Leonov’s nonlocal reduction technique for nonlinear integro-differential equations

Starting from pioneering works by Lur’e, Popov and Zames, global stability theory for nonlinear control systems has been primarily focused on systems with only one equilibrium. Global stability criteria for other kinds of attractors (such as e.g. infinite sets of equilibria) are not well studied and typically require special tools, primarily based on the Lyapunov method. Analysis of stability becomes especially complicated for infinite-dimensional dynamical systems with multiple equilibria, e.g. systems described by delay or more general convolutionary equations. In this paper, we propose novel stability criteria for infinite-dimensional systems with periodic nonlinearities, which have infinite sets of equilibria and describe dynamics of phase-locked loops and other synchronization circuits. Our method combines Leonov’s nonlocal reduction technique with the idea of Popov’s “integral indices” and allows to obtain new frequency-domain conditions, ensuring the convergence of every solution to one of the equilibria points

Phase limitations of Zames-Falb multipliers

Phase limitations of both continuous-time and discrete-time Zames-Falb multipliers and their relation with the Kalman conjecture are analysed. A phase limitation for continuous-time multipliers given by Megretski is generalised and its applicability is clarified; its relation to the Kalman conjecture is illustrated with a classical example from the literature. It is demonstrated that there exist fourth-order plants where the existence of a suitable Zames-Falb multiplier can be discarded and for which simulations show unstable behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is developed. Its application is demonstrated with a second-order counterexample to the Kalman conjecture. Finally, the discrete-time limitation is used to show that there can be no direct counterpart of the off-axis circle criterion in the discrete-time domain

Variance-constrained multiobjective control and filtering for nonlinear stochastic systems: A survey

The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixed H 2 / H ∞ control and filtering problems are reviewed in great detail. Subsequently, some latest results on the variance-constrained multi-objective control and filtering problems for the nonlinear stochastic systems are summarized. Finally, conclusions are drawn, and several possible future research directions are pointed out