using this model by solving online a numerical optimization problem.

    On one side, the main advantage of the use of the fuzzy control is its ability to handle several practical problems that cannot be adequately solved by classical control methods. Thus, the most important problem of fuzzy control systems is getting a system design with the guarantee of control performance, robustness and stability[1-3]. In these works, a nonlinear plant is approximated by a Takagi-Sugeno (TS) fuzzy model. Most previous control applications of fuzzy logic have been formulated using a set of IF-THEN rules. These rules involve linguistic values and concepts in order to describe the control strategy.

    On the other side, industrial systems are generally composed of a number of subsystems, which are interconnected and characterized by several interactions. Due to their nonlinearity, modern control strategies are becoming more and more complex and difficult to design.

    To overcome this problem, different solutions can be developed. The decentralized control method represents the most used control strategy in industrial processes[4, 5].

    During the last decade, there has been a quickly growing interest in fuzzy control of nonlinear systems such as interconnected systems using the TS fuzzy model[6], and they have been used in many successful applications. Typically, MPC is implemented in a centralized fashion[7]. However, for interconnected systems, it is necessary to introduce a decentralized approach to solve the problem of interaction between subsystems[8, 9].

    Results obtained by the decentralized MPC were compared to those obtained by the classical centralized MPC[10]. The stability of a networked decentralized MPC strategy was studied and developed[7]. In [11], a Lyapunov function is developed to ensure the stability of interconnected systems. In this context, the properties of interconnected systems have been widely studied[12-14] and many different approaches have been proposed to stabilize the interconnected linear systems[15-18].

    Many researchers have been interested to study the decentralized control scheme[19-21]. In [3, 22, 23], a set of model-based fuzzy controllers have been synthesized to stabilize nonlinear interconnected systems. The developed approach in [24] consists of designing each control rule based on its corresponding rule of TS model. The decentralized control for interconnected large-scale systems with time-delay has received more attention[25, 26].

    Decentralized control was also used as an adaptive approach to assure robustness for interconnected systems[6, 27, 28]. In addition, LMI-based procedure has been determined for the design of decentralized dynamic output controllers for systems composed of linear subsystems coupled by uncertain nonlinear interconnections satisfying quadratic constraints[29].

    To the best of our knowledge, the association of the fuzzy models with the predictive control and the interconnected systems has not been considered so far. Thus, our contribution in this work is the development of fuzzy model based predictive control strategies fuzzy model predictive control (FMPC) for interconnected nonlinear systems using two proposed assumptions to develop the cost function.

    The remainder of the paper is organized as follows. In Section 2, the problem of fuzzy model associated to nonlinear interconnected systems is presented. In Section 3, the proposed FMPC strategies are detailed. In Section 4, numerical results are provided to demonstrate the effectiveness of our proposed strategies.

    Two examples have been studied: The first one is based on two-machines interconnected system while the second deals with two inverted pendulums connected by a spring mounted on two carts. Finally, a conclusion has been presented.

    Notations. The following notations will be used throughout this paper. $A_{i}^{\rm T}$ and $A_{i}^{-1}$ denote, respectively, the transpose and the inverse of any square matrix $A_{i}$ where this matrix is positive define $A_{i}>0$.

    ${\bf {R}}^{n_{i}}$ denotes the $n_{i}$-dimensional Euclidian space.

    $x_{i}(k) \in {\bf {R}}^n_{i}$, $u_{i}(k) \in {\bf {R}}^m_{i}$ and $y_{i}(k) \in {\bf {R}}^p_{i}$ are the state vector, the control vector and the output vector, respectively, of the $i$-th subsystem, where $i$=1, $\cdots$, $N$ and $N$ is the number of subsystems.

    $A_{i}$, $B_{i}$ and $C_{i}$ mean, the state matrix, the control matrix and the output matrix, respectively, of each subsystem.

    $A_{ij}$ represents the interconnection matrix between the $i$-th and $j$-th subsystems.

    $A_{i_{r}}$ denotes the matrix associated to the $r$-th rule for the $i$-th subsystem, where $r=1, \cdots, R^{i}$: $R^{i}$ indicates the number of rules for each subsystem.

    Finally, $\|\cdot\|$ denotes the Euclidean norm or its induced norm. $I$ is the identity matrix.

  • Consider a nonlinear interconnected system $S$ which is composed of $N$ nonlinear subsystems ($ S_{i}, i=1, \cdots, N$) represented by (1).

    $ \begin{equation}\label{1} ~~S_{i}=\left\{ \begin{array}{ll} x_{i}(k+1)=F_{i}\Big(~x_{i}(k), x_{j}(k), u_{i}(k), u_{j}(k), w_{i}(k)~\Big) \\ y_{i}(k)= G_{i}\Big(~x_{i}(k), x_{j}(k), v_{i}(k)~\Big) \end{array} \right. \end{equation} $

    (1)

    where:

    $x_{l}(k)$ is the state vector of the $l$-th subsystem at time $k$ and $l=i, j$.

    $u_{l}(k)$ is the control signal of the $l$-th subsystem at time $k$ and $l=i, j$.

    $y_{i}(k)$ is the output vector of the $i$-th subsystem at time $k$.

    $w_{i}(k)$ is the external disturbance of the $i$-th subsystem at time $k$.

    $v_{i}(k)$ is the measurement noise at time $k$.

    $F_{i}$ and $G_{i}$ are two nonlinear function vectors.

    Each subsystem ($S_{i}$) will be described by a fuzzy Takagi-Sugeno model[30, 31], where the conclusion part of each fuzzy rule is composed of a local linear state model. Based on this fuzzy model, a fuzzy predictive control strategy will be developed.

    The $r$-th rule, for $r=1, \cdots, R^{i}$, of the fuzzy model attributed to the subsystem ($S_{i}$) has the following form.

    Rule $r$:

    IF $z_{i_{1}}(k)$ is $M^{r}_{i_{1}}$ and $\cdots$ and $z_{i_{g}}(k)$ is $M^{r}_{i_{g}}$

    THEN

    $ \begin{align*} \left\{ \begin{array}{lcccl} x_{i_{r}}(k+1)&=A_{i_{r}}x_{i_{r}}(k)+B_{i_{r}}u_{i_{r}}(k)+\\ &\quad\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij_{r}}x_{j_{r}}(k)+w_{i_{r}}(k)\\ y_{i_{r}}(k)&=C_{i_{r}}x_{i_{r}}(k)+v_{i_{r}}(k)~~~~~~ \end{array} \right. \end{align*} $

    (2)

    where:

    $M^{r}_{i_{1}}, \cdots, M^{r}_{i_{g}}$ are the fuzzy sets associated to inputs $z_{i}=[z_{i_{1}}(k), \cdots, z_{i_{g}}(k)]^{\rm T}$ of the fuzzy model.

    $A_{ir}$, $B_{ir}$ and $C_{ir}$ are matrices assumed to be known for every subsystem for the $r$-th rule.

    $A_{ijr}$ denotes the interconnection between the $i$-th and $j$-th subsystems for the $r$-th rule.

    $R_{i}$ is the number of rules of each subsystem.

    The overall fuzzy model of subsystem ($S_{i}$) can be deduced as

    $ \begin{equation}\label{3} ~~~~\left\{ \begin{array}{lcccl} x_{i}(k+1)=\frac{\textstyle 1}{\textstyle\sum \limits_{r=1}^{R^{i}}\alpha_{r}(z_{i}(k) )}\sum \limits _{r=1}^{R^{i}}\alpha_{r}(z_{i}(k))\bigg (A_{i_{r}}x_{i_{r}}(k)+~~~~~~~~~~~\\ ~~~~~~~~~~~~~~~~B_{i_{r}}u_{i_{r}}(k)+\sum\limits_{\underset{j \neq i}{j=1}}^{N}A_{ij_{r}}x_{j_{r}}(k)+w_{i_{r}}(k)\bigg )\\ y_{i}(k)=\frac{\textstyle \sum \limits _{r=1}^{R^{i}}\alpha_{r}(z_{i}(k))\bigg ( C_{i_{r}}x_{i_{r}}(k)+v_{i_{r}}(k)\bigg )}{\textstyle\sum \limits_{r=1}^{R^{i}}\alpha_{r}(z_{i}(k) )}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{array} \right. \end{equation} $

    (3)

    where

    $ \begin{eqnarray}\label{951} \alpha_{r}(z_{i}(k)) &=& \prod _{h=1}^{g}M^{r}_{i_{h}}(z_{i_{h}}(k)) \end{eqnarray} $

    (4)

    $M^{r}_{i_{h}}(z_{i_{h}}(k))$ is the grade of membership of $z_{i_{h}}(k)$ to $M^{r}_{i_{h}}$. Suppose that

    $ \begin{eqnarray*} \nonumber \beta_{r}(z_{i}(k)) &=& \frac{\textstyle \alpha_{r}(z_{i}(k))}{\sum \limits_{r=1}^{R^{i}}\alpha_{r}(z_{i}(k))}, \, r=1, 2, \cdots, R^{i}. \end{eqnarray*} $

    So, (3) becomes

    $ \begin{equation*}\label{4} ~~~~~~\left\{ \begin{array}{lcccl} x_{i}(k+1)=\sum \limits _{r=1}^{R^{i}}\beta_{r}(z_{i}(k))\bigg (A_{i_{r}}x_{i_{r}}(k)+\\ ~~~~~~~~~~~~~~~~B_{i_{r}}u_{i_{r}}(k)+\sum\limits_{\underset{j \neq i}{j=1}}^{N}A_{ij_{r}}x_{j_{r}}(k)+w_{i_{r}}(k)\bigg )\\ y_{i}(k)=\sum \limits _{r=1}^{R^{i}}\beta_{r}(z_{i}(k))\bigg ( C_{i_{r}}x_{i_{r}}(k)+v_{i_{r}}(k)\bigg ). \end{array} \right. \end{equation*} $

    (5)
  • In MPC, also called receding horizon control, the control input is obtained by solving a discrete-time optimal control problem over a given horizon, producing an optimal open-loop control input sequence. The first control action of this sequence is applied. At the next sampling instant, a new optimal control problem is formulated and solved based on the new measurements.

