A new nonlinear adaptive control law for a class of uncertain nonlinear systems is proposed. The proposed control law is designed by a modified adaptive control Lyapunov function (ACLF) which satisfies a Hamilton-Jacobi-Bellman (HJB) equation. The modified ACLF is derived from transformation of an ACLF. The proposed control law is different from the inverse optimal one in decreasing the value of a cost function specified by a designer. In this paper, we show a transformation coefficient for an ACLF and a design method of a nonlinear adaptive controller. Finally, it is shown by a numerical simulation that the proposed control law decreases the value of a given cost function and achieves the desirable trajectory. 1. Introduction Design of control laws considering stability and optimality is the central issue in control theory [1]. For stability, Lyapunov theory is a strong tool to design controllers and to assure the stability of systems. For optimality, a value function which is the solution to a Hamilton-Jacobi-Bellman (HJB) equation is derived from dynamic programming. If a value function and an optimal control law can be found, then the closed system possesses robustness such as gain margin, phase margin, and low sensitivity against parameter variations [2, 3]. However, a general approach to find the value function has not been shown and it is not easy to design the optimal control. Due to the difficulty, the inverse optimal control problem which minimizes a meaningful cost function was proposed by Freeman and Kokotovic. If the inverse optimal problem is solved, namely, a control Lyapunov function (CLF) is found, it is possible to design a control law with the good characteristics mentioned above by applying a CLF to the Pointwise Min-Norm (PMN) control law [4]. But the minimized cost function and the trajectory may not be desirable. In order to improve this problem, a locally approximate approach around the origin by numerical calculation and transformation of a CLF was proposed. The approach gives characteristics of local optimality without loss of characteristics of the global inverse optimality [5, 6], and it is based on the fact that the PMN control law can minimize the desired cost function if a CLF has the same level sets as the value function [7]. The inverse optimal control law was applied to robust control and adaptive control [8, 9]. Moreover, a control law in which a Sontag type control law and a PMN control law were generalized has been proposed [10–12]. Also, many approaches to approximate the value function have been proposed. They are
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