Natalia V. Utina

Frequency-domain estimates for transient attributes of discrete phase systems

A multidimensional discrete control system with periodic nonlinearity is investigated. Its two important charac- teristics: the transient time and a number of slipped cycles are considered. By means of Lyapunov direct method and Yakubovich-Kalman theorem certain estimates of these two attributes are obtained. They are formulated as frequency- domain criteria.

Frequency estimates for the number of cycle slippings in a phase system with nonlinear vector function

Two classes of phase control systems with vector nonlinearities are considered: systems described by ordinary differential equations and system described by difference equations. They are characterized by the presence of a periodic vector nonlinearity in the mathematical description of the system. The problem of the number of cycle slippings is investigated. For both classes of control systems, frequency estimates of the deviation of each angular coordinate from its initial value are obtained. The estimation technique is based on the direct Lyapunov method with periodic Lyapunov functions. With the use of the Yakubovich-Kalman lemma, all results are formulated in terms of the transfer function of the linear part of the system. The results obtained have the form of frequency inequalities with variable parameters, which satisfy some algebraic inequalities.

Coordinatewise estimates for vector outputs of multivariable phase control systems

Two classes of multidimensional phase control systems with differentiable vector periodic functions are considered, the class of continuous systems described by ordinary differential equations and the class of discrete systems described by difference equations. The number of cycle slips for angular coordinates in phase systems with differentiable nonlinearities is studied. The study is based on the direct Lyapunov method and uses periodic Lyapunov functions, extensions of the phase space of the system, and the Yakubovich-Kalman lemma. This lemma provides necessary and sufficient conditions for the existence of Lyapunov functions by using the transfer matrix of the linear part of the system. As a result, for phase systems possessing global asymptotics, frequency criteria making it possible to sharpen estimates for the deviation of angular coordinates from their initial values are obtained. These criteria contain multiparameter frequency inequalities with variable parameters satisfying certain algebraic inequalities.

Cycle slipping in nonlinear circuits under periodic nonlinearities and time delays

Phase-locked loops (PLL), Costas loops and other synchronizing circuits are featured by the presence of a nonlinear phase detector, described by a periodic nonlinearity. In general, nonlinearities can cause complex behavior of the system such as multi-stability and chaos. Even if the phase locking is guaranteed for any initial conditions, the transient behavior of the circuit can still be unsatisfactory due to the cycle slipping. Growth of the phase error caused by cycle slipping is undesirable, leading e.g. to demodulation and decoding errors. This makes the problem of estimating the phase error oscillations and number of slipped cycles in nonlinear PLL-based circuits extremely important for modern telecommunications. Most mathematical results in this direction, available in the literature, focus on the phase jitter and cycle slipping under random noise and examine the relations between the probabilistic characteristics of the noise and of the phase error, e.g. the expected number of slipped cycles. At the same time, cycle slipping occurs also in deterministic systems with periodic nonlinearities, depending on the initial conditions, properties of the linear part and the periodic nonlinearity and other factors such as delays in the loop. In the present paper we give analytic estimates for the number of slipped cycles in PLL-based systems, governed by integro-differential equations, allowing to capture effects of high-order dynamics, discrete and distributed delays. We also consider the effects of singular small-parameter perturbations on the cycle slipping behavior.

Problem of cycle-slipping for infinite dimensional systems with MIMO nonlinearities

The concept of cycle slipping was introduced by J.J. Stocker for the mathematical pendulum with friction. After an impact, the pendulum make several revolutions around the suspension point before settling down in the lower stable equilibrium. The number of those revolutions is referred to as the number of cycles, slipped by the solution. In general, the number of slipped cycles may be defined for any system with periodic nonlinearity and gradient-like behavior, being an important characteristic of the transient process. In phase-locked loop (PLL) based systems, this number shows how large may be the phase error before the locking, which makes the problem of cycle slipping important for telecommunications and electronics. The problem addressed in the present paper is how to estimate the number of slipped cycles in infinite-dimensional systems, consisting of linear Volterra-type equation in the interconnection with a periodic MIMO nonlinearity. The techniques developed in the paper stem from the Popovs approach of “a priori integral indices”, which was proposed originally as a tool for proving stability of nonlinear systems and was the prototype for the method of integral quadratic constraints (IQC). Employing novel types of Popov-type quadratic constraints, we obtain new frequency-domain estimates for the number of cycles slipped.

