Complexity

Volume 2017 (2017), Article ID 5813192, 11 pages

https://doi.org/10.1155/2017/5813192

## Parameter Optimization of MIMO Fuzzy Optimal Model Predictive Control By APSO

Laboratory of Engineering of Industrial Systems and Renewable Energy, National High School of Engineers of Tunis (ENSIT), 5 Av. Taha Hussein, BP 56-1008, Tunis, Tunisia

Correspondence should be addressed to Adel Taieb

Received 16 April 2017; Revised 18 June 2017; Accepted 22 August 2017; Published 9 October 2017

Academic Editor: Carla Pinto

Copyright © 2017 Adel Taieb et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper introduces a new development for designing a Multi-Input Multi-Output (MIMO) Fuzzy Optimal Model Predictive Control (FOMPC) using the Adaptive Particle Swarm Optimization (APSO) algorithm. The aim of this proposed control, called FOMPC-APSO, is to develop an efficient algorithm that is able to have good performance by guaranteeing a minimal control. This is done by determining the optimal weights of the objective function. Our method is considered an optimization problem based on the APSO algorithm. The MIMO system to be controlled is modeled by a Takagi-Sugeno (TS) fuzzy system whose parameters are identified using weighted recursive least squares method. The utility of the proposed controller is demonstrated by applying it to two nonlinear processes, Continuous Stirred Tank Reactor (CSTR) and Tank system, where the proposed approach provides better performances compared with other methods.

#### 1. Introduction

Predictive control is a member of advanced discrete-time process control algorithms. This control algorithm is based on the use of an explicit process model to predict the manipulated variables and thus the future control actions are optimized throughout a finite horizon. To obtain a good performance, a process model describing the effects of all the different inputs on all the outputs must be developed. Although the linear model predictive control is suitable for processes that are not highly nonlinear, this strategy has been applied in the control of nonlinear systems, whether for SISO systems [1–3] or for MIMO systems [4–7]. But many industrial processes have strong nonlinearities and predictive control is applied in order to provide satisfactory control results. Two problems have appeared because of the introduction of nonlinearities in the predictive control.(i)The first of the problems is that the modeling of processes is much more difficult and complex than the linear case. fuzzy logic is among the most used strategies in all areas [4]. Despite the fact that this strategy has been developed in the last few years, the fuzzy models of the TS type remain among the most used methods to deal with the MPC control for nonlinear systems (NMPC), because of their capacity to give an accurate approximation of the complex nonlinear MIMO systems.(ii)The second important problem in nonlinear predictive control is the solving of the optimization problem. Conventional optimization methods, such as the gradient search method, used for designing the state feedback controller are restricted to the eigenvalues of the linear system matrix that not only increases the difficulty but also takes long time to find the global optimum solution [8]. Hence, evolutionary computation (EC) can be considered an alternative method to solve this type of optimization problem. In literature, plenty of works have been reported to solve the control optimization problems using EC techniques because they do not require explicit gradient information for optimization.

Particle Swarm Optimization (PSO), introduced by [9], is a population based metaheuristic search (MS) algorithm, which emulates the collaborative behaviour of bird flocking and fish schooling in searching for foods. In addition, unlike other heuristic optimization methods, PSO has a flexible and well-balanced mechanism to enhance the global and local exploration abilities. Since the introduction of PSO, many works based on MPC have been solved using PSO because it is not largely affected by the size and nonlinearity of the problem. Reference [10] introduces an approach for designing an adaptive fuzzy model predictive control using the PSO algorithm (AFMPC-PSO). In [11], a type of MPC is proposed using Chaos Particle Swarm Optimization (CPSO). First, for the modeling phase, the TS fuzzy model is employed to approximate the nonlinear system. Second, we introduce CPSO into MPC using a modified performance criterion in order to provide less computational controller’s expression.

Although these methods have represented effective solutions to the problem of the MPC control of nonlinear systems, there is the problem of choice of control parameters. Several studies have shown the influence of these parameters on the quality of the responses of the systems treated. To overcome this problem, this paper presents a method of tuning the weight parameters of the performance function according to the output quantities detected from the system. One of the challenges in MPC is how control parameters can be tuned for various target systems, and the use of APSO for automatic tuning is one of the solutions. Firstly, for the modeling phase, the TS fuzzy model is employed to approximate the nonlinear system. In the second step, we used the principle of optimal control to calculate the control law of each linear subsystem. Then, we introduce APSO algorithm to determine the best weight parameters of the performance function using a performance criterion in order to improve the quality of the response with a minimum of energy.

The rest of the paper is organized as follows. Section 2 presents the influence of weights existing in the objective function on the quality of system performances. Section 3 reviewed the TS fuzzy model and the OMPC design method. The main contribution of this paper is presented in Section 4. In order to show the good performance of the proposal approach, simulation results are given in Section 5. Finally, Section 6 concludes.

#### 2. Statement

The MPC control is a method of designing process control systems with feedback. This designing is carried out by the online repetition of a procedure that includes inputting data to a system of the initial input values. The principle of calculating the control law is the resolution of an optimal control problem using the present output values. So, the MPC control is a special case of the optimal control whose horizon is finite. It is recalled that this command minimizes the function described above. However, when the horizon is infinite, we speak of optimal control. The objective function is written as follows:where , are the weight values of the outputs and inputs. are the control instructions that are defined beforehand based on an expertise of the system. The partial derivative of the objective function described by (1) is as follows:By applying the principle of the optimal control, we obtain the following:The structure of the loop system based on OMPC illustrating this method is shown in Figure 1. According to (1), the criterion depends strongly on the weights . These weights directly affect the performances index (Pi) of the system studied, such as overshoot (Ov%), settling time (Ts), rise time (Tr), and static error (Es).