Derive Stability Conditions Research Articles

Tipping prediction of a class of large-scale radial-ring neural networks

Understanding the emergence and evolution of collective dynamics in large-scale neural networks remains a complex challenge. This paper seeks to address this gap by applying dynamical systems theory, with a particular focus on tipping mechanisms. First, we introduce a novel (n+mn)-scale radial-ring neural network and employ Coates’ flow graph topological approach to derive the characteristic equation of the linearized network. Second, through deriving stability conditions and predicting the tipping point using an algebraic approach based on the integral element concept, we identify critical factors such as the synaptic transmission delay, the self-feedback coefficient, and the network topology. Finally, we validate the methodology’s effectiveness in predicting the tipping point. The findings reveal that increased synaptic transmission delay can induce and amplify periodic oscillations. Additionally, the self-feedback coefficient and the network topology influence the onset of tipping points. Moreover, the selection of activation function impacts both the number of equilibrium solutions and the convergence speed of the neural network. Lastly, we demonstrate that the proposed large-scale radial-ring neural network exhibits stronger robustness compared to lower-scale networks with a single topology. The results provide a comprehensive depiction of the dynamics observed in large-scale neural networks under the influence of various factor combinations.

STABILIZATION OF DISCRETE 2D LINEAR SYSTEMS IN FORNASINI-MARCHESINI MODEL

This paper is concerned with the stabilization problem of discrete 2D linear systems described by the second Fornasini-Marchesini model. A necessary and sufficient condition involving the characteristic polynomial is first quoted by which the unforced system is structurally or exponentially stable. On the basis of the derived stability condition, a tractable condition is formulated in the form of linear matrix inequality for obtaining the controller gain of a desired stabilizing state-feedback controller.

Bidirectional coupling in fractional order maps of incommensurate orders

We study the stability of bidirectionally coupled integer and fractional-order maps. The system is further generalized to the case where both the equations have fractional order difference operators. We derive stability conditions for the synchronized fixed point in both cases. We show that this formalism can be extended to inhomogeneous systems of N coupled map where any map can be of arbitrary fractional order or integer order. We give a solution to a specific case of a system with periodic disorder where alternate maps are of integer and fractional order or different fractional orders.

A Unified Distributed Digital Control Architecture for Secondary Control in Islanded Microgrids

Distributed control theory has gaining momentum for the global stability of modern Micro-grids (MGs), which can be hence regarded as a cyber-physical energy systems in a networked control systems perspective. Besides the benefits, the control over networks poses several issues and, among them, how to preserve stability despite the unavoidable sampling process is one of the most crucial to be addressed for the implementation in real-world scenarios. The paper addresses this concern by introducing a unified digital control architecture for secondary control of inverter-based MGs working in islanded mode, where both frequency and voltage regulations are ensured via fully-distributed sampled-data control protocols. By choosing a small-enough sampling period h > 0, the proposed strategies can ensure a reduction of the communication burden by avoiding a continuous interactions among electrical entities, thus enhancing the efficiency of the MG. Exponential stability of frequency and voltage closed-loop systems is analytically proven via Lyapunov-Krasovskii theory. The derived stability conditions are in the form of Linear Matrix Inequalities (LMIs) whose solution provides the maximum sampling period preserving the global stability. Simulation results, carried out in MATLAB/SimPower Systems, confirm the theoretical derivation.

Relaxed Stability Criteria for Delayed Memristor-Based Neural Network Systems via a Novel Matrix-Separation Legendre Inequality.

This article studies the issue of stability in memristor-based neural network (MNN) systems with time-varying delays. First, a novel matrix-separation Legendre inequality is proposed to achieve a tight hierarchical bound on augmented-type integral terms. To derive implementable inequality conditions, several delay-dependent matrices are introduced to eliminate the reciprocal terms associated with time-varying delay. Furthermore, a new Lyapunov-Krasovskii (L-K) functional is proposed by incorporating augmented-type double integrals and delay-product terms. A series of free-weighting matrices are introduced into the L-K functional, leveraging the zero-sum equations and the S-procedure pertaining to both the delay and its derivative. Based on the proposed matrix-separation Legendre inequality and L-K functional, the derived stability conditions exhibit reduced conservatism, as validated by three numerical cases and simulation results.