    In the last research works on fuzzy control, the parallel distributed compensation (PDC) is adopted to associate a fuzzy control to each model rule of subsystem ($S_{i}$), where the conclusion is a predictive linear control law. To elaborate this later, a cost function is associated to each subsystem.

    In this context, two methods will be investigated to formulate the decomposition based fuzzy model predictive control.

    It is to be noted that in the two approaches of FMPC, local criteria associated to each subsystem are the bases of the computation of the control sequences.

    Thus, sub-optimal solutions are developed.

  • The resolution of MPC problem is based on the minimization of a cost function associated to a local model of subsystem ($S_{i}$). To simplify all the expressions, the problem will be formulated and presented for a local linear model associated to each subsystem. Considering this assumption, the cost function is given by the following expression:

    $ \begin{align}\label{222} %\begin{array}{lccc} &J_{i}(k)=\sum \limits_{k_{j}=h_{i}^{i}}^{h_{p}^{i}}\parallel{\hat{y_{i}}(k+k_{j})-y^{d}_{i}(k+k_{j})}\parallel_{Q_{i}}+\nonumber\\ &\qquad \quad \lambda^{i}\sum \limits_{k_{j}=0}^{h_{c}^{i}-1}\parallel\Delta u_{i}(k+k_{j})\parallel_{R_{i}} % \end{array} \end{align} $

    (6)

    where:

    $h_{p}^{i}$ is the prediction horizon.

    $h_{c}^{i}$ is the control horizon.

    $h_{i}^{i}$ is the initial horizon.

    $\lambda^{i}$ is the weighting factor.

    $\hat{y}_{i}(k+k_{j})$ is the $j$-th step of predicted output.

    ${y}_{i}^{d}(k+k_{j})$ is the $j$-th step of desired output.

    $\Delta u_{i}(k+k_{j})$ is the $j$-th step of the control increment at the instant $k+k_{j}$.

    $Q_{i}$ is a symmetric and positive definite weighting matrix.

    $R_{i}$ is a symmetric and positive definite weighting matrix.

    $\parallel \textbf{X}\parallel_{\Lambda}^{2}=\textbf{X}^{\rm T}\Lambda \textbf{X}$.

    The basic idea is to decompose the control action into two terms. Denote with $u_{ic}(k)$ and $u_{id}(k)$, the centralized and decentralized part of the control input for the $i$-th subsystem respectively, the control input is given by (7).

    $ \begin{equation}\label{14} u_{i}(k)=u_{id}(k)+u_{ic}(k). \end{equation} $

    (7)

    Based on (7), it is clear that

    $ \begin{equation}\label{104} \Delta u_{i}(k)=\Delta u_{id}(k)+\Delta u_{ic}(k). \end{equation} $

    (8)

    Due to the complexity to resolve the problem given by (6), a strategy has to be formulated where the prediction horizon has to be chosen equal to three ($h_{p}=3$).

    The problem of MPC will be determined for a local linear model associated to each subsystem. The strategy is undertaken without the index "$r$" to simplify the process and will be generalized to the formulation of fuzzy model predictive control by considering a nonlinear interconnected system. For $k=1, \cdots, h_{p}^{i} $, the expression of the state for the local model of the $i$-th subsystem is given by the relation:

    $ \begin{equation}\label{18} x_{i}(k)=x_{id}(k)+x_{ic}(k) \end{equation} $

    (9)

    where $x_{id}(k)$ depends only on local variables (the state $x_{i}(k)$ and the input $u_{i}(k)$) of the local model of subsystem ($S_{i}$), it is denoted by decentralized part of the state. $x_{ic}(k)$ depends on the states $x_{j}(k)$ and the inputs $u_{j}(k)$ of the subsystems ($S_{j}$), and is denoted by centralized part of the state.

    For each subsystem $(S_{i})$, suppose that $j\neq i$.

    To simplify the calculation, suppose that the external disturbance and the measurement noise of the $i$-th subsystem are zeros at time $k$, ($w_{i}(k)=0$ and $v_{i}(k)=0$).

    In another way, suppose using a perfect predictor where

    $ \begin{equation} \nonumber \hat{x}_{i}(k)=x_{i}(k). \end{equation} $

    So,

    $ \begin{equation}\label{9876} \begin{array}{lcccl} \hat{x}_{i}(k+1)&=&\underbrace{A_{i}x_{i}(k)+B_{i}u_{i}(k)}_{\hat{x}_{id}(k+1)}+\underbrace{\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij}x_{j}(k)}_{\hat{x}_{ic}(k+1)}. \end{array} \end{equation} $

    (10)

    Using (10), the matrix form is as follows:

    $ \begin{equation}\label{19} \begin{array}{lcccl} \hat{X}_{i}(k)=&\underbrace{A_{id}\hat{X}_{d}(k)+B_{id}u_{i}(k-1)+E_{id}\Delta U_{id}(k)}_{\hat{X}_{id}(k)}+\\ &\underbrace{A_{ic}\hat{X}_{c}(k)+B_{ic}u_{j}(k-1)+E_{ic}\Delta U_{ic}(k)}_{\hat{X}_{ic}(k)} \end{array} \end{equation} $

    (11)

    where

    $ \begin{equation} \label{420} \hat{X}_{i}(k)= \left( \begin{array}{c} \hat{x}_{i}(k+1) \\ \hat{x}_{i}(k+2) \\ \hat{x}_{i}(k+3) \\ \end{array} \right) \end{equation} $

    (12)

    $ {\hat X_d}(k) = \left( {\begin{array}{*{20}{c}} {{{\hat x}_i}(k)}\\ {{{\hat x}_i}(k)}\\ {{{\hat x}_i}(k)} \end{array}} \right), {\hat X_c}(k) = \left( {\begin{array}{*{20}{c}} {{{\hat x}_j}(k)}\\ {{{\hat x}_j}(k)}\\ {{{\hat x}_j}(k)} \end{array}} \right) $

    (13)

    $ \Delta U_{id}(k) = \left( \begin{array}{c} \Delta u_{i}(k) \\ \Delta u_{i}(k+1) \\ \Delta u_{i}(k+2) \end{array} \right) \\ \Delta U_{ic}(k) = \left( \begin{array}{c} \Delta u_{j}(k) \\ \Delta u_{j}(k+1) \\ \Delta u_{j}(k+2) \end{array} \right) $

    (14)

    $ \begin{equation}\label{741} A_{id} = \left( \begin{array}{c} A_{i} \\ A_{i}^{2}+\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij}A_{ji} O_{i} \\ \end{array} \right) \end{equation} $

    (15)

    $ \begin{align}\label{80} %\begin{array}{lcccl} & O_{i}=A_{i}^{3}+\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{i}A_{ij}A_{ji}+\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij} \Big (A_{j}A_{ji}+A_{ji}A_{i} \Big )+\nonumber\\ & ~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq j \neq i}{l=1}}^{N}A_{ij}A_{jl}A_{li} % \end{array} \end{align} $

    (16)

    $ \begin{equation} \label{209} B_{id}= \left( \begin{array}{c} B_{i} \\ A_{i}B_{i}+B_{i} \\ T_{i} \\ \end{array} \right) \end{equation} $

    (17)

    $ \begin{equation} \label{720} E_{id} = \left( \begin{array}{ccc} B_{i} & 0 & 0 \\ A_{i}B_{i}+B_{i} & B_{i} & 0 \\ T_{i} & A_{i}B_{i}+B_{i} & B_{i} \\ \end{array} \right) \end{equation} $

    (18)

    $ \begin{equation}\label{81} T_{i}=\sum \limits_{m=0}^{h_{p}^{i}-1}A_{i}^{m}B_{i}+\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij}A_{ji}B_{i} \end{equation} $

    (19)

    $ \begin{equation} \label{820} A_{ic} = \left( \begin{array}{c} \overset{N}{\underset{j=1, i\neq j}{\sum}}A_{ij} \\ V_{j} \\ W_{j} \\ \end{array} \right) \end{equation} $

    (20)

    $ \begin{equation}\label{82} V_{j}=A_{i}\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij}+\sum \limits_{\underset{j \neq i}{j=1}}^{N}A_{ij}A_{j} +\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq j \neq i}{l=1}}^{N}A_{il}A_{lj} \end{equation} $

