Narrow Results

The approach to research the uncertainties and the regularities of nonlinear system behavior is developed in the paper. The peculiarity of the approach is connected with modifying the experimental bifurcation diagrams that allows to identify and to analyze certain aspects of the nonlinear dynamics evolution. The variety and the interrelation of the modified bifurcation diagrams are shown. Attention is focused on estimating the limits of the zone within which the nonlinear system behavior is characterized by the uncertainty and on making visible the particular tendencies of dynamics evolution within this zone. The results obtained on the experimental setup of PWM buck converter are used for illustrations. In particular, the uncertainty zone is assumed to be the unit of measurement to estimate the operating process stability margin. It is demonstrated that the properties corresponding to the practical experience appear as per this assumption.

The paper is devoted to the novel logic (SUC-logic) of the nonlinear dynamics forecasting. The SUC-logic is based on three main points: the special sections (S) to build the 2D projections of multidimensional spaces without the loss of useful information; the special units (U) of measurement to estimate the nonlinear dynamics evolution; the special consecutions (C) to realize the nonlinear dynamics forecasting step-by-step. The fractal approach to forecasting the nonlinear dynamics in real-time together with the approach to build the modified bifurcation diagrams to research the regularities and the uncertainties of the evolution scenarios are developed with the SUC-logic. The physical meaning of the uncertainty zone, the stability margin, the risk estimation, the farthest forecasting and the earliest forecasting are considered from the viewpoint of the nonlinear dynamics aspect. Reasonings and discussions are based on experimental and computational results.

The popularity of systems of pulse energy conversion (PEC-systems) for practical applications is due to the heightened efficiency of energy conversion processes with comparatively simple realizations. Nevertheless, a PEC-system represents a nonlinear object with a variable structure, and the bifurcation analysis remains the basic tool to describe PEC dynamics evolution. The paper is devoted to the discussion on whether the scientific viewpoint on the natural nonlinear dynamics evolution can be involved in practical applications. We focus on the problems connected with stability boundaries of an operating regime. The results of both small-signal analysis and computational bifurcation analysis are considered in the parametrical space in comparison with the results of the experimental identification of the zonal heterogeneity of the operating process. This allows to propose an adapted stability margin as a sufficiently safe distance before the point after which the operating process begins to lose the stability. Such stability margin can extend the permissible operating domain in the parametrical space at the expense of using cause-and-effect relations in the context of natural regularities of nonlinear dynamics. Reasoning and discussion are based on the experimental and computational results for a synchronous buck converter with a pulse-width modulation. The presented results can be useful, first of all, for PEC-systems with significant variation of equivalent inductance and/or capacity. We believe that the discussion supports a viewpoint by which the contemporary methods of the computational and experimental bifurcation analyses possess both analytical abilities and experimental techniques for promising solutions which could be practice-oriented for PEC-systems.

The paper continues the discussion on bifurcation analysis for applications in practice-oriented solutions for pulse energy conversion systems (PEC-systems). Since a PEC-system represents a nonlinear object with a variable structure, then the description of its dynamics evolution involves bifurcation analysis conceptions. This means the necessity to resolve the conflict-of-units between the notions used to describe natural evolution (i.e. evolution of the operating process towards nonoperating processes and vice versa) and the notions used to describe a desirable artificial regime (i.e. an operating regime). We consider cause-effect relations in the following sequence: nonlinear dynamics-output signal-operating characteristics, where these characteristics include stability and performance. Then regularities of nonlinear dynamics should be translated into regularities of the output signal dynamics, and, after, into an evolutional picture of each operating characteristic. In order to make the translation without losses, we first take into account heterogeneous properties within the structures of the operating process in the parametrical (P-) and phase (X-) spaces, and analyze regularities of the operating stability and performance on the common basis by use of the modified bifurcation diagrams built in joint PX-space. Then, the correspondence between causes (degradation of the operating process stability) and effects (changes of the operating characteristics) is decomposed into three groups of abnormalities: conditionally unavoidable abnormalities (CU-abnormalities); conditionally probable abnormalities (CP-abnormalities); conditionally regular abnormalities (CR-abnormalities). Within each of these groups the evolutional homogeneity is retained. After, the resultant evolution of each operating characteristic is naturally aggregated through the superposition of cause-effect relations in accordance with each of the abnormalities. We demonstrate that the practice-oriented bifurcation analysis has fundamentally specific purposes and tools, like for the computer-based bifurcation analysis and the experimental bifurcation analysis. That is why, from our viewpoint, it seems to be a rather novel direction in the general context of bifurcation analysis conceptions. We believe that the discussion could be interesting to pioneer research intended for the design of promising systems of pulse energy conversion.

The paper continues the discussion on the bifurcation analysis conceptions for applications in practice-oriented solutions for pulse energy conversion systems (PEC-systems). This viewpoint means an attempt to resolve so-called conflict-of-units between the notions used to describe natural evolution (i.e. evolution of the operating process towards nonoperating processes and vice versa) and the notions used to describe a desirable artificial regime (i.e. the operating regime). In this connection, the correspondence between causes (degradation of the operating process stability) and effects (changes of the operating characteristics) is established in the following sequence: nonlinear dynamics output signal operating characteristics, where these characteristics include stability and performance. Two starting points follow from the previous parts. First, there are distinguishable thresholds of evolutional degradation, between which the operating process loses stability in a particular manner; and these particularities can be systematized by means of so-called uncertainty zones. Second, multi-D integrating translations of phase images with regular parametrical variation are combined with the corresponding boundaries of the operating stability and performance by means of the modified bifurcation diagrams. Then we focus on the basic characteristics of transients and first demonstrate and discuss some unified form of the modified bifurcation diagrams, to which a solution on the operating performance can be reduced. Namely, we show such solutions for an overshoot and a settling time in comparison with a control error. We believe that the practice-oriented bifurcation analysis could be interesting to pioneer research intended for the design of promising systems of pulse energy conversion.

