Lyapunov Inequality Research Articles

Fault detection and isolation based on space projection operator for hypersonic re-entry vehicles with concurrent actuator faults

This article investigates fault detection, isolation and fault-tolerant control for an over-actuated system in hypersonic re-entry vehicle with concurrent actuator faults and disturbance. A series of residuals developed by spatial projection operators is only sensitive to certain faults that can decouple the effects of faults in different directions. Threshold intervals designed through sliding-time windows and a hypothesis test are used to detect faults. Single and concurrent fault isolation can be achieved by utilizing different residual combinations. Subsequently, an augmented observer is introduced to estimate the faults and satisfies the [Formula: see text]-gain constraint to reduce the effect of disturbances. Finally, an adaptive backstepping fault-tolerant control algorithm is designed to achieve stable attitude tracking. The stability of the proposed schemes is proved by Lyapunov and linear matrix inequality theories. Numerical simulation results demonstrate the effectiveness of the proposed methods.

Finite-Time Spatial Sampled-Data Control for Reaction–Diffusion Systems

The finite-time control is considered for reaction–diffusion systems (RDSs) under a designed spatial sampled-data controller (SSDC). Firstly, an SSDC is designed and the finite-time stability is investigated for RDSs, based on the designed controller. In terms on Lyapunov functional method and inequality techniques, a sufficient condition is obtained to ensure the finite-time stability. Then, uncertain RDSs are studied and the robust finite-time stability is considered. Under the designed SSDC, a criterion is obtained to ensure the uncertain RDSs to achieve robust finite-time stability. When there exist external disturbances, the $$H_\infty $$ performance of RDSs is investigated. Both distributed disturbances and boundary disturbances are considered, and sufficient conditions are obtained to guarantee the finite-time $$H_\infty $$ performance with the designed SSDC. Finally, examples are given to verify the effectiveness of our theoretical results.

An Analysis on the Finite Volume Schemes and the Discrete Lyapunov Inequalities for the Chemotaxis System

We analyze two finite volume schemes, linear and nonlinear, for the chemotaxis system in two-dimensional domain, which preserve the mass conservation and positivity without the CFL condition. For the nonlinear scheme, the well-posedness is proved by using Brouwer’s fixed point theory, and we show the convergence of the Picard iteration. We also investigate two discrete Lyapunov functionals, the asymptotic stability of equilibrium and the local stability. Moreover, we apply the discrete semi-group theory to error analysis and obtain the convergence rate $$O(\tau +h)$$ in $$L^p$$ norm. The theoretical results are confirmed by numerical experiments.

Event-generator-based H∞ control of fuzzy distributed parameter systems

This paper studies the adaptive event-based H∞ control problem for a class of T-S fuzzy distributed parameter system with parameter uncertain and actuator faults. To overcome the drawback of the period control scheme, an event-triggered control scheme with adaptive threshold is proposed to minimize the number of unnecessary sampled data transmission and to reduce the update frequency of the controller. Furthermore, a T-S fuzzy controller based on the adaptive event-triggered sampled data is designed to ensure the stochastic exponential stability of the distributed parameter system with H∞ disturbance attenuation performance. Based on a new Lyapunov functional and inequality technique, the stochastic exponential stability criterion of the closed-loop system is obtained, and the controller parameters are designed. The new developed inequality can reduce the conservatism of the stability criterion. Finally, the effectiveness of the theoretical calculation results is verified by numerical simulation, and the results are compared with the relevant literature in the simulation, showing that the methods and results in this paper are less conservative.

Control Minkowski–Lyapunov functions

This article considers the control Lyapunov inequality with a priori prescribed decrease over the space of Minkowski functions generated by compact convex sets containing the origin in the interior within the setting of unconstrained and constrained linear discrete time systems. Within the unconstrained setting, the article derives a novel necessary and sufficient condition, which characterizes all Minkowski functions that satisfy the considered control Lyapunov inequality. This novel necessary and sufficient condition is provided for Minkowski functions of general compact convex sets containing the origin in the interior, and it is also refined for Minkowski functions of polytopic sets containing the origin as an interior point. In addition, the article characterizes, and establishes topological properties of, the related stabilizing set-valued control map. The novel results derived for the unconstrained setting are also extended to the constrained setting.

Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed, and applications to hybrid dynamical systems are included.

