Design of Low Order Controllers for Decoupled MIMO Systems With Time Response Specifications

1. Introduction

Recent years, several inspiring control system approaches have been proposed, such as optimal control, nonlinear feedback control, adaptive control, sliding mode control, nonlinear dynamics, chaos control, chaos synchronization control, fuzzy logic control, fuzzy adaptive control, fractional order control, and robust control and their integrations (Azar & Vaidyanathan, 2015a,b,c, 2016; Azar & Zhu, 2015; Billel et al, 2017, 2016; Boulkroune et al, 2016a,b; Ghoudelbourk et al., 2016; Meghni et al, 2017a,b,c; Azar et al., 2017a,b,c,d; Azar 2010a,b, 2012; Mekki et al., 2015; Vaidyanathan & Azar, 2015a,b,c,d, 2016a,b,c,d,e,f,g, 2017a,b,c; Zhu & Azar, 2015; Grassi et al., 2017; Ouannas et al., 2016a,b, 2017a,b,c,d,e,f,g,h,I,j; Singh et al., 2017; Vaidyanathan et al, 2015a,b,c; Wang et al., 2017; Soliman et al., 2017; Tolba et al., 2017). In the industrial environment, engineers are in several applications facing multivariable systems, such as refinery process, chemical reactor (Skogestad & Postlethwaite, 2005). Because of interactions between the input/output variables, MIMO processes present, usually, difficulties to design controllers. So in order to deal with this problem, control engineers have used the decoupling system techniques. These techniques have been discussed in the literature over the years (Wang, 2003; Ogunnaike & Harmor, 1994). The choice of a decoupling method is not obvious because each technique possesses some benefits and limitations. It should be noted that in practice the simplified decoupling is the most used method. In fact, its principal advantage is the simplicity of its elements. Although it facilitates the determination of the controller transfer matrix, the ideal decoupling is rarely implemented in real applications. In addition, the inverted decoupling is uncommonly employed in practice. It has the principal benefits of the simplified and ideal decoupling techniques. Some research works have already made comparisons between different decoupling types. Luyben (1970) and Weischedel and McAvoy (1980) have made a comparison between ideal and simplified decoupling methods. They concluded that simplified decoupling is more robust with regards to ideal decoupling. Shinskey (1988) described both simplified and inverted decoupling techniques which are also detailed by Seborg et al., (1989). Gagnon et al., (1998) proposed a comparative study between simplified, ideal and inverted decoupling. They also demonstrated some drawbacks such as the implementation problem of inverted decoupling compared to other techniques. A centralized multivariable control based on simplified decoupling was presented by Garrido et al., (2012). Jevtovic and Matausek (2010) proposed the design of PID controller for TITO system based on ideal decoupler. Thanks to its former advantages, the simplified decoupling the simplified decoupling will be exploited in this work. In this research, the objective is to design a controller for MIMO systems with time response specifications. It should be noted that the controller parameters are obtained by resolving a non-convex optimization problem. In fact, the optimization is found at the heart of several real problem-solving processes. Therefore, the resolution of optimization problems has attracted the attention of many researchers in various fields. Toksari (2009) proposed an ant colony optimization algorithm to find the global minimum. This algorithm was tested on some standard functions and it was compared with other algorithms. Zhou, et al. (2013) used the PSO in the control algorithm in order to allow robots to navigate towards the remote frontier after exploring the region. The PSO is also applied by Abu-Seada, et al. (2013) to obtain an optimal tuning of proportional integral derivative controller parameters for an automatic voltage regulator system of a synchronous generator. Mousa, et al. (2015) exploited the PSO in order to determine PI and PID controllers parameters. These controllers with a feed forward gain are used with a reduced linear quadratic regulator for stabilizing swinging-up the inverted pendulum. Shahin, et al. (2014) proposed the improvement of the steady state and dynamic performance of the power grids by using the advanced flexible AC transmission systems based on evolutionary computing methods. The control of the electric power system can be achieved by using the PSO method applied to this subject to enhance the characteristics of the controller performance. Bahgaat, et al. (2014) used different methods such as the PSO, the adaptive weight PSO, the adaptive acceleration coefficients based PSO and the adaptive neuro fuzzy inference system to determine the PID controller parameters. They concluded that time performances such as the overshoots and settling times with their proposed controllers are better than the outputs of the conventional PID controllers. GA with hierarchically structured population was applied by Toledo, Oliveira and França (2014) with the aim of solving unconstrained optimization problems. The implementation of GA in an embedded microcontroller based polarization control system was proposed by Mamdoohi et al. (2012). The controller measures the signal intensity. These measures will be exploited in the estimation of the genetic value. Then, the GA controls this process. To attain the optimum performance, the best genetic parameters optimize the code such that the fastest execution time can be obtained. Valdez et al., (2014) presented a hybrid approach for optimization, combining PSO and GAs using fuzzy logic so as to integrate the results. They affirmed that the proposed method, in their work, combines the advantages of PSO and GA to get an improved FPSO and FGA hybrid method. Fuzzy logic is employed with the objective of combining the results of the GA and PSO in the best possible way. Fuzzy logic is also used with an aim of adjusting parameters in FPSO and FGA. Jiang et al., (2014) proposed a hybrid approach in order to solve economic emission load dispatch problems considering various practical constraints by employing Hybrid PSO and GSA. Their algorithm provided a combination between the PSO and the GSA and adopted co-evolutionary technique to update its particle position in the swarm with the cooperation of PSO and GSA. With the aim of finding the global minimum in the numerical, Servet Kiran et al., (2012) proposed a hybrid algorithm which is based on PSO and ACO. This algorithm is named hybrid ant particle optimization algorithm. The ACO and the PSO work, at each iteration, independently and give rise to their solutions. The best solution is, subsequently, chosen as the global best solution of the system and its parameters are used to choose the new position of particles and ants at the next iteration. Tabakhi et al., (2014) presented a method based on ACO, manned unsupervised feature selection method based on ACO so as to find an optimal solution to the feature selection problem. Geometric branch and bound techniques are usually known solution algorithms for non-convex continuous global optimization problems. Several approaches can be found in the literature. Schöbel & Scholz (2014) proposed an extension of geometric branch and bound methods for mixed integer optimization problems. As a matter of fact, they presented some general bounding operations. Hence, with the objective of solving mixed integer optimization problems they provided a general algorithm.