Stability Of Nonlinear Dynamical Systems Research Articles

Lyapunov stability of fuzzy dynamical systems based on fuzzy-number-valued function granular differentiability

Lyapunov stability theory provides a powerful technique for stability analysis of dynamical systems, particularly for the stability of nonlinear dynamical system. This paper deals with the stability of fuzzy dynamical systems using a novel notion called granular fuzzy Lyapunov function. In order to analyze the stability, some new notions are introduced such as fuzzy equilibrium point, the ball of fuzzy state space, granular fuzzy 2-norm and etc. Based on the concept of fuzzy-number-valued function granular differentiability, two theorems are proved for calculation. Moreover, using granular fuzzy Lyapunov function, the granular fuzzy local stability theorem is proposed, and two definitions of exponential stability and like-Lyapunov stability are generalized. Finally, several examples are given to illustrate the proposed theorems.

Frequency criterion for stability «as a whole» of modulation-type pulse voltage converters according to Lyapunov

Background. The use of methods for researching and designing stable nonlinear dynamic systems by constructing the frequency characteristics of the loop gain of the negative feedback of a pulsed voltage converter through time characteristics obtained by numerical methods for solving nonlinear differential equations that describe electromagnetic processes in the converter takes a lot of time and has low accuracy in calculating optimal modes, when you need to calculate hundreds of options.
 Aim. Therefore, it seems necessary to further develop the theory of stability of modulation-type pulse voltage converters using frequency stability criteria, Lyapunov functions, amplitude and phase frequency characteristics of the converter loop gain.
 Methods. Stability in the «small» is solved using linearized models, which are constructed using the so-called first Lyapunov method - the «first approximation» method. The second method for studying the stability of nonlinear dynamic systems is the harmonic linearization method, when the nonlinear element is replaced by a harmonically linearized link with a transmission coefficient only for the first harmonic.
 Results. The Lyapunov function for a pulse voltage converter with nonlinear relay-type elements was found. Conclusion. The results obtained in the work make it possible to simplify the design of Lyapunov-stable pulse voltage converters through the use of the method of amplitude-frequency characteristics used for stationary linear systems.

Generalized Lyapunov stability theory of continuous-time and discrete-time nonlinear distributed-order systems and its application to boundedness and attractiveness for networks models

The Lyapunov direct method is a powerful tool for analyzing the stability of nonlinear dynamical systems. However, most existing research has primarily focused on asymptotic stability, neglecting the crucial aspect of boundedness. In light of this, this paper aims to address the issues concerning the generalized Lyapunov stability theory of distributed-order systems and its application to ensure uniform boundedness and global attractiveness for distributed-order Hopfield neural networks (DOHNNs) with time-varying external input. To achieve this, several essential inequalities that form the foundation of our main work are proposed. Subsequently, two stability criteria in the form of generalized distributed-order Lyapunov theorem are derived by employing Laplace transform techniques. Meanwhile, building upon the proposed inequalities and Lyapunov theorem, sufficient conditions that guarantee the uniform boundedness and global attractiveness of DOHNNs with time-varying external input are established. By the way, all our proposed conclusions are presented in both continuous-time and discrete-time cases to present the distinction and relationship between these two cases. Finally, various simulation examples are designed meticulously to validate the correctness of our proposed theorems and the effectiveness to real-world applicability of our approach.

Stability Analysis of Recurrent-Neural-Based Controllers Using Dissipativity Domain

This paper proposes a method for the stability analysis of dynamic neural networks. The stability analysis of dynamic neural networks is a challenging task due to internal feedback connections. In this research work, we propose an algorithm based on the Reduction of Dissipativity Domain (RODD) algorithm. The RODD algorithm is a numerical technique for the detection of the stability of nonlinear dynamic systems. The method works by using an approximation of the reachable set. This paper proposes linear and quadratic approximations of reachable sets. RODD-LB uses a linear approximation, RODD-EB uses a quadratic approximation, and the RODD-Hybrid algorithm uses a combination of the linear and quadratic approximations. The accuracy and convergence of these algorithms were derived through numerical dynamic systems.

Estimation and stochastic control of nonlinear dynamic systems over the AWGN channel: Application in tele‐presence and tele‐operation of autonomous vehicles

AbstractThis paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Stabilization of nonlinear dynamical systems via intermittent control with non-instantaneous impulses and actuator saturation

In many actual discontinuous control systems, the state might undergo some abrupt changes when the impulsive effects appear and the impulsive effect will stay active for a period of time which cannot be ignored. Obviously, the traditional instantaneous impulsive system models cannot describe the above process. Then in this paper, considering the actuator saturation and non-instantaneous impulse existing in the real environment, we formulate the new models to describe the nonlinear intermittent control system subject to actuator saturation with non-instantaneous impulsive effects occurring at the moments of system switching. Some exponential stability criteria of this considered intermittent control systems in the form of matrix inequalities are first put forward. Meanwhile, some LMI optimization problems are also proposed to design the control gain for estimating the attraction domain with a fixed decay rate. Finally, simulation results demonstrate the feasibility of the obtained theoretical results.

