Research Article A new optimal sliding mode controller design using scalar sign function Mithun Singla a,n , Leang-San Shieh a , Gangbing Song b , Linbo Xie c , Yongpeng Zhang d a Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204, United States b Department of Mechanical Engineering, University of Houston, Houston, TX 77204, United States c Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Wuxi, Jiangsu 214122, China d Department of Engineering Technology, Prairie View A & M University, Prairie View, TX 77446, United States article info Article history: Received 27 December 2012 Received in revised form 13 August 2013 Accepted 6 September 2013 This paper was recommended for publication by Dr. Q.-G. Wang Keywords: Chattering Optimal control Sliding mode control Scalar sign function abstract This paper presents a new optimal sliding mode controller using the scalar sign function method. A smooth, continuous-time scalar sign function is used to replace the discontinuous switching function in the design of a sliding mode controller. The proposed sliding mode controller is designed using an optimal Linear Quadratic Regulator (LQR) approach. The sliding surface of the system is designed using stable eigenvectors and the scalar sign function. Controller simulations are compared with another existing optimal sliding mode controller. To test the effectiveness of the proposed controller, the controller is implemented on an aluminum beam with piezoceramic sensor and actuator for vibration control. This paper includes the control design and stability analysis of the new optimal sliding mode controller, followed by simulation and experimental results. The simulation and experimental results show that the proposed approach is very effective. & 2013 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Sliding mode controllers are known for their robustness and stability. The concept of Sliding Mode Control (SMC) has received much attention in the past few decades. The concept of sliding mode was first proposed by Utkin [1] as a Variable Structure System (VSS) controller and showed that a sliding mode could be achieved by changing the controller structure. In the sliding mode controller, the system state trajectory is forced to move along a chosen stable manifold, called the sliding manifold, in the state space. The sliding manifold is always chosen in such a manner that derived control law guarantees the system stability. Young et al. [2] presented a guide for control engineers to design different sliding mode controllers. Since SMC's inception, many different techniques have been proposed for choosing the sliding manifold, sometimes referred to as a sliding mode. Xu et al. [3] proposed an optimal sliding mode controller to solve the infinite time optimal control problem. An LQR approach was used to calculate the optimal gain for the sliding mode controller and to deal with uncertainties stochasti- cally. Laghrouche et al. [4] proposed another higher order sliding mode control based on an optimal LQR approach. Many complex hybrid sliding mode controller structures also have been proposed in association with other techniques, such as adaptive control techniques and fuzzy control techniques [5–10]. These techniques ensure asymptotical stability and the reduction of chattering. However, most of these hybrid controllers require complex imple- mentation algorithms. Nikkah et al. [11] proposed a novel method based on nonlinear predictive control to design optimal linear sliding surfaces for control of under-actuated systems. In this method, the proposed sliding surface is a combination of the classic linear surface and an adaptive time varying linear component. In their approach, even if optimization of the system is not feasible, the controller has to be implemented, making this approach a bit cumbersome. On the other hand, Niu et al. [12] proposed an improved sliding mode control algorithm for discrete time systems. They proposed a new reaching law for the sliding surface; however, the proposed reaching law still has some conditions to satisfy. Shieh et al. [13] proposed a robust sliding mode control approach for magnetic levitation systems. In their approach, integral sliding mode control with a robust optimal approach was developed to achieve high performance in position tracking. Li et al. [14] implemented a PD-sliding mode hybrid controller to control the speed of perma- nent magnet synchronous motor robustly. Sliding Mode control algorithm has been used extensively by researchers in vibration control of flexible structures. Song et al. [15] has implemented sliding mode based controller to suppress the vibrations of a flexible beam. Later, Gu et al. [16] has imple- mented a fuzzy logic based adaptive sliding mode controller for Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.09.007 n Corresponding author. Tel.: þ1 713 410 05 92. E-mail address: mithunsingla@gmail.com (M. Singla). Please cite this article as: Singla M, et al. A new optimal sliding mode controller design using scalar sign function. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.007i ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