Adaptive neural network control for strict-feedback nonlinear systems using backstepping design (Q5926187)

An adaptive neural network control scheme is proposed for nonlinear systems transformable into the strict-feedback canonical form NEWLINE\[NEWLINE\dot x_i= f_i(x_1,\dots, x_i)+ g_i(x_1,\dots, x_i) x_{i+1},\;i= 1\leq i\leq n-1,\quad \dot x_n= f_n(x)+ g_n(x)u,\quad y= x_1,NEWLINE\]NEWLINE where \(f_i(\cdot)\), \(g_i(\cdot)\), \(i= 1,\dots, n\), are unknown smooth functions that cannot be linearly parameterized and \(x= [x_1,\dots, x_n]^T\in \mathbb{R}^n\), \(u\in\mathbb{R}\), \(y\in\mathbb{R}\) denote the state variables, system input and output, respectively. The initial step of the design procedure uses an integral-type Lyapunov function to provide a smooth and singularity-free adaptive controller for a first-order plant. Then, an extension is made to high-order nonlinear systems, which exploits neural network approximation and adaptive backstepping techniques. The uniform ultimate boundedness of the closed-loop adaptive system is proved and it is also shown that the tracking error converges to a small residual set, adjustable by the design parameter timing. Such a tuning can be guided by a relationship formulated between the transient performance and the design parameters. Several simulation experiments illustrate the effectiveness of the proposed adaptive controller.