Application of the variational method for studying stability of the 3D Lotka — Volterra system

The paper investigates the stability of the Lotka — Volterra dynamic system ≪predator — two preys≫ with the help of variational method. This method is applicable for studying equilibrium stability non-linear dynamical systems of any kind and in any dimension. It approves its effectiveness even in cases where the use of the Lyapunov function causes great difficulties (for example, when solving tasks of large dimension). The proposed method is based on a variation method for estimating the rate of phase-state metric change for the perturbed system when the variable t → ∞. The disadvantage of the variation method is the fact that this method provides only the necessary conditions of stability. However, its it forms the first step in studying the stability of non-linear dynamical systems of any kind and any dimension.

Estimating the region of attraction based on a polynomial lyapunov function

This paper deals with the problem of obtaining estimates of the domains of attraction and stability in nonlinear dynamical systems. It is based on a special recursive procedure for constructing polynomial Lyapunov functions. The procedure is tested on several examples used in previous studies by other authors, with a comparison of the results. The approach is also applied to estimate the domain of attraction for single DoF mechanical systems with uncertain parameters (Van der Pol and Duffing oscillators).

Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

DYNAMICS OF INNOVATIVE COMPETITION

The subject of this work is the problem of the dynamic interaction of innovative products in a competitive market. Sustainable economic growth is currently impossible without increasing the competitiveness of enterprises and industries, largely determined by the strengthening of the competitive environment. As a result of the competition of economic agents for achieving various advantages in the markets, new products are created and improved with the corresponding development of technologies. In the analysis of previous publications, the role of the national innovation policy of the state in the system of forming priorities of investment activity in market conditions is noted. The purpose of the work is to study the features of building competition models of innovative processes. It should be emphasized that economic systems have a complex and heterogeneous structure with a hierarchical pattern of interaction of endogenous and exogenous factors. A common feature of many balance models of a market economy is the presence of autocatalytic components that determine the growth mechanisms of an innovative product. The article solves the problem of constructing and analyzing the behavioral properties of a mathematical model of competition between two economic entities in a common market. As a prototype for the model of endogenous homeodynamic system, the widely known mathematical model "predator-prey" is used in population dynamics. This model is a system of two ordinary differential equations with quadratic nonlinearities, which has several equilibrium positions and has a behavior with a change in the nature of stability. Research methods are based on the mathematical apparatus of economic synergetics and the theory of stability of nonlinear dynamic systems. As a result, the conditions under which the self-oscillating mode is implemented in the vicinity of the equilibrium state with the appearance of one or more limit cycles are specified. It has been established that the maximum number of limit cycles in the system under study around the equilibrium position is three. The subject analysis of the bifurcation properties of the cyclic dynamics of competitive interaction is carried out, the boundaries of stability loss by the equilibrium position are determined. Conclusions. The behavioral nature of the dynamics of innovation processes is changing significantly and it is possible to implement a leap transition from monotonous economic growth to relaxation fluctuations.

Power-law stability of Hausdorff derivative nonlinear dynamical systems

ABSTRACTThe power law decay is widely observed in lab experiments and field observations. The Power-law stability, however, has little been reported in the literature. In this study, the definition of the Power-law stability is proposed, and then via the Lyapunov direct method, the Power-law stability of nonlinear dynamical systems based on the Hausdorff derivative is investigated. Furthermore, the fractal comparison principle is introduced to obtain the stability conditions for the dynamical systems of this type. Finally, two examples are given to elucidate the notion of Power-law stability.

Stabilization of Nonlinear Dynamic Systems Over Limited Capacity Communication Channels

This article addresses the stabilization of noiseless nonlinear dynamic systems over limited capacity communication channels. It is shown that the stability of nonlinear dynamic systems over memory-less communication channels implies an inequality condition between the Shannon channel capacity and the summation of the positive equilibrium Lyapunov exponents of the dynamic system or, equivalently, the logarithms of the magnitude of the unstable eigenvalues of system Jacobian. Furthermore, we propose an encoder, decoder, and a controller to prove that scalar nonlinear dynamic systems are stabilizable under the aforementioned inequality condition over the digital noiseless and the packet erasure channels, respectively, in sure and almost sure senses. The performance of the proposed coding scheme is illustrated by computer simulations.

Logarithmic norm-based analysis of robust asymptotic stability of nonlinear dynamical systems

In this paper, we address the issue of robust asymptotic stability of zero solution for the nonlinear dynamical systems x˙=A(t)x+g(x,t). Using the approach based on the logarithmic norm and combining with the variation constant formula we show the persistence of local or global “attractiveness” of the origin for a wide class of perturbations.