    (21)

    $ \begin{align}\label{83} %\begin{array}{lcccl} & ~~W_{j}=\sum \limits_{\underset{j \neq i}{j=1}}^{N} \Big ( A_{ij}A_{j}^{2}+A_{i}^{2}A_{ij}+A_{i}A_{ij}A_{j}\Big )+\nonumber\\ & ~~~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq i}{l=1}}^{N}A_{il}A_{li}A_{ij} +A_{i}\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq j \neq i}{l=1}}^{N}A_{il}A_{lj}+\nonumber\\ & ~~~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq j \neq i}{l=1}}^{N}\Bigg (A_{il} \Big ( A_{lj}A_{j}+A_{l}A_{lj} \Big)+A_{ij}A_{jl}A_{lj} \Bigg)+\nonumber\\ & ~~~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{N}\sum \limits_{\underset{l \neq j \neq i}{l=1}}^{N}\sum \limits_{\underset{q \neq l \neq j \neq i}{q=1}}^{N}A_{il}A_{lq}A_{qj} % \end{array} \end{align} $

    (22)

    $ \begin{equation} \label{1120} B_{ic}= \left( \begin{array}{c} 0 \\ \overset{N}{\underset{i=1, j\neq i}{\sum}}A_{ij}B_{j} \\ T_{j} \\ \end{array} \right) \end{equation} $

    (23)

    $ \begin{equation}\label{2120} E_{ic} = \left( \begin{array}{ccc} 0 & 0 & 0 \\ \overset{N}{\underset{i=1, j\neq i}{\sum}}A_{ij}B_{j} & 0 & 0 \\ T_{j} & \overset{N}{\underset{i=1, j\neq i}{\sum}}A_{ij}B_{j} & 0 \\ \end{array} \right) \end{equation} $

    (24)

    $ \begin{matrix} {{T}_{j}}=\sum\limits_{\underset{j\ne i}{\mathop{j=1}}\,}^{N}{\sum\limits_{\underset{l\ne j\ne i}{\mathop{l=1}}\,}^{N}{{{A}_{il}}}}{{A}_{lj}}{{B}_{j}}+\sum\limits_{\underset{j\ne i}{\mathop{j=1}}\,}^{N}{{{A}_{ij}}}{{A}_{j}}{{B}_{j}}+ \\ ~\sum\limits_{\underset{j\ne i}{\mathop{j=1}}\,}^{N}{{{A}_{i}}}{{A}_{ij}}{{B}_{j}}+\sum\limits_{\underset{j\ne i}{\mathop{j=1}}\,}^{N}{{{A}_{ij}}}{{B}_{j}}. \\ \end{matrix} $

    (25)

    For each subsystem ($S_{i}$), we suppose to use the following notations:

    $ \begin{array}{l} {u_i}(k - 1) = {u_{id}}(k - 1)\\ {u_j}(k - 1) = {u_{ic}}(k - 1). \end{array} $

    Based on (11), the matrix form is as follows:

    $ \begin{equation}\label{21} \hat{X}_{i}(k)=\hat{X}_{id}(k)+\hat{X}_{ic}(k). \end{equation} $

    (26)

    So

    $ \begin{equation}\label{421} \hat{Y}_{i}(k)=\hat{Y}_{id}(k)+\hat{Y}_{ic}(k) \end{equation} $

    (27)

    where

    $ \hat{X}_{i*}(k)=A_{i*}\hat{X_{*}}(k)+B_{i*}u_{i*}(k-1)+E_{i*}\Delta U_{i*}(k) $

    (28)

    and

    $ \begin{equation}\label{721} \hat{Y}_{i*}(k)=C_{i}\hat{X}_{i*}(k). \end{equation} $

    (29)

    The index "$*$" can be "$c$" or "$d$".

    Based on the same idea, the criterion will be decomposed into two parts namely, respectively, $J_{ic}(k)$ the centralized part and $J_{id}(k)$ the decentralized one corresponding to the $i$-th subsystem. In this context, two criteria are defined

    $ \begin{equation}\label{22} J_{i*}(k)=\|\hat{Y}_{i*}(k)-Y^{d}_{i}(k)\|^{2}_{Q_{i*}}+\|\Delta U_{i*}(k)\|^{2}_{R_{i*}}. \end{equation} $

    (30)

    Define the following functions:

    $ \begin{equation}\label{40} \varphi _{i*}(k) = Y^{d}_{i}(k)-C_{i}A_{i*}\hat{X}_{*}(k)-C_{i}B_{i*}u_{i}(k-1). \end{equation} $

    (31)

    So, the defined criteria will be expressed as follows:

    $ \begin{align}\label{2002} & J_{i*}(k)=\|C_{i}E_{i*}\Delta U_{i*}(k)-\varphi _{i*}(k)\|^{2}_{Q_{i*}}+\nonumber\\ & ~~~~~~~~~~~~\|\Delta U_{i*}(k)\|^{2}_{R_{i*}}. \end{align} $

    (32)

    The development of (32) using (57) gives

    $ \begin{align}\label{42} & J_{i*}(k)=\Delta U_{i*}^{\rm T}(k)\Phi_{i*}\Delta U_{i*}(k)-\nonumber\\ & ~~~~~~~~~~~~\Delta U_{i*}^{\rm T}(k)\Psi_{i*}(k)+\Omega_{i*}(k) \end{align} $

    (33)

    where

    $ \begin{equation}\label{883} \Phi_{i*} = \delta_{i*}^{\rm T} Q_{i*}\delta_{i*} + R_{i*} \end{equation} $

    (34)

    $ \begin{equation}\label{182} \Psi_{i*}(k) = 2\delta_{i*}^{\rm T} Q_{i*}\varphi _{i*}(k) \end{equation} $

    (35)

    $ \begin{equation}\label{882} \delta_{i*} = C_{i}E_{i*} \end{equation} $

    (36)

    $ \begin{equation}\label{282} \Omega_{i*}(k)=\varphi _{i*}^{\rm T}(k)Q_{i*}\varphi _{i*}(k). \end{equation} $

    (37)

    The sub-optimal sequences of control action increments are obtained by setting the criteria to zero

    $ \begin{equation}\label{equa34} \frac{\partial J_{i*}(k)}{\partial\Delta U_{i*}(k)}=0. \end{equation} $

    (38)

    It is clear that

    $ \begin{eqnarray} \nonumber \frac{\partial \Omega_{i*}}{\partial\Delta U_{i*}(k)}=0. \end{eqnarray} $

    Then, the optimal sequences of control action increments are given by the following equation

    $ \begin{equation}\label{equa735} \Delta U_{i*}(k)=\frac{1}{2}\Phi_{i*}^{-1}\Psi_{i*}(k). \end{equation} $

    (39)

    Only the first element of the optimal sequence given by (39), denoted as $\Delta u_{i*}(k)$, is applied to the considered subsystem. So,

    $ \begin{equation}\label{equa1754} \Delta u_{i*}(k)=[1, \underbrace{0, \cdots, 0}_{h_{c}^{i}-1}]\Delta U_{i*}(k). \end{equation} $

    (40)

    The control action to be applied to the local model of subsystem $S_{i}$ is expressed by (7), where $u_{ic}(k)$ and $u_{id}(k)$ are given by (41).

    $ \begin{equation}\label{equa745} u_{i*}(k)=\Delta u_{i*}(k)+u_{i*}(k-1). \end{equation} $

    (41)
  • The second approach is based on the same idea as the first one (decomposition).

    In this case, same as in the first method, suppose that the criterion based on the centralized predicted output $\hat{Y}_{ic}(k)$ is noted the centralized criterion part namely $J_{ic}(k)$, while the criterion based on the decentralized predicted output $\hat{Y}_{id}(k)$ is noted the decentralized one namely $J_{id}(k)$. These criteria are associated to the $i$-th subsystem. Note that the predictor output is the same one formulated using the first approach (see (10) to (29)).

