Conditions For Stability Analysis Research Articles

Stability and Dynamic Performance Improvement of Adaptive Full-Order Observers for Sensorless PMSM Drive

Stability and good dynamic performances of adaptive full-order observers are of utmost importance for the sensorless PMSM drive. In this paper, to accomplish both requirements, the analytical stability conditions are firstly derived to provide a general framework for the feedback gain design. Closed-form solutions of the stabilizing feedback gains are consequently given, and are used in the zero and pole placement design to obtain an adequate and constant damping factor along with the stability at all operating frequencies. New design rules for the adaptation of PI gains to satisfy the required performances are also proposed. The robustness of the adaptive full-order observer against stator resistance and inductance variations is also investigated. Validity of the designed sensorless control is confirmed by simulation and experiment.

Codimension‐Two Bifurcations of Fixed Points in a Class of Discrete Prey‐Predator Systems

The dynamic behaviour of a Lotka‐Volterra system, described by a planar map, is analytically and numerically investigated. We derive analytical conditions for stability and bifurcation of the fixed points of the system and compute analytically the normal form coefficients for the codimension 1 bifurcation points (flip and Neimark‐Sacker), and so establish sub‐ or supercriticality of these bifurcation points. Furthermore, by using numerical continuation methods, we compute bifurcation curves of fixed points and cycles with periods up to 16 under variation of one and two parameters, and compute all codimension 1 and codimension 2 bifurcations on the corresponding curves. For the bifurcation points, we compute the corresponding normal form coefficients. These quantities enable us to compute curves of codimension 1 bifurcations that branch off from the detected codimension 2 bifurcation points. These curves form stability boundaries of various types of cycles which emerge around codimension 1 and 2 bifurcation points. Numerical simulations confirm our results and reveal further complex dynamical behaviours.

Controlling synchrony by delay coupling in networks: From in-phase to splay and cluster states

We study synchronization in delay-coupled oscillator networks using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of supercritical or subcritical Hopf bifurcation), we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.

Robust stability of the new general 2D model of a class of continuous-discrete linear systems

The problems of asymptotic stability and robust stability of the new general 2D model of scalar linear dynamic continuous-discrete systems, standard and positive, are considered. Simple analytic conditions for asymptotic stability and for robust stability are given. These conditions are expressed in terms of coefficients of the model. The considerations are illustrated by numerical examples. The methods proposed can be generalized to scalar Fornasini-Marchesini and Roesser models of 2D continuous-discrete systems.

Analytical Stability Condition of the Latency Insertion Method for Nonuniform GLC Circuits

The latency insertion method (LIM) is a transient simulation technique for circuits and is based on a finite-difference formulation, like the well-known finite-difference time-domain (FDTD) method for solving Maxwells equations. The LIM, like the FDTD method, is only conditionally stable resulting in an upper bound for the time step of the transient simulation. This bound on the time step is a function of the circuit topology and the circuit element values. It is critical to know this bound analytically for a given circuit. However, stability conditions of the LIM have been proven only for 1-D, infinitelylong, distributed uniform RLC circuits, employed in transmission line modeling. For nonuniform circuits, these conditions have been predicted and have been observed experimentally as well but have not been possible to prove using the existing stability analysis techniques. Recently, analytical stability conditions of the LIM for nonuniform RLC circuits have been proven using the Lyapunovs direct method (LDM). However, when a conductance to ground (G) is added to a node of an LC or RLC circuit, the stability conditions cannot be derived using the Lyapunov function proposed. In this brief, analytical stability condition of the LIM is derived for the first time for nonuniform GLC circuits using the LDM with a new Lyapunov function.

