Cycles of the second kind for uncertain pendulum-like systems with several nonlinearities
The authors study the pendulum-like system with several nonlinearities \[ \begin{aligned}\dot{\sigma}&=\xi,\\ \dot{\xi}&=-\varphi(\xi,\sigma)+x^T C f(\sigma)+g(\sigma),\\ \dot{x}&=Ax+Bf(\sigma)\xi,\end{aligned} \] where \(A\in \mathbb R^{n\times n}\), \(B,C\in \mathbb R^{n\times m}\) are constant matrices, \(f\colon \mathbb R\rightarrow \mathbb R^m\) and \(g\colon \mathbb R\rightarrow \mathbb R\) are \(\Delta\)-periodic functions, and \(\varphi\colon \mathbb R\times \mathbb R\rightarrow \mathbb R\) is a function satisfying \[ \begin{aligned}&\varphi(\xi,\sigma+\Delta)=\varphi(\xi,\sigma)\\ &\mu_1\xi^2\leq \varphi(\xi,\sigma)\xi\leq \mu_2 \xi^2\end{aligned} \] for all \((\xi,\sigma)\in \mathbb R\times \mathbb R\) with constants \(\mu_1,\mu_2\). Using the Kalman-Yakubovich-Popov lemma, the authors derive sufficient conditions for the existence of cycles of the second kind for the above system in the form of matrix inequalities. These results are applied to robust analysis and synthesis of uncertain pendulum-like systems with norm-bounded parameter uncertainties. In particular, it is shown that the static state feedback controller and dynamic output feedback controller problems for a system with uncertainties are equivalent to the same problems for a system without uncertainties.
scientific article; zbMATH DE number 5290572