Concept Of Practical Stability Research Articles

Research of direct and inverse problems of sensitivity using methods of practical stability by direction

The general statements of the problems of construction of extreme ranges of tolerances on parameters for a linear system of differential equations in the presence of dynamic restrictions on the spread of the state vector are considered. It was found that such sensitivity problems are closely related to the problems of estimating the maximum area of stability in the corresponding function space. In order to obtain upper estimates of the sought values, the problem of estimating tolerances from the positions of stability in a single direction is formalized. The cases of linear and nonlinear restrictions on the spread of the phase coordinate vector with initial linear initial conditions relative to the vector of the system parameters are studied. Necessary and sufficient conditions are formulated in the form of criteria for estimating tolerances on parameters covering the case of fixed initial conditions. The tasks of estimating the maximum volume of the stability region with respect to the deviations of the state vectors and system parameters are considered. For the case of compatible admissible constraints, the paper gives the corresponding numerical calculations for the upper estimates. In particular, for fixed initial conditions, the maximum set of tolerances for system parameters with compatible dynamic restrictions on fluctuations of state vectors and parameters are determined according to these evaluation criteria. From the standpoint of practical stability in direction, direct sensitivity problems are investigated, numerical estimates of the spread of the phase coordinate vector of the system in the presence of linear and fixed initial conditions relative to the parameter vector are given. It has been demonstrated that in the given specification of the problem of maximum estimation of tolerances it is completely covered by problem statements of practical stability by direction. Thus, to calculate areas of guaranteed sensitivity, the concept of practical stability in a single direction is extended to the space of sensitivity functions at the initial moment of time. The above algorithms are used to estimate the range of tolerances for the adjustment parameters of the discrete model of the induction acceleration system with the given limitations on the spread of the quality criterion; the variation of the quality indicator within the tolerance field is presented in the form of a linear form relative to the spread of the parameter vector.

Impulsive Controllers Design for the Practical Stability Analysis of Gene Regulatory Networks with Distributed Delays

This paper studies gene regulatory networks (GRNs) with distributed delays. The essential concept of practical stability of the genes is introduced. We investigate the problems of practical stability and global practical exponential stability of the GRN model under an impulsive control. New practical stability criteria are proposed by designing appropriate impulsive controllers via the Lyapunov functions approach. In the design of the impulsive controller, we consider the effect of impulsive perturbations at fixed times and distributed delays on the stability of the considered GRNs. Several numerical examples are also presented to justify the proposed criteria.

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results.

Neural Networks in Engineering Design: Robust Practical Stability Analysis

Abstract In recent years, we are witnessing artificial intelligence being deployed on embedded platforms in our everyday life, including engineering design practice problems starting from early stage design ideas to the final decision. One of the most challenging problems is related to the design and implementation of neural networks in engineering design tasks. The successful design and practical applications of neural network models depend on their qualitative properties. Elaborating efficient stability is known to be of a high importance. Also, different stability notions are applied for differently behaving models. In addition, uncertainties are ubiquitous in neural network systems, and may result in performance degradation, hazards or system damage. Driven by practical needs and theoretical challenges, the rigorous handling of uncertainties in the neural network design stage is an essential research topic. In this research, the concept of robust practical stability is introduced for generalized discrete neural network models under uncertainties applied in engineering design. A robust practical stability analysis is offered using the Lyapunov function method. Since practical stability concept is more appropriate for engineering applications, the obtained results can be of a practical significance to numerous engineering design problems of diverse interest.

Adaptive control of free-floating manipulator in presence of stochastic input disturbances and unknown dynamical parameters

The trajectory tracking problem of a free-floating manipulator with dynamical uncertainties and stochastic input disturbances is solved in this study. First, the free-floating manipulator is mapped to a conventional fixed base dynamically equivalent manipulator. Then, by using the well-known properties of a revolute joint manipulator and taking into account the random disturbances with unknown power spectral density in control inputs, an adaptive controller scheme is developed. The proposed technique uses the exponential practical stability concept which guarantees that the tracking error and its derivative converge to an arbitrarily small neighborhood of zero by appropriate tuning of the controller’s parameters. It is noteworthy that the proposed controller does not need any physical parameters of the robot. Simulation studies demonstrate the effectiveness and capability of the proposed method for trajectory tracking in the presence of unknown stochastic input disturbances and dynamical uncertainties.

