Arabian Journal for Science and Engineering https://doi.org/10.1007/s13369-018-3616-1 RESEARCH ARTICLE - ELECTRICAL ENGINEERING Optimal Nonlinear Model Reference Controller Design for Ball and Plate System Hazem I. Ali 1 · Haider M. Jassim 1 · Amjad F. Hasan 1 Received: 7 May 2018 / Accepted: 28 October 2018 © King Fahd University of Petroleum & Minerals 2018 Abstract This paper presents a new design procedure of an optimal nonlinear controller using the model reference approach. The ball and plate system is used as nonlinear, uncertain, and MIMO system to verify the effectiveness of the proposed controller. The main goal of the proposed design is to assure a desirable performance despite the presence of the coupling among control loops and uncertainty. The invasive weed optimization (IWO) method, which is one of the metaheuristic optimization algorithms, is used to obtain the optimal parameters of the proposed controller. The feasibility and efficiency of the proposed nonlinear controller are illustrated experimentally using real ball and plate system. Keywords Nonlinear control · Optimal control · Model reference · Ball and plate · IWO 1 Introduction Ball and plate is an electromechanical device that has been constructed to simulate the behaviour of certain types of mul- tivariable systems. Due to its high nonlinearity, uncertainties, coupling, and under actuation properties, it is recently being used as a benchmark problem in control system design. The ball and plate system is considered as an extension of the one-dimensional ball and beam problem. It is com- posed of a metallic ball rolled freely on a flat plate by the two-dimensional deflection caused by electromechani- cal actuators. The plate must be mounted on a special type of spherical joint to facilitate this sort of movement, which usually approximates the ball and plate system application to the robotics field [1]. Although the system was first introduced back in the 1990s [2], the studies tend to diverge from the initial assumptions about the system design principles. In [3], a general proce- dure for manufacturing a ball and plate system prototype has been introduced. The system was based on a resistive touch B Haider M. Jassim 60168@uotechnology.edu.iq Hazem I. Ali 60143@uotechnology.edu.iq Amjad F. Hasan Afhmr6@mail.missouri.edu 1 University of Technology, Baghdad, Iraq screen for sensing the position of the ball and electrical servo- motors entangled to the plate to generate the plate deflection angles. Another research extracted the ball location infor- mation from a continuous stream of images acquired by a top-positioned camera [4–6]. A similar approach used a camera and a computer algorithm that can identify coloured ball on a black plate surface then tracks its location which can acquire the ball location faster than other shape-based algorithm [7]. The research in [8] utilized dual pneumatic rotary cylinders as an actuator to generate the plate deflec- tion angles. It offered a safe, low cost, and good low-speed control accuracy to the system; however, it over complicates the controller design because of the high nonlinearities intro- duced by the pneumatic actuators. In addition, most designs employed some sort of metallic ball with associated, rel- atively heavy, known mass. In contrast, [6] used a hollow plastic ball with negligible mass which significantly affects the system modelling and automatically decoupled the two inputs parameters. A nonsymmetric gimbal design intro- duced in [7]. The design offered a wider range of the input deflection angles than the traditional designs since the second motor was mounted on the plate itself which allows greater flexibility in each axis rotation. A similar, but more restricted, system which involves stabilizing a ball on a rigid triangle is found in [9]. The main objective of this system is to examine the effectiveness of disturbance rejection algorithms. The stabilization of the ball at a specified location on the plate represents the task of many research studies. One 123