Applied Computational Intelligence and Soft Computing

Volume 2015, Article ID 437943, 15 pages

http://dx.doi.org/10.1155/2015/437943

## Constrained Fuzzy Predictive Control Using Particle Swarm Optimization

^{1}SET Laboratory, Electronics Department, University of Blida 1, Route de Soumaa, BP 270, 09000 Blida, Algeria^{2}High School of Computer Sciences (HEB-ESI), Rue Royale 67, 1000 Brussels, Belgium

Received 26 September 2014; Revised 24 April 2015; Accepted 24 April 2015

Academic Editor: Shyi-Ming Chen

Copyright © 2015 Oussama Ait Sahed 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

A fuzzy predictive controller using particle swarm optimization (PSO) approach is proposed. The aim is to develop an efficient algorithm that is able to handle the relatively complex optimization problem with minimal computational time. This can be achieved using reduced population size and small number of iterations. In this algorithm, instead of using the uniform distribution as in the conventional PSO algorithm, the initial particles positions are distributed according to the normal distribution law, within the area around the best position. The radius limiting this area is adaptively changed according to the tracking error values. Moreover, the choice of the initial best position is based on prior knowledge about the search space landscape and the fact that in most practical applications the dynamic optimization problem changes are gradual. The efficiency of the proposed control algorithm is evaluated by considering the control of the model of a 4 × 4 Multi-Input Multi-Output industrial boiler. This model is characterized by being nonlinear with high interactions between its inputs and outputs, having a nonminimum phase behaviour, and containing instabilities and time delays. The obtained results are compared to those of the control algorithms based on the conventional PSO and the linear approach.

#### 1. Introduction

In the late seventies, a new class of advanced control algorithms, grouped under the denomination Model Predictive Control (MPC), has emerged. The MPC strategy is based on the use of an explicit model to predict the process future behaviour over a finite horizon and then to compute a sequence of future control actions by minimizing a given cost function. The MPC is known to be a very powerful control strategy for many industrial processes [1–3]. Its attraction is due to its ability to handle complex control problems which involve multivariable process interactions, constraints in the system variables, nonminimum phase behaviour, and variable or unknown time delays. Linear MPC (LMPC) techniques, which use a linear model for the process, have been successfully used in many industrial applications, such as refining, chemicals, petrochemicals, air and gas processing, and food processing industries [4–6].

Although LMPC algorithms provide satisfactory performance in many applications, against highly nonlinear process, severe degradation in the control performance can occur, unless of course the operating conditions are very close to the steady state around which the model is linearized [7]. Since most practical processes are nonlinear by nature, new efficient nonlinear MPC (NMPC) techniques have to be derived to incorporate nonlinearities and ensure higher control performance. The introduction of nonlinear models will enhance the overall controller performance. However, the relatively simple optimization problem in LMPC algorithms will be transformed to a nonlinear and a nonconvex optimization problem which requires complex and time consuming procedures [8–10]. Thus, the challenge is to derive an efficient optimization algorithm that allows the real-time implementation of the control law.

An important step in the design of NMPC algorithms based controllers is the construction of the explicit nonlinear model of the system. The model choice is a critical task; the controller efficiency and performance depend extremely on the accuracy of the used model. In many cases it is even impossible to obtain a suitable model for the underlying process due to its complexity or lack of knowledge of its critical parameters. Different approaches that can provide satisfactory models exist; one which is widely used is the fuzzy logic technique [11]. Fuzzy inference systems (FIS) are universal approximators capable of approximating any continuous function with a certain level of accuracy [12]. Takagi-Sugeno (TS) models, a subdivision of fuzzy models, are particularly suitable for NMPC algorithms [13]. These models are able to express the dynamic nature of systems with characteristics of randomness, large delay time, and strong nonlinearity [14, 15].