Application of the method of Lyapunov periodic functions

The paper is concerned with asymptotic behavior of continuous and discrete phase control systems involving periodic differentiable nonlinear vector functions and featuring nonunique equilibrium state.Two stability problems are examined in sequence: the problem of global asymptotics and the problem on the number of cycle slippings. A conventional tool in attacking such problems is the second Lyapunov method. However, Lyapunov functions of the kind “quadratic form” or “quadratic form plus the integral of nonlinear function,” which are conventional in the control theory, are not capable of solving these problems for the class of systems under consideration. As a result, several new methods were put forward in the 1960s and 1970s to deal with phase control systems in the framework of the second Lyapunov method. In this work, one of these methods is exploited; namely, the method of Lyapunov periodic functions. Extensions of well-known Lyapunov periodic functions and sequences are proposed permitting us to refine estimates of regions of global asymptotics in the space of parameters of a phase system. Multiparameter frequency criteria for global asymptotics are formulated; they allow for improvement of the aforementioned estimates. The frequency criteria obtained are used as well to establish estimates for the number of cycle slippings.

Stability and oscillations of singularly perturbed phase synchronization systems with distributed parameters

In this paper we examine singularly perturbed systems, described by integro-differential Volterra equations with periodic nonlinearities and a small parameter at the higher derivative. This type of equations describes many “pendulumlike” systems as well as phase-locked loops and other synchronization circuits. Such systems usually have infinite sequence of equilibrium points. The main problem for this class of control systems is the problem of gradient-like behavior, i.e. whether any solution converges to one of the equilibria. In this paper we propose a frequency-algebraic criterion of gradient-like behavior for singular perturbed systems. If the system is not gradient-like it may have periodic regimes. We give constructive frequency-domain conditions, which guarantee that all periodic solutions in the system, if they exist, have frequencies lower than some predefined constant. An important property of this estimate is its uniformity with respect to the small parameter.

Transient processes in synchronization systems governed by singularly perturbed Volterra equations

Many natural phenomena and engineering applications are based on synchronization between several periodic processes. A commonly known example is a phase-locked loop (PLL), that is, a feedback circuit providing synchronization between the endogenous oscillator and exogenous periodic signal in phase. Imprecise recovery of the signals phase leads to decoding and demodulation errors, so the properly designed PLL must provide the fast “phase-locking”, i.e. convergence of the phase shift (or error) to a steady value. In the simplest situation the error converges to the nearest equilibrium; in general, due to the initial conditions or disturbances, it can leave this basin of attraction and converge to another equilibrium. During this undesirable transient process, referred to as the cycle slipping, the phase shift is increased by a multiple of the period. The mechanical analog of a slipping PLL is a pendulum, making several turns around the point of suspension before stabilization at the lower equilibrium. In this paper we estimate the number of slipped cycles for a class of synchronization systems, governed by integro-differential Volterra equation with a scalar periodic nonlinearity and including, particularly, PLLs with discrete or distributed delays in the loop. These estimates are extended to singularly perturbed Volterra equations, having a small parameter at the higher derivative. Such models naturally arise if the system has “slow” and “fast” dynamics.

Conditions for the absence of cycles of the second kind in continuous and discrete systems with cylindrical phase space

Continuous and discrete indirect control systems with periodic vector nonlinearities are considered. One of the key problems concerning the asymptotic behavior of such systems is studied, namely, the existence of cycles in a cylindrical phase space. Multiparameter frequency-algebraic criteria for the absence in a cylindrical phase space of a system of cycles with given frequency are obtained.