Showing papers on "Lyapunov function published in 2012"

Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Abstract: Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, Hybrid Dynamical Systems unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

Strict Lyapunov Functions for the Super-Twisting Algorithm

Abstract: A method to construct a family of strict Lyapunov functions, i.e., with negative definite derivative, for the super-twisting algorithm, without or with perturbations, is provided. This second order sliding modes algorithm is widely used to design controllers, observers and exact differentiators. The proposed Lyapunov functions ascertain finite time convergence, provide an estimate of the convergence time, and ensure the robustness of the finite-time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm. Since the Lyapunov functions and their derivatives are quadratic forms, the operation with them is as simple as for linear time invariant systems.

Command Filtered Adaptive Backstepping

Abstract: Implementation of adaptive backstepping controllers requires analytic calculation of the partial derivatives of certain stabilizing functions. It is well documented that, as the order of a nonlinear system increases, analytic calculation of these derivatives becomes prohibitive. Therefore, in practice, either alternative control approaches are used or the derivatives are neglected in the implementation. Neglecting the derivatives results in the loss of all guarantees proven by Lyapunov methods for the adaptive backstepping approach and may result in instability. This paper presents a new implementation approach for adaptive backstepping control. The main objectives are to facilitate the derivation and implementation of the adaptive backstepping approach, with performance guarantees proven by Lyapunov methods, for applications that were prohibitively difficult using the standard analytic implementation approach. The new approach uses filtering methods to produce certain command signals and their derivatives which eliminates the requirement of analytic differentiation. The approach also introduces filters to generate certain compensating signals necessary to compute compensated tracking errors suitable for adaptive parameter estimation. We present a set of Lemmas and Theorems to analyze the performance both during the initialization and the operating phases. We show that the initialization phase is of finite duration that can be controlled by selection of a design parameter. We also show that all signals within the system are bounded during this short initialization phase. During the operating phase, we show that the command filtered implementation approach has theoretical properties identical to those of the conventional approach. The general approach is presented and analyzed for systems in generalized parameter strict feedback form. Extensions of the approach are presented to demonstrate the application of the method to a land vehicle trajectory following application. Application and effectiveness of the proposed method is shown by simulation results.

Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs

Abstract: This paper presents three design techniques for cooperative control of multiagent systems on directed graphs, namely, Lyapunov design, neural adaptive design, and linear quadratic regulator (LQR)-based optimal design. Using a carefully constructed Lyapunov equation for digraphs, it is shown that many results of cooperative control on undirected graphs or balanced digraphs can be extended to strongly connected digraphs. Neural adaptive control technique is adopted to solve the cooperative tracking problems of networked nonlinear systems with unknown dynamics and disturbances. Results for both first-order and high-order nonlinear systems are given. Two examples, i.e., cooperative tracking control of coupled Lagrangian systems and modified FitzHugh-Nagumo models, justify the feasibility of the proposed neural adaptive control technique. For cooperative tracking control of the general linear systems, which include integrator dynamics as special cases, it is shown that the control gain design can be decoupled from the topology of the graphs, by using the LQR-based optimal control technique. Moreover, the synchronization region is unbounded, which is a desired property of the controller. The proposed optimal control method is applied to cooperative tracking control of two-mass-spring systems, which are well-known models for vibration in many mechanical systems.

Economic model predictive control of nonlinear process systems using Lyapunov techniques

