IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 25, NO. 5, SEPTEMBER 2017 1833 Nonlinear Suboptimal Tracking Controller Design Using State-Dependent Riccati Equation Technique Yazdan Batmani, Mohammadreza Davoodi, and Nader Meskin Abstract— In this brief, a new technique for solving a suboptimal tracking problem for a class of nonlinear dynamical systems is presented. Toward this end, an optimal tracking problem using a discounted cost function is defined and a control law with a feedback-feedforward structure is designed. A state-dependent Riccati equation (SDRE) framework is used in order to find the gains of both the feedback and the feedforward parts, simultaneously. Due to the significant properties of the SDRE technique, the proposed method can handle the presence of input saturation and state constraint. It is also shown that the tracking error converges asymptotically to zero under mild conditions on the discount factor of the corresponding cost func- tion and the desired trajectory. Two simulation and experimental case studies are also provided to illustrate and demonstrate the effectiveness of our proposed design methodology. Index Terms— Input saturation, linear quadratic tracking, optimal control, state-dependent Riccati equation (SDRE), time-varying desired trajectory. I. I NTRODUCTION O PTIMAL control deals with the problem of finding a control law in order to achieve the best possible behavior with respect to a predefined criterion. The optimal quadratic regulation problem for linear systems was solved in the 1960s [1] and the obtained results were also extended to the optimal tracking problem for linear systems [1], [2]. Nevertheless, in many practical engineering problems, the system to be controlled is nonlinear. Due to the complexity of the arising Hamilton–Jacobi–Bellman (HJB) equation, which is too difficult or even impossible to be solved, various methods were developed to find approximate solutions of the nonlinear optimal regulation problem (see [3]–[5]). Although some methods were proposed to solve the optimal tracking problem for nonlinear systems [6], [7], it can be said that much less attention has been paid to this problem. The state-dependent Riccati equation (SDRE) technique was originally proposed by Pearson in 1962 to approxi- mately solve the optimal regulation problem for nonlinear systems [8]. Representing a nonlinear system dynamics as a state-dependent linear system, called the pseudo-linearization or extended linearization [9], is the main idea of the SDRE technique. Since then several methods have been Manuscript received February 10, 2016; revised July 20, 2016; accepted September 29, 2016. Date of publication October 31, 2016; date of current version August 7, 2017. Manuscript received in final form October 9, 2016. This work was supported by NPRP under Grant NPRP 5-045-2-017 from the Qatar National Research Fund (a member of Qatar Foundation). Recommended by Associate Editor A. G. Aghdam. Y. Batmani is with the Department of Electrical Engineering, University of Kurdistan, Sanandaj, Iran (e-mail: y.batmani@uok.ac.ir). M. Davoodi and N. Meskin are with the Department of Electrical Engineering, Qatar University, Doha, Qatar (e-mail: davoodi.mr@qu.edu.qa; nader.meskin@qu.edu.qa). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2016.2617285 developed based on the pseudo-linearization framework to solve different problems such as robust H ∞ filter design [10], suboptimal sliding mode control design for delayed sys- tems [11], observer design for nonlinear delayed systems [12], and so on. These methods were effectively applied in a wide variety of applications, such as drug administration in cancer treatment [13] and dive plane control of autonomous underwater vehicles (AUVs) [14]. Two complete surveys of the SDRE techniques and the related theories can be found in [8] and [9]. For the set-point tracking problem, the SDRE technique is developed based on the integral action method [8]. However, to the best of our knowledge, the optimal tracking control problem for nonlinear systems, which is practically very important, has not been solved using the SDRE technique. The main reason for this shortage is that the quadratic cost function used in the SDRE technique is only valid for the desired trajec- tories generated by an asymptotically stable system. However, many of desired trajectories, such as steps and sinusoidal signals, are not generated by such systems. This problem and interesting properties of the SDRE method, such as simplicity and flexibility of the SDRE design procedure, ability to con- sider input saturation, and maintaining the nonlinear charac- teristics of the system, motivate us to develop an SDRE-based control design method for the nonlinear tracking problem. Toward this end, a discounted cost function is used to tackle the above-mentioned problem and define an optimal tracking problem for more general desired trajectories. Then, the opti- mal nonlinear tracking problem is converted into an optimal nonlinear regulation problem and the SDRE technique is used to find a suboptimal solution of the obtained optimal regulation problem or equivalently a solution of the original optimal tracking problem. The proposed method inherits almost all of the interesting properties of the SDRE technique such as ability to consider input saturation, robustness with respect to parametric uncertainties and unmodelled dynamics, and so on. The preliminary result of this brief is presented in [15]. In this brief, the stability of the proposed tracking controller is investigated and a theorem is also presented to find proper values of the discount factor. The results of applying the proposed method to two simulation and experimental case studies are also presented to illustrate the effectiveness and capabilities of the proposed design methodology. The remainder of this brief is organized as follows. In Section II, we first define an optimal tracking problem for a broad class of nonlinear dynamical systems and then using the pseudo-linearization technique, a method is proposed to find a suboptimal solution of the tracking problem. The asymptotic stability of the closed-loop system is also investigated in this section. In Section III, results of applying the proposed method 1063-6536 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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