Event-Triggered Second-Order Sliding Mode Controller Design and Implementation

The paper presents an event-triggered higher-order sliding mode controller design. The event-triggering technique is the alternative approach to real-time controller execution, unlike the classic time-triggering technique, which is not time-dependable and is governed by the triggering policy. The technique is suitable for system resource relaxation in case of computation burden or network usage mitigation. The paper describes the stability analysis of the super-twisted sliding mode controller based on input-to-state stability notation. The stability analysis introduces a triggering policy related directly to the ultimate boundness of the system states and preselected sliding variables. The controller time execution with the selected triggering condition prevents the exhibition of the Zeno phenomena, where the minimal inter-event time of the controller has a positive non-zero lower bound. The minimal value of the inter-event time is related directly to the controller parameters and triggering bound, the selection of which is given with the derived stability conditions regarding the designer’s objective. Preventing the fast nonlinear controller execution, especially close to the sliding manifold, also alleviates the chattering phenomena effectively, which is a primal drawback, and limits the usage of the controller on various systems. The method’s efficiency is verified with the hardware-in-the-loop system, where the dynamic and robustness of the triggering approach are compared to the standard time-triggered execution technique.

Exploring models of running vacuum energy with viscous dark matter from a dynamical system perspective

Running vacuum models and viscous dark matter scenarios beyond perfect fluid idealization are two appealing theoretical strategies that have been separately studied as alternatives to solve some problems rooted in the ΛCDM cosmological model. In this paper, we combine these two notions in a single cosmological setting and investigate their cosmological implications, paying particular attention in the interplay between these two constituents in different cosmological periods. Specifically, we consider a well-studied running vacuum model inspired by renormalization group, and a recently proposed general parameterization for the bulk viscosity ξ. By employing dynamical system analysis, we explore the physical aspects of the new phase space that emerges from the combined models and derive stability conditions that ensure complete cosmological dynamics. We identify four distinct classes of models and find that the critical points of the phase space are non-trivially renewed compared to the single scenarios. We then proceed, in a joint and complementary way to the dynamical system analysis, with a detailed numerical exploration to quantify the impact of both the running parameter and the bulk viscosity coefficient on the cosmological evolution. Thus, for some values of the model parameters, numerical solutions show qualitative differences from the ΛCDM model, which is phenomenologically appealing in light of cosmological observations.

Novel stability criteria of generalized neural networks with time-varying delay based on the same augmented LKF and bounding technique

This paper researches the stability issue of generalized neural networks (GNN) with time-varying delay. For the delay, its derivative has an upper bound or is unknown. Firstly, the augmented Lyapunov-Krasovskii functional (LKF) is constructed based on the state vectors of the third order integral inequalities. Then, by introducing two sets of state vectors, the LKF derivative is presented as the quadratic and quintic polynomials of the delay, respectively. Next, the new quadratic and quintic polynomial negative definite conditions (NDCs) are proposed to set up the linear matrix inequalities (LMIs). In addition, based on the same LKF and third order integral inequalities, this paper proves that the introduction of extra state vectors increases the conservatism of the derived stability conditions. Eventually, the advantages of the provided conditions are illustrated by several numerical examples.

Event-triggered [formula omitted]-gain control for networked systems with time-varying delays and round-robin communication protocol

This paper studies the event-triggered (ET) L2-gain control issues for time-delay networked control systems (NCSs) subject to round-robin protocol. The considered NCS consists of n sensor nodes, where the communication from the sensors to the controller is scheduled by round-robin protocol. This protocol requires that, at every sampling instant, only one sensor is chosen to use the sensor/controller network and transmits its measurement information toward the controller. On the other hand, the ET strategy is adopted in the controller/actuator network, which means that the actuator will receive the updated control input signal in the situation that the predefined ET conditions are satisfied. The utilization of round-robin protocol and ET strategy will help to relieve the burden of network transmission. The network resources and bandwidth will be saved. By developing appropriate Lyapunov functional, sufficient conditions in matrix inequalities are established in order to achieve the stability and L2-gain of the studied systems. Then, a novel iterative algorithm is presented to solve the derived stability conditions so as to design the controllers and obtain the minimal L2-gain simultaneously. Two examples are executed at last to demonstrate the validness and benefits of our results.

