Showing papers on "Lyapunov function published in 2014"
TL;DR: A new lemma for the Caputo fractional derivatives, when 0 α 1 , is proposed, which has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
1,010 citations
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13 Feb 2014TL;DR: In this paper, the authors identify two potential sources of excessive control effort in Lyapunov design techniques and show how such effort can be greatly reduced, and present a variety of control design methods suitable for systems described by low-order nonlinear ordinary differential equations.
Abstract: Presenting advances in the theory and design of robust nonlinear control systems, this volume identifies two potential sources of excessive control effort in Lyapunov design techniques and shows how such effort can be greatly reduced. Within the framework of Lyapunov design techniques the authors develop a variety of control design methods suitable for systems described by low-order nonlinear ordinary differential equations. There is an emphasis on global controller designs, that is designs for the entire region of model validity.
732 citations
TL;DR: It is proved that consensus tracking in the closed-loop multi-agent systems with a fixed topology having a directed spanning tree can be achieved if the feedback gain matrix and the coupling strength are suitably selected.
Abstract: Distributed consensus tracking is addressed in this paper for multi-agent systems with Lipschitz-type node dynamics. The main contribution of this work is solving the consensus tracking problem without the assumption that the topology among followers is strongly connected and fixed. By using tools from M-matrix theory, a class of consensus tracking protocols based only on the relative states among neighboring agents is designed. By appropriately constructing Lyapunov function, it is proved that consensus tracking in the closed-loop multi-agent systems with a fixed topology having a directed spanning tree can be achieved if the feedback gain matrix and the coupling strength are suitably selected. Furthermore, with the assumption that each possible topology contains a directed spanning tree, it is theoretically shown that consensus tracking under switching directed topologies can be achieved if the control parameters are suitably selected and the dwell time is larger than a positive threshold. The results are then extended to the case where the communication topology contains a directed spanning tree only frequently as the system evolves with time. Finally, some numerical simulations are given to verify the theoretical analysis.
705 citations
01 Dec 2014
TL;DR: A control methodology that unifies control barrier functions and control Lyapunov functions through quadratic programs is developed, which allows for the simultaneous achievement of control objectives subject to conditions on the admissible states of the system.
Abstract: This paper develops a control methodology that unifies control barrier functions and control Lyapunov functions through quadratic programs. The result is demonstrated on adaptive cruise control, which presents both safety and performance considerations, as well as actuator bounds. We begin by presenting a novel notion of a barrier function associated with a set, formulated in the context of Lyapunov-like conditions; the existence of a barrier function satisfying these conditions implies forward invariance of the set. This formulation naturally yields a notion of control barrier function (CBF), yielding inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the simultaneous achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). These formulations are illustrated in the context of adaptive cruise control, where the control objective of achieving a desired speed is balanced by the minimum following conditions on a lead car and force-based constraints on acceleration and braking.
703 citations
10 Sep 2014
TL;DR: In this paper, the authors use a state-space approach and focus on stability analysis and the synthesis of stabilizing control laws in both local and global contexts, and propose methods and algorithms based on the use of linear programming and linear matrix inequalities for computing estimates of the basin of attraction.
Abstract: This monograph details basic concepts and tools fundamental for the analysis and synthesis of linear systems subject to actuator saturation and developments in recent research. The authors use a state-space approach and focus on stability analysis and the synthesis of stabilizing control laws in both local and global contexts. Different methods of modeling the saturation and behavior of the nonlinear closed-loop system are given special attention. Various kinds of Lyapunov functions are considered to present different stability conditions. Results arising from uncertain systems and treating performance in the presence of saturation are given. The text proposes methods and algorithms, based on the use of linear programming and linear matrix inequalities, for computing estimates of the basin of attraction and for designing control systems accounting for the control bounds and the possibility of saturation. They can be easily implemented with mathematical software packages.
