NEW RESULTS ON CYCLE–SLIPPING IN PENDULUM–LIKE SYSTEMS

In this paper, we examine dynamics of multidimensional control systems obtained as feedback interconnec-tions of stable linear blocks and periodic nonlinearities. The simplest of such systems is the model of mathematical pendulum (with viscous friction), so we call such systems pendulum-like . Other examples include, but are not limited to, coupled vibrating units, networks of oscillators, Josephson junction arrays and numerous synchronization circuits used in radio and telecommunica-tion engineering. Typically, a pendulum-like system has inﬁnite sequence of equilibria, and one of the central problems addressed in the theory of such systems is to ﬁnd the conditions of “global stability”, or gradient-like behavior ensuring that every solution converges to one of the equilibria points. If a system is gradient-like, another problem arises, being the main concern of this paper: can we ﬁnd the terminal equilibrium, given the initial condition of the system? It is well known that solutions do not converge, in general, to the nearest equilibrium; this phenomenon is known as cycle-slipping . For a pendulum, cycle-slipping corresponds to multiple rotations of the pendulum about its suspension point. In this paper, we estimate the number of slipped cycles for general pendulum-like systems by means of periodic Lyapunov functions and the Kalman-Yakubovich-Popov lemma.


Introduction
A number of systems, arising in natural sciences and engineering, can be represented by a feedback interconnection of an asymptotically stable linear stationary system and periodic nonlinearity. The simplest example of such a system is a pendulum with viscous damping, inspiring thus the term "pendulumlike". Sometimes, systems with periodic nonlinearities are also called synchronization (synchronous control) systems [Leonov, 2006;Hoppensteadt, 1983] in view of numerous applications to synchronization circuits such as phase,frequency and delay-locked loops (PLL/FLL/DLL) [Margaris, 2004;Best, 2003;Leonov and Kuznetsov, 2014]. Other examples include, but are not limited to, electric motors, power generators, single and coupled Josephson junctions, networks of coupled oscillators and neurons [Baker and Blackburn, 2005;Stoker, 1950;Blekhman, 2000;Monteiro et al., 2003;Hoppensteadt and Izhikevich, 2000;Imry and Schulman, 1978;Qin and Chen, 2004].
Pendulum-like systems are featured by complex multistable dynamics with infinite sequences of stable and unstable equilibria points (in fact, their natural phase space is a cylindric or toric manifold [Kudrewicz and Wasowicz, 2007;Leonov et al., 1996]). Many effects in such systems, e.g. oscillations, hidden attractors and "cycle slipping" [Chicone and Heitzman, 2013;Leonov et al., 2015b;Leonov et al., 2015a;Best et al., 2016;Dudkowski et al., 2016] cannot be examined by tools of classical nonlinear control and require special techniques. One of the central problems, concerned with dynamics of systems, is the convergence of all solutions to equilib-ria points. This counterpart of global asymptotic stability in pendulum-like systems with unique equilibrium is referred to as the gradient-like behavior [Leonov, 2006;Duan et al., 2007]; the gradient-like behavior excludes, in particular, limit cycles and other hidden attractors.
Another important problem concerned with pendulum-like systems is to find the equilibrium to which the solution is attracted. In general, the solution does not converge to the nearest equilibrium, as exemplified by the mathematical pendulum with friction or the stepper motor [Stoker, 1950]. The pendulum can make several turns around the suspension point before calming down at the lower equilibrium. Similarly, a stepper motor can skip steps. This phenomenon is known as cycle slipping. Cycle slipping in synchronization circuits is usually considered to be undesired behavior as the continuous changing of the phase error leads to demodulation and decoding errors. This motivated extensive research on the cycle slipping phenomena in the engineering community. For more than 50 years (since the seminal paper [Viterbi, 1963]) these studies have mainly focused on stochastic cycle slipping, caused by random noise [Ascheid and Meyr, 1982;Sancho et al., 2014]. However, solutions of a deterministic system with periodic nonlinearities can also slip several cycles under some initial conditions; obtaining non-conservative estimates for the phase error increment is a non-trivial problem even for a low-order dynamics.
In this paper, the frequency-domain estimates for the number of slipped cycles are extended to the systems with external disturbances. Obviously, if such a disturbance persistently excites the solution (being e.g. harmonic or other periodic oscillatory signal), the solution no longer converges to an equilibrium point but rather oscillates. In synchronization systems, such disturbances are typically modeled as combinations of stationary random signals and polyharmonic signals [Hill and Cantoni, 2000;Cataliotti et al., 2007;Schilling et al., 2010] to be rejected or damped. In this paper, we deal with other type of disturbances that have finite limit at infinity (being thus combinations of constant and decaying signals), enabling thus the disturbed system to have equilibria. Frequency-algebraic criteria for gradient-like behavior of synchronization systems with disturbances have been proved in [Smirnova et al., 2018a;Smirnova et al., 2018b]. In this paper, we obtain several frequencyalgebraic estimates for the number of cycles slipped un-der the influence of an external force.

The problem setup
Consider a phase synchronization system with external disturbances described by the equations (1) Here A ∈ R m×m , b, c ∈ R m , ρ ∈ R are constant, the symbol * stands for Hermitian conjugation.
Henceforth the following assumptions are adopted. A1. The pair (A, b) is controllable, the pair (A, c) is observable, the matrix A is a Hurvitz matrix.
In this paper we go on with stability investigation of forced synchronization systems, addressing the problem of cycle-slipping. We extend the results from [Perkin et al., 2013] to forced synchronization systems. The estimates are formulated in terms of the transfer function of the linear part: In case f (t) ≡ 0 the estimates become more complicated. They depend on parameters of external disturbance, namely, the constants

The case of non-differentiable nonlinearity
In this section we consider the case when there is no information about the derivative ψ (σ). So we have to waive the C 1 -smoothness of ψ(σ) and confine ourselves to the assumption of ψ(σ) ∈ C(R). Then we cannot use the auxiliary construction (12) and the quadratic form G(y, η) with τ > 0. We have to admit that τ = 0. Consequently the frequency inequality (13) takes the form The inequality (63) guarantees [Gelig et al., 2004] the existence of matrix H 0 = H * 0 such that Since (64) implies that we conclude that H 0 is positive definite. Theorem 3. Suppose there exist k ∈ N, κ = 0 and ε, δ > 0 such that the following requirements are true: 1) the frequency-domain inequality (63) holds; 2) there exists a matrix H 0 = H * 0 , satisfying (64) such that inequalities where r j are defined by (16) and M 0 is taken from (19), are valid.
Proof. The proof of this theorem is alike that of Theorem 1. Let and ε 0 be so small that the inequalities are valid. We introduce the Lyapunov-type functions where and F j are defined by (23). Then we compute the derivatives of Lyapunov-type functions in virtue of system (8): It follows from (64) and (8) thaṫ where U (κ, ε, τ ; t) is defined by (18). From (71) by virtue of (67) we get the estimate Suppose that Then Notice that Since H 0 is positive definite one has which contradicts (72). So Using the function V 1 (t) we can easily prove that σ(t) < σ(0) + k∆.