Sufficient Conditions For Asymptotic Stability Research Articles

Iterative Dissipativity of Partial Difference Equation Dynamics in Open-Loop Iterative Learning Control Mode

Complex physical processes, which could evolve in both spatial and temporal dimensions and be represented by partial difference equations, could also operate in a repetitive mode with iterative learning methods as suitable control laws. For these three-dimensional systems (of the spatial, temporal, and iterative dimensions), the stability in the iterative direction is critical for many applications, which can be analyzed and synthesized under the proposed concept of iterative dissipativity. The definition of iterative dissipativity, which is first introduced in this paper, encapsulates the dominant information in both the spatial and temporal dimensions, while also placing a particular emphasis on the iteration improvement. This property allows for the derivation of sufficient conditions for asymptotic stability in the iteration direction, which are represented by linear matrix inequality criteria that can be readily solved. Performance in both the spatial and temporal dimensions can also be satisfied under this iterative dissipativity concept, even in absence of real-time feedback. Moreover, the optimization solutions of the control parameters can be determined. Finally, a thermal process and a numeric example are presented to illustrate the effectiveness of the proposed iteratively dissipative learning control approach.

Stability of short memory fractional-order hybrid systems

The stability theory of short memory fractional-order hybrid systems is studied in this article. Sufficient conditions of asymptotic stability and ultimately bounded stability for the nonlinear short memory fractional-order hybrid systems are proposed respectively. For the convenience of application, a convenient criterion of stability for the nonlinear short memory fractional-order hybrid systems is derived in the study. Demonstrative examples are given to verify the correctness of the proposed theorems in the part of simulation.

Observer-based robust preview tracking control for a class of continuous-time Lipschitz nonlinear systems

<p>In this paper, a novel observer-based robust preview tracking controller design method is proposed for a class of continuous-time Lipschitz nonlinear systems with external disturbances and unknown states. First, a state observer is designed to reconstruct unknown system states. Second, using differentiation, the state lifting technique, the differential mean value theorem, and several ingenious mathematical manipulations, an augmented error system (AES) containing the previewable information of a reference signal is constructed, thereby transforming the tracking control problem into a robust $ H_{\infty} $ control problem. Based on linear parameter-varying (LPV) system theory, a sufficient condition for asymptotic stability of a closed-loop system with a robust $ H_{\infty} $ performance level is established in terms of the linear matrix inequality (LMI). Furthermore, a tracking controller, which includes observer-based feedback control, integral control, and preview feedforward compensation, is established for the original system. In particular, the tracking controller design is simplified by computing the observer and tracking controller gains simultaneously via only a one-step LMI algorithm. Finally, numerical simulation results demonstrate that the proposed controller leads to superior improvement in the output tracking performance compared with the existing methods.</p>

Asymptotic Stability of Delayed Boolean Networks With Random Data Dropouts.

In real networks, communication constraints often prevent the full exchange of information between nodes, which is inevitable. This brief investigates the problem of time delay and randomly missing data in Boolean networks (BNs). A Bernoulli random variable is assigned to each node to characterize the probability of data packet dropout. Time delay and missing data are modeled by independent random variables. A novel data-sending rule that incorporates both communication constraints is proposed. An augmented system, comprising current states, delayed information, and successfully transmitted data, is established for theoretical analysis. Using the semitensor product (STP), the necessary and sufficient condition for asymptotic stability of delayed BNs with random data dropouts is derived. The convergence rate is also obtained.

Synchronization for fractional-order neural networks with unbounded delays

This paper deals with synchronization analysis problem for a class of fractional-order neural networks with unbounded delays. Using the Lyapunov function method combined with fractional Halanay inequality, we derive a novel sufficient condition for asymptotic stability of the error system resulting in two neural networks are synchronized. The obtained conditions are given in terms of linear matrix inequalities, which therefore can be efficiently checked. A numerical example is proposed to illustrate the effectiveness of the obtained results.

