Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture

The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were constructed in such a way that one locally stable limit cycle (hidden oscillation) co-exists with a locally stable equilibrium. In this paper we demonstrate a multistable configuration of three co-existing hidden oscillations (limit cycles) and a locally stable equilibrium in the phase space of the fourth-order system, which provides a new class of counterexamples to the Kalman conjecture.peerReviewe

Hold-in, Pull-in and Lock-in Ranges for Phase-locked Loop with Tangential Characteristic of the Phase Detector

In the present paper the phase-locked loop (PLL), an electric circuit widely used in telecommunications and computer architectures is considered. A new modification of the PLL with tangential phase detector characteristic and active proportionally-integrating (PI) filter is introduced. Hold-in, pull-in and lock-in ranges for given circuit are studied rigorously. It is shown that lock-in range of the new PLL model is infinite, compared to the finite lock-in range of the classical PLL.peerReviewe

Analysis of oscillations in discontinuous Lurie systems via LPRS method

We discuss advantages and limitations of the harmonic balance method and the locus of a perturbed relay system (LPRS) method in the problem of finding periodic oscillations. In this paper we present the results of using harmonic balance method and LPRS method while investigating a 3rd order dynamic system in Lurie form. In this system a symmetric periodic oscillation is found, while other two asymmetric periodic motions are not found using both methods.peerReviewe

Counterexamples to the Kalman Conjectures

In the paper counterexamples to the Kalman conjecture with smooth nonlinearity basing on the Fitts system, that are periodic solution or hidden chaotic attractor are presented. It is shown, that despite the fact that Kalman’s conjecture (as well as Aizerman’s) turned out to be incorrect in the case of n > 3, it had a huge impact on the theory of absolute stability, namely, the selection of the class of nonlinear systems whose stability can be studied with linear methods.peerReviewe