Positive Definite Lyapunov Function Research Articles

Fully Distributed Fault-Tolerant Leaderless Consensus of Multi-Agent Systems Under Directed Communication Graph

This paper investigates a fault-tolerant leaderless consensus problem for general linear multi-agent systems (MASs) under a strongly connected directed communication graph. The influences of the actuator bias faults and loss of actuator effectiveness faults on MASs are considered in this paper. The designed fully distributed adaptive fault-tolerant consensus protocols (FTCPs) can make all agents' states achieve consensus, asymptotically. Using the features of the loss of actuator effectiveness faults and the strongly connected graph, novel positive definite Lyapunov functions are constructed skillfully to analyse the validity of the proposed protocols. Finally, simulation examples are presented to demonstrate the effectiveness of the proposed protocols.

Isolated periodic wave trains in a generalized Burgers–Huxley equation

We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.

Converse Lyapunov Theorem for Nabla Asymptotic Stability Without Conservativeness

This article focuses on the conservativeness issue of the existing Lyapunov method for linear time-invariant (LTI) nabla fractional-order systems and proposes a converse Lyapunov theorem to overcome the conservative problem. It is shown that the LTI nabla fractional-order system is asymptotically stable if and only if there exist a positive-definite Lyapunov function whose first-order difference is negative definite. After developing a systematic scheme to construct such Lyapunov candidates, the Lyapunov indirect method is derived for the nonlinear system. Finally, the effectiveness and practicability of the proposed methods are substantiated with four examples.

Lyapunov approach based design of a gain adaptive interval type-2 fuzzy controller for servo systems

This paper focuses on the development of a stable Mamdani type-2 fuzzy logic based controller for automatic control of servo systems. The stability analysis of the fuzzy controller has been done by employing the concept of Lyapunov. The Lyapunov approach results in the derivation of an original stability analysis that can be used for designing the rule base of our proposed online gain adaptive Interval Type-2 Fuzzy Proportional Derivative controller (IT2-GFPD) for servo systems with assured stability. In this approach a quadratic positive definite Lyapunov function is used and sufficient stability conditions are satisfied by the adaptive type-2 fuzzy logic control system. Illustrative simulation studies with linear and nonlinear models as well as experimental results on a real-time servo system validate the stability and robustness of the developed intelligent IT2-GFPD. A comparative performance study of IT2-GFPD with other controllers in presence of noise and disturbance also proves the superiority of the proposed controller.

Phase distribution control of a population of oscillators

The collective behavior of biological oscillators has been recognized as an important problem for several decades, but its control has come into limelight only recently. Much of the focus for control has been on desynchronization of an oscillator population, motivated by the pathological neural synchrony present in essential and parkinsonian tremor. Other applications, such as the beating of the heart and insulin secretion, require synchronization, and recently there has been interest in forming clusters within an oscillator population as well. In this article, we use a formulation that allows us to devise control frameworks to achieve all of these distinct collective behaviors observed in biological oscillators. This is based on the Fourier decomposition of the partial differential equation governing the evolution of the phase distribution of a population of identical, uncoupled oscillators. Our first two control algorithms are Lyapunov-based, which work by decreasing a positive definite Lyapunov function towards zero. Our third control is an optimal control algorithm, which minimizes the control energy consumption while achieving the desired collective behavior of an oscillator population. Motivated by pathological neural synchrony, we apply our control to desynchronize an initially synchronized neural population. Given the proposed importance of enhancing spike time dependent plasticity to stabilize neural clusters and counteract pathological neural synchronization, we formulate the phase difference distribution in terms of the phase distribution, and prove some of its fundamental properties, and in turn apply our control to transform the neural phase distribution to form clusters. Finally, motivated by eliminating cardiac alternans, we apply our control to phase shift a synchronous cardiac pacemaker cell population.

A new control method for global stabilisation of translational oscillator with rotational actuator

ABSTRACTA translational oscillator with a rotational actuator (TORA) is an underactuated mechanical system with two degrees of freedom. This paper presents a new method to solve the global stabilisation control problem for this underactuated system. First, a set of suitable state variables is chosen for the TORA. It transforms the system into a cascade affine nonlinear system. Then, a positive-definite Lyapunov function is constructed based on the structural characteristics of the system. A control law is designed to guarantee that the derivative of the Lyapunov function is non-positive. Finally, the global stability of the closed-loop control system is analysed using the Lasalle invariance principle. The condition on the control parameters is presented to globally stabilise the TORA at the origin. The effectiveness and robustness of the method are verified by a numerical example.