639 citations
TL;DR: It is proved that the proposed adaptive neural network (NN) consensus control method guarantees the convergence on the basis of Lyapunov stability theory.
Abstract: Because of the complicity of consensus control of nonlinear multiagent systems in state time-delay, most of previous works focused only on linear systems with input time-delay. An adaptive neural network (NN) consensus control method for a class of nonlinear multiagent systems with state time-delay is proposed in this paper. The approximation property of radial basis function neural networks (RBFNNs) is used to neutralize the uncertain nonlinear dynamics in agents. An appropriate Lyapunov–Krasovskii functional, which is obtained from the derivative of an appropriate Lyapunov function, is used to compensate the uncertainties of unknown time delays. It is proved that our proposed approach guarantees the convergence on the basis of Lyapunov stability theory. The simulation results of a nonlinear multiagent time-delay system and a multiple collaborative manipulators system show the effectiveness of the proposed consensus control algorithm.
528 citations
TL;DR: The purpose of this paper is to provide a comprehensive review of the research on stability of continuous-time recurrent Neural networks, including Hopfield neural networks, Cohen-Grossberg neural networks and related models.
Abstract: Stability problems of continuous-time recurrent neural networks have been extensively studied, and many papers have been published in the literature The purpose of this paper is to provide a comprehensive review of the research on stability of continuous-time recurrent neural networks, including Hopfield neural networks, Cohen-Grossberg neural networks, and related models Since time delay is inevitable in practice, stability results of recurrent neural networks with different classes of time delays are reviewed in detail For the case of delay-dependent stability, the results on how to deal with the constant/variable delay in recurrent neural networks are summarized The relationship among stability results in different forms, such as algebraic inequality forms, \(M\) -matrix forms, linear matrix inequality forms, and Lyapunov diagonal stability forms, is discussed and compared Some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed Concluding remarks and future directions of stability analysis of recurrent neural networks are given
515 citations
TL;DR: The present paper introduces memristor-based fractional-order neural networks and establishes the conditions on the global Mittag-Leffler stability and synchronization are established by using Lyapunov method.
459 citations
TL;DR: This paper deals with the problem of output-feedback H∞ control for a class of active quarter-car suspension systems with control delay with Lyapunov theory and linear matrix inequality approach, and the existence of admissible controllers is formulated in terms of LMIs.
Abstract: This paper deals with the problem of output-feedback H∞ control for a class of active quarter-car suspension systems with control delay. The dynamic system of the suspension systems is first formed in terms of the control objectives, i.e., ride comfort, road holding, suspension deflection, and maximum actuator control force. Then, the objective is to the design of the dynamic output-feedback H∞ controller in order to ensure asymptotic stability of the closed-loop system with H∞ disturbance attenuation level and the output constraints. Furthermore, using Lyapunov theory and linear matrix inequality (LMI) approach, the existence of admissible controllers is formulated in terms of LMIs. With these satisfied conditions, a desired dynamic output-feedback controller can be readily constructed. Finally, a quarter-vehicle model is exploited to demonstrate the effectiveness of the proposed method.
455 citations
TL;DR: By using Lyapunov analysis, it is proven that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded in probability and the system output tracks the reference signal to a bounded compact set.
Abstract: This paper studies an adaptive tracking control for a class of nonlinear stochastic systems with unknown functions. The considered systems are in the nonaffine pure-feedback form, and it is the first to control this class of systems with stochastic disturbances. The fuzzy-neural networks are used to approximate unknown functions. Based on the backstepping design technique, the controllers and the adaptation laws are obtained. Compared to most of the existing stochastic systems, the proposed control algorithm has fewer adjustable parameters and thus, it can reduce online computation load. By using Lyapunov analysis, it is proven that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded in probability and the system output tracks the reference signal to a bounded compact set. The simulation example is given to illustrate the effectiveness of the proposed control algorithm.