Abstract: In this work, we develop model predictive control (MPC) designs, which are capable of optimizing closed-loop performance with respect to general economic considerations for a broad class of nonlinear process systems. Specifically, in the proposed designs, the economic MPC optimizes a cost function, which is related directly to desired economic considerations and is not necessarily dependent on a steady-state—unlike conventional MPC designs. First, we consider nonlinear systems with synchronous measurement sampling and uncertain variables. The proposed economic MPC is designed via Lyapunovbased techniques and has two different operation modes. The first operation mode corresponds to the period in which the cost function should be optimized (e.g., normal production period); and in this operation mode, the MPC maintains the closed-loop system state within a predefined stability region and optimizes the cost function to its maximum extent. The second operation mode corresponds to operation in which the system is driven by the economic MPC to an appropriate steady-state. In this operation mode, suitable Lyapunov-based constraints are incorporated in the economic MPC design to guarantee that the closed-loop system state is always bounded in the predefined stability region and is ultimately bounded in a small region containing the origin. Subsequently, we extend the results to nonlinear systems subject to asynchronous and delayed measurements and uncertain variables. Under the assumptions that there exist an upper bound on the interval between two consecutive asynchronous measurements and an upper bound on the maximum measurement delay, an economic MPC design which takes explicitly into account asynchronous and delayed measurements and enforces closed-loop stability is proposed. All the proposed economic MPC designs are illustrated through a chemical process example and their performance and robustness are evaluated through simulations. V C 2011 American Institute of Chemical Engineers AIChE J, 58: 855–870, 2012

Robust Exponential Stability of Uncertain Delayed Neural Networks With Stochastic Perturbation and Impulse Effects

Abstract: This paper focuses on the hybrid effects of parameter uncertainty, stochastic perturbation, and impulses on global stability of delayed neural networks By using the Ito formula, Lyapunov function, and Halanay inequality, we established several mean-square stability criteria from which we can estimate the feasible bounds of impulses, provided that parameter uncertainty and stochastic perturbations are well-constrained Moreover, the present method can also be applied to general differential systems with stochastic perturbation and impulses

Zero-Gradient-Sum Algorithms for Distributed Convex Optimization: The Continuous-Time Case

Abstract: This technical note presents a set of continuous-time distributed algorithms that solve unconstrained, separable, convex optimization problems over undirected networks with fixed topologies. The algorithms are developed using a Lyapunov function candidate that exploits convexity, and are called Zero-Gradient-Sum (ZGS) algorithms as they yield nonlinear networked dynamical systems that evolve invariantly on a zero-gradient-sum manifold and converge asymptotically to the unknown optimizer. We also describe a systematic way to construct ZGS algorithms, show that a subset of them actually converge exponentially, and obtain lower and upper bounds on their convergence rates in terms of the network topologies, problem characteristics, and algorithm parameters, including the algebraic connectivity, Laplacian spectral radius, and function curvatures. The findings of this technical note may be regarded as a natural generalization of several well-known algorithms and results for distributed consensus, to distributed convex optimization.

A differential Lyapunov framework for contraction analysis

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.

A Lyapunov-Function-Based Control for a Three-Phase Shunt Hybrid Active Filter

Abstract: In this paper, an energy-based Lyapunov function control technique is developed for a three-phase shunt hybrid active filter (SH-AF) to compensate harmonics generated by nonlinear loads and is applied for balanced operation. The method provides compensation for harmonic load current components. The strategy determines the control law that makes the derivative of the Lyapunov function always negative for all values of the states. The dc bus voltage of the SH-AF is maintained to 50 V, which is significantly lower than that of the conventional hybrid active filter. The rating of the active filter in the SH-AF system is much smaller than the one used in the conventional shunt active power filter because the passive filter takes care of the major burden of compensation. The SH-AF performances, during both nominal and severe operating conditions, are then evaluated using a dSPACE DS1104 controller board, supported by a Matlab/Simulink Real-Time Workshop environment. A significantly high correlation between the experimental results and the theoretical model, implemented with Simulink/Matlab, is obtained.

Online Optimal Control of Affine Nonlinear Discrete-Time Systems With Unknown Internal Dynamics by Using Time-Based Policy Update

Abstract: In this paper, the Hamilton-Jacobi-Bellman equation is solved forward-in-time for the optimal control of a class of general affine nonlinear discrete-time systems without using value and policy iterations. The proposed approach, referred to as adaptive dynamic programming, uses two neural networks (NNs), to solve the infinite horizon optimal regulation control of affine nonlinear discrete-time systems in the presence of unknown internal dynamics and a known control coefficient matrix. One NN approximates the cost function and is referred to as the critic NN, while the second NN generates the control input and is referred to as the action NN. The cost function and policy are updated once at the sampling instant and thus the proposed approach can be referred to as time-based ADP. Novel update laws for tuning the unknown weights of the NNs online are derived. Lyapunov techniques are used to show that all signals are uniformly ultimately bounded and that the approximated control signal approaches the optimal control input with small bounded error over time. In the absence of disturbances, an optimal control is demonstrated. Simulation results are included to show the effectiveness of the approach. The end result is the systematic design of an optimal controller with guaranteed convergence that is suitable for hardware implementation.

