Continuous Lyapunov Research Articles

A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function W, we say that the system is globally asymptotically controllable to the target withW-regulated cost, whenever, starting from any point z, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than W(z). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a Minimum Restraint Function. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.

An enhanced looped-functional framework for stability analysis of sampled-data systems

This paper enhances the looped-functional framework for sampled-data systems by using continuous Lyapunov functionals instead of xT(t)Px(t). The continuous Lyapunov functionals exploit the maximum allowable sampling interval compared with the traditional Lyapunov function xT(t)Px(t) in the conventional looped-functional framework. The enhanced looped-functional framework ensures the negativeness of the continuous Lyapunov functionals concerning the sampling instants and guarantees the stability of the sampled-data systems. Based on these continuous Lyapunov functionals, this paper constructs novel Lyapunov functionals with the looped-functional which not only exploits all available vectors related to sampling intervals but also introduces an additional interval which is zero at t=tk+1. Also, this paper proposes the stability criteria based on the enhanced looped-functional framework for the sampled-data systems with the asynchronous and synchronous samplings, respectively. The numerical examples show the effectiveness of the proposed framework.

On a Class of Piecewise Continuous Lyapunov Functions and Uniform Eventual Stability of Nonlinear Impulsive Caputo Fractional Differential Equations via New Generalized Dini Derivative

On a Class of Piecewise Continuous Lyapunov Functions and Uniform Eventual Stability of Nonlinear Impulsive Caputo Fractional Differential Equations via New Generalized Dini Derivative

NONAN GaitPrint: An IMU gait database of healthy young adults

An ongoing thrust of research focused on human gait pertains to identifying individuals based on gait patterns. However, no existing gait database supports modeling efforts to assess gait patterns unique to individuals. Hence, we introduce the Nonlinear Analysis Core (NONAN) GaitPrint database containing whole body kinematics and foot placement during self-paced overground walking on a 200-meter looping indoor track. Noraxon Ultium MotionTM inertial measurement unit (IMU) sensors sampled the motion of 35 healthy young adults (19–35 years old; 18 men and 17 women; mean ± 1 s.d. age: 24.6 ± 2.7 years; height: 1.73 ± 0.78 m; body mass: 72.44 ± 15.04 kg) over 18 4-min trials across two days. Continuous variables include acceleration, velocity, position, and the acceleration, velocity, position, orientation, and rotational velocity of each corresponding body segment, and the angle of each respective joint. The discrete variables include an exhaustive set of gait parameters derived from the spatiotemporal dynamics of foot placement. We technically validate our data using continuous relative phase, Lyapunov exponent, and Hurst exponent—nonlinear metrics quantifying different aspects of healthy human gait.

Identifiability in Continuous Lyapunov Models

Identifiability in Continuous Lyapunov Models

Structural Spectral Methods for Solving Continuous Lyapunov Equations

Structural Spectral Methods for Solving Continuous Lyapunov Equations

Stability and stabilization analysis of periodic switching stochastic systems based on Lyapunov function with continuous time-varying matrix polynomials

In this paper, a Lyapunov function with continuous time-varying Lyapunov matrix polynomials is constructed to ensure the exponential stability in mean square of a periodic switched linear stochastic system, where the Lyapunov matrix of this Lyapunov function is no longer a constant matrix, but a continuous time-varying matrix polynomial. Then, based on this stability study, we design an aperiodic intermittent control for unstable periodic switching stochastic systems to solve the influence of unstable subsystems and switching rules on stability, and the corresponding control gains are continuous time-varying matrix polynomial. Finally, an example is demonstrated to illustrate the effectiveness of the proposed control strategy.

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Predefined-time fault-tolerant attitude control for tailless aircraft considering actuator input saturation

This paper proposes a predefined-time fault-tolerant attitude control methodology to tailless aircraft in the presence of external disturbance, input saturation, and possibly the infinite number of time-varying actuator faults. First, a novel predefined-time performance function is designed, subsequently, a smooth function and bound-based adaptation laws are developed to compensate for the adverse influence caused by actuator fault and input saturation simultaneously. Such a design is less constrained to faults, requires less computation burden, and can be applied to a larger scope of fault scenarios. Besides, a piecewise continuous Lyapunov function is introduced to proceed with the stability analysis of the closed-loop system, and it has been analytically proved that the attitude tracking errors can converge to a predefined residual set within a predefined time while guaranteeing that actuator fault and input saturation are well handled. Finally, comparative simulations are carried out to validate the effectiveness and superiority of the developed control scheme.