One of the most illustrative examples of promising applications of the practice-oriented bifurcation analysis concerns forecasting. We focus on emergency forecasting for pulse energy conversion systems (PEC-systems). In this case it becomes necessary to integrate regularities from both phase and parametrical spaces and demands on the operating stability and performance, taking into account the critical demand to uninterrupted analytics during steady states and transients. Thus so-called conflict-of-units between the notions used to understand natural evolution (for example, the evolution connected with the operating process) and the notions used to describe desirable artificial regimes (for example, the operating regime) should be resolved. With this purpose, we attract a specific integrating analytics (bifurcation-fractal analytics), the basis of which is provided by modified bifurcation diagrams and fractal regularities. Here the fractal regularities mirror geometrical similarities between shapes of limit cycles and mirror regular dimensional modifications of these shapes with parametrical variation. Modified bifurcation diagrams provide the conciliation of the practicing and scientific notions without distortions and losses of the useful information from parametrical and phase spaces. Then multi-D conflict-free correspondence between causes (degradation of the operating process stability) and effects (changes of the operating regime characteristics) is established, and empirical recommendations on the operating performance can be substituted by clear nonlinear regularities. Fractal methods of real-time forecasting during transients are included in the discussion and their adaptation to the emergency forecasting is proposed. It opens a novel way on how to forecast the operating stability and performance on the common basis of nonlinear regularities which indicate the operating changes towards emergencies. The discussion is illustrated by computer-based and experimental examples. We believe that the results seem to be interesting to researchers in the field of the practice-oriented bifurcation analysis.

The paper presents a discussion on an opinion about the stability margin towards an emergency in local climate dynamics from the bifurcation analysis viewpoint. With this purpose we propose to attract the practice-oriented bifurcation analysis, where the conflict-of-units between notions used to understand natural evolution processes and notions used to describe desirable artificial regimes is resolved by integrating analytics on the basis of modified bifurcation diagrams. The discussion focuses on the phenomenon of interannual temperature variability, where local annual maximums and minimums are analyzed with daily details in both time and temperature coordinates. This phenomenon is considered via the probable, periodical and regulator conceptions. Advantages of the regulator conception are verified by results of processing the data of temperature meteorological observations on daily means over the last 135 years. This conception is based on the HDS-hypothesis, in accordance to which local climate dynamics is determined by the natural competition between the amplitude quantization (restricted by the temperature Hysteresis) and time quantization (caused by the Double Synchronization). Thus an alternation between three elementary processes with the same period (year) and different patterns of annual warming–cooling cycles is supposed as a typical behavior for local climate systems, and the idea on high-dynamic local climate ensembles is developed instead of the conventional opinion on quasi-static local climate norms. Mechanisms of temperature changes due to abrupt shifts (so-called change-points) of the HDS-regulator parameters are distinguished from mechanisms of temperature changes due to bifurcations. The notion of a stability margin is used as a distance to an emergency and is visualized in the parametrical space. So, in spite of the mechanisms of temperature changes with/without bifurcations are different, their conflict-free sewing becomes conceptually possible in the context of the stability margin towards emergencies determined relatively bifurcation boundaries in the parametrical space. Since the discussed dynamics is not supposed to exist in terms of the traditional estimations concerning the observed local climate changes, then we believe that the paper would be interesting for scientists in the field of bifurcation analysis as well as for scientists and specialists, activity areas of which relate to the contemporary challenges connected with climate changes.

A new method is proposed for the design of a biquadratic (biquad) allpass phase-compensating system whose stability-margin satisfies an arbitrarily preset stability criterion. The design technique utilizes both the generalized stability-triangle (GST) and the bilinear (BL) phase-error function of the all-pass biquad system, which are derived by the author. Based on the GST condition, it is shown that the original coefficients of the allpass biquad system can be converted into the functions of other two new variables, while the two variables have no limits (bounds) on their values to satisfy the GST condition. That is, arbitrary values of the two new variables can meet the GST condition. Based on the above variable conversions, the design technique further employs a nonlinear optimization method to find the optimum values of the two variables to approximate a prescribed ideal phase. Thanks to the variable conversitions, the resulting values of the two variables never violate the GST condition. As a result, the resulting biquad allpas system not only can always satisfy a prescribed stability margin, but also can best approximate a given ideal phase in the minimax sense. To demonstrate the guaranteed stability margin as well as the system accuracy, a biquad system design example is included.

This paper concerns the control of an autonomous high mobility wheel-legged rover crossing uneven terrains. A new control strategy, using active redundancies of the robot, leads to elaborate a posture control based on the potential field approach of the stability measurement. Then a decoupled posture and trajectory control algorithm based on the velocity model of the robot is proposed. Last, simulation results showing performance of the control algorithm are presented.