Bifurcations in a Plant-Pollinator Model with Multiple Delays

The plant-pollinator model is a common model widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as a class of delay differential-difference equations. In this paper, based on several early plant-pollinator models, we consider a plant-pollinator model with two combined delays to further describe the mutual constraints between the two populations under different time delays and qualitatively analyze its stability and Hopf bifurcation. Specifically, by selecting different combinations of two delays as branch parameters and analyzing in detail the distribution of roots of the corresponding characteristic transcendental equation, we investigate the local stability of the positive equilibrium point of equations, derive the sufficient conditions of asymptotic stability, and demonstrate the Hopf bifurcation for the system. Under the condition that two delays are not equal, some explicit formulas for determining the direction of Hopf bifurcation and some conditions for the stability of periodic solutions of bifurcation are obtained for delay differential equations by using the theory of norm form and the theorem of center manifold. In the end, some examples are presented and corresponding computer numerical simulations are taken to demonstrate and support effectiveness of our theoretical predictions.

Asymptotic behavior of fractional-order nonlinear systems with two different derivatives

This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.

Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique

In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples.

Asymptotic stability for nonlinear time-varying systems on time scales

This paper is concerned with nonlinear time-varying systems on time scales, which can cover not only trivial continuous and discrete dynamical systems, but also some other nontrivial cases. Some less conservative sufficient conditions for asymptotic stability of nonlinear systems are derived. Compared with the traditional stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. Two examples about trivial and nontrivial systems are included to illustrate the effectiveness of the results obtained.

The stability of differential-difference systems with linearly increasing delay. II. Systems with additive right side

The article considers an uncontrolled system of differential-difference equations with a homogeneous additive right side and linearly increasing delay. Sufficient conditions for asymptotic stability are known for a number of special cases of such systems. Razumikhin's theorem on the asymptotic stability of homogeneous systems with proportional delay is formulated. Sufficient conditions for asymptotic stability are obtained basing on the asymptotic stability of the initial system without delay and constructing the Lyapunov function.

Data-Driven Output-Feedback Control for Unknown Switched Linear Systems

This paper proposes a data-driven control method to stabilize unknown switched linear systems under arbitrary switching. We consider the case where the system state is not measurable and design an output feedback controller only using measured input-output data. First, the system with multiple outputs is transformed into a single-output system with observability preserved. Then, a data-based state-space representation that has the same input-output relationship as the original system is constructed using the input-output data of the single-output system, based on which the data-driven controller is designed. Sufficient conditions for asymptotic stability of the closed-loop system under arbitrary switching are established in terms of linear matrix inequalities (LMIs). Compared with the existing method, the proposed method decreases the dimension of the constructed data-based state-space representation, which may reduce the computational burden of the controller design. The effectiveness of the proposed controller is illustrated by an example.

EXPONENTIAL STABILITY IN DELAY LOTKA-VOLTERRA MODELS WITH SMALL INTERACTIONS BETWEEN SPECIES

The linearisation of the Lotka-Volterra system with time delay is considered. Any type of interaction between two spices is under consideration. The sufficient conditions of asymptotic stability for this linear system are obtained

On Stability Analysis of Riemann-Liouville Fractional Singular Systems with Delays

In this study, two lagged fractional order singular neutral differential equations are considered. Using the advantage of the association property of the Riemann -Liouville derivative, the derivative of the appropriate Lyapunov function is calculated. Then, with the help of LMI, sufficient conditions for asymptotic stability of zero solutions are obtained.

A necessary and sufficient condition of asymptotic stability for a class of Fornasini-Marchesini models

In this paper, we study the asymptotic stability of a particular class of linear 2D discrete Fornasini-Marchesini models. The solutions of the model can be expressed in terms of doubly indexed sequences in a simple way only when the matrices describing the model commute. In this situation, we are able to analyse directly the limit of all the trajectories. By doing so, we propose the first necessary and sufficient condition for asymptotic stability of Fornasini-Marchesini matrix models. This condition is not computationally challenging as it is ultimately based on the eigenvalues of the matrices describing the model. We support our result with numerical simulations.