An improved sampled-data synchronization criterion for delayed neural networks with two-type transmission delays

This paper investigates the sampled-data synchronization problem for delayed neural networks (DNNs) with two-type transmission delays by input delay approach. We propose novel Lyapunov functionals consisting of continuous Lyapunov functionals and looped-functionals by exploiting the transmission delays induced by neurons and the communication network and the input delay induced by the sampler. The proposed Lyapunov functionals utilize the transmission delays and a mixed delay of the network transmission delay and the input delay into the augmented vector and the integral functionals. The proposed looped-functionals utilize an additional interval [t−ηk,tk−τt] which is zero at t=tk+1 into the integral functionals and include an additional function by exploiting the state vectors related to sampling pattern. Furthermore, this paper proposes a zero equality with a slack variable for the elements of the augmented vector and relaxes the positive conditions of some matrices in the proposed Lyapunov functionals. From the proposed Lyapunov functionals and the relaxations, we derive the stabilization criteria represented as linear matrix inequalities. The simulation result presents the superiority and validity of the proposed criteria.

Resilient Sampled-Data Control for Stabilization of T--S Fuzzy Systems via Interval-Dependent Function Method: Handling DoS Attacks

This paper is concerned with the resilient sampled-data control for stabilization of T-S fuzzy systems in the presence of denial-of-service (DoS) attacks. To describe the effects of DoS attacks, an appropriate mathematical model is established to characterize the complicated dynamical behaviors of DoS attacks and periodic sampling mechanism. On this basis, a resilient sampled-data control scheme including communication protocols and security controller gains is proposed to account for the effects of DoS attacks. Meanwhile, a novel index is developed to measure the anti-attack ability and security level. By utilizing the information of DoS attacks and resilient sampling intervals, a continuous interval-dependent Lyapunov function is constructed which does not increase at the resilient sampling instants. Then, according to the designed interval-dependent function, several inequalities estimation techniques, convex combination approach, and together with discrete-time Lyapunov theory, two sufficient criteria are derived to guarantee that the closed-loop T-S fuzzy systems are globally asymptotically stable with prespecified security levels subject to DoS attacks. The advantages of designed function are well analyzed in contrast with the previous discontinuous Lyapunov functionals. At last, two simulation examples are given to demonstrate the validity and effectiveness of the obtained results.

Designing Distributed Impulsive Controller for Networked Singularly Perturbed Systems

This paper develops a novel synthesis approach for synchronization of a network of singularly perturbed systems (SPSs) with a small singular perturbation parameter (SPP) <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> via distributed impulsive control. First, a decoupling method in the setting of directed networks is employed to decompose networked SPSs related to complex eigenvalues of the Laplacian matrix. Then, based on an improved piecewise continuous Lyapunov function, an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent synchronization criterion is established. The relationship among the impulse interval, the impulse gain matrix and <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> is revealed. By employing the newly-obtained synchronization criterion, sufficient conditions on the existence of an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent impulse gain matrix are derived. Finally, an example is simulated to verify the effectiveness of the theoretical results.

Worst-case error bounds for the super-twisting differentiator in presence of measurement noise

The super-twisting differentiator, also known as the first-order robust exact differentiator, is a well known sliding mode differentiator. In the absence of measurement noise, it achieves exact reconstruction of the time derivative of a function with bounded second derivative. This note proposes an upper bound for its worst-case differentiation error in the presence of bounded measurement noise, based on a novel Lipschitz continuous Lyapunov function. It is shown that the bound can be made arbitrarily tight and never exceeds the true worst-case differentiation error by more than a factor of two. A numerical simulation illustrates the results and also demonstrates the non-conservativeness of the proposed bound.