Disturbance Suppression and System Design Based on Parallel-Equivalent-Input-Disturbance Approach

The conventional equivalent-input-disturbance (EID) approach exhibits two estimation errors that degrade the disturbance-suppression performance. This article aims to suppress the effects of estimation errors and improve disturbance-suppression performance. Taking the errors to be an artificial disturbance, several other EID compensators are connected to suppress the disturbance. Simplifying the connected EID compensators yields the design of a parameter, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$</tex-math> </inline-formula> , in the conventional EID estimator. Based on such an idea, this article presents a parallel-EID (PEID) approach, which has many merits. The disturbance-suppression performance of this approach is better than that of previous related studies, while its configuration is simple and universal. The physical meaning of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$</tex-math> </inline-formula> is clear and explains the reason why the disturbance-suppression performance improves. Furthermore, this article designs a novel virtual filter for transforming the PEID-based control system into an equivalent configuration that is easy to derive stability conditions. Finally, the PEID approach is applied in controlling the speed of a permanent magnet synchronous motor. Additionally, comparisons between the conventional EID, improved EID, PEID, and modified EID approaches show the validity and superiority of the PEID approach.

Stability of fixed points in generalized fractional maps of the orders $$0&lt; \alpha &lt;1$$

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders $0< \alpha <1$ that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point - asymptotically period two bifurcation point.

Numerical solutions to the fractional-order wave equation

New numerical solution to the linear fractional-order wave equation is presented. The Liouville–Caputo sense fractional-derivative operator and Crank–Nicholson finite difference method (CN-FDM) algorithm are employed. The stability of the present technique is considered by the fractional Von Neumann stability analysis method. Special example as an application of the method is provided. The obtained results are examined to check the derived stability condition of the proposed algorithm. Computational results indicate that the present numerical algorithm is efficient and applicable for the problem under study and many other problems.

An implicit representation for the analysis of piecewise affine discrete-time systems

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems.

Some aspects of dynamic buckling and dynamic response of thin plate under in-plane compression

The paper is devoted to investigating the response of thin orthotropic square plates subjected to in-plane time-dependent compressive load related to static buckling load. Two types of loads have been considered: pulse load with a duration corresponding to a period of natural frequencies of a plate; harmonic load with a different mean value. It has allowed us to analyse the plate behaviour under dynamic load and present the application of phase portraits, Poincare maps and Lyapunov exponents for dynamic buckling analysis. The system’s response is analysed based on the FEM model and one degree of freedom oscillator derived from the full continuous model. The research has been performed to check the thesis that the methods used in the analysis of the stability of motions of a rigid body can be used in dynamic buckling determinations and dynamic response analysis of thin-walled deformable structures. The results show that both considered approaches give reliable outcomes, and derived stability conditions are matched.

Adaptive control for reliable cooperative intersection crossing of connected autonomous vehicles

AbstractRapid advances in vehicle automation and communication technologies enable connected autonomous vehicles (CAVs) to cross intersections cooperatively, which could significantly improve traffic throughput and safety at intersections. Virtual platooning, designed upon car‐following behavior, is one of the promising control methods to promote cooperative intersection crossing of CAVs. Nevertheless, demand variation raises safety and stability concerns when CAVs adopt a virtual platooning control approach. Along this line, this study proposes an adaptive vehicle control method to facilitate the formation of a virtual platoon and the cooperative crossing of CAVs, factoring demand variations at an isolated intersection. This study derives the stability conditions of virtual CAV platoons depending on the time‐varying traffic demand. Based on the derived stability conditions, an optimization model is proposed to adaptively control CAVs dynamics by balancing approaching traffic mobility and safety to enhance the reliability of cooperative crossing at intersections. The simulation results show that, compared to the nonadaptive control, our proposed method can increase the intersection throughput by 18.2%. Also, time‐to‐collision results highlight the advantages of the proposed adaptive control in securing traffic safety.

