On robust ADDPS of nonlinear uncertain systems (Q2725167)

Based on the backstepping method, the uncertain version of the robust almost disturbance decoupling problem with stability is defined and investigated for the nonlinear uncertain disturbed SISO systems NEWLINE\[NEWLINE\left\{\begin{aligned} \dot z&=\Phi(x_1,z)+\Psi^T(x_1,z)w,\\ \dot x_1&=x_2+p_1^T(x_1,z)w,\\ \dot x_2&=x_3+p_2^T(x_1,x_2,z)w,\\ &\dots\\ \dot x_r&=v+p_r^T(x_1,x_2,\dots,x_r,z)w+\Delta f(x_1,x_2,\dots,x_r,z,t),\\ y&=x_1,\end{aligned}\right.NEWLINE\]NEWLINE where \(z\in\mathbb R^{n-r}\), \(x_i\in\mathbb R\), \(i=1,2,\dots,r< n\), are state variables; \(v\), \(y\in\mathbb R\), \(w\in\mathbb R^p\) are control input, output, and disturbance input, respectively; \(\Phi(\cdot)\), \(\Psi(\cdot)\), \(p_i(\cdot)\) are smooth mappings, \(\Phi(0,0) = 0; \Delta f\) represents the system uncertainty, \(\Delta f(0,\dots,0,t) = 0\), \(\forall t\geqslant 0.\) The analysis provides a practical constructive procedure for these systems, guaranteeing that the influence from the disturbance on the controlled output can be arbitrarily small and the closed-loop systems are internally stable. The uncertainty here is of time-varying feature, and has a time-varying bound, therefore the controller is time-varying. The controller design need not solve any Hamilton-Jacobi equation.