# A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems
## 1 Introduction  
For designing an optimal controller, the Hamiltonian model is frequently used to formulate the main equations of state and co-state models, which are required for the controller formulation in a logical approach. Recently, many research works on the development of optimal controllers, which can improve aspects of control performances such as reducing the cost function (pertaining to control action) or leading to enhanced system stability, have been undertaken. In this aspect, the objective of this work is to introduce a new design method to achieve adaptive gains of the optimal controller for the linear time-varying system. In addition, several optimal controllers, which are robust to parameter uncertainties and disturbances, have been proposed by many researchers. An adaptive optimal control for unknown constrained input systems was formulated in [1] where the Hamilton-Jacobi-Bellman (HJB) was used, and then combined with neural networks for treating of uncertain parameters of the system. Another study of adaptive optimal control was presented in [2] with the Hamilton-Jacobi-Isaacs (HJI) function. The combination of Hamiltonian function and Riccati-like equation for decentralized system was also studied in [3]. It is known that the HJB has been successfully used for optimal controlof unknown systems with disturbances [4]. The linear quadratic regulator (LQR) was also applied for underactuated problem [5], and the capacity in synchronizing of control and plant [6]. An optimal control based on game theory was presented in [7], in which the control model was a combination of the game theory with the term of the Pareto model. There was an optimal control law with a multiphase cost function and the inverse model [8], in which the recovery matrix theory was used to change the matrix in the cost function. On the other hand, the Bolza-Meyer criterion associated with the Hamiltonian function is frequently used to formulate advanced optimal control laws [9; 10; 11; 12; 13; 14]. In these studies, the Hamiltonian function is used as a basis to formulate new optimal controllers. In addition, the application of sign function of the sliding mode control logic is undertaken to design a new advanced optimal control law, which can bring several benefits such as less control effort and higher stability. In recent years, several optimal sliding mode controllers based on Riccati-like equations have been proposed in [15; 16; 17; 18], where the sliding mode control and fuzzy neural networks models are integrated. From the above literature survey, various types of optimal control laws have been developed to enhance control performances, and the robustness of the systems subjected to unknown parameters and disturbances. However, the development of new optimal control laws, which can provide the superior control performances to existing optimal control laws, is still attractive. In particular, an optimal controller based on criteria such as the Meyer, Bolza-Meyer, HJB and HJI term needs to be improved since the value of definite symmetry matrices (gains) is 1 or less than 1. This value limits both control performances and robustness of the optimal controllers [9; 10; 11; 12; 13; 14]. This problem becomes more serious when conventional models of optimal control laws are used. The chosen matrices in [9; 10; 11; 12; 13; 14] were 1 or less than 1 because of the inconsequential method (i.e., shooting method) to solve the differential equation of the optimal control. The value of matrix function [9; 10; 11; 12; 13; 14] was predicted, and then the shooting method was applied following the value of gain function at a final time. It normally takes a long time to finish the calculation. Consequently, the main technical contribution with the originality of this work is to develop a new optimal control law, which can eliminate the drawback of conventional optimal controllers, based on a certain criterion used in [9; 10; 11; 12; 13; 14].