An Overview of LPV Systems

Abstract: The framework of Linear Parameter Varying (LPV) systems concerns linear dynamical systems whose state-space representations depend on exogenous nonstationary parameters. Since its introduction by Shamma and Athans in 1988 to model gain-scheduling, the LPV paradigm has become a standard formalism in systems and controls, with many papers devoted to analysis, controller synthesis, and system identification of LPV models. This chapter reviews basic concepts and presents a representative selection of analytical approaches for LPV systems.

Identification and Learning Control of Ocean Surface Ship Using Neural Networks

Abstract: This paper presents the problems of accurate identification and learning control of ocean surface ship in uncertain dynamical environments. Thanks to the universal approximation capabilities, radial basis function neural networks (NNs) are employed to approximate the unknown ocean surface ship dynamics. A stable adaptive NN tracking controller is first designed using backstepping and Lyapunov synthesis. Partial persistent excitation (PE) condition of some internal signals in the closed-loop system is satisfied during tracking control to a recurrent reference trajectory. Under the PE condition, the proposed adaptive NN controller is shown to be capable of accurate identification/learning of the uncertain ship dynamics in the stable control process. Subsequently, a novel NN learning control method which effectively utilizes the learned knowledge without re-adapting to the unknown ship dynamics is proposed to achieve closed-loop stability and improved control performance. Simulation studies are performed to demonstrate the effectiveness of the proposed method.

A Survey on Delay-Aware Resource Control for Wireless Systems—Large Deviation Theory, Stochastic Lyapunov Drift, and Distributed Stochastic Learning

Abstract: In this paper, a comprehensive survey is given on several major systematic approaches in dealing with delay-aware control problems, namely the equivalentrate constraint approach, the Lyapunov stability drift approach, and the approximate Markov decision process approach using stochastic learning. These approaches essentially embrace most of the existing literature regarding delay-aware resource control in wireless systems. They have their relative pros and cons in terms of performance, complexity, and implementation issues. For each of the approaches, the problem setup, the general solution, and the design methodology are discussed. Applications of these approaches to delay-aware resource allocation are illustrated with examples in single-hop wireless networks. Furthermore, recent results regarding delay-aware multihop routing designs in general multihop networks are elaborated. Finally, the delay performances of various approaches are compared through simulations using an example of the uplink OFDMA systems.

Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay

Abstract: In this paper, a new impulsive control strategy, namely pinning impulsive control, is proposed for the stabilization problem of nonlinear dynamical networks with time-varying delay. In this strategy, only a small fraction of nodes is impulsively controlled to globally exponentially stabilize the whole dynamical network. By employing the Lyapunov method combined with the mathematical analysis approach as well as the comparison principle for impulsive systems, some criteria are obtained to guarantee the success of the global exponential stabilization process. The obtained criteria are closely related to the proportion of the controlled nodes, the impulsive strength, the impulsive interval and the time-delay. Numerical examples are given to demonstrate the effectiveness of the designed pinning impulsive controllers.

Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems

Abstract: In this paper we consider the class of discrete-time switched systems switching between p autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common function. Secondly, stabilizability is introduced and characterized. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and mutually related, too. Finally, some properties of the patterns of the stabilizing switching sequences are investigated, and the relationship between a sufficient condition for stabilizability (the existence of a Schur convex combination of the subsystem matrices) and an equivalent condition for stabilizability (the existence of a Schur matrix product of the subsystem matrices) is explored.