On Second-Moment Stability of Discrete-Time Linear Systems With General Stochastic Dynamics

This article provides a new unified framework for second-moment stability of discrete-time linear systems with stochastic dynamics. Relations of notions of second-moment stability are studied for systems with general stochastic dynamics, and associated Lyapunov inequalities are derived. The system dynamics may depend on any type of stochastic process in our framework. Our results for the unified framework can immediately lead us to more specific and tractable stability conditions when the underlying stochastic process is restricted to a more definite one. Usefulness of the developed framework is demonstrated through three selected applications.

Quasi-synchronization of coupled neural networks with reaction-diffusion terms driven by fractional brownian motion

This paper investigates the quasi-synchronization of reaction-diffusion neural networks with hybrid coupling and parameter mismatches via sampled-data control technology. First, the models of neural networks with switching parameter and fraction Brownian motion are given. As a result of parameter mismatches, synchronization is normally not possible to realize directly, then the improved Halanay’s inequality is introduced, which is an important lemma to prove that the considered networks realize quasi-synchronization. Furthermore, based on stochastic theory, Lyapunov function method and inequality techniques, some sufficient conditions are derived to guarantee the quasi-synchronization of hybrid coupled neural networks with reaction-diffusion terms driven by fractional Brownian motion. Finally, two simulation examples are given to prove the efficiency of the developed criteria.

HMM-based quantized dissipative control for 2-D Markov jump systems

In this paper, the dissipative quantized control problem is addressed for Markov jump two-dimensional systems based on Roesser model, in which both asynchronous phenomenon and signal quantization between system modes and controller modes are taken into consideration simultaneously. Moreover, the hidden Markov model (HMM) is adopted to tackle such an asynchronous phenomenon. The principal goal is to devise a state feedback controller, which guarantees that the established closed-loop system achieves asymptotic mean square stability as well as satisfies a prescribed extended dissipative property. Drawing support from Lyapunov function approach and inequality technique, some less conservative criteria ensuring the implementability of the desired controller are derived. Ultimately, the availability and practicability of the developed results are certified through a simulation example.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Output Synchronization of Complex Dynamical Networks With Multiple Output or Output Derivative Couplings.

In this paper, the output synchronization problem for complex dynamical networks (CDNs) with multiple output or output derivative couplings is discussed in detail. Under the help of Lyapunov functional and inequality techniques, an output synchronization criterion is presented for CDNs with multiple output couplings (CDNMOCs). To ensure the output synchronization of CDNMOCs, an adaptive control scheme is also devised. Similarly, we also take into account the adaptive output synchronization and output synchronization of CDNs with multiple output derivative couplings. At last, several numerical examples are designed to testify the effectiveness of the proposed results.

Robust control Minkowski–Lyapunov functions

This article introduces, and studies, the robust control Minkowski–Lyapunov inequality, which is a natural generalization of the recently introduced control Minkowski–Lyapunov inequality. The generalization considers the setting of parametrically uncertain linear discrete time systems. The necessary and sufficient conditions for the characterization and existence of Minkowski functions that verify the robust control Minkowski–Lyapunov inequality are derived. The article also characterizes, and establishes topological properties of, the corresponding robustly stabilizing set-valued control map.

On finite-time stability and stabilization of nonlinear hybrid dynamical systems

<abstract> Finite time stability involving dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time have been studied for both continuous-time and discrete-time systems. For continuous-time systems, finite time stability is defined for equilibria of continuous but non-Lipschitzian nonlinear dynamics, whereas discrete-time systems can exhibit finite time stability even when the system dynamics are linear, and hence, Lipschitz continuous. Alternatively, for impulsive dynamical systems it may be possible to reset the system states to an equilibrium state achieving finite time stability without requiring a non-Lipschitz condition for the continuous-time part of the hybrid system dynamics. In this paper, we develop sufficient Lyapunov conditions for finite time stability of impulsive dynamical systems using both a scalar differential Lyapunov inequality on the continuous-time dynamics as well as a scalar difference Lyapunov inequality on the discrete-time resetting dynamics. Furthermore, using our proposed finite time stability results, we design universal hybrid finite time stabilizing control laws for impulsive dynamical systems. Finally, we present several numerical examples for finite time stabilization of network impulsive dynamical systems. </abstract>

Adaptive Predefined-Time Synchronization of Two Different Fractional-Order Chaotic Systems With Time-Delay

This paper devotes to the adaptive globally synchronization within predefined-time of two time-delayed fractional-order chaotic systems. Firstly, through fractional calculus, two novel different fractional-order systems with time-delay are proposed, whose convergence is guaranteed and phase trajectory is given. Secondly, by exploiting the non-negative Lyapunov function and inequality theorem, a novel global predefined-time stability theorem is proposed, which can ensure the settling time tunable. And the upper bound of the settling time estimation is more accurate compared with the classical results. With the help of novel predefined-time stability theorem, two active controllers are designed, namely the fixed-time synchronization controller and predefined-time synchronization controller, to achieve the fixed-time synchronization and the predefined-time synchronization of two different time-delayed fractional-order chaotic systems respectively. Finally, several numerical simulations are presented in order to show the effectiveness of the proposed methods.