A note on sufficient conditions of asymptotic stability in distribution of stochastic differential equations with [formula omitted]-Brownian motion

This paper investigates a sufficient condition of asymptotic stability in distribution of stochastic differential equations driven by G-Brownian motion (G-SDEs). We define the concept of asymptotic stability in distribution under sublinear expectations. Sufficient criteria of the asymptotic stability in distribution based on sublinear expectations are given. Finally, an illustrative example is provided.

Asymptotic stability of general linear dynamical networks under distributed relative-state feedback control

This paper investigates general linear dynamical networks (GLDNs), distributed relative-state feedback control, and pinning control. For symmetric GLDNs under distributed relative-state feedback control, some necessary and sufficient conditions for asymptotic stability are proposed. While for the asymmetric case, some sufficient conditions are derived. If the obtained stability conditions are not satisfied, one can design some pinning controllers to asymptotically stabilize the GLDNs. Compared with the existing results, the considered dynamical network model is more general, and the obtained theoretical results are novel.

Energy-to-peak combined switching bumpless transfer control for switched interval type-2 fuzzy delayed systems

This paper addresses the energy-to-peak bumpless transfer control for switched interval type-2 fuzzy delayed systems using combined switching. The non-increasing condition of the Lyapunov functional at the switching instants which is commonly required in the existing results is removed by adopting a generalized combined switching strategy. A novel bumpless transfer performance associated with combined switching is established. By using an improved dwell-time-dependent Lyapunov–Krasovskii functional and the adopted switching strategy, sufficient conditions for asymptotic stability with energy-to-peak bumpless transfer performance are obtained. Finally, a tunnel diode circuit model is exploited to illustrate the applicability and effectiveness of the developed method.

Stability of Discrete-Time Systems Under Restricted Switching via Logic Dynamical Generator and STP-Based Mergence of Hybrid States

In this article, we propose a novel technique for the stability analysis of discrete-time switched systems under constrained switching based on the semitensor product (STP) of matrices and vector-representation of logic. We assume that the admissible switching signals are generated by a logic dynamical system, referred to as the logic dynamical generator (LDG) of the admissible switching. We demonstrate that switching with minimum dwell-time and restricted successors represent two special cases of this type, by constructing their respective LDGs. First, we transfer the LDG into algebraic form, based on the vector-representation of logic. Next, we merge the logic state of the LDG and the continuous state of the original switched system using the STP and derive the dynamics of the merged state; a switched system without constraints on switching. The stability of the switched system under constrained switching is equivalent to the stability of the merged system under arbitrary switching. Based on this, sufficient conditions for the asymptotical stability of nonlinear switched systems with analytic subsystems under constrained switching are obtained. Specifically, in the case of all subsystems being linear, necessary and sufficient conditions for asymptotical stability under constrained switching are proposed.

On the stability of Lotka-Volterra model with a delay

The paper examines the stability problem of biological, economic and other processes modeled by the Lotka-Volterra equations with delay. The difference between studied equations and the known ones is that the adaptability functions and the coefficients of the relative change of the interacting subjects or objects are non-linear and take into account variable delay in the action of factors affecting the number of subjects or objects. Moreover, these functions admit the existence of equilibrium positions’ set that is finite in a bounded domain. The stability study of three types of equilibrium positions is carried out using direct analysis of perturbed equations and construction of Lyapunov functionals that satisfy conditions of well-known theorems. Corresponding sufficient conditions for asymptotic stability including global stability are derived, as well as instability and attraction conditions of these positions.

Stability and stabilization of short memory fractional differential equations with delayed impulses

This paper concentrates on the stability and stabilization of short memory fractional differential equations with delayed impulses. The sufficient conditions for asymptotic stability of short memory fractional differential equations with two kinds of delayed impulses are derived, respectively. The results show that the delayed impulses in short memory fractional differential equations exhibit double effects on system performance. For an unstable system, one can stabilize the system by inputting delays in impulses; for a stable system, the stability would be destroyed if the delays were too long. Further, a class of fractional chaotic systems is presented to test the validity of the established theoretical results, some criteria for impulsive synchronization of fractional chaotic systems are derived, and the corresponding impulsive controllers are designed. Finally, a fractional Chua chaotic oscillator is presented to illustrate the practicability of the established impulsive controllers.