Solitons and collapses: two evolution scenarios of nonlinear wave systems

Abstract: Two alternative scenarios pertaining to the evolution of nonlinear wave systems are considered: solitons and wave collapses. For the former, it suffices that the Hamiltonian be boundedfrombelow(orabove),andthenthesolitonrealizingits minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a pro- cess absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrequation and the three-wave system is used to show how the boundedness of the Hamiltonian—andhencethestabilityofthesolitonminimizing it—can be proved rigorously using the integral estimate meth- od based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.

Adaptive sliding mode control of dynamic system using RBF neural network

Abstract: This paper presents a robust adaptive sliding mode control strategy using radial basis function (RBF) neural network (NN) for a class of time varying system in the presence of model uncertainties and external disturbance. Adaptive RBF neural network controller that can learn the unknown upper bound of model uncertainties and external disturbances is incorporated into the adaptive sliding mode control system in the same Lyapunov framework. The proposed adaptive sliding mode controller can on line update the estimates of system dynamics. The asymptotical stability of the closed-loop system, the convergence of the neural network weight-updating process, and the boundedness of the neural network weight estimation errors can be strictly guaranteed. Numerical simulation for a MEMS triaxial angular velocity sensor is investigated to verify the effectiveness of the proposed adaptive RBF sliding mode control scheme.

Stability Analysis of Isolated Bidirectional Dual Active Full-Bridge DC–DC Converter With Triple Phase-Shift Control

Abstract: This paper proposes a new method for stability analysis of a bidirectional dual full-bridge dc-dc converter with triple phase-shift control under arbitrary parameter changes. The present analysis makes the stability determination of these power converters more systematic and precise than the existing methods in this field, which are largely based on simulation. Nonlinear and periodic operation of the bidirectional converter is presented including the control circuit. Using the working theory, the converter operation is separated into several stages. Equivalent circuits and state equations are developed for each stage. The Lyapunov function method is used to determine the stability of the converter in every stage. Justification is provided for the absence of abrupt changes of the state variables or infinite noise at the interface of different stages. The stability of the bidirectional converter is determined theoretically by integrating these concepts. Some simulation results are provided to validate the developments.

New Energy Analytical Results for the Regulation of Underactuated Overhead Cranes: An End-Effector Motion-Based Approach

Abstract: In this paper, we present a novel end-effector (payload) motion-based control development approach for the regulation of underactuated overhead cranes, which is efficient even in the presence of external disturbance and system parameter variations/uncertainties. The control system is elegantly constructed so that the problem of simultaneously regulating the trolley motion and suppressing the payload swing is successfully addressed by stabilizing a newly defined payload motion signal. Specifically, we first couple the actuated trolley motion and the unactuated payload swing via the defined payload motion signal, based on which a new energy storage function is established. Consequently, a payload motion-based control law is constructed straightforwardly, and the equilibrium point of the resulting closed-loop system is proven to be asymptotically stable by Lyapunov techniques and LaSalle's invariance theorem. Unlike traditional energy-based controllers, the proposed control law takes a much simpler structure independent of the system parameters. Both simulation and experimental results are included to demonstrate the superior performance of the proposed control method over some traditional controllers and its robustness against parameter variations, which illuminates the promising practical application potentiality of the designed crane control system.

LMI Solution for Robust Static Output Feedback Control of Discrete Takagi–Sugeno Fuzzy Models

Abstract: This paper deals with the stabilization problem of discrete-time Takagi-Sugeno (T-S) fuzzy systems via static output controller (SOFC). The proposed method uses the descriptor approach to study this problem and leads to strict linear matrix inequality (LMI ) formulation. In contrast with the existing results, the method allows coping with multiple output matrices, as well as uncertainties. Moreover, the new proposed method can lead to less conservative results by introducing slack variables and considering multiple Lyapunov matrices. A robust SOFC for uncertain T-S fuzzy models is also derived in strict LMI terms. Numerical examples are given to illustrate the effectiveness of the proposed design results.

Compensation of Time-Varying Input and State Delays for Nonlinear Systems

Abstract: We consider general nonlinear systems with time-varying input and state delays for which we design predictor-based feedback controllers. Based on a time-varying infinitedimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization. [DOI: 10.1115/1.4005278]