Synthesis of Controllers for MIMO Systems with Time Response Specifications

Introduction

The majority of industrial processes, by nature, have multiple outputs and states variables, hence the need to control them. Multi-Input Multi-Output (MIMO) systems are made up of several measurement and control signals which often present complex couplings between them. To deal with this problem, control engineers have resorted to decoupling system techniques. Decoupling control has been approached in the literature over the years (Wang, 2003; Ogunnaike & Harmor, 1994). The selection of a decoupling method is relatively complicated because each technique presents some advantages and also some drawbacks. Simplified decoupling is extremely the most used method in practice. Its principal asset is the simplicity of its elements. Ideal decoupling which is not often implemented eases enormously the focus of the controller transfer matrix. Inverted decoupling is also infrequently employed. Certain authors have previously made comparisons between ideal, inverted and simplified decoupling. Luyben (1970) and Weischedel and McAvoy (1980) have both compared simplified and ideal decoupling methods by the use of the distillation column simulators. Seborg, Edgar and Mellichamp (1989) detailed both simplified and inverted decoupling methods which are also described by Shinskey (1988). A comparative study between simplified, ideal and inverted decoupling is presented by Gagnon, Pomerleau and Desbiens (1998). Centralized multivariable PI controllers for interacting MIMO processes are proposed by Vijay Kumar, Rao and Chidambaram (2012). Garrido, Vazquez and Morilla (2012) presented centralized multivariable control by simplified decoupling. Because of its advantages enumerated beforehand, the simplified decoupling will be chosen in this work. The synthesis of a fixed low-order controller for linear time invariant, Single Input Single Output (SISO) systems with some step response specifications such as the settling time and the overshoot was presented in previous work (Ben Hariz, Bouani & Ksouri, 2012). In this paper, an extension of this work in the case of MIMO systems will be proposed. The controller design is formulated as an optimization problem which takes in account the desired closed loop performances. Kim, Keel and Bhattacharyya (2003) presented the methodology to fix the desired closed loop characteristic equation by the user. Since, the controller parameters are obtained by minimizing a non convex optimization problem, the use of global optimization method is suggested. 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 Practical Swarm Optimization (PSO) in the control algorithm in order to allow robots to navigate towards 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. Genetic algorithm with hierarchically structured population was applied by Toledo, Oliveira and França (2014) with the aim of solving unconstrained optimization problems. In this work, the Generalized Geometric Programming (GGP) will be applied as a global optimization method in order to resolve the optimization problem. The principle of this method is to transform a non convex optimization problem to a convex one by means of variable transformations. The outline of this paper is as follows. In section 2, the decoupling system technique is presented. The exposition of problem statement is given in section 3. In section 4, the controller’s design procedure is developed. Then the GGP method is introduced in section 5. In order to illustrate the effectiveness of the proposed controllers some simulation results are given in section 6. Then section 7 discusses and analyzes the simulation results. Finally, conclusion and future work are presented.