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Showing papers on "Lyapunov function published in 2010"


Book
20 Sep 2010
TL;DR: In this article, the authors present a modern theory of analysis, control, and optimization for dynamic networks, including wireless networks with time-varying channels, mobility, and randomly arriving traffic.
Abstract: This text presents a modern theory of analysis, control, and optimization for dynamic networks. Mathematical techniques of Lyapunov drift and Lyapunov optimization are developed and shown to enable constrained optimization of time averages in general stochastic systems. The focus is on communication and queueing systems, including wireless networks with time-varying channels, mobility, and randomly arriving traffic. A simple drift-plus-penalty framework is used to optimize time averages such as throughput, throughput-utility, power, and distortion. Explicit performance-delay tradeoffs are provided to illustrate the cost of approaching optimality. This theory is also applicable to problems in