    The first method uses the decomposition approach of the state and so it involves the formulation of two cost functions. But the second method starts with the global state and so the formulation of a global cost function, is given by the following equation

    $ \begin{equation}\label{Jicd} J_{i}(k)=\|\hat{Y}_{i}(k)-Y^{d}_{i}(k)\|^{2}_{{Q}_{i}}+\|\Delta U_{i}(k)\|^{2}_{{R}_{i}}. \end{equation} $

    (42)

    The development of (42) gives

    $ \begin{align}\label{Jicddeveloped} %\begin{array}{lcccl} & J_{i}(k)=\hat{Y}_{i}(k)^{\rm T}Q_{i}\hat{Y}_{i}(k)-2\hat{Y}^{\rm T}_{i}(k)Q_{i}Y^{d}(k)+\nonumber\\ & ~~~~~~~~~~~ Y^{d^{\rm T}}(k)Q_{i}Y^{d}(k)+\Delta U_{i}^{\rm T}(k)R_{i}\Delta U_{i}(k). % \end{array} \end{align} $

    (43)

    Based on (7) and (27), (42) becomes

    $ \begin{align}\label{Jideveloped1} %\begin{array}{lcccl} & J_{i}(k)=\|\{\hat{Y}_{ic}(k)+\hat{Y}_{id}(k)\}-Y^{d}(k)\|^{2}_{Q_{i}}+\nonumber\\ & ~~~~~~~~~~~\|\Delta U_{ic}(k)+\Delta U_{id}(k)\|^{2}_{{R}_{i}}. %\end{array} \end{align} $

    (44)

    In the other case, based on (7), (27) and (43)

    $ \begin{equation}\label{Jideveloped2} \begin{array} {lcccl} J_{i}(k)=\{\hat{Y}_{ic}^{\rm T}(k)+\hat{Y}_{id}^{\rm T}(k)\}Q_{i}\{\hat{Y}_{ic}(k)+\hat{Y}_{id}(k)\}-\\ ~~~~~~~~~~2\{\hat{Y}_{ic}^{\rm T}(k)+\hat{Y}_{id}^{\rm T}(k)\}Q_{i}Y^{d}(k)+{Y^{d}}^{\rm T}(k)Q_{i}Y^{d}(k)+\\ ~~~~~~~~~~\{\Delta U_{ic}^{\rm T}(k)+\Delta U_{id}^{\rm T}(k)\}R_{i}\{\Delta U_{ic}(k)+\Delta U_{id}(k)\}. \end{array} \end{equation} $

    (45)

    The development of (45) gives

    $ \begin{align} %\begin{array} {lcccl} &J_{i}=\hat{Y}_{ic}^{\rm T}Q_{i}\hat{Y}_{ic}-2\hat{Y}_{ic}^{\rm T}Q_{i}Y^{d}+{Y^{d}}^{\rm T}Q_{i}Y^{d}+\nonumber\\ &~~~~~~\Delta U_{ic}^{\rm T}R_{i}\Delta U_{ic}+\hat{Y}_{id}^{\rm T}Q_{i}\hat{Y}_{id}-2\hat{Y}_{id}^{\rm T}Q_{i}Y^{d}+\nonumber\\ &~~~~~~{Y^{d}}^{\rm T}Q_{i}Y^{d}+\Delta U_{id}^{\rm T}R_{i}\Delta U_{id}+2\Delta U_{ic}^{\rm T}R_{i}\Delta U_{id}+\nonumber\\ &~~~~~~2\hat{Y}_{ic}^{\rm T}Q_{i}Y_{id}-{Y^{d}}^{\rm T}Q_{i}Y^{d}. % \end{array} \end{align} $

    (46)

    Define three new functions as follows:

    $ \begin{eqnarray}\label{treenewfunction1} \begin{array} {lcccl} J_{i*}(k)&=&\|\hat{Y}_{i*}(k)-Y^{d}(k)\|^{2}_{\mathbb{Q}_{i*}}+\|\Delta U_{i*}(k)\|^{2}_{{\bf R}_{i*}} \end{array} \end{eqnarray} $

    (47)

    and

    $ \begin{align}\label{treenewfunction3} & J_{i0}(k)=2\Delta U_{ic}(k)^{\rm T}R_{i}\Delta U_{id}(k)+\nonumber\\ & ~~~~~~~~~~~ 2\hat{Y}_{ic}(k)^{\rm T}Q_{i}Y_{id}(k)-{Y^{d}}(k)^{\rm T}Q_{i}Y^{d}(k) \end{align} $

    (48)

    so

    $ \begin{eqnarray}\label{treenewfunction} J_{i}(k)=J_{ic}(k)+J_{id}(k)+J_{icd}(k). \end{eqnarray} $

    (49)

    The development of (47) and (48) gives

    $ \begin{align}\label{treenewfunction11} %\begin{array} {lcccl} & J_{i*}(k)=\hat{Y}_{i*}^{\rm T}(k)Q_{i*}\hat{Y}_{i*}(k)-2\hat{Y}^{\rm T}_{ic}(k)Q_{i*}Y^{d}(k)+\nonumber\\ & ~~~~~~~~~~~~ Y^{d^{\rm T}}(k)Q_{i*}Y^{d}(k)+\Delta U_{i*}^{\rm T}(k)R_{i*}\Delta U_{i*}(k) % \end{array} \end{align} $

    (50)

    and

    $ \begin{align}\label{treenewfunction33} %\begin{array} {lcccl} & J_{i0}(k)=2\Delta U_{ic}^{\rm T}(k)R_{i}\Delta U_{id}(k)+\nonumber\\ & ~~~~~~~~~~~~2\hat{Y}_{ic}^{\rm T}(k)Q_{i}Y_{id}(k)-{Y^{d}}^{\rm T}(k)Q_{i}Y^{d}(k). % \end{array} \end{align} $

    (51)

    It is clear that

    $ \begin{equation}\label{fi} \begin{array} {lcccl} \frac{\textstyle\partial J_{ic}(k)}{\textstyle\partial \Delta U_{id}(k)} &=& 0 \end{array} \end{equation} $

    (52)

    and

    $ \begin{equation}\label{fi2} \begin{array} {lcccl} \frac{\textstyle\partial J_{id}(k)}{\textstyle\partial \Delta U_{ic}(k)} &=& 0.\ \end{array} \end{equation} $

    (53)

    The control vector is obtained by setting the derivative of the criterion to zero:

    $ \begin{equation*} \frac{\partial J_{i}(k)}{\partial\Delta U_{ic}(k)}={\frac{\partial J_{ic}(k)}{\partial\Delta U_{ic}(k)}}+{\frac{\partial J_{i0}(k)}{\partial\Delta U_{ic}(k)}}=0 \Rightarrow \Delta U_{ic}(k) \end{equation*} $

    (54)

    and

    $ \begin{equation*} \frac{\partial J_{i}(k)}{\partial\Delta U_{id}(k)}={\frac{\partial J_{id}(k)}{\partial\Delta U_{id}(k)}}+{\frac{\partial J_{i0}(k)}{\partial\Delta U_{id}(k)}}=0 \Rightarrow \Delta U_{id}(k). \end{equation*} $

    (55)

    Define the following functions:

    $ \begin{equation*} \frac{\partial J_{i}(k)}{\partial\Delta U_{id}(k)}={\frac{\partial J_{id}(k)}{\partial\Delta U_{id}(k)}}+{\frac{\partial J_{i0}(k)}{\partial\Delta U_{id}(k)}}=0 \Rightarrow \Delta U_{id}(k). \end{equation*} $

    (56)

    where

    $ \begin{array}{l} {u_c}(k - 1) = {u_j}(k - 1)\\ {u_d}(k - 1) = {u_i}(k - 1). \end{array} $

    So,

    $ \begin{array}{l} {J_{ic}}(k) = \Delta U_{ic}^{\rm{T}}(k){\Phi _{ic}}\Delta {U_{ic}}(k) - \Delta U_{ic}^{\rm{T}}(k){\Psi _{ic}} + {\Omega _{ic}}\\ {J_{id}}(k) = \Delta U_{id}^{\rm{T}}(k){\Phi _{id}}\Delta {U_{id}}(k) - \Delta U_{id}^{\rm{T}}(k){\Psi _{id}} + {\Omega _{id}}\\ {J_{i0}}(k) = \Delta U_{ic}^{\rm{T}}(k){\Phi _{i0}}\Delta {U_{id}}(k) + \\ \Delta U_{ic}^{\rm{T}}(k){\Psi _{i0}} + {\Omega _{i0}}\Delta {U_{id}}(k) + {\Upsilon _{i0}}(k) \end{array} $

    (57)

    where

    $ \begin{eqnarray} \nonumber \Phi_{ic}= E_{ic}^{\rm T}C_{i}^{\rm T} Q_{ic}C_{i}E_{ic} + R_{ic}\\ \nonumber \Psi_{ic}(k)= 2E_{ic}^{\rm T}C_{i}^{\rm T} Q_{ic}\varphi _{ic}(k)~~\\ \Omega_{ic}(k)=\varphi _{ic}^{\rm T}(k)Q_{ic}\varphi _{ic}(k)~~~~ \end{eqnarray} $

    (58)

    and

    $ \begin{eqnarray} \nonumber \Phi_{id}= E_{id}^{\rm T}C_{i}^{\rm T} Q_{id}C_{i}E_{id} + R_{id}\\ \nonumber \Psi_{id}(k) =2E_{id}^{\rm T}C_{i}^{\rm T} Q_{id}\varphi _{id}(k)~~\\ \Omega_{id}(k)=\varphi _{id}^{\rm T}(k)Q_{id}\varphi _{id}(k)~~~~ \end{eqnarray} $

    (59)

    and

    $ \begin{eqnarray} \nonumber \Phi_{i0}= 2E_{ic}^{\rm T}C_{i}^{\rm T}Q_{i}C_{i}E_{id} + 2R_{i}~~~~~~~~~~~~~~~~~~~~~\\ \nonumber \Psi_{i0}(k)= 2E_{ic}^{\rm T}C_{i}^{\rm T} Q_{i}\varphi _{idd}(k)~~~~~~~~~~~~~~~~~~~~~~~\, \\ \nonumber \Omega_{i0}(k)=2\varphi _{icc}(k)^{\rm T} Q_{i}C_{i}E_{id}~~~~~~~~~~~~~~~~~~~~~~~\\ \Upsilon_{i0}(k)=2\varphi _{icc}(k)^{\rm T} Q_{i}\varphi _{idd}-Y^{d^{\rm T}}(k)Q_{i}Y^{d}(k). \end{eqnarray} $