Stability and Accuracy of a Finite-Difference Time-Domain Scheme for Modeling Double-Negative Media With High-Order Rational Constitutive Parameters

This paper introduces an extension of the original finite-difference time-domain (FDTD) method for modeling double-negative media characterized by high-order frequency-dependent permittivity and permeability. The approach basically consists of adding electric and magnetic current densities to Maxwell's curl equations and considering Ohm's law as a constitutive relationship. Current densities are discretized by using a weighted average in time and Ohm's law by applying the Mobius transformation technique. The extended FDTD formulation is validated and its numerical features are carefully examined. More specifically, analytical stability conditions are derived for several types of double-negative media and the numerical dissipation issue is discussed. In addition, the numerical dispersion equation for general high-order double-negative media is given and the order of accuracy of the scheme is studied. Finally, the definition of numerical refractive index is addressed and it is shown that, when the discretization parameters of the problem are not properly chosen, a negative refractive index may become a positive one in the discrete world, thus changing the physics of the problem.

Robust stability analysis of adaptive control based on recurrent ANN

Adaptive control by means of neural networks for non-linear dynamical systems is an open issue. For real world applications, practitioners have to pay attention to external disturbances, parameters uncertainty and measurement noise, as long as these factors will influence the stability of the closed loop system. As a consequence, robust stability of the closed loop controlled by neural network is an important issue that must be considered. Our contribution concerns the robustness analysis and synthesis of adaptive indirect control scheme. This scheme is based on fully connected neural networks, and inspired from the standard real time recurrent learning. This analysis is concerned by combining Lyapunov approach and linearisation around the nominal parameters to establish analytical sufficient conditions for the global robust stability of adaptive neural network controller. Advantages of the proposed algorithm are suggested according to simulation examples.

Repeated games and direct reciprocity under active linking

Direct reciprocity relies on repeated encounters between the same two individuals. Here we examine the evolution of cooperation under direct reciprocity in dynamically structured populations. Individuals occupy the vertices of a graph, undergoing repeated interactions with their partners via the edges of the graph. Unlike the traditional approach to evolutionary game theory, where individuals meet at random and have no control over the frequency or duration of interactions, we consider a model in which individuals differ in the rate at which they seek new interactions. Moreover, once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. Whenever the active dynamics of links is sufficiently fast, population structure leads to a simple transformation of the payoff matrix, effectively changing the game under consideration, and hence paving the way for reciprocators to dominate defectors. We derive analytical conditions for evolutionary stability.

The stability of internal equilibria in predator-prey models with breeding suppression

We present two models that represent the suppression of breeding by prey in response to short-term increases in predation pressure. For both of these models, we have been able to produce analytic conditions for the local stability of the interior steady state, in terms of the values of combinations of these parameters. Although our models are as simple as possible to capture the effect of breeding suppression, the expressions for local stability, even in their simplest form, are complex. Thus, we come to the important conclusion that there is no simple and general rule for the effect of the behaviours described here (anti-predatory breeding suppression and prey switching by predators) on the stability of population dynamics. Rather, effects will be system specific. However, we hope that the results and methodological framework outlined here will provide a useful tool for others to investigate the consequences for particular real systems.

Analysis of the extended operation range of electrostatic actuators by current-pulse drive

A new driving mode of electrostatic actuators has been examined. It is based on charge drive rather than on voltage drive. Charge drive is accomplished by a pulsed current source delivering a given amount of charge to the actuator. It is theoretically shown that if the drive is ideal, any position across the gap is stable. The main reason underlying this result is that the electrostatic force does not depend on the remaining gap of the actuator and hence, the pull-in instability is avoided. These findings are supported on simulations using an analog high level description language allowing to represent the actuator by its differential equations. The effect of parasitic capacitances has also been examined and the stability conditions analyzed. A simple analytical stability condition can be written relating the value of the parasitic capacitance to the maximum deflection allowed for a stable equilibrium point. These results are also supported by simulation. Finally, the current leakage effects on the stability and drift of the equilibrium position have been analyzed. A trade-off between the relative loss of charge and the minimum refresh frequency required has been formulated.

