Abstract—Despite previous efforts to solve linear and nonlinear deadbeat control systems, a need still exists for better methodology in terms of performance and stability. This paper proposes a new design methodology for deadbeat control of nonlinear systems in discrete-time. The proposed methodology is based on partitioning the solution into two components; each with different sampling time. The proposed control can be divided into two sub-controllers: one uses state feedback and the other uses the Diophantine equations. The complete nonlinear design guarantees the convergence to a neighborhood of origin from any initial state in finite time; thus, providing a stable deadbeat performance. Results shows that the ripple-free deadbeat controller is able to track the input signal and the error decays to zero in a finite number of sampling times. Index Terms—Deadbeat control, diophantine equations, multi-rate, output-feedback linearization. I. INTRODUCTION Digital deadbeat controller offers the fastest settling time in control theory. Thus, deadbeat controller ensures that the error sequence vanishes at the sampling instants after a finite time [1, 2]. Due to the nonlinearity nature of plants and processes, the deadbeat control technique must be improved in order to overcome the nonlinearity and discretization. All digital deadbeat techniques for linear systems have a common property: all poles of the closed-loop transfer function should are moved to the origin of the z-plane either by using state-feedback [3-4], Diophantine equations design methods [5, 6], or any other technique. Paz [7] proposed a two-degree-of freedom controller design for a well-known transfer function addressing performance and robustness specifications for linear systems. The controller is given in terms of the solution of two Diophantine equations. Shifting closed loop poles of nonlinear system to the origin may not be acceptable, thus; using full state feedback to deadbeat nonlinear system is not a good technique. Salgado and Oyarzun [8] proposed two objective optimal multivariate ripple free deadbeat controls with simple parameterization. The designed controller dealt with step input for linear system. Yamada [9] proposed a parameterization of all multivariable ripple-free deadbeat tracking controller that handled various input signals for linear systems. Elaydi and Albatsh [10] and Albatsh [11] solved the multirate ripple-free deadbeat control for linear Manuscript received March 31, 2012; revised July 17, 2012. H. A. Elaydi is with the Electrical Engineering Department at the Islamic University of Gaza, Gaza, Palestine (e-mail: helaydi@iugaza.edu.ps). Mohammed Elamassie is with University College of Applied Science, Gaza, Palestine (e-mail: melamassie@ucas.edu.ps). systems using the Diophantine equations. In this paper, multi-rate deadbeat control for nonlinear system is proposed based on evaluating the solution of the two independent Diophantine equations for second order approximated model of a linearized nonlinear system. Nonlinear system will be linearized using full state-feedback linearization. This paper will show simulation results of the designed controller on the nonlinear plant. This paper is organized as follow: section 2 talks about material and methods where it states the problem formulation and talks about state feedback and Diophantine equations control designs, section 3 covers results and discussion by stating the constraints and design steps and solving examples to show the effectiveness of the proposed method, section 4 concludes this paper. II. MATERIALS AND METHODS A. Problem Formulation Controlling a nonlinear system in a ripple-free deadbeat manner is quite challenging. Typical procedures that are normally followed in linear system are no longer valid here. The problem here is the nonlinearity region and the robustness of the controller around this region. The ripple-free deadbeat controller for nonlinear system, shown in Fig. 1, consists of the following two design steps: First step is concerned with time-domain approach such as state and output feedbacks with integral controller that is used to linearize and stabilize nonlinear system with sampling time T1 to make the response of nonlinear system closely equal the reference signal. The second step is concerned with polynomial approach namely the Diophantine equations design methods based on the internal model principle are utilized and applied to the linearized and stabilized nonlinear system with sampling time T2 to make the response of the system exactly equal the reference signal and provide some robustness. Fig. 1. Multi-rate ripple-free deadbeat controller for nonlinear system The designed controller is based on using different sampling times in order to ensure that we can use different sampling times in two sub controllers (1-full-state feedback, and 2- Diophantine equations) and to decrease the processing time by decreasing sampling rate of one sub-controller if we can. Multi-rate Ripple-Free Deadbeat Control for Nonlinear Systems Using Diophantine Equations Hatem Elaydi and Mohammed Elamassie IACSIT International Journal of Engineering and Technology, Vol. 4, No. 4, August 2012 489