447 citations
TL;DR: An integral-barrier Lyapunov function (IBLF)-based control is proposed to suppress the undesirable vibrations of the flexible crane system with the boundary output constraint.
Abstract: In this paper, a flexible cable with a payload attached at the bottom is considered to be the model of a crane system used for positioning the payload. The dynamics of the flexible cable coupled with the tip payload contribute to a hybrid system represented by partial-ordinary differential equations. An integral-barrier Lyapunov function (IBLF)-based control is proposed to suppress the undesirable vibrations of the flexible crane system with the boundary output constraint. Adaption laws are developed for handling parametric uncertainties. A novel IBLF is adopted to guarantee the uniform stability of the closed-loop systems without the violation of the boundary constraint. All closed-loop signals are ensured to be bounded. Extensive simulations are demonstrated to illustrate the performance of the control system.
TL;DR: The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers.
Abstract: This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models-systems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.
TL;DR: It is shown that the settling time of the proposed new class of finite-time consensus protocols is upper bounded for arbitrary initial conditions, which makes it possible for network consensus problems that the convergence time is designed and estimated offline for a given undirected information flow and a group volume of agents.
Abstract: This paper is devoted to investigating the finite-time consensus problem for a multi-agent system in networks with undirected topology. A new class of global continuous time-invariant consensus protocols is constructed for each single-integrator agent dynamics with the aid of Lyapunov functions. In particular, it is shown that the settling time of the proposed new class of finite-time consensus protocols is upper bounded for arbitrary initial conditions. This makes it possible for network consensus problems that the convergence time is designed and estimated offline for a given undirected information flow and a group volume of agents. Finally, a numerical simulation example is presented as a proof of concept.
TL;DR: A sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed, using a discretized Lyapunov function approach in the framework of dwell time.
TL;DR: A method based on second order sliding mode control (2-SMC) is proposed to design controllers for a small quadrotor UAV by using Lyapunov theory, which guarantees that all system state trajectories reach and stay on the sliding surfaces.
Abstract: A method based on second order sliding mode control (2-SMC) is proposed to design controllers for a small quadrotor UAV. For the switching sliding manifold design, the selection of the coefficients of the switching sliding manifold is in general a sophisticated issue because the coefficients are nonlinear. In this work, in order to perform the position and attitude tracking control of the quadrotor perfectly, the dynamical model of the quadrotor is divided into two subsystems, i.e., a fully actuated subsystem and an underactuated subsystem. For the former, a sliding manifold is defined by combining the position and velocity tracking errors of one state variable, i.e., the sliding manifold has two coefficients. For the latter, a sliding manifold is constructed via a linear combination of position and velocity tracking errors of two state variables, i.e., the sliding manifold has four coefficients. In order to further obtain the nonlinear coefficients of the sliding manifold, Hurwitz stability analysis is used to the solving process. In addition, the flight controllers are derived by using Lyapunov theory, which guarantees that all system state trajectories reach and stay on the sliding surfaces. Extensive simulation results are given to illustrate the effectiveness of the proposed control method.
TL;DR: This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strict-feedback form and achieves smoother parameter adaption, better accuracy, and improved performance.
Abstract: This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strict-feedback form. Dynamic surface control technique is incorporated into radial-basis-function neural networks (NNs)-based control framework to eliminate the problem of explosion of complexity. To avoid the analytic computation, the command filter is employed to produce the command signals and their derivatives. Different from directly toward the asymptotic tracking, the accuracy of the identified neural models is taken into consideration. The prediction error between system state and serial-parallel estimation model is combined with compensated tracking error to construct the composite laws for NN weights updating. The uniformly ultimate boundedness stability is established using Lyapunov method. Simulation results are presented to demonstrate that the proposed method achieves smoother parameter adaption, better accuracy, and improved performance.
TL;DR: The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems and the generalized Lyapunov theorems for stability analysis and convergence time estimation are presented.
Abstract: The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory.
