Research Article Linear optimal control of continuous time chaotic systems Kaveh Merat, Jafar Abbaszadeh Chekan, Hassan Salarieh n , Aria Alasty Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran article info Article history: Received 11 June 2013 Received in revised form 4 December 2013 Accepted 17 January 2014 This paper was recommended for publica- tion by Mohammad Haeri. Keywords: Chaos Optimal control Continuous time system Rossler system AFM system abstract In this research study, chaos control of continuous time systems has been performed by using dynamic programming technique. In the first step by crossing the response orbits with a selected Poincare section and subsequently applying linear regression method, the continuous time system is converted to a discrete type. Then, by solving the Riccati equation a sub-optimal algorithm has been devised for the obtained discrete chaotic systems. In the next step, by implementing the acquired algorithm on the quantized continuous time system, the chaos has been suppressed in the Rossler and AFM systems as some case studies. & 2014 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The nonlinear dynamical systems which exhibit chaos phenom- enon are appeared in many fields of science such as engineering, economy, ecology, and engineering. The concept of chaos has been introduced in 1975 by Li and Yorke for the first time [1]. The elimination of chaos has been studied in many researches and investigations, and different techniques including optimal approaches, and advanced nonlinear methods were utilized for chaos stabilization. The primary technique which is well known as the OGY method was proposed by Ott et al. for stabilizing the unstable periodic orbits embedded in a chaotic attractor [2]. In addition, some approaches based on OGY method such as SOGY have been presented to enhance the performance of control [3,4]. Pyragas [5] also proposed the delayed feedback technique to stabilize the unstable periodic orbits. Further- more, the model free control of the Lorenz chaotic system is performed by utilizing an approach based on an approximate optimal control in [6]. In [7] an optimal control policy has been introduced to control a chaotic system via state feedback. In the mentioned study, at first the system has been converted to an uncertain piecewise linear model and then an optimal controller has been designed which minimizes the upper bound on cost function under constraints in the form of bilinear matrix inequality. In some other case studies, synchronization of chaos is concerned instead of controlling chaotic systems. For instance in [8], Jayaram and Tadi utilized State Dependent Riccati Equation (SDRE) method to synchronize chaotic systems. Furthermore, Park synchronized two chaotic systems using a non- linear controller designed based on the Lyapunov stability theory [9]. In some other studies, Robust, adaptive and nonlinear control theory, also were applied. Cao introduced a nonlinear adaptive method for controlling a chaotic oscillator [10]. In [11] Layeghi et al. stabilized periodic orbits of chaotic systems by applying fuzzy adaptive sliding mode control. The Adaptive Lyapunov-based control which is another nonlinear control approach was hired by Salarieh and Shahrokhi in [12] to suppress the chaotic motion. Also, Fuh et al. introduced a Robust controller which combines feedback linearization and distur- bance observer to suppress chaotic motion in a nonlinear system which is under external excitation [13]. Zhang and Tang [14] studied the dynamics of a new chaotic system containing two system parameters with nonlinear terms, using the Lyapunov exponents. They both stabilized and synchronized the mentioned chaotic system globally, using a linear state feedback controller, designed through the simple sufficient conditions resulted from the Lyapunov stability criteria. Dynamics of a new three dimensional chaotic system contain- ing a nonlinear term in the form of arc-hyperbolic sine function was studied in [15], in which the system has been converted to an uncertain piecewise linear system. Using piecewise quadratic Lyapu- nov function method, the chaos phenomenon has been controlled globally in this system with alpha-stability constraint and via piece- wise linear state feedback. The same authors in [16], by applying Lyapunov stability criteria, synchronized chaos in a chaotic system with a nonlinear term which does not satisfy Lipschitz continuity but satisfies 1/2- Hölder continuity. Since the target of our study is stabilizing chaotic systems via dynamic programming algorithm, reviewing some relevant works is useful. Dynamic programming concept was introduced in 1957 by Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2014.01.003 0019-0578 & 2014 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding Author. Tel: þ982166165538. E-mail address: salarieh@sharif.edu (H. Salarieh). Please cite this article as: Merat K, et al. Linear optimal control of continuous time chaotic systems. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.003i ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