On the Aizerman Problem for Systems of Two Differential Equations

The stability of equilibria of systems of nonlinear ordinary differential equations is studied. Acriterion for the reducibility of a second-order linear systemto a scalar differential equation is given. Both positive definite and semidefinite Lyapunov functions are used to obtain sufficient conditions for the asymptotic stability (global stability) of second-order nonlinear differential equations. It is proved that the Aizerman problem has a positive solution with respect to the roots of the characteristic equation of two-dimensional systems of differential equations.

Finite-time convergence of an nth nonlinear system with unmatched disturbance and its application in integrated guidance and control system

The finite-time convergence problem of an [Formula: see text]th nonlinear system with unmatched disturbance is primarily studied in this paper. During the recursive procedure, a new finite-time controller is designed and proven by adding a sign function and a power integrator. Meanwhile, a [Formula: see text] positive definite and proper Lyapunov function, which satisfies the finite-time Lyapunov stability law, is designed. Finally, the designed finite-time controller is applied to some examples and an application of integrated guidance and control system to test and verify its advantage and practicability.

An Expanded Two-Dimensional Proportional–Derivative Command to Line-of-Sight Guidance Law

This paper deals with an expanded two-dimensional (2D) proportional-derivative (PD) command to line-of-sight (CLOS) guidance law which fights the spiral type trajectory of an anti-tank guided missile (ATGM) in the plane perpendicular to the line-of-sight (LOS) or the so called picture plane, in order to improve the transition process performance while putting the missile onto the LOS. The guidance law acts as a classical PD control law within a small predetermined area around the LOS while at missile deviations pointing a position outside this area the control law includes additional nonlinear components connected with the derivatives of the missile position vector in the plane perpendicular to the LOS. The global asymptotic stability of the guidance loop is established by a specific positive definite Lyapunov function. The effectiveness of the proposed approach is illustrated by simulation results. The guidance law enables to decrease the near-field boundary of the missile operational range.

A Lyapunov Function for an Extended Super-Twisting Algorithm

Recently, an extension of the super-twisting algorithm for relative degrees <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq {}1$</tex-math></inline-formula> has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m=1$</tex-math></inline-formula> . Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained—and hence possibly inexact—sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> $m&gt;1$</tex-math></inline-formula> is outlined.

Optimal preview control for a class of linear continuous-time large-scale systems

In this paper, the problem of optimal preview control is studied for a class of linear continuous-time large-scale systems. We first construct an augmented system including the error signal and the reference signal to transform the tracking problem into the regulator problem. Then, the controllers are designed for isolated augmented subsystems, which also constitute the controller of large-scale systems. On the basis of proving the asymptotic stability of closed-loop large-scale systems and the existence of the controller, sufficient conditions for reaching optimal preview control are given. In particular, the limiting condition of the correlation matrices is determined by the fact that the total derivative of a positive definite Lyapunov function is negative definite. The numerical simulation indicates that the controller can drive the large-scale systems to track the reference signal without steady-state error, and the tracking effect is improved with the increasing preview length.

Adaptive Asymptotic Tracking Control of Uncertain Time-Driven Switched Linear Systems

This technical note establishes a novel result for adaptive asymptotic tracking control of uncertain switched linear systems. The result exploits a recently proposed stability condition for switched systems. In particular, a time-varying positive definite Lyapunov function is used to develop a novel piecewise continuous model-reference adaptive law and a dwell-time switching law. In contrast with previous research, where asymptotic tracking was possible only in the presence of a common Lyapunov function for the reference models, in this work asymptotic tracking is shown in a more general setting. Additionally, in the presence of persistence of excitation, the controller parameter estimation errors will converge to zero asymptotically. The main contribution of this work consists in establishing a symmetry between adaptive control of classical non-switched linear systems and adaptive control of switched linear systems. A practical example with an electro-hydraulic system is adopted to illustrate the results.

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the -functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state are established. Examples are worked out to illustrate the results.

On the consensus of homogeneous multi-agent systems with arbitrarily switching topology

In this paper we investigate the consensus problem under arbitrary switching for homogeneous multi-agent systems with switching communication topology, by assuming that each agent is described by a single-input stabilizable state–space model and that the communication graph is connected at every time instant. Under these assumptions, we construct a common quadratic positive definite Lyapunov function for the switched system describing the evolution of the disagreement vector, thus showing that the agents always reach consensus. In addition, the proof leads to the explicit construction of a constant state-feedback matrix that allows the multi-agent system to achieve consensus.