TL;DR: A novel CAFTFTC scheme is proposed to guarantee that all follower nodes asymptotically synchronize a leader node with tracking errors converging to a small adjustable neighborhood of the origin in spite of actuator faults.
Abstract: In this paper, the cooperative adaptive fault tolerant fuzzy tracking control (CAFTFTC) problem of networked high-order multiagent with time-varying actuator faults is studied, and a novel CAFTFTC scheme is proposed to guarantee that all follower nodes asymptotically synchronize a leader node with tracking errors converging to a small adjustable neighborhood of the origin in spite of actuator faults. The leader node is modeled as a higher order nonautonomous nonlinear system. It acts as a command generator giving commands only to a small portion of the networked group. Each follower is assumed to have nonidentical unknown nonlinear dynamics, and the communication network is also assumed to be a weighted directed graph with a fixed topology. A distributed robust adaptive fuzzy controller is designed for each follower node such that the tracking errors are cooperative uniform ultimate boundedness (CUUB). Moreover, these controllers are distributed in the sense that the controller designed for each follower node only requires relative state information between itself and its neighbors. The adaptive compensation term of the optimal approximation errors and external disturbances is adopted to reduce the effects of the errors and disturbances, which removes the assumption that the upper bounds of unknown function approximation errors and disturbances should be known. Analysis of stability and parameter convergence of the proposed algorithm are conducted that are based on algebraic graph theory and Lyapunov theory. Comparing with results in the literature, the CAFTFTC scheme can minimize the time delay between fault occurrence and accommodation and reduce its adverse effect on system performance. In addition, the FTC scheme requires no additional fault isolation model, which is necessary in the traditional active FTC scheme. Finally, an example is provided to validate the theoretical results.
TL;DR: The obtained simulation results show that the synthesis control method has good performance in terms of position and attitude tracking when faced with external disturbances.
Abstract: A synthesis control method is proposed to perform the position and attitude tracking control of the dynamical model of a small quadrotor unmanned aerial vehicle (UAV), where the dynamical model is underactuated, highly-coupled and nonlinear. Firstly, the dynamical model is divided into a fully actuated subsystem and an underactuated subsystem. Secondly, a controller of the fully actuated subsystem is designed through a novel robust terminal sliding mode control (TSMC) algorithm, which is utilized to guarantee all state variables converge to their desired values in short time, the convergence time is so small that the state variables are acted as time invariants in the underactuated subsystem, and, a controller of the underactuated subsystem is designed via sliding mode control (SMC), in addition, the stabilities of the subsystems are demonstrated by Lyapunov theory, respectively. Lastly, in order to demonstrate the robustness of the proposed control method, the aerodynamic forces and moments and air drag taken as external disturbances are taken into account, the obtained simulation results show that the synthesis control method has good performance in terms of position and attitude tracking when faced with external disturbances.
TL;DR: In this article, an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle is provided, which is an essential tool for stability analysis of differential equations.
Abstract: Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.
TL;DR: This survey addresses stability analysis for stochastic hybrid systems (SHS) by reviewing many of the stability concepts that have been studied, including Lyapunov stability, Lagrange stability, asymptotic stability, and recurrence.
TL;DR: This paper analyzes distributed control protocols for first- and second-order networked dynamical systems and proposes a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of non linear interaction functions.
Abstract: This paper analyzes distributed control protocols for first- and second-order networked dynamical systems. We propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of nonlinear interaction functions. By using integral Lyapunov functions, we prove the stability of the proposed control protocols, and explicitly characterize the equilibrium set. We also propose a distributed proportional-integral (PI) controller for networked dynamical systems. The PI controllers successfully attenuate constant disturbances in the network. We prove that agents with single-integrator dynamics are stable for any integral gain, and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double-integrator dynamics. Throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites, power systems and building temperature control.
