Providing the invariance property on the basis on oscillation modes (Q393884)

The authors study the problem of zero asymptotic output regulation for a linear system of the form \[ \begin{aligned} &\dot x_1 = A_{11}x_1+A_{10}x_0 + Q_1f(t),\\ &\dot x_0 = A_{01}x_1+A_{00}x_0 + Q_0f(t)+Bu,\\ & y=Dx_1,\end{aligned} \] where the pair \((A_{11},A_{10})\) is controllable, \(\operatorname{Im} Q_1 \subseteq \operatorname{Im} A_{10}\), \(B\) is of maximal rank and \(f(t)\) represents the vector of the (in general, unmatched) disturbances. The function \(f(t)\) is unknown, but it is bounded together with its first and second derivatives. The paper contains three results, with some differences both in the assumptions and in the conclusions, based on the explicit construction of discontinuous feedback laws, generating nonlinear oscillations in the system.