Internal Model Control of MIMO Discrete Under-Actuated Systems Via Squaring Matrix Transforms Islem Bejaoui #1 , Imen Saidi *2 , Dhaou Soudani #3 # Automatic Control Research Laboratory, ENIT, University of Tunis El Manar BP 37, Le Belvédére,1002 Tunis, Tunisia 1 islem.bejaoui@enit.utm.tn 2 imen.saidi@gmail.com 3 Dhaou.soudani@enit.rnu.tn Abstract— Internal model control (IMC) is an established technique in continuous-time linear control for SISO an MIMO fully-actuated systems but has not been developed for discrete- time under-actuated systems. In this paper we present a new IMC structure to the multivariable under-actuated systems, which is based on a specific inversion principle of the model plant. Simulated examples are presented to prove the effectiveness of the proposed control method to ensure set-point tracking, stability and disturbance rejection. Keywords— Internal model control; under-actuated systems; stability; disturbance rejection; specific inversion ; I. INTRODUCTION Under-actuated systems offer challenging control problems to solve operational inconveniences with great interest from theoretical point of view. Non-square system is a common industrial process in fields of the practical engineering. The number of the input variables does not equal to that of the outputs, e.g. This class of systems are abundant in real life; examples of such systems include, but are not limited to, surface vessels, spacecraft, underwater vehicles, helicopters, road vehicles, mobile robots, space robots and under-actuated manipulators[12,13] . In dispite of their generality in industry, the analysis and control for under-actuated systems need further research. In recent years, much works focus these field [2,6,7]. In order to contribute to this research area, we propose in this paper to apply an interesting Internal Model Control approach (IMC), to a class of discrete multivariable under-actuated systems. Control methodologies such as dynamic inversion and Moore- Penrose control require an inversion of the input influence matrix. However, if the transfer function system matrix is non-square direct inversion is not possible [16]. During the early to mid 1970, internal model control was an active research area starting with [5]. Specifically, treats the disturbance rejection problem and ensure stability. The specificity of this IMC structure resides in the use of a special controller which is an approximate inverse of the model plant. The use of this controller ensures a high level of robustness [1,3,4,5,6]. The analysis of the stability of elements of the internal model control has been conducted in the literature by numerous fundamental researches that depend on the type of systems considered and the scope. There are many methods studying the stability of linear discrete multivariable systems. These stability criteria can be classified into two main categories namely the frequency criterion using the notion of the characteristic equations and the time criterion based on Lyapunov theory. All of the results in [1-5] on internal model control are confined to continuous-time systems. Analogous results for discrete-time systems are not available in the literature. The purpose of this paper is to propose a new IMC method control for the discrete-time under-actuated systems. In the controller procedure, a simple design is presented such that initial conditions are taken into account; the controller has a good performance of tracking ability, excellent robustness and good control performance [8]. The influence of the model parameters and external disturbances will be also discussed. In the present paper, we develop an alternative approach to internal model control that is directly applicable to discrete- time under-actuated systems . Using this approach, we simultaneously solve the command following and disturbance rejection problem in discrete-time. We also present simulations examples to prove the effectiveness of the proposed control method. II. PROBLEM FORMULATION The basic IMC structure was designed for linear SISO systems and afterwards for linear MIMO fully-actuated systems [3,11,14]. The IMC structure of multivariable International Conference on Automation, Control Engineering and Computer Science (ACECS ) Proceedings of Engineering and Technology – PET Vol.20 pp.10 -14 Copyright IPCO-2017 ISSN 2356-5608