Another important factor in driving the NMPC based controllers is the resolution of the relatively difficult and time consuming nonlinear and nonconvex optimization problem. Seeing that the analytical approach is impractical, several other suboptimal solutions were proposed [7, 16–18]. Some of them are based on numerical optimization algorithms, which are known to have a slow convergence rate and could easily be trapped in local minima. Other more interesting approaches are the metaheuristic based optimization algorithms, such as the powerful but computationally complex genetic algorithms (GA) and the particle swarm optimization strategy. PSO algorithms require much less computational effort than the GA and their implementation is relatively easy [19]. A lot of papers have considered the PSO to solve the optimization problem of NMPC [20–23]. In these papers, a large number of particles and iterations were used which increases the computational burden of the control algorithm. Recently, several PSO algorithms have been proposed in the literature to solve different optimization problems [24–26]. In most of these algorithms, improving the accuracy and the convergence proprieties of the algorithm is obtained at the cost of increasing its computing complexity. For example, in [26] more than one equation is used to update the particles positions at the same time. Obviously, using several update equations will increase the algorithm computing time and will limit its use in real time applications.

In this work, a PSO based constrained fuzzy predictive controller is proposed. The aim is to develop a simple and efficient controller capable of handling the relatively complex optimization problem with minimal computational time. This can be achieved using a reduced population size and a small number of iterations. The proposed control algorithm uses the normal distribution law to distribute the initial particles positions within the area around the best position. The radius limiting this area is adaptively changed according to the tracking error values. Moreover, the choice of the initial best position is based on prior knowledge about the search space landscape and the fact that in most practical applications the dynamic optimization problem changes are gradual. The algorithm efficiency is evaluated by considering the control of the model of a 4 × 4 Multi-Input Multi-Output (MIMO) industrial boiler [27, 28]. This model is characterized by being nonlinear with high interactions between its inputs and outputs, having a nonminimum phase behaviour, containing instabilities and time delays, and including constraints on its variables, disturbances, and uncertainties.

The paper is organized as follows. In Section 2 the design stages of Takagi-Sugeno fuzzy models are presented. Section 3 presents and describes the fuzzy predictive control problem. The PSO algorithm, used to solve the fuzzy MPC optimization problem, is introduced in Section 4. Section 5 deals with the proposed control algorithm and Section 6 gives the simulation results. Finally, conclusions are drawn in the last section.

#### 2. Takagi-Sugeno Fuzzy Modelling

Fuzzy inference systems (FIS) are universal approximators capable of approximating any continuous function with certain level of accuracy [12]. The FIS are based on the so-called IF-THEN rules and are classified into two main categories: Mamdani models and Takagi-Sugeno models. The rules of TS fuzzy models, used in this work, have the following form:where is the set of premise values, is the set of inputs and outputs values used in the consequent regressors, is the set of membership functions associated with the antecedents of the rule, and is the parameter vector of the submodel while is its output (; is the number of fuzzy rules).

It is clear, from (1), that the Takagi-Sugeno fuzzy models have fuzzy propositions only in their antecedents while their consequences are linear functions of the antecedents or their variables.

Using input/output data extracted from a nonlinear process, a fuzzy model can be constructed to mimic the process behaviour, by defining the parameter matrix asand the normalized membership grade vector asThe inferred output of the TS fuzzy model is given byUsing the input/output representation, the TS model given by (4) can be rewritten as follows:where , , and .

The global output of the fuzzy model can be written aswhere is the output of the th submodel andAssuming that a set of input-output data pairs is available and defining , a regression matrix which has the following expression can be constructed:The vector can be calculated by solving the following least square problem:where .

The resolution of (9), which is the last step in TS fuzzy model construction, deals with the definition of the consequences part of the rules. However, the antecedent, which involves in part the number, the position, the shape, and the distribution of the membership functions, must be first selected. A lot of techniques for the construction of fuzzy models from input-output data are described in the literature; in Espinosa et al. [16] different approaches are well explained.

#### 3. Fuzzy Predictive Control Principle

We can always distinguish two common features present in any given MPC based algorithm. The explicit model used to predict the future process behaviour and the optimization problem from which a control sequence is derived.

The main steps of the MPC strategy are given below:(1)The constructed model is used to predict the future behaviour of the process over a given prediction horizon.(2)The reference trajectory must be defined over the prediction horizon.(3)The control sequence is obtained by minimizing a given cost function.(4)Only the first element of the control sequence is applied on the system.

The previous steps are repeated, at every sampling time, according to the receding horizon idea. These steps are illustrated by the block diagram of Figure 1.