Robust adaptive control for nonlinear uncertain systems
The authors develop a direct robust adaptive control framework for nonlinear uncertain systems \[ \dot x(t)= f(x(t)) + \Delta f(x(t)) + G(x(t))u(t),\qquad x(0)=x_0,\quad t\geq 0, \] where \(x(t)\in\mathbb R^n\) is the state vector, \(u(t)\in\mathbb R^m\) is the control input, \(f:\mathbb R^n \to\mathbb R^n\) and satisfies \(f(0) = 0\), \(\Delta f:\mathbb R^n \to\mathbb R^n\) and satisfies \(\Delta f(0) = 0\), and \(G:\mathbb R^n \to\mathbb R^{n\times m}\), \(t\geqslant 0\). It is assumed that \(f(\cdot)\) and \(\Delta f(\cdot)\) are uncertain. In particular, the uncertainty in \(f(\cdot)\) is captured by a constant linearly parametrized system uncertainty model of a known structure but unknown variation, while \(\Delta f(\cdot)\) denotes structured (possibly nonlinear) parametric uncertainty with bounded variation. The structured uncertainty \(\Delta f(\cdot)\) can effectively capture the residual approximation error inherent in the linear parametrization of the system uncertainty \(f(\cdot)\) as well as capture structured nonlinear (possibly state-dependent) uncertainty in the system of known bounded variation. The proposed framework is Lyapunov-based and guarantees partial asymptotic robust stability of the closed-loop system; that is, asymptotic robust stability with respect to part of the closed-loop system states associated with the plant. A numerical example is provided to demonstrate the efficacy of the proposed approach.
scientific article; zbMATH DE number 1900996