Robust neural adaptive tracking control for nonlinear systems (Q2785105)

The present note is devoted to the second-order nonlinear system NEWLINE\[NEWLINE\begin{aligned} \dot x_1 &= g_1(x_1)x_2+ f_1(x_1),\\ \dot x_2 &= g_2(x)u+ f_2(x),\tag{2}\\ y &= x_1,\end{aligned}NEWLINE\]NEWLINE where \(x= [x_1,x_2]^T\in \mathbb{R}^2\) is the state-vector, \(u\in \mathbb{R}\) and \(y\in\mathbb{R}\) denote the control-input and output, respectively, and \(f_i(\cdot)\), \(g_i(\cdot)\) \((i= 1,2)\) are unknown (possibly not linearly parameterizing) nonlinear functions.NEWLINENEWLINENEWLINEIt is assumed that, (i) the sign of the function \(g_i\) \((i=1,2)\) is known and there exist a constant \(g_{i0}> 0\) and a known function \(g_{ib}(\cdot)\) such that \(g_{ib}(\cdot)\geq|g_i(\cdot)|\geq g_{i0}> 0\); (ii) the expected tracking vector \(x_d= [y_d,\dot y_d]^T\) can be obtained as a bounded continuous vector.NEWLINENEWLINENEWLINEThe authors establish a robust neural adaptive output tracking control design for system (1) by using some quadratic Lyapunov functions. They shown that, under the proposed adaptive control, all the signals in the closed-loop system are bounded and the chosen neural network weight updating law can prevent drift of the adaptive network weights. A simulation example is also given.