Stability Analysis of Sliding Whirl in a Nonlinear Jeffcott Rotor with Cross-Coupling Stiffness Coefficients

An analytical study is carried out on the stability of the fullannular rub solutions of an externally excited, modified Jeffcott rotorwith a given rotor/stator clearance and cross-coupling influences. Theobtained analytical stability conditions provide an opportunity for abetter understanding of the dynamical phenomena of rotor/stator systemswith rubs, such as jump phenomena and the transition between periodicand quasi-periodic full rub responses as well as between the fullannular rubs and the partial rubs. A systematic study on the influenceof the system parameters on these phenomena is carried out. It is foundthat the simultaneous presence of the coefficient of friction and thecross-coupling stiffness coefficient with a proper value may benefit thedynamics of the rotor/stator system with rubs.

Stability of the tubular layer of a deformed material in a rotating horizontal cylinder

The stability conditions for the steady-state motion of the tubular layer of a treated deformable material in a rotating horizontal cylinder are determined analytically. With allowance for the accepted similarity criteria, universal diagrams of the boundaries of transition of modes of motion of liquid and loose materials in the cylinder are obtained on the basis of experimental data. Analysis of the diagrams shows the identity of the stability conditions for a liquid layer and a loose medium, which can be regarded as a Newtonian liquid upon fast relative motions. It is shown also that the analytical stability conditions for the liquid layer correspond to the experimental data for large Reynolds numbers when the mode hysteresis occurs and do not correspond to these data for small Reynolds numbers when secondary circulating flows form.

Analytical stability condition for the ideal m=n=1 kink mode in a toroidal plasma with elliptic cross section

An analytical stability condition for the ideal kink mode with toroidal mode number n=1 and (dominating) poloidal mode number m=1 in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity (e) is treated as a small parameter independent of the inverse aspect ratio (ε), and the expansion is performed up to order ε2e in the potential energy δW. It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when Δq=1−q0 is small (q0 is the safety factor at the magnetic axis), whereas an ellipticity of opposite sign (horizontal elongation) is stabilizing. By means of an additional expansion, in Δq, this effect is shown to be due to a Δq-independent term in δW proportional to the ellipticity and to the poloidal beta value (βp) at the q=1 surface. Since the leading order term in δW in the absence of ellipticity is of order Δq [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)], the ellipticity term dominates δW for sufficiently small values of Δq. The analytical result is in agreement with previous, numerical results [Lütjens et al., Nucl. Fusion 32, 1625 (1992)], and leads, already for rather moderately vertically elongated plasmas, to stability limits in βp much lower than the value 0.3 valid in the limit Δq→0 for a plasma with circular cross section.

Dynamic Stability of Linear Misaligned Journal Bearings Via Lyapunov's Direct Method©

One of the most important problems of rotor-bearing systems, the stability problem, especially in the case that the rotor is in misaligned position in respect with the bearing, is examined in this paper. Lyapunov's direct method was preferred in this analysis instead of a classical eigenvalue analysis in order to obtain the stability conditions in the form of analytical, expressions. The eight dynamic coefficients (four stiffness and four damping) for the aligned, journal bearing or the 32 dynamic coefficients (16 stiffness and, 16 damping) for the misaligned case can be used here, in order to obtain, the analytical stability conditions and the stability charts. The results of the present investigation are compared with the results and. experiments in the literature and are found to be in good agreement. The rigid, rotor model was chosen for work in order to concentrate on the misaligned bearing behavior, thus avoiding the flexible shaft influence on the system. The model used is comprised of a rigid rotor mounted on two linear isotropic misaligned bearings and is used to develop closed-form stability criteria. Using the method presented, in this paper, the analytical expressions for the stability of misaligned journal bearings are obtained in the closed form of inequality constraints. The influence of the misalignment angles on the stability of the system is demonstrated as a function of the slenderness ratio and the dimensionless mass. Finally, the comparison of the stability ranges, using first the eight and then the 32 dynamic coefficients, gives the degree of importance of all these terms, and from total point of view of the proposed method.

Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

Apoint mappinganalysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system's dynamics, and its dependence on system's parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

STABILITY ANALYSIS OF PERIODIC SYSTEMS BY TRUNCATED POINT MAPPINGS

An approach is presented deriving analytical stability and bifurcation conditions for systems with periodically varying coefficients. The method is based on a point mapping(period to period mapping) representation of the system's dynamics. An algorithm is employed to obtain an analytical expression for the point mapping and its dependence on the system's parameters. The algorithm is devised to derive the coefficients of a multinominal expansion of the point mapping up to an arbitrary order in terms of the state variables and of the parameters. Analytical stability and bifurcation condition are then formulated and expressed as functional relations between the parameters. To demonstrate the application of the method, the parametric stability of Mathieu's equation and of a two-degree of freedom system are investigated. The results obtained by the proposed approach are compared to those obtained by perturbation analysis and by direct integration which we considered to the “exact solution”. It is shown that, unlike perturbation analysis, the proposed method provides very accurate solution even for large valuesof the parameters. If an expansion of the point mapping in terms of a small parameter is performed the method is equivalent to perturbation analysis. Moreover, it is demonstrated that the method can be easily applied to multiple-degree-of-freedom systems using the same framework. This feature is an important advantage since most of the existing analysis methods apply mainly to single-degree-of-freedom systems and their extension to higher dimensions is difficult and computationally cumbersome.

Modulational stability in dispersive optical systems with cascaded nonlinearity

We analyze the stability of continuous-wave (cw) solutions in the system of two equations with quadratic nonlinearities, describing a dispersive nonlinear optical medium. We obtain full analytical stability conditions for the solutions. The analytical results are confirmed by systematic numerical simulations. The stability boundries that we obtain are necessary conditions for the formation of simultaneous dark solitons (‘simultons’).

Lyapunov’s Stability of Nonlinear Misaligned Journal Bearings

In this paper the stability of nonlinear misaligned rotor-bearing systems is investigated, using the Lyapunov direct method. A finite element formulation is used to determine the journal bearing pressure distribution. Then the linear and nonlinear stiffness, damping, and hybrid (depending on both displacement and velocity) coefficients are calculated. A general method of analysis based on Lyapunov’s stability criteria is used to investigate the stability of nonlinear misaligned rotor bearing systems. The equations of motion of the rigid rotor on the nonlinear bearings are used to find a Lyapunov function using some of these coefficients, which depend on L/D ratio and the misalignment angles ψx, ψy. The analytical conditions for the stability or instability of some examined cases are given and some examples for the orbital stability are also demonstrated.

Stability and Bifurcations of Towed Underwater Vehicles in the Dive Plane

The problem of static and dynamic loss of stability in the vertical plane in steady towing of underwater vehicles is considered. Bifurcations of steady-state equilibria are studied using singularity theory techniques and all qualitatively different bifurcation diagrams that occur locally are revealed. Analytical conditions for stability of straight-line motion are derived. Bifurcations to periodic solutions are analyzed and shown to provide paths to complicated dynamics. The incorporation of the techniques used in this work, with related studies in cable dynamics, can lead to a design methodology for safer and more efficient operations.

Nonlinear dynamics of feedback multilayer perceptrons.

We study the nonlinear dynamics of multilayer perceptrons with feedback and propose their application to the analysis of signals with complex time dependence. We show that their dynamics provides a built-in time-warping invariance, as, e.g., required for presentation speed fluctuations in speech recognition. We suggest an appropriate learning rule (open-loop learning), give an analytical stability condition for the resulting multistable states, and determine their basins of attraction. To demonstrate their utility for possible applications, we consider the example of a three-stage feedback multilayer perceptron that is trained to detect words in a sequence of letters and does it with perfect invariance with respect to presentation speed fluctuations.