Inverse Mittag-Leffler stability of structural derivative nonlinear dynamical systems

Abstract The logarithmic time evolution is widely observed in laboratory and field measurements. However, logarithmic stability has not been well considered till now. In this paper, the definition of Lyapunov stability, including logarithmic and inverse Mittag-Leffler stabilities, is proposed. And via the Lyapunov direct method, the stability of nonlinear dynamical systems based on the structural derivative is investigated. Furthermore, the comparison principle based on the structural derivative is presented in order to obtain the stability conditions for nonlinear dynamical systems. Finally, two demonstrative examples are given to test the proposed stability concept.

On the global polynomial stabilization of nonlinear dynamical systems

This paper presents new criteria for the rational stability of dynamical systems. We introduce basic and useful results on how to construct control laws to assure rational stability. Both Coron and Sontag examples were re-visited. A particular attention is paid to time-varying systems where sufficient Lyapunov conditions are developed guaranteeing the rational stability of this class of systems. As application, the example of double integrator is treated where new time-varying feedbacks rationally stabilizing this system are reconstructed.

Stability analysis of nonlinear dynamic system with linear observer for a multilink flexible manipulator

In this paper, the stability of nonlinear dynamic system for a multilink flexible manipulator (MLFM) is investigated by designing of linear observer to construct a closed-loop system such that for all admissible parameter uncertainties, unknown nonlinearities, and exogenous disturbances input. In a Lagrangian approach, one closed-loop augmented system is unfolded and proven to be asymptotically stable in the mean square (ASMS) by utilizing a Lyapunov theory. Further, if the stability of dynamic closed-loop system of the manipulator is maintained, a desired linear observer will be obtained by solving several algebraic Riccati inequalities. Simulation studies are used to verify the effectiveness of the proposed approach.One of the main contributions of this work is in the derivation of dynamic system stability with a suitable linear observer for the MLFM, based on Lyapunov theory, which: (1) explains physical mechanism for flexible-link movements and dynamics in a Lagrangian approach and (2) sufficient and necessary conditions of the dynamic system stability are derived through seeking the optimal feasible solutions of several algebraic Riccati matrix inequalities.

Structural stability of nonlinear population dynamics.

In population dynamics, the concept of structural stability has been used to quantify the tolerance of a system to environmental perturbations. Yet, measuring the structural stability of nonlinear dynamical systems remains a challenging task. Focusing on the classic Lotka-Volterra dynamics, because of the linearity of the functional response, it has been possible to measure the conditions compatible with a structurally stable system. However, the functional response of biological communities is not always well approximated by deterministic linear functions. Thus, it is unclear the extent to which this linear approach can be generalized to other population dynamics models. Here, we show that the same approach used to investigate the classic Lotka-Volterra dynamics, which is called the structural approach, can be applied to a much larger class of nonlinear models. This class covers a large number of nonlinear functional responses that have been intensively investigated both theoretically and experimentally. We also investigate the applicability of the structural approach to stochastic dynamical systems and we provide a measure of structural stability for finite populations. Overall, we show that the structural approach can provide reliable and tractable information about the qualitative behavior of many nonlinear dynamical systems.

Stability Analysis of an Underactuated Autonomous Underwater Vehicle Using Extended-Routh’s Stability Method

In this paper, a new approach to stability analysis of nonlinear dynamics of an underactuated autonomous underwater vehicle (AUV) is presented. AUV is a highly nonlinear robotic system whose dynamic model includes coupled terms due to the hydrodynamic damping factors. It is difficult to analyze the stability of a nonlinear dynamical system through Routh's stability approach because it contains nonlinear dynamic parameters owing to hydrodynamic damping coefficients. It is also difficult to analyze the stability of AUVs using Lyapunov's criterion and LaSalle's invariance principle. In this paper, we proposed the extended-Routh's stability approach to verify the stability of such nonlinear dynamic systems. This extended-Routh's stability approach is much easier as compared to the other existing methods. Numerical simulations are presented to demonstrate the efficacy of the proposed stability verification of the nonlinear dynamic systems, e.g., an AUV system dynamics.

Word length perturbations in certain symmetric presentations of dihedral groups

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities called λ1 and λ2 have been defined that allow for a precise measure of how stable a group is under certain types of small perturbations in the generating expressions for the elements of the group. These quantities provide a means to measure differences among all possible paths in a Cayley graph for a group, establish a group theoretic analog for the notion of stability in nonlinear dynamical systems, and play an important role in the application of groups to computational genomics. In this paper, we further expose the fundamental properties of λ1 and λ2 by establishing their bounds when the underlying group is a dihedral group. An essential step in our approach is to completely characterize so-called symmetric presentations of the dihedral groups, providing insight into the manner in which λ1 and λ2 interact with finite group presentations. This is of interest independent of the study of the quantities λ1,λ2. Finally, we discuss several conjectures and open questions for future consideration.

Stabilization of Nonlinear Dynamical Systems by Delayed Feedback Control or by Delayed Connections

Stabilization of Nonlinear Dynamical Systems by Delayed Feedback Control or by Delayed Connections