    (60)

    The resolution of (54) and (55) gives

    $ \begin{align}\label{equa1134} % \begin{array} {lcccl} &\frac{\textstyle \partial J_{ic}(k)}{\textstyle \partial\Delta U_{ic}(k)}=\textstyle 2\Phi_{ic}\Delta U_{ic}(k)-\Psi_{ic}\nonumber\\ &\frac{\textstyle \partial J_{i0}(k)}{\textstyle \partial\Delta U_{ic}(k)}=\Phi_{i0}\Delta U_{id}(k)+\Psi_{i0}. %\hfill(61)\setcounter{equation}{61} %\end{array} \end{align} $

    (61)

    And

    $ \begin{align}\label{equa11134} % \begin{array} {lcccl} &\frac{\textstyle \partial J_{id}(k)}{\textstyle \partial\Delta U_{id}(k)}=\textstyle 2\Phi_{id}\Delta U_{id}(k)-\Psi_{id}\nonumber\\ &\frac{\textstyle \partial J_{i0}(k)}{\textstyle \partial\Delta U_{id}(k)}=\textstyle \Phi_{i0}\Delta U_{ic}(k)+\Omega_{i0}.~~~~~~~~~~~ %\hfill(62)\setcounter{equation}{62} %\end{array} \end{align} $

    (62)

    So, from (61),

    $ \begin{equation}\label{equa305} \nonumber \frac{\partial J_{ic}(k)}{\partial\Delta U_{ic}(k)}+\frac{\partial J_{i0}(k)}{\partial\Delta U_{ic}(k)}=0 \\ \Delta U_{ic}(k)=\frac{1}{2}\Phi_{ic}^{-1} \big (\Psi_{ic}-\Phi_{i0}\Delta U_{id}(k)-\Psi_{i0} \big). \\ \end{equation} $

    (63)

    And from (62),

    $ \begin{equation}\label{equa1035} \nonumber \frac{\partial J_{id}(k)}{\partial\Delta U_{id}(k)}+\frac{\partial J_{i0}(k)}{\partial\Delta U_{id}(k)}=0 \\ \Delta U_{id}(k)=\frac{1}{2}\Phi_{id}^{-1} \big (\Psi_{id}-\Phi_{i0}\Delta U_{ic}(k)-\Omega_{i0} \big). \\ \end{equation} $

    (64)

    Using the system defined by (63) and (64),

    $ \begin{equation}\label{112} \left\{ \begin{array}{ll} \Delta U_{ic}(k)=\dfrac{1}{2}\Phi_{ic}^{-1} \big (\Psi_{ic}-\Phi_{i0}\Delta U_{id}(k)-\Psi_{i0} \big) \\[2mm] \Delta U_{id}(k)=\dfrac{1}{2}\Phi_{id}^{-1} \big (\Psi_{id}-\Phi_{i0}\Delta U_{ic}(k)-\Omega_{i0} \big). \end{array} \right. \end{equation} $

    (65)

    So, the resolution of system defined by relation (65) gives

    $ \begin{equation}\label{1012} \Delta U_{ic}(k)=\big (I-G_{ic}\big)^{-1}F_{ic} \end{equation} $

    (66)

    $ \begin{equation}\label{1013} \Delta U_{id}(k)=\big (I-G_{id}\big)^{-1}F_{id} \end{equation} $

    (67)

    where

    $ \begin{equation}\label{312} \begin{array} {lcccl} F_{ic}=\dfrac{1}{2}\Phi_{ic}^{-1}\Big(\Psi_{ic}-\Psi_{i0}\Big)-\dfrac{1}{4}\Phi_{ic}^{-1}\Phi_{i0}\Phi_{id}^{-1}\Big(\Psi_{id}-\Omega_{i0}\Big)\\[2mm] F_{id}=\dfrac{1}{2}\Phi_{id}^{-1}\Big(\Psi_{id}-\Omega_{i0}\Big)-\dfrac{1}{4}\Phi_{id}^{-1}\Phi_{i0}\Phi_{ic}^{-1}\Big(\Psi_{ic}-\Psi_{i0}\Big)\\[2mm] G_{ic}=\dfrac{1}{4}\Phi_{ic}^{-1}\Phi_{i0}\Phi_{id}^{-1}\Phi_{i0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\[2mm] G_{id}=\dfrac{1}{4}\Phi_{id}^{-1}\Phi_{i0}\Phi_{ic}^{-1}\Phi_{i0}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{array} \end{equation} $

    (68)

    Suppose that

    $ \begin{equation}\label{equa1004} \Delta u_{ic}(k)=[1, \underbrace{0, \cdots, 0}_{h_{c}^{i}-1}]\Delta U_{ic}(k) \end{equation} $

    (69)

    $ \begin{equation}\label{equa10004} \Delta u_{id}(k)=[1, \underbrace{0, \cdots, 0}_{h_{c}^{i}-1}]\Delta U_{id}(k). \end{equation} $

    (70)

    Finally,

    $ \begin{align}\label{equa45} % \begin{array} {lcccl} &u_{ic}(k)=\Delta u_{ic}(k)+u_{ic}(k-1)\nonumber\\ &u_{id}(k)=\Delta u_{id}(k)+u_{id}(k-1). %\end{array} \end{align} $

    (71)
  • The basic idea of the proposed approach consists of associating a sub-FMPC for each subsystem. The development of the last one is accomplished using the parallel distributed compensation (PDC) since each subsystem is described by a fuzzy model. Each fuzzy rule conclusion of the sub-FMPC will be computed by the minimizing of a performance criterion related to the considered subsystem (see Sections 3.1 and 3.2).

    Fuzzy model predictive control is an approach where the fuzzy model of the system is used to predict the future evolution of the system. Complete system$'$s dynamics is obtained by local models interpolation, which is described by the following equation:

    $ \begin{equation}\label{3333} ~~~~\left\{ \begin{array}{lcccl} x_{i}(k+1)=\frac{\textstyle 1}{\textstyle\sum \limits_{r=1}^{R^{i}}\alpha_{r}(z_{i}(k) )}\sum \limits _{r=1}^{R^{i}}\alpha_{r}(z_{i}(k))\bigg (A_{i_{r}}x_{i_{r}}(k)+\\ ~~~~~~~~~~~~~~~~B_{i_{r}}u_{i_{r}}(k)+\sum\limits_{\underset{j \neq i}{j=1}}^{N}A_{ij_{r}}x_{j_{r}}(k)+w_{i_{r}}(k)\bigg )\\ y_{i}(k)=\frac{\textstyle \sum \limits _{r=1}^{R^{i}}\alpha_{r}(z_{i}(k))\bigg ( C_{i_{r}}x_{i_{r}}(k)+v_{i_{r}}(k)\bigg )}{\textstyle\sum \limits_{r=1}^{R^{i}}\alpha_{r}(z_{i}(k) )}. \end{array} \right. \end{equation} $

    (72)

    System (72) can be simplified as

    $ \begin{equation}\label{3332} S_{i}=\left\{ \begin{array}{ll} \hat{x}_{i}(k+1)=\frac{\textstyle \sum \limits _{r=1}^{R^{i}}\alpha_{r}\hat{x}_{i_{r}}(k+1)}{\textstyle\sum \limits_{r=1}^{R^{i}}\alpha_{r}}\\ \hat{y}_{i}(k+1)= \frac{\textstyle \sum\limits_{r=1}^{R^{i}}\alpha_{r}\hat{y}_{i_{r}}(k+1)}{\textstyle\sum\limits_{r=1}^{R^{i}}\alpha_{r}} \end{array} \right. \end{equation} $

    (73)

    where $\hat{x}_{i_{r}}(k+1)$ and $\hat{y}_{i_{r}}(k+1)$ are respectively the predicted local state and the predicted local output for the $i$-th subsystem associated to the $r$-th rule given by relations (28) and (29).

    The control action is obtained by solving, at each sampling instant, an optimization problem in order to minimize a performance criterion.

    The $r$-th control rule, with $r=1, \cdots, R^{i}$, of the fuzzy model predictive control (FMPC) of the subsystem $S_{i}$ has the following form.

    Rule $r$:

    IF $z_{i_{1}}(k)$ is $A^{r}_{i_{1}}$ and $\cdots$ and $z_{i_{g}}(k)$ is $A^{r}_{i_{g}}$

    THEN

    $ \begin{equation}\label{9512} u_{i_{r}}(k)=u_{ic_{{r}}}(k)+u_{id_{{r}}}(k), r=1, \cdots, R^{i} \end{equation} $

    (74)

    where the expressions of $u_{ic_{{r}}}(k)$ and $u_{id_{{r}}}(k)$ are presented by (71). So, the control action associated to each subsystem can be expressed by the following expression:

    $ \begin{equation}\label{9} u_{i}(k+j)=\frac{\overset{R^{i}}{\underset{r=1}{\sum}}\alpha_{i_{r}}u_{i_{r}}(k+j)}{\overset{R^{i}}{\underset{r=1}{\sum}}\alpha_{i_{r}}} \end{equation} $

    (75)

    where $u_{i_{r}}(k)$ is the local control for the $r$-th rule associated to the $i$-th subsystem at the instant $k$, and is given by (74) and $\alpha_{i_{r}}$ is given by (4).

