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 non-linearities. 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 global 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 frequency-domain 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 non-trivial problem. In this paper, we demonstrate that a previously derived frequency-domain conditions for stability and absence of oscillations can be substantially simplified, which allows to tighten the estimates on the system’s parameters ensuring the corresponding asymptotic property. We demonstrate the efficiency of new criteria on specific synchronization systems


Introduction
In this paper, we go on with developing multiparametric frequency-algebraic criteria for the asymptotic behavior of infinite-dimension synchronization systems described by integral-differential equations (see [Smirnova and Proskurnikov, 2019] and bibliography there).
The simplest example of the synchronization system is the model of mathematical pendulum with viscous friction.So such systems are often called pendulum-like.Synchronization (or pendulum-like) control systems are featured by the properties like those of the mathematical pendulum.Their linear parts have zero eigenvalues, their nonlinear parts are periodic.
The peculiarities of synchronization systems imply the specific character of their asymptotic behaviorThese systems have, as a rule, denumerable sets of equilibria, both Lyapunov stable and unstable ones.The global stability of the synchronization system is usually treated as the convergence of every solution.Correspondingly, the methods of stability investigation devised for control systems with the unique equilibrium have turned out to be of no good for synchronization systems.These methods had to be adjusted for a new kind of asymptotic behavior.That is why the special tool has been developed [Gelig et al., 2004;Leonov et al., 1996] within the framework of traditional methods.
If a synchronization system is not stable, it may have periodic oscillations.Since high-frequency oscillations are undesired, the problem appeared to establish the estimates for the frequencies of possible oscillations.
The special procedures developed for stability investigation of synchronization systems turned out to be rather fruitful here.These procedures were combined with the tool of Fourier series and in such a way a number of frequency-algebraic criteria for the absence of highfrequency oscillations was obtained ( see [Perkin et al., 2015] and bibliography there).The criteria were formulated in terms of the transfer function of the linear part of the system.They had the form of frequency-domain inequalities with varying parameters.The criteria give the opportunity to obtain estimates for domains with low-frequency oscillations.The more is a number of varying parameters the more "flexible" is the frequencyalgebraic criteria and the more tight are the estimates for such domains in the space of parameters of the system.
In this paper, being based of results of [Proskurnikov and Smirnova, 2020] we choose the optimal algebraic conditions for varying parameters.
The optimal frequency-algebraic criterion for the absence of high-frequency oscillations is appllied to phaselocked loops with the proportional-integrating filter.
The main problem for synchronization system is to provide its global stability.
Definition 1.We say that equation ( 1) is globally stable (is gradient-like) if its any solution converges to a certain equilibrium.
Another significant problem is to guarantee the absence of high-frequency oscillations.
Definition 2. We say that equation ( 1) has a periodic solution σ(t) if there exist a number T > 0 and an integer I such that The number ω = 2π T is called the frequency of the periodic solution.In case I = 0 we fix a periodic solution of the first kind.If I ̸ = 0 we call σ(t) a periodic solution of the second kind.
In [Smirnova and Proskurnikov, 2019] a number of frequency-algebraic theorems have been proved both for global stability and for the absence of periodic solutions.In this paper, we examine the problem of the absence of high-frequency periodic solutions.Our goal is to get the most accurate upper estimates for the frequencies of possible oscillations.

The absence of high frequency periodic solutions
Frequency-domain criteria are based on fundamental characteristics of linear and nonlinear parts of the control system.
The linear part of synchronization system (1) is entirely defined by its transfer function from the input φ to the output (− σ): The periodic function φ(σ) is characterized by two constants

It is clear that
We introduce then a non-negative accompanying function where , and constants A 2 be varying parameters.We shall need the functions: From [Smirnova and Proskurnikov, 2019] we deduce, for a SISO system the following assertion.
Let us analyze the conditions of Theorem 1.Notice first that if κ = 0 the inequality ( 12) is violated for ω = 0 and if κ ̸ = 0 one can scale the parameters ε > 0 and τ > 0 by |κ|.So we can put Consider algebraic condition 1).It is clear that one should choose the value of a in such a way that the value of δ is the minimal one.It follows from 1) that δ > 1 4 It is easy to calculate that the minimal value of the right-hand part of ( 16) is achieved for and The value a 0 is "the best" for the fixed set {κ, ε, τ, α} since it corresponds to the minimal possible value of δ.So inequality (16) should be substituted by δ > M (ε, τ, α). (18) Compare now the right-hand parts of inequalities ( 14) and ( 18).Notice that Proof.It is sufficient to prove that Consider the function Notice that and So by Cauchy-Bunyakovsky-Schwarz inequality Thus ( 20) is true and consequently the Proposition is proved.
It follows that Theorem 1 can be substituted by the following assertion.
Theorem 2 implies that a triple {ε, τ, α} of varying parameters is the element of the set If Ξ is an empty set (i.e.Ξ = Ø) then Theorem 2 is inapplicable.
Next we determine Notice that every triple of varying parameters (ε, τ, α) ∈ Ξ defines "its own" value of ω.
The transfer function of (28) from φ to (− σ) is as follows System ( 28) can be easily reduced to the integraldifferential equation (1) with 4.1 The undelayed PLL with a sine-shaped characteristic of phase detector (PD) In this case System ( 28), (32) has three parameters: s, T, β.
Then we fix the value of δ small enough and choose a step h ω = h ω (δ) in such a way that the inequalities Thus we find the value of ω such that inequality (40) is true.Changing the values of m ∈ (0; 1], ε ∈ (0; 1) and τ ∈ (0; 1 − ε) with small steps we define the value of ω 0 as the minimal of all possible ω.
In Fig. 2 the domains with ω0 T equal to 2, 4, 7 are shown.They are situated under the curves.The leftmost one corresponds to the gradient-like system.We confine ourselves to the variant of It is easy to calculate that Inequality ( 24) has the form (36).In Fig. 3 for s = 0.2 the domains of slow oscillations with ω < ω 0 = 3, 8, 10 are shown.The leftmost curve corresponds to a globally stable system (ω 0 = 0).

Conclusion
This paper is devoted to the asymptotic behavior of infinite-dimensional synchronization (pendulum-like) systems.In particular, the problem of the absence of high-frequency oscillations is considered.Some frequency-algebraic criteria for the absence of oscillations of the prescribed frequency are analyzed, and the optimal one is applied to phase-locked loops with the proportionally-integrating filter.

Figure 1 .
Figure 1.Domains where the oscillations with the frequency more than ω 0 are absent for undelayed PLL with PIF and a sine-shaped characteristic of PD

Figure 2 .
Figure 2. Domains of slow oscillations for PLL with PIF, a sineshaped characteristic of PD, and delay h = 0.1T

Figure 3 .
Figure 3. Domains where the oscillations with the frequency more than ω 0 are absent for undelayed PLL with PIF and a triangle characteristic of PD