Adaptive control for discontinuous variable-order fractional systems with disturbances

This paper deals with the global asymptotic stabilization and finite-time stabilization issues for variable-order fractional systems with partial a priori bounded disturbances by designing an appropriate adaptive controller. Via the inductive method and Arzela-Ascoli theorem, the existence and uniqueness of the solution for the considered system is firstly verified under the proposed control strategy. By applying Lyapunov stability theory, non-smooth analysis and inequality technique, sufficient stabilization criteria are established under the framework of variable-order fractional Filippov differential inclusion. Finally, two numerical simulations are given to demonstrate the validity of the proposed method.

Exponential Synchronization of Delayed Memristor-Based Uncertain Complex-Valued Neural Networks for Image Protection.

This article solves the exponential synchronization issue of memristor-based complex-valued neural networks (MCVNNs) with time-varying uncertainties via feedback control. Compared with the traditional control methods, a more practical and general control scheme with the available uncertain information of the parameters is newly developed for MCVNNs. Our approach considers the proposed neural networks as two dynamic real-valued systems. Then, the less conservative exponential synchronization criteria are proposed by incorporating the framework of the Lyapunov method and inequality techniques. Under the proposed algorithm, not only can the stability of MCVNNs be guaranteed but also the behavior of such a system is appropriate for image protection. Meanwhile, the sensitive measure of the encryption and decryption can be converted into synchronization error. When monitoring the secure mechanism as a whole, the influence of error feasible domain on image decryption is analyzed. Simulation examples are provided to verify the efficacy of the proposed synchronization criterion and the results of practical application on image protection.

Globally projective synchronization for Caputo fractional quaternion-valued neural networks with discrete and distributed delays

&lt;abstract&gt;&lt;p&gt;This paper is devoted to discussing the globally projective synchronization of Caputo fractional-order quaternion-valued neural networks (FOQVNNs) with discrete and distributed delays. Without decomposing the FOQVNNs into several subsystems, by employing the Lyapunov direct method and inequality techniques, the algebraic criterion for the globally projective synchronization is derived. The effectiveness of the proposed result is illustrated by the MATLAB toolboxes and numerical simulation.&lt;/p&gt;&lt;/abstract&gt;

Exponential stability of Hopfield neural networks of neutral type with multiple time-varying delays

<abstract> This paper investigates the problem for exponential stability of Hopfield neural networks of neutral type with multiple time-varying delays. Different from the existing results, the states of the neurons involve multiple time-varying delays and time derivative of states of neurons also include multiple time-varying delays. The exponential stability of such neutral-type system has not been received enough attention since it is not easy to construct a suitable Lyapunov-Krasovskii functional to analyze the exponential stability of this type of neural system. Novel sufficient conditions of the exponential stability are established by using Lyapunov method and inequality techniques. Compared with some references, the mathematical expression of the neutral-type system is more general and the established algebraic conditions are less conservative. Three examples are given to demonstrate the effectiveness of the theoretical results and compare the established stability conditions to the previous results. </abstract>

H ∞ state estimation for discrete memristive neural networks with signal quantization and probabilistic time delay

In this paper, the problem of state estimation is discussed for a class of delayed discrete memristive neural networks with signal quantization. A random variable obeying the Bernoulli distribution is used to describe the probabilistic time delay. A switching function is introduced to reflect the state dependence of memristive connection weight on neurons. Our aim is to design a state estimator to ensure that the specified disturbance attenuation level is guaranteed. By using Lyapunov stability theory and inequality scaling techniques, the specific explicit expression of gain parameter is given. Finally, a numerical example is given to verify the effectiveness of the proposed estimation method.

On the stability of port-Hamiltonian descriptor systems

We study stable differential-algebraic equations and stabilizable differential-algebraic systems. Besides characterizing the stability in terms of a generalized Lyapunov inequality, we show that these systems can always be rewritten as port-Hamiltonian systems on the subspace where the solutions evolve.