  • In order to test and validate the proposed approaches, two simulation examples will be considered. The first one is two-machines interconnected system[3], while the second one deals with two pendulums connected by a spring mounted on two carts[32].

  • Consider the system composed of two interconnected machine subsystems $S_{i}$ and described as follows[3]:

    $ \begin{equation} S_{i}=\left\{ \begin{array}{lcccl} \dot{x}_{i_{1}}(t)= \textstyle x_{i_{2}}(t) \\[2mm] \dot{x}_{i_{2}}(t)= -\frac{\textstyle H_{i}}{\textstyle P_{i}}x_{i_{2}}(t)+\frac{\textstyle 1}{\textstyle P_{i}}u_{i}(t)+\\[2mm] ~~~~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{2}\frac{\textstyle T_{i}T_{j}Z_{ij}}{\textstyle P_{i}}\bigg ({\rm cos}_{1}-{\rm cos}_{2}\bigg )+w_{i_{2}}(t) \\ y_{i}(t)= x_{i_{1}}(t)+v_{i}(t) \end{array} \right. \end{equation} $

    (76)

    where

    $ \begin{array}{l} {\rm{co}}{{\rm{s}}_1} = {\rm{cos}}(\delta _{ij}^0 - {\theta _{ij}})\\ {\rm{co}}{{\rm{s}}_2} = {\rm{cos}}({x_{{i_1}}}(t) - {x_{{j_1}}}(t) + \delta _{ij}^0 - {\theta _{ij}}). \end{array} $

    Assume that the absolute parameters of the machines are the following ones:

    $ \begin{align*} & T_{1 }=1.017, T_{2 }=1.005 \\[3mm] & P_{1 }=1.03, P_{2 }=1.25 \\[3mm] &H_{1 }=0.8, H_{2 }=1.2 \\[3mm] &Z_{12 }=Z_{21 }=1.98 \\[3mm] &\theta_{12 }=-\theta_{21 }=1.5 \\[3mm] &\delta^{0}_{12} =-\delta^{0}_{21}=1.2 \end{align*} $

    where:

    $x_{i_{1}}$, $x_{i_{2}}$ are the absolute rotor angle and the angular velocity of the $i$-th machine, respectively.

    $Z_{ij}$ is the module of the transfer admittance between the $i$-th and $j$-th machines.

    $\theta_{ij}$ is the phase angle of the transfer admittance between the $i$-th and $j$-th machines.

    $w_{i2}(k)$ is the external disturbance which is assumed to be sinusoidal with amplitude $1$ and period $\pi$.

    $v_{i}(k)$ is the measurement noise which is assumed to be zero mean white noise with variance $0.1$ for $i, j=1, 2~(i\neq j$).

    $P_{i}$ is the inertia coefficient.

    $H_{i}$ is the damping coefficient.

    $T_{i}$ is the internal voltage.

    Since the developed methods are for the discrete state, it is necessary to discretize the system given by (76) using Euler method. In this context, suppose that the sampling period is chosen to be equal to $T$=0.01 s. So, the matrices given in Appendix A are equivalent to system in the discrete state.

    The main goal is based on the discretization of conclusions of $R_{i}$ rules.

    The formulation of the fuzzy model predictive control for the considered system is performed in two steps:

    Step 1. It consists of decomposing the interconnected system (76) into a set of Takagi-Sugeno models. This step gives a set of linear fuzzy local models.

    Rule $r$:

    IF $x_{i_{1}}(k)$ is $I_{r}$ and $x_{i_{2}}(k)$ is $O_{r}$

    THEN

    $ \begin{align*}\left\{ \begin{array}{lccl} x_{i}(k+1)=A_{i_{r}}x_{i}(k)+B_{i_{r}}u_{i}(k)+\\ ~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{~2}A_{ij_{r}}x_{j}(k)+w_{i}(k)\\ y_{i}(k)= C_{i_{r}}x_{i}(k)+v_{i}(k)\\ ~~~~r=1, \cdots, 9 \end{array} \right. \end{align*} %\end{equation*} $

    (77)

    where:

    $I_{r} \in \{ {\rm {positive, zero, negative}} \}$.

    $O_{r} \in \{ {\rm {positive, zero, negative}} \}$.

    $A_{i_{r}}$, $B_{i_{r}}$ and $C_{i_{r}}$ are given in Appendix A.

    $w_{i}(k)=[0, w_{i_{2}}]^{\rm T}$.

    $C_{i_{r}}(k)=[1,0]^{\rm T}$ for $i=1, 2$ and $r=1, \cdots, 9$.

    Consider a reference model for the $i$-th subsystem described as

    $ \begin{equation}\label{5} x_{iref}(k+1)=A_{iref}x_{iref}(k)+r_{i}(k) \end{equation} $

    (78)

    where:

    $x_{iref}(k)$ denotes reference state.

    $A_{iref}$ denotes a specific asymptotically stable matrix.

    $r_{i}(k)$ denotes bounded reference input.

    The reference models are given as

    $ \begin{eqnarray} \nonumber A_{iref} &=& \left( \begin{array}{cc} 0 & 0.01 \\ -1 & -1.01 \\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber r_{1}(k) = \left( \begin{array}{c} 0 \\ {\rm cos}(0.5k) \\ \end{array} \right), r_{2}(k)= \left( \begin{array}{c} 0 \\ {\rm sin}(0.5k) \\ \end{array} \right), i=1, 2. \end{eqnarray} $

    Step 2. $R_{i}$ fuzzy local rules are formulated for each subsystem $(S_{i})$. Thus, fuzzy model predictive control will be designed for the interconnected systems.

    The membership functions for $x_{i_{1}}(k)$ and $x_{i_{2}}(k)$, shown in Fig. 1, are used from Rules 1 to 9 for two subsystems.

    Figure 1.  Membership function for $x_{i_{1}}$ and $x_{i_{2}}$

    The Gaussian function used for membership is defined by (79).

    $ \begin{equation}\label{5505} \mu_{r} \{z(k)\}=\exp - \frac{(z(k)-\varsigma_{r}^{i})^{^{2}}}{2(\sigma_{r}^{i})^{2}} \end{equation} $

    (79)

    where:

    $z(k)=\{x_{i_{1}}(k), x_{i_{2}}(k), i=1, 2\}$, $\varsigma_{r}^{i}$ and $\sigma_{r}^{i}$ are respectively the center and the width of the used function.

    For simulation, suppose that

    $\varsigma_{r}^{i}\in \{-\frac{\pi}{2}, 0, \frac{\pi}{2}\}$

    $\sigma_{r}^{i}=1.5$

    $h_{p}^{i}=3$

    $h_{c}^{i}=3$

    $i=1, 2$

    $r=1, \cdots, 9$.

    Matrices $Q_{ic}, Q_{id}, R_{ic}$ and $R_{id}$ have been chosen after several manual tests.

    $ \begin{eqnarray} \nonumber R_{1c}=10^{-2}\times\left( \begin{array}{ccc} 1 & 0& 0\\ 0 & 1& 0\\ 0 & 0& 1\\ \end{array} \right), R_{2c}=10^{-2}\times\left( \begin{array}{ccc} 5 & 0& 0\\ 0 & 5& 0\\ 0 & 0& 5\\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber Q_{ic} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right), R_{1d} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber R_{2d} = \left( \begin{array}{ccc} 0.5 & 0& 0\\ 0 & 0.5& 0\\ 0 & 0& 0.5\\ \end{array} \right), Q_{id} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right). \end{eqnarray} $

  • Using the first method developed in Section 3.1, the trajectories of the state variable $x_{1_{1}}(k)$ and its reference $x_{r_{1_{1}}}(k)$ are shown in Fig. 2 (a). The trajectories of the state variable $x_{1_{2}}(k)$ and its reference $x_{r_{1_{2}}}(k)$ are shown in Fig. 2 (b).

    Figure 2.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line)

    The trajectories of the state variable $x_{2_{1}}(k)$ and its reference $x_{r_{2_{1}}}(k)$ are shown in Fig. 3 (a). The trajectories of the state variable $x_{2_{2}}(k)$ and its reference $x_{r_{2_{2}}}(k)$ are shown in Fig. 3 (b). The trajectories of the control actions $u_{1}(k)$ and $u_{2}(k)$ are shown in Fig. 4.

    Figure 3.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line)

    Figure 4.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line)

    To test the robustness of the developed method, parameter variations will be considered for two machines. In the first step, absolute parameters $P_{1}$ and $P_{2}$ will be varied from the sampling instant $k=500$. In the second step, starting from the instant $k=1\, 500$, a second variation will be considered: the variations of $H_{1}$ and $H_{2}$. Suppose that the new parameters are as follows: $P_{1 }=1.5$, $P_{2 }=2$, $H_{1 }=0.1$ and $H_{2 }=0.5$.

    The simulation results after this variation, by keeping the same parameters already used ($h_{p}^{i} h_{c}^{i}, \varsigma_{r}^{i}, \sigma_{r}^{i}, Q_{i} $ and $ R_{i}$, $i=1, 2$), are illustrated in Figs. 5-7.