Super Twisting Based Lyapunov Redesign for Uncertain Linear Delay Systems

We present a new continuous Lyapunov redesign (LR) methodology for the robust stabilization of a class of uncertain time-delay systems that is based on the so-called super twisting algorithm. The main feature of the proposed approach is that allows one to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneously</i> adjust the chattering effect and achieve asymptotic stabilization of the uncertain system, which is lost when continuous approximation of the unit control is considered. At the basis of the super twisting based LR methodology is a well-known class of Lyapunov–Krasovskii functionals. The particular form of the time derivative of this class of functionals allows one to define a delay-free sliding manifold on which some class of smooth uncertainties are compensated.

Mode Switching Control for Drag-Free Satellite Based on Region of Attraction

This paper aims to propose a switching rule to improve the efficiency and stability of the mode switching process for the drag-free satellite. The switching rules will ensure the stability of the different controllers during the switching process. Unlike traditional satellite switching control, the inner loop and the outer loop of the drag-free satellite are strongly coupled. The drag-free satellite not only needs to consider the controller design in the inner loop but also the controller design of the outer loop. In the outer loop, a Proportion Integration Differentiation control method is adopted to design the controller. In the inner loop, considering the release error effect of the test mass, a nonlinear sliding-mode control is employed as a controller before the mode switch. The H∞ mixed-sensitivity controller, to improve the robustness of the system and solve the problem of controller saturation, is designed after the mode switch. In the stability analysis of the switching system, the piecewise continuous Lyapunov function method is adopted. The region of attraction, which is used as the switching rule, is calculated based on the sum of squares. The obtained results demonstrate that the proposed switching rule satisfies the control accuracy and system stability.

Solution Bounds and Numerical Methods of the Unified Algebraic Lyapunov Equation

In this paper, applying some properties of matrix inequality and Schur complement, we give new upper and lower bounds of the solution for the unified algebraic Lyapunov equation that generalize the forms of discrete and continuous Lyapunov matrix equations. We show that its positive definite solution exists and is unique under certain conditions. Meanwhile, we present three numerical algorithms, including fixed point iterative method, the acceleration fixed point method and the alternating direction implicit method, to solve the unified algebraic Lyapunov equation. The convergence analysis of these algorithms is discussed. Finally, some numerical examples are presented to verify the feasibility of the derived upper and lower bounds, and numerical algorithms.

Disturbance-term-based switching event-triggered synchronization control of chaotic Lurie systems subject to a joint performance guarantee

This work is dedicated to the exponential synchronization of chaotic Lurie systems subject to a joint performance guarantee via a disturbance-term-based switching event-triggered control. For the sake of network resources saving, a novel event-triggering scheme is devised for chaotic Lurie systems. Therein, the disturbance term is additionally integrated into the threshold function of the scheme, which is capable of enlarging the time interval between two successively triggering instants and then lessening the triggering times compared with certain previous schemes. By adopting a time-dependent continuous Lyapunov functional, a sufficient condition is proposed via linear matrix inequalities to ensure that the Lurie synchronization-error system is exponentially stable and has a joint L2−L∞ and H∞ performance guarantee. Based on the condition, a co-design algorithm of the desired control gain and the event-trigger matrix is developed by means of a matrix decoupling method. In the end, a numerical simulation is employed to exemplify the validity of the designed controller and the superiority of the devised disturbance-term-based switching event-triggering scheme.

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.

Computation of the Solutions of Lyapunov Matrix Equations with Iterative Decreasing Dimension Method

The existence of a solution of continuous and discrete-time Lyapunov matrix equations was studied. Both Lyapunov matrix equations are transformed into a matrix-vector equation and the solution of the obtained new system was examined. The iterative decreasing dimension method (IDDM) was implemented for solving the generated matrix-vector equation. Computations have been done with Maple procedures that run the constituted algorithms.

Lyapunov theorems for stability and semistability of discrete-time stochastic systems

This paper develops Lyapunov and converse Lyapunov theorems for discrete-time stochastic semistable nonlinear dynamical systems expressed by Itô-type difference equations possessing a continuum of equilibria. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose difference operator involves a discrete-time analog of the infinitesimal generator for continuous-time Itô dynamical systems and decreases along the dynamical system sample solution sequences satisfying an inequality involving the average distance to the set of the system equilibria.