Self-synchronization of unbalanced rotors and the swing equation

We consider a problem of self-synchronization in a system of vibro-exciters (rotors) installed on a common oscillating platform. This problem was studied by I.I. Blekhman and later by L. Sperling. Extending their approach, we derive the equations for a system of n rotors and show that, separating the slow and fast motions, the “slow” dynamics of this systems reduces to a special case of a so-called swing equation that is well studied in theory of power networks. On the other hand, the system may be considered as “pendulum-like” system with multidimensional periodic nonlinearities. Using the theory of such systems developed in our previous works, we derive an analytic criteria for synchronization of two rotors. Unlike synchronization criteria available in mechanical literature, our criterion ensures global convergence of every trajectory to the synchronous manifold

Constructive Estimates of the Pull-In Range for Synchronization Circuit Described by Integro-Differential Equations

The pull-in range, known also as the acquisition or capture range, is an important characteristics of synchronization circuits such as e.g. phase-, frequency- and delay-locked loops (PLL/FLL/DLL). For PLLs, the pull-in range characterizes the maximal frequency detuning under which the system provides phase locking (mathematically, every solution of the system converges to one of the equilibria). The presence of periodic nonlinearities (characteristics of phase detectors) and infinite sequences of equilibria makes rigorous analysis of PLLs very difficult in spite of their seeming simplicity. The models of PLLs can be featured by multi-stability, hidden attractors and even chaotic trajectories. For this reason, the pull-in range is typically estimated numerically by e.g. using harmonic balance or Galerkin approximations. Analytic results presented in the literature are not numerous and primarily deal with ordinary differential equations. In this paper, we propose an analytic method for pull-in range estimation, applicable to synchronization systems with infinite-dimensional linear part, in particular, for PLLs with delays. The results are illustrated by analysis of a PLL described by second-order delay equations

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

The sunflower equation: novel stability criteria

In this paper, we consider a delayed counterpart of the mathematical pendulum model that is termed sunflower equation and originally was proposed to describe a helical motion (circumnutation) of the apex of the sunflower plant. The “culprits” of this motion are, on one hand, the gravity and, on the other hand, the hormonal processes within the plant, namely, the lateral transport of the growth hormone auxin. The first mathematical analysis of the sunflower equation was conducted in the seminal work by Somolinos (1978) who gave, in particular, a sufficient condition for the solutions’ boundedness and for the existence of a periodic orbit. Although more than 40 years have passed since the publication of the work by Somolinos, the sunflower equation is still far from being thoroughly studied. It is known that a periodic solution may exist only for a sufficiently large delay, whereas for small delays the equation exhibits the same qualitative behavior as a conventional pendulum, and every solution converges to one of the equilibria. However, necessary and sufficient conditions for the stability of the sunflower equation (ensuring the convergence of all solutions) are still elusive. In this paper, we derive a novel condition for its stability, which is based on absolute stability theory of integro-differential pendulum-like systems developed in our previous work. As will be discussed, our estimate for the maximal delay, under which the stability can be guaranteed, improves the existing estimates and appears to be very tight for some values of the parameters

Convergent and oscillatory solutions in infinite-dimensional synchronization systems

Control systems that arise in phase synchronization problems are featured by infinite sets of stable and unstable equilibria, caused by presence of periodic nonlinearities. For this reason, such systems are often called “pendulum-like”. Their dynamics are thus featured by multi-stability and cannot be examined by classical methods that have been developed to test the lobal stability of a unique equilibrium point. In general, only sufficient conditions for the solution convergence are known that are usually derived for pendulum-like systems of Lurie type, that is, interconnections of stable LTI blocks and periodic nonlinearities, which obey sector or slope restrictions. Most typically, these conditions are written as multi-parametric frequency-domain inequalities, which should be satisfied by the transfer function of the system’s linear part. Remarkably, if the frequencydomain inequalities hold outside some bounded range of frequencies, then the absence of periodic solutions with frequencies in this range is guaranteed, which can be considered as a weaker asymptotical property. It should be noticed that validation of the frequency domain stability condition for a given structure of the known linear part of the system is a self-standing nontrivial problem. In this paper, we demonstrate that a previously derived frequency-domain conditions for stability and absence of oscillations can be substantially simplified, parameters ensuring the corresponding asymptotic property. We demonstrate the efficiency of new criteria on specific synchronization systems

The development of Lyapunov direct method in application to synchronization systems

The paper is devoted to asymptotic behavior of synchronization systems, i.e. Lur'e–type systems with periodic nonlinearities and infinite sets of equilibrum. This class of systems can not be efficiently investigated by standard Lyapunov functions. That is why for synchronization systems several new methods have been elaborated in the framework of Lyapunov direct method. Two of them: the method of periodic Lyapunov functions and the nonlocal reduction method, proved to be rather efficient. In this paper we combine these two methods and the Kalman-Yakubovich-Popov lemma to obtain new frequency–algebraic criteria ensuring Lagrange stability and the convergence of solutions

Leonov’s method of nonlocal reduction for pointwise stability of phase systems

In this paper we go on with the analysis of the asymptotic behavior of Lur'e-type systems with periodic nonlinearities and infinite sets of equilibria. It is well known by now that this class of systems can not be efficiently investigated by the second Lyapunov method with the standard Lur'e-Postnikov function ("a quadratic form plus an integral of the nonlinearity"). So several new methods have been elaborated within the framework of Lyapunov direct method. The nonlocal reduction technique proposed by G.A. Leonov in the 1980s is based on the comparison principle. The feedback system is reduced to a low-order system with the same nonlinearity and known asymptotic behavior. Its trajectories are injected into Lyapunov function of the original system. In this paper we develop the method of nonlocal reduction. We propose a new Lyapunov-type function which involves both the trajectories of the comparison system and a modified Lur'e-Postnikov function. As a result a new frequency-algebraic criterion ensuring the convergence of every solution to some equilibrium point is obtained