    Figure 5.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line): Robustness test

    Figure 6.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line): Robustness test

    Figure 7.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line): Robustness test

  • Using the second method developed in Section 3.2, the trajectories of the state variable $x_{1_{1}}(k)$ and its reference $x_{r_{1_{1}}}(k)$ are shown in Fig. 8 (a). The trajectories of the state variable $x_{1_{2}}(k)$ and its reference $x_{r_{1_{2}}}(k)$ are shown in Fig. 8 (b).

    Figure 8.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line)

    The trajectories of the state variable $x_{2_{1}}(k)$ and its reference $x_{r_{2_{1}}}(k)$ are shown in Fig. 9 (a). The trajectories of the state variable $x_{2_{2}}(k)$ and its reference $x_{r_{2_{2}}}(k)$ are shown in Fig. 9 (b).

    Figure 9.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line)

    The trajectories of the control actions $u_{1}(k)$ and $u_{2}(k)$ are shown in Fig. 10.

    Figure 10.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line)

    To test the robustness, the same conditions of the first approach will be considered for the second one.

    The simulation results after this variation, by keeping the same parameters already used ($h_{p}^{i}, h_{c}^{i}, \varsigma_{r}^{i}, \sigma_{r}^{i}, Q_{i} $ and $ R_{i}$), are illustrated in Figs. 11 to 13: To test the robustness, the same conditions of the first approach will be considered for the second one.

    Figure 11.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line): Robustness test

    Figure 12.  (a) The trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line): Robustness test

    Figure 13.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line): Robustness test

    The simulation results after this variation, by keeping the same parameters already used ($h_{p}^{i}, h_{c}^{i}, \varsigma_{r}^{i}, \sigma_{r}^{i}, Q_{i} $ and $ R_{i}$), are illustrated in the following figures.

  • In this section, the problem of two inverted pendulums connected by a spring mounted on two carts will be considered (see Fig. 14)[32].

    Figure 14.  Configuration of two inverted pendulums connected by a spring mounted on two carts

    The state variables are defined as follows:

    $x_{1_{1}}(t)=\theta_{1}(t)$

    $x_{1_{2}}(t)=\dot{\theta}_{1}(t)$

    $x_{2_{1}}(t)=\theta_{2}(t)$

    $x_{2_{2}}(t)=\dot{\theta}_{2}(t)$.

    The considered system is described by (80)

    $ \begin{equation}\label{8} ~~~~~S_{i}=\left\{ \begin{array}{ll} \dot{x}_{i_{1}}(t)= x_{i_{2}}(t) \\ \dot{x}_{i_{2}}(t)=\big(\rho_{1}-\rho_{2}\big)x_{i_{1}}(t)-\rho_{3}sin(x_{i_{1}}(t))x^{2}_{i_{2}}(t)+\\ ~~~~~~~~~~~\rho_{2}x_{_{j}1}(t) +\rho_{4}u_{i}(t)+0.1w_{i_{2}}(t)\\ y_{i}(t)= x_{i_{1}}(t)+0.01v_{i}(t) \end{array} \right. \end{equation} $

    (80)

    where:

    $\rho_{1}=\frac{\textstyle g}{\textstyle cl}$

    $\rho_{2}=\frac{\textstyle ka(a-cl)}{\textstyle cml^{2}}$

    $\rho_{3}=\frac{\textstyle m}{\textstyle M}$

    $\rho_{4}=\frac{\textstyle 1}{\textstyle cml^{2}}$.

    The system parameters are given by the following values:

    $M$=5 kg is the mass of the pendulum

    $m=$1 kg is the mass of the cart

    $a$ = 0.2 m

    $c=\frac{\textstyle m}{\textstyle m+M}$

    $l=1\, {\rm m}$ is the length of the pendulum

    $k=1\, N$/m is the spring constant (stiffness)

    $g=9.8\, {\rm m}/s^{2} $ is the gravity constant

    $w_{22}(t)=5{\rm cos}(10t){\rm e}^{-0.5t}$

    $v_{1}(t)=10{\rm sgn}({\rm sin}(5t)){\rm e}^{-0.5t}$

    $v_{2}(t)=10{\rm sgn}({\rm cos}(5t)){\rm e}^{-0.5t}$.

    Before the formulation of the fuzzy model predictive control for the considered system, it is necessary to use the discretized form based on Euler method to resolve the problem. In this context, suppose that the sampling period is chosen to be equal to $T$=0.01 s. So, the matrices given in Appendix B are equivalent to system in the discrete state. Based on Euler method, all equations are in the discrete form.

    Therefore, the main goal is based on discretization of conclusions of $R_{i}$ rules.

    As a definition, suppose that:

    $x_{i_{1}}(k)$ and $x_{i_{2}}(k)$ are the angle (in radians) and the angular velocity (in radians per second) of the $i$-th pendulum, respectively.

    $u_{i}(k)$ is the control force of the $i$-th pendulum.

    $y_{i}(k)$ is the output of the $i$-th pendulum.

    $w_{i2}(k)$ denotes the external disturbance.

    $v_{i}(k)$ denotes measurement noise with $i, j=1, 2~ (i\neq j)$.

    To formulate the fuzzy model predictive control for the considered system, two steps are to be followed:

    Step 1. It consists of decomposing of the interconnected system (80) into a set of Takagi-Sugeno models. This step gives a set of linear fuzzy local models.

    Rule $r$:

    IF $x_{i_{1}}$ is $I$ and $x_{i_{2}}$ is $O$

    $ \begin{equation*}\label{7} ~~~~\textbf{THEN}\left\{ \begin{array}{lll} x_{i}(k+1)=A_{i_{r}}x_{i}(k)+B_{i_{r}}u_{i}(k)+\\ ~~~~~~~~~~~~~~~\sum \limits_{\underset{j \neq i}{j=1}}^{2}A_{ij_{r}}x_{j}(k)+w_{i}(k)\\ y_{i}(k)= C_{i_{r}}x_{i}(k)+v_{i}(k)\\ r=1, \cdots, 9 \end{array} \right. \end{equation*} $

    (81)

    where:

    $I \in \{ {\rm {positive~~large, small, negative~~large}} \}$

    $O \in \{ {\rm {positive fast, slow, negative fast}} \}$

    All matrices $A_{i_{r}}, B_{i_{r}}, A_{ij_{r}}, D_{i_{r}}, C_{i_{r}}$ and $ E_{i_{r}}$ are listed in Appendix B.

    $w_{i}(k)=[0, w_{i_{2}}]^{\rm T}$, for $i=1, 2$ and $r=1, \cdots, 9$.

    Step 2. The $R_{i}$ fuzzy local rules are formulated for each subsystem ($S_{i}$). Thus, the fuzzy model predictive control will be designed for the interconnected systems.

    The membership functions for $x_{i_{1}}(k)$ and $x_{i_{2}}(k)$, shown in Fig. 15, are used from Rules 1 to 9 to formulate the global model for each subsystem. The Gaussian function used for membership is defined by (79).

    Figure 15.  Membership function for $x_{i_{1}}$ and $x_{i_{2}}$

    For simulation, suppose that $\varsigma_{r}^{1}\in \{-\frac{\pi}{2}, 0, \frac{\pi}{2}\}$, $\varsigma_{r}^{2}\in \{-10, 0, 10\}$, $\sigma_{r}^{1}=1.5$, $\sigma_{r}^{2}=5$, $h_{p}^{i}=3$, $h_{c}^{i}=3$ and $i=1, 2$.

    $ \begin{eqnarray} \nonumber R_{1c} = \left( \begin{array}{ccc} 0.01 & 0& 0\\ 0 & 0.01& 0\\ 0 & 0& 0.01\\ \end{array} \right), R_{1d} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber R_{2c} = \left( \begin{array}{ccc} 0.05 & 0& 0\\ 0 & 0.05& 0\\ 0 & 0& 0.05\\ \end{array} \right), Q_{1c} = \left( \begin{array}{ccc} 1 & 0& 0\\ 0 & 1& 0\\ 0 & 0& 1\\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber Q_{2c} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right), R_{2d} = \left( \begin{array}{ccc} 0.5 & 0& 0\\ 0 & 0.5& 0\\ 0 & 0& 0.5\\ \end{array} \right) \end{eqnarray} $

    $ \begin{eqnarray} \nonumber Q_{1d} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right), Q_{2d} = \left( \begin{array}{ccc} 0.1 & 0& 0\\ 0 & 0.1& 0\\ 0 & 0& 0.1\\ \end{array} \right). \end{eqnarray} $

  • The trajectories of the state variable $x_{1_{1}}(k)$ and its reference $x_{r_{1_{1}}}(k)$ are shown in Fig. 16 (a). The trajectories of the state variable $x_{1_{2}}(k)$ and the reference state variable $x_{r_{1_{2}}}(k)$ are shown in Fig. 16 (b). The trajectories of the state variable $x_{2_{1}}(k)$ and its reference $x_{r_{2_{1}}}(k)$ are shown in Fig. 17 (a). The trajectories of the state variable $x_{2_{2}}(k)$ and its reference $x_{r_{2_{2}}}(k)$ are shown in Fig. 17 (b). The trajectories of the control actions $u_{1}(k)$ and $u_{2}(k)$ are shown in Fig. 18.