Practical stability with respect to initial time difference for Caputo fractional differential equations

Practical stability with initial data difference for nonlinear Caputo fractional differential equations is studied. This type of stability generalizes known concepts of stability in the literature. It enables us to compare the behavior of two solutions when both initial values and initial intervals are different. In this paper the concept of practical stability with initial time difference is generalized to Caputo fractional differential equations. A definition of the derivative of Lyapunov like function along the given nonlinear Caputo fractional differential equation is given. Comparison results using this definition and scalar fractional differential equations are proved. Sufficient conditions for several types of practical stability with initial time difference for nonlinear Caputo fractional differential equations are obtained via Lyapunov functions. Some examples are given to illustrate the results.

Practical stability assessment of distributed synchronous generators under variations in the system equilibrium conditions

This paper proposes a method to assess the practical stability of power distribution systems with synchronous generators subject to changes in the system equilibrium conditions due to fast varying loads. The concept of practical stability deals with two known state-space regions Ω1 (which contains all the initial conditions reflecting the perturbations at which the system is subject during its operation) and Ω2 (which represents the operating security region of the power distribution system) satisfying Ω1⊂Ω2. The practical stability problem and the focus of this paper is to determine under which conditions the system trajectories will be confined into a security region of operation for a certain time interval of interest, as the equilibrium point of the model changes. This study was carried out using a mathematical model of the distribution system with synchronous generators in the form of a switched affine system. This proposed model is capable of describing the system behavior over a certain period within which changes on the equilibrium conditions of the system can occur. Sufficient conditions for the power distribution system with synchronous generators described as a switched affine system to be practically stable with respect to its operating security region Ω2 are given in the form of matrix inequalities constraints. The results, obtained for the model of a cogeneration plant of 10MW added to a distribution network constituted by a feeder and six buses, show that the less stringent properties of the concept of practical stability can be very well-suited to the security analysis of power systems subjected to frequent variations in the load level.

Insufficiencies of practical BIBO stable n-D systems

This paper is focused on analyzing the practical bounded input bounded output (BIBO) stability concept originally introduced by Agathoklis and Bruton with respect to its applicability in n-D signal processing applications. By comparing the frequency response expected on basis of the transfer function to the Fourier transform of the actually measured impulse response of exemplary systems it is shown that the concept of practical stability cannot be a general n-D stability criterion for systems, in particular for signal processing applications, if roots of the given transfer function are not guaranteed to be close to the respective n-D BIBO stability region.

Dewetted Bridgman crystal growth: practical stability over a bounded time period in a forced regime

The concept of practical stability is applied to the case of dewetted Bridgman crystal growth on Earth, in a simplified configuration. Taking into account both heat transfer and capillarity, it is formally demonstrated that the process is stable in case of convex menisci, provided that pressure fluctuations remain in a range which can be computed. The theoretical concepts are illustrated by examples involving InSb and GaSb crystal growth. It is concluded that practical stability gives valuable knowledge of the dynamics of the studied technique and could be usefully applied to other crystal-growth processes, especially those involving capillary shaping.