Nonlinear boundary output feedback stabilization of reaction–diffusion equations

This paper studies the design of a finite-dimensional output feedback controller for the stabilization of a reaction-diffusion equation in the presence of a sector nonlinearity in the boundary input. Due to the input nonlinearity, classical approaches relying on the transfer of the control from the boundary into the domain with explicit occurrence of the time-derivative of the control cannot be applied. In this context, we first demonstrate using Lyapunov direct method how a finite-dimensional observer-based controller can be designed, without using the time derivative of the boundary input as an auxiliary command, in order to achieve the boundary stabilization of general 1-D reaction-diffusion equations with Robin boundary conditions and a measurement selected as a Dirichlet trace. We extend this approach to the case of a control applying at the boundary through a sector nonlinearity. We show from the derived stability conditions the existence of a size of the sector (in which the nonlinearity is confined) so that the stability of the closed-loop system is achieved when selecting the dimension of the observer to be large enough.

Stability analysis of reaction–diffusion PDEs coupled at the boundaries with an ODE

This paper addresses the derivation of generic and tractable sufficient conditions ensuring the stability of a coupled system composed of a reaction–diffusion partial differential equation (PDE) and a finite-dimensional linear time invariant ordinary differential equation (ODE). The coupling of the PDE with the ODE is located either at the boundaries or in the domain of the reaction–diffusion equation and takes the form of the input and output of the ODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well as in-domain Dirichlet/Neumann couplings. The adopted approach relies on the spectral reduction of the problem by projecting the trajectory of the PDE into a Hilbert basis composed of the eigenvectors of the underlying Sturm–Liouville operator and yields a set of sufficient stability conditions taking the form of LMIs. We propose numerical examples, consisting of an unstable reaction–diffusion equation and an unstable ODE, such that the application of the derived stability conditions ensure the stability of the resulting coupled PDE–ODE system.

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

Designing low-pass filter in equivalent-input-disturbance compensator for improving disturbance-rejection performance

This paper presents a high-order low-pass filter for the equivalent-input-disturbance (EID) approach to improving the disturbance-rejection performance. The configuration characteristic of the presented filter clearly explains the reason why the disturbance-rejection performance is improved and provides a guideline to design it. Using the presented filter to replace the conventional filter derives a high-order EID (HEID) approach. It is easy to apply the small-gain theorem in deriving stability conditions of the HEID-based control system. Moreover, the presented filter is proved to be better than the conventional one. Finally, a comparison shows the validity and superiority of the presented method. And a simulation result shows that the HEID approach is easily extended in a multiple-input, multiple-output system even with effects of a white noise and parameter uncertainties.

Static output-feedback peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays

In this article, the problem peak-to-peak gain control (also known as [Formula: see text]-gain control) via static output-feedback scheme is studied for discrete-time positive linear systems with diverse interval delays. By novel comparison techniques involving steady states of upper- and lower-scaled systems with peak values of exogenous disturbances, a characterization for [Formula: see text]-induced norm of the input–output operator is formulated. The obtained characterization is then utilized to derive necessary and sufficient conditions subject to [Formula: see text]-induced performance with prescribed attenuation level. On the basis of the analysis results, and based on a vertex optimization technique, a complete solution to the synthesis problem of a static output-feedback controller that minimizes the worst case amplification from disturbances to regulated outputs subject to peak-to-peak gain is addressed. The derived stabilization conditions are formulated in terms of tractable linear programming conditions, which can be effectively solved by various convex algorithms. Numerical examples and simulations are given to illustrate the effectiveness of the proposed method.