Power-Efficient Algorithm for the Spacecraft Reorientation Control by Noisy Measurements

Traditionally, the problem of the satellite control, its reorientation from one angular position into another, is solved, as a rule, with the use of Pontryagin maximum principle. An example of its application is the algorithm for the control reorientation of the Resurs-P satellite equipped with the inertial effectors (control momentum gyros). On the whole, the algorithm implements the programmed rotations around the Euler axis. The whole interval of the given turn consists of three phases, including "speedup", motion at a constant angular velocity and "slowdown". "Speedup" and "slowdown"phases, in their turn, are also divided into three sub-phases. At the first of them the motion goes on with acceleration, increasing linearly in the absolute terms, at the second sub-phase the acceleration is constant, and at the third one, it decreases (in absolute terms) down to zero. Solution to a two-point boundary problem is a precondition for building of the optimal program for control of a satellite 's angular motion in its reorientation. In order to eliminate the necessity for solving of the two-point boundary problem and to build a closed system for the satellite motion control in its reorientation in a feedback form, the present article poses and solves the task of synthesis of a discrete system, which is linear (both in state and control) and based on the power-efficient algorithm for control by the noisy measurements, developed by the author. According to the algorithm, the spur minimum of the covariance control matrix at each step was chosen as the system's quality index. The author introduced an auxiliary dynamic system and set its parameters. The optimal closed-loop control law was chosen in a linear form - control at the current step was defined as an algebraic sum of the control at the previous step and the weighted difference of the state vectors of the specified and auxiliary systems at the current step. The measurable state vector of the set system was fully observable and comprised random measurement errors, such as a centered discrete white noise. The reorientation time was not explicitly defined. There were no limitations for the control parameters. The power-efficient control algorithm proved to be quite constructive, since the specified quality index of the system was, at the same time, a positive definite Lyapunov function, which added a condition of the asymptotic stability to the whole system. The author developed a discrete model of the undisturbed angular motion of the Resurs-P spacecraft equipped with three pairs of the control momentum gyros and mathematically modeled the power-efficient control algorithm for the spacecraft reorientation. The results of modeling confirmed the efficiency of the proposed algorithm. For simplicity reasons, the angular position of the satellite is defined by the angles of rotation around the body axes, and not by the quaternions.

Anti-synchronization of nonlinear dynamical systems based on Gardano’s method

This letter derives new results for the design of nonlinear controller for the anti-synchronization of two non-identical 4D hyperchaotic systems. The stability results are established using two analytical approaches; Lyapunov’s second method and Gardano’s method. Since the constructing positive definite Lyapunov functions are not required for the Gardano method, the nonlinear control strategy is very effective and convenient to realize the anti-synchronization of non-identical hyperchaotic systems. Furthermore, Gardano method is used to overcome some problem in Lyapunov method which helps to achieve the convergence of error dynamics easily. This proposed method has certain significances for reducing the effort and complexity of controller implementation. Finally, numerical simulations are given to illustrate and verify the results.

Finite-Time Stabilization of Homogeneous Non-Lipschitz Systems

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback, which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system finite-time stable. The proof is based on a recursive design algorithm developed recently to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz nonlinear systems.

Low Model Analysis and Synchronous Simulation of the Wave Mechanics

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

Globally Stable Adaptive Backstepping Neural Network Control for Uncertain Strict-Feedback Systems With Tracking Accuracy Known &lt;italic&gt;a Priori&lt;/italic&gt;

This paper addresses the problem of globally stable direct adaptive backstepping neural network (NN) tracking control design for a class of uncertain strict-feedback systems under the assumption that the accuracy of the ultimate tracking error is given a priori. In contrast to the classical adaptive backstepping NN control schemes, this paper analyzes the convergence of the tracking error using Barbalat's Lemma via some nonnegative functions rather than the positive-definite Lyapunov functions. Thus, the accuracy of the ultimate tracking error can be determined and adjusted accurately a priori, and the closed-loop system is guaranteed to be globally uniformly ultimately bounded. The main technical novelty is to construct three new n th-order continuously differentiable functions, which are used to design the control law, the virtual control variables, and the adaptive laws. Finally, two simulation examples are given to illustrate the effectiveness and advantages of the proposed control method.

On the construction of the Lyapunov function with sign-definite derivative with the help of auxiliary functions with sign-constant derivatives

The short survey of studies of the asymptotic stability with the help of auxiliary functions (positive definite and nonnegative) whose derivatives are nonpositive by virtue of the equations of perturbed motion and the construction of positive definite Lyapunov functions with negative definite derivatives on the basis of these auxiliary functions is given. The example of the construction of the Lyapunov function with the use of an auxiliary nonnegative function with nonpositive derivative is presented.