Robustness Analysis for Terminal Phases of Reentry Flight

A NOVEL approach to analyze the robustness of a flight control system (FCS) with respect to parametric uncertainties is presented, which specifically applies to gliding vehicles in the terminal phases of reentry flight. Robustness analyses are particularly challenging for these systems. Their reference trajectories are appreciably time-varying and encompass a broad variety of flight regimes. Furthermore, significant uncertainties on some critical design parameters affect the vehicle model, most notably those related to the aerodynamic behavior [1]. Current practice in FCS robustness analysis for this kind of application mainly relies on the theory of linear time-invariant (LTI) systems. In this approach, the original nonlinear system is linearized around a limited number of representative time-varying trajectories, including the nominal one. Then thewell-known frozentime approach [2] is applied, yielding multiple LTI models. In this way, classical stability margins [3] or more sophisticated LTI-based robustness criteria, such as analysis [4] and D-stability analyses [5], can be evaluated. Recently, a Lyapunov-based criterion coupled to interval analysis techniques [6] has been proposed for establishing robustness of a FCS. This approach does not resort to linearization of the system dynamics, but still requires the introduction of fictitious equilibrium points obtained by a frozen-time approach. Even if the flight experience demonstrated that frozen-time approaches are indeed operative, they are widely recognized as inefficient [7]. In fact, because the nominal trajectory may not be an equilibrium trajectory for the system in offnominal conditions, frozen-time analyses can provide only indicative, and often heavily conservative, results. To overcome such problems, further investigations are usually performed to identify a limited set of worst-case combinations of uncertain parameters to be used for FCS design refinement. In this case, nonlinear simulations in specific offnominal conditions, selected using sensitivity analysis and designer’s experience, represent the current practice. Optimization-based worst-case search has also been proposed [8], which may disclose the mutual effects of multiple uncertainties, but to a limited extent. In fact, the complexity of reentry dynamics under multiple uncertainties implies that actual worst cases relevant for FCS design refinement are difficult to identify. In any case, worst-case analysis can select only a limited number of test cases, hiding possible further causes of requirement violations, thus driving wrong refinement strategies that would not solve (or even worsen) FCS robustness problems. Monte Carlo (MC) analysis is, in practice, the only tool that is capable of investigating the combined effect of all uncertainties with a reasonable effort. However, being only a verification tool, when unsatisfactory robustness is discovered at this stage, the identification of its causes can require considerable postprocessing effort [9]. This yields one of the major limitations of this approach: that is, the limited support to the FCS design refinement when a requirement violation occurs due to poor robustness. As a result, in these cases, one is forced to iterate the design with scarce additional information. The present paper contributes toward advancing the current practice used in robustness analysis for FCS design refinement by introducing a method that takes into account nonlinear effects of multiple uncertainties over the whole trajectory, to be used before robustness is finally assessed withMC analysis. Themethod delivers feedback on the causes of requirement violation and adopts robustness criteria directly linked to the original mission or system requirements, such as those employed in MC analyses. The first objective is achieved estimating the region of requirement compliance in the space of the uncertain parameters. In this way, the approach provides an exhaustive coverage of the uncertainty’s effects on the FCS robustness. To translate mission requirements into robustness criteria over the whole trajectory, rather than at isolated points as in frozen-time approaches, we make use of the practical stability concept [10], which, to the authors’ knowledge, has never been applied to robustness analyses of atmospheric reentry vehicles.

Practical stability analysis for transient system dynamics

The stability of multi-body systems in transient conditions, such as vehicles under braking, is considered. Stability in this case is not univocal, because according to the widely used classical definitions of Lyapunov and Malkin, nearly all motions taking place over a finite time are stable. Here, the use of the concept of practical stability is proposed, which is concerned with a limited growth of perturbations expected to be present in a real system. A viable calculation procedure for applying this concept to multi-body systems is proposed, in which the equations of motion and their linearizations are generated and analysed in a symbolic program, AutoSim. Applications are made to the acceleration of a non-linear Jeffcott rotor through its critical speed and the braking of a motorcycle. The rotor remains stable, provided that the acceleration is sufficiently large and the mass eccentricity sufficiently small. For the motorcycle, braking mainly enhances the wobble mode, whereas locking of the rear wheel may lead to a fall.

Quantitative analysis of hybrid parabolic systems with Markovian regime switching via practical stability

In this paper, the concept of practical stability is investigated for parabolic partial differential equations under Markovian switching. The concept of vector-Lyapunov-like functionals coupled with the comparison principle is utilized to establish sufficient conditions for various types of practical stability criteria in the p th moment and in probability of the equilibrium state of the system under Markovian regime switchings. An example is provided to demonstrate the obtained results.