    Figure 16.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line)

    Figure 17.  (a) The trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line)

    Figure 18.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line)

    To test the robustness of the first method, parameter variations will be considered for two pendulums. In the first step, the length of pendulum ($l$) will be varied from the sampling instant $k=500$. In the second step, and starting from the instant $k=1\, 500$, a second variation will be considered: the variation of absolute parameters $m$ and $M$. Suppose that the new parameters are as follows: the mass of cart ($m$= 2 kg), the mass of pendulum ($M$ = 6 kg) and the length of pendulum ($l$ = 0.9 m).

    The simulation results after this variation, by keeping the same parameters already used ($h_{p}^{i}, h_{c}^{i}, \varsigma_{r}^{i}, \sigma_{r}^{i}, Q_{i} $ and $ R_{i}$), are illustrated in the following Figs. 19 to 21.

    Figure 19.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line): Robustness test

    Figure 20.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line): Robustness test

    Figure 21.  Fuzzy model predictive control $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line): Robustness test

  • Using the second method developed in Section 3.2, the trajectories of the state variable $x_{1_{1}}(k)$ and its reference $x_{r_{1_{1}}}(k)$ are shown in Fig. 22 (a). The trajectories of the state variable $x_{1_{2}}(k)$ and its reference $x_{r_{1_{2}}}(k)$ are shown in Fig. 22 (b).

    Figure 22.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line)

    The trajectories of the state variable $x_{2_{1}}(k)$ and its reference $x_{r_{2_{1}}}(k)$ are shown in Fig. 23 (a). The trajectories of the state variable $x_{2_{2}}(k)$ and its reference $x_{r_{2_{2}}}(k)$ are shown in Fig. 23 (b).

    Figure 23.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line)

    The trajectories of the control actions $u_{1}(k)$ and $u_{2}(k)$ are shown in Fig. 24.

    Figure 24.  Evolution of actions $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line)

    Figure 25.  (a) Trajectories of the state variable $x_{1_{1}}(k)$ (solid line) and reference state variable $x_{r_{1_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{1_{2}}(k)$ (solid line) and reference state variable $x_{r_{1_{2}}}(k)$ (dashed line): Robustness test

    To test robustness, the same conditions of the first approach will be considered for the second one.

    The simulation results after this variation, by keeping the same parameters already used ($h_{p}^{i}, h_{c}^{i}, \varsigma_{r}^{i}, \sigma_{r}^{i}, Q_{i} $ and $ R_{i}$), are illustrated in the following figures.

  • The simulation results show the effectiveness of Takagi-Sugeno fuzzy systems for modeling nonlinear interconnected systems. Indeed, the developed fuzzy models have been used to formulate local controller in order to develop model predictive control for the nonlinear interconnected systems. In this context, based on the presented simulation results, it is clear that the two proposed methods based on the decomposition approach give satisfactory results.

    Therefore, for both considered strategies where the first one is based on the development of two local cost functions for each subsystem, whereas the second one proposes the use of the global cost function decomposed to get three sub-cost functions and good tracking and robustness properties have been shown.

  • In this paper, two strategies of fuzzy predictive control have been developed for nonlinear interconnected systems. The interconnected nonlinear system is composed of a set of subsystems where each one is described by a fuzzy model. The proposed control approach consists of associating a sub-FMPC for each subsystem based on the parallel distributed compensation (PDC). Each fuzzy rule conclusion of the sub-FMPC is computed based on the minimization of a performance criterion related to the considered subsystem. The two proposed approaches have been developed based on the decomposition approach of the cost function of each subsystem. Indeed, two ways have been studied. In order to validate the developed control methods, two examples have been considered. The first one is a two machines interconnected system, while the second example deals with two invented pendulums connected by a spring mounted on two carts. The obtained results show the effectiveness of the developed control strategies.

    Figure 26.  (a) Trajectories of the state variable $x_{2_{1}}(k)$ (solid line) and reference state variable $x_{r_{2_{1}}}(k)$ (dashed line), (b) Trajectories of the state variable $x_{2_{2}}(k)$ (solid line) and reference state variable $x_{r_{2_{2}}}(k)$ (dashed line): Robustness test

    Figure 27.  Fuzzy model predictive control $u_{1}(k)$ (solid line) and $u_{2}(k)$ (dashed line): Robustness test

  • $ \begin{eqnarray} A_{1_{1}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.007\, 04 & 0.992\, 2 \\ \end{array} \right]\\ A_{1_{2}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.014\, 81 & 0.992\, 2 \\ \end{array} \right]\end{eqnarray} $

    $ \begin{eqnarray} \\ A_{1_{3}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.014\, 54 & 0.992\, 2 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{1_{4}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.010\, 47 & 0.992\, 2 \\ \end{array} \right]\\ A_{1_{5}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.005\, 1 & 0.992\, 2 \\ \end{array} \right]\\ A_{1_{6}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.015\, 48 & 0.992\, 2 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{1_{7}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.011\, 26 & 0.992\, 2 \\ \end{array} \right]\\ A_{1_{8}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.007\, 68 & 0.992\, 2 \\ \end{array} \right]\\ A_{1_{9}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.005\, 06 & 0.992\, 2 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{2_{1}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.002\, 77 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{2}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.006\, 47 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{3}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.009\, 53 & 0.990\, 4 \\ \end{array} \right] \end{eqnarray} $

    $ \begin{eqnarray} A_{2_{4}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.012\, 53 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{5}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.003\, 2 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{6}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.009\, 13 & 0.990\, 4 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{2_{7}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.012\, 13 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{8}}&= &\left[ \begin{array}{cc} 1 & 0.01 \\ 0.012\, 21 & 0.990\, 4 \\ \end{array} \right]\\ A_{2_{9}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ -0.005\, 1 & 0.990\, 4 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{12_{1}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.005\, 08 & 0 \\ \end{array} \right]\\ A_{12_{2}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.015\, 48 & 0 \\ \end{array} \right]\\ A_{12_{3}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.014\, 55 & 0 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{12_{4}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.007\, 67 & 0 \\ \end{array} \right]\\ A_{12_{5}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.005\, 2 & 0 \\ \end{array} \right]\\ A_{12_{6}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.014\, 81 & 0 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{12_{7}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.011\, 29 & 0 \\ \end{array} \right]\\ A_{12_{8}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.010\, 48 & 0 \\ \end{array} \right]\\ A_{12_{9}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.007\, 017 & 0 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{21_{1}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.004\, 41 & 0 \\ \end{array} \right]\\ A_{21_{2}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.009\, 44 & 0 \\ \end{array} \right]\\ A_{21_{3}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.009\, 28 & 0 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{21_{4}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.012\, 18 & 0 \\ \end{array} \right]\\ A_{21_{5}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.002\, 91 & 0 \\ \end{array} \right]\\ A_{21_{6}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.006\, 47 & 0 \\ \end{array} \right]\\ A_{21_{7}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.012\, 10 & 0 \\ \end{array} \right] \end{eqnarray} $

    $ \begin{eqnarray} A_{21_{8}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.012\, 67 & 0 \\ \end{array} \right]\\ A_{21_{9}}&=& \left[ \begin{array}{cc} 0 & 0 \\ -0.003\, 51 & 0 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} B_{1_{r}}&=& \left[ \begin{array}{c} 0 \\ 0.009\, 7 \\ \end{array} \right]\\ B_{2_{r}}&=& \left[ \begin{array}{c} 0 \\ 0.008 \\ \end{array} \right]\\ C_{i_{r}}&=&\left[ \begin{array}{c} 1 \\ 0 \\ \end{array} \right]^{\rm T}, r=1, \cdots, 9 , i=1, 2. \end{eqnarray} $

  • $ \begin{eqnarray} A_{i_{1}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.586\, 2 & 0.998\, 5 \\ \end{array} \right]\\ A_{i_{2}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.587\, 6 & 1 \\ \end{array} \right]\\ A_{i_{3}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.586\, 2 & 1.001\, 5 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{i_{4}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.585\, 3 & 1 \\ \end{array} \right]\\ A_{i_{5}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.587\, 6 & 1 \\ \end{array} \right]\\ A_{i_{6}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.585\, 3 & 1 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} A_{i_{7}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.586\, 2 & 1.001\, 5 \\ \end{array} \right]\\ A_{i_{8}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.587\, 6 & 1 \\ \end{array} \right]\\ A_{i_{9}}&=& \left[ \begin{array}{cc} 1 & 0.01 \\ 0.586\, 2 & 0.998\, 5 \\ \end{array} \right]\\ \end{eqnarray} $

    $ \begin{eqnarray} B_{i_{r}}&=& \left[ \begin{array}{c} 0 \\ 0.006 \\ \end{array} \right]\\ A_{ij_{r}}&=& \left[ \begin{array}{cc} 0 & 0 \\ 0.000\, 4 & 0 \\ \end{array} \right]\\ D_{i_{r}} &=& \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0.001 \\ \end{array} \right]\\ C_{i_{r}}&=&\left[ \begin{array}{c} 1 \\ 0 \\ \end{array} \right]^{\rm T}, E_{i_{r}}=0.000\, 1, i=1, 2. \end{eqnarray} $

Reference (32)

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