TL;DR: In this article, an adaptive position control for a pump-controlled electrohydraulic actuator (EHA) based on an adaptive backstepping control scheme is presented, which combines a modified back-stepping algorithm with a special adaptation law to compensate all nonlinearities and uncertainties in EHA system.
Abstract: This paper presents an adaptive position control for a pump- controlled electrohydraulic actuator (EHA) based on an adaptive backstepping control scheme. The core feature of this paper is the combination of a modified backstepping algorithm with a special adaptation law to compensate all nonlinearities and uncertainties in EHA system. First of all, the mathematical model of the EHA is developed. The position control is then formulated using a modified backstepping technique and the uncertainties in hydraulic system are adapted by employing a special Lyapunov function. The control signal consists of an adaptive control signal to compensate the uncertainties and a simple robust structure to ensure the robustness corresponding to a bounded disturbance. Experimental results proved strongly the ability of the proposed control method.
TL;DR: In the present tutorial, introduction to Lyapunovbased methods for stability of time-delay systems is given together with some advanced results on the topic.
TL;DR: This paper investigates the problem of Hankel-norm output feedback controller design for a class of T-S fuzzy stochastic systems and proposes the fuzzy-basis-dependent Lyapunov function approach and the conversion on theHankel- norm controller parameters.
TL;DR: The pest’s structured population dynamic model is employed to illustrate the effectiveness of the proposed method, and necessary and sufficient conditions for stochastic stability and l 1 -gain performance of the positive discrete-time MJLS.
TL;DR: It is illustrated in this paper that some classical types of Lyapunov functions can be seen as special cases of the proposed CPLF, and it is proved that the conservativeness of the obtained criteria can be further reduced as the degree of the LyAPunov function increases.
TL;DR: Experimental results validate the theoretical analysis and demonstrate the very good performance of the proposed discrete-time sensorless vector control.
Abstract: This paper proposes a sensorless vector control that combines two discrete-time observers to estimate the rotor speed and position of permanent magnet synchronous machines (PMSM). The first one is a sliding mode (DSM) current observer and the second one is an adaptive electromotive force (EMF) observer. Initially, the sliding conditions that assure the sliding motion around the sliding surface are derived and a design procedure to the DSM current observer is developed. Moreover, using discrete-time adaptive Lyapunov based EMF observer the rotor speed and position are obtained. Experimental results validate the theoretical analysis and demonstrate the very good performance of the proposed discrete-time sensorless vector control.
TL;DR: In this paper, the authors considered a class of impulsive Caputo fractional-order cellular neural networks with time-varying delays and provided sufficient conditions for global Mittag-Leffler stability.
Abstract: In this paper we consider a class of impulsive Caputo fractional-order cellular neural networks with time-varying delays. Applying the fractional Lyapunov method and Mittag-Leffler functions, we give sufficient conditions for global Mittag-Leffler stability which implies global asymptotic stability of the network equilibrium. Our results provide a design method of impulsive control law which globally asymptotically stabilizes the impulse free fractional-order neural network time-delay model. The synchronization of fractional chaotic networks via non-impulsive linear controller is also considered. Illustrative examples are given to demonstrate the effectiveness of the obtained results.
TL;DR: In this paper, the induced l 2 dynamic output feedback controller (DOFC) design problem for discrete-time Markovian jump repeated scalar nonlinear systems was addressed by employing both the switching-sequence dependent Lyapunov function approach and the positive definite diagonally dominant LyAPF technique.
Abstract: This paper is concerned with the induced l2 dynamic output feedback controller (DOFC) design problem for discrete-time Markovian jump repeated scalar nonlinear systems. By employing both the switching-sequence dependent Lyapunov function approach and the positive definite diagonally dominant Lyapunov function technique, a sufficient condition is first established, which guarantees the underlying system to be stochastically stable with an induced l2 disturbance attenuation performance. Then the desired full- or reduced-order DOFCs are designed by using projection approach. Cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem. Finally, a numerical example is presented to show the effectiveness of the proposed methods.