Practical stability of nonlinear differential equation with initial time difference

This paper develops the concept of practical stability to nonlinear differential equations with solutions starting with different initial times. Several practical stability criteria of nonlinear dynamical systems relative to initial time difference are presented by employing two Lyapunov-like functions. The method of two Lyapunov-like functions enables us to obtain results under weaker assumptions.

Practical stability of closed-loop descriptor systems

Practical stability is of significant practical importance in scientific and engineering problems but less investigated for descriptor systems. This paper develops the concepts of practical stability to descriptor systems and by using Lyapunov functions and comparison principle, presents some techniques to specify admissible control sets, such that the motion of the closed-loop descriptor systems is practically stable and impulsive-free. Finally, as an application, this paper discusses linear time-invariant descriptor systems and derives a sufficient condition. To illustrate our approach, a simple circuit network and a numerical example are analysed.

Practical stability of descriptor systems with time delays in terms of two measurements

This paper introduces the concepts of practical stability for descriptor systems with time delays in terms of two measurements. Some results of practical stability parallel to Lyapunov stability theorems are given. Based on Lyapunov functions and the comparison principle, a criterion, by which the problem of a descriptor system with time delay is reduced to that of a standard state-space system without time delays, is derived. Finally, the application of the results obtained is demonstrated through an analysis of a class of linear time-invariant descriptor systems with time delay.

PRACTICAL STABILITY ANALYSIS FOR QUANTIZED CONTROL SYSTEMS VIA SMALL-GAIN THEOREM 1

The paper deals with the concept of practical stability of discrete-time linear systems under nonlinear static output feedback. The main result provides a sufficient condition of practical stability based on a generalization of the small- gain theorem to this class of systems. Examples are given concerning the control synthesis of practically stabilizing quantized controllers.

Practical L2 disturbance attenuation for nonlinear systems

In this work, a notion of generalized L 2-gain for nonlinear systems, where the gain is considered as a function of the state instead of a (global) constant, is presented. This new notion seems to be adequate to characterize the gain properties of several nonlinear systems which do not possess a uniform L 2-gain property (i.e. the L 2-gain depends on the operating point). Moreover, a notion of practical L 2-gain attenuation, which extends the standard definition and parallels (mutatae mutandis) the concepts of practical stability, is also proposed.

Practical stability of chaotic attractors

In this paper we introduce the concept of practical stability and practical stability in finite time for chaotic attractors. The connection between practical and asymptotic stability is discussed.

Analysis of practical stability and Lyapunov stability of linear time-varying neutral delay systems

Practical stability guarantees trajectories of a dynamical system being bounded within a prespecified region during a specified time interval and is of great interest in many applications. For a class of linear time-varying systems described by delay differential equations of neutral type, concepts of practical stability involving both variations of the state and its derivative are introduced in terms of given estimate sets. Sufficient conditions of practical stability are established on the basis of the mixed differential difference comparison principle presented in this paper, in terms of coefficients of the systems and the given allowable trajectory bounds. For the case of time-invariant estimate sets, these conditions are also independent of delay. These results are then applied to the Lyapunov stability and positively invariant tubes of the corresponding homogeneous systems. Specifically, an algebraic condition of globally exponentially asymptotical stability is derived, which is related to the well known property of the M-matrix and is independent of delay. Finally, an illustrative example is given.

Stabilization and de-stabilization of predator-prey systems under harvesting and nutrient enrichment

A predator-prey system is modelled by a pair of ordinary differential equations, and the qualitative effects of prey nutrient enrichment and predator harvesting on the stability of the system are studied. Some theoretical analysis concerning the existence of equilibrium points and limit cycles is included. Three models are examined as examples and the concept of practical stability is introduced. For one of the models, computer simulations are included to illustrate the changes in qualitative behaviour under nutrient enrichment and harvesting. The results indicate that care must be taken in harvesting to prevent extinction of the predator and a resulting collapse of the entire ecosystem. This study should, therefore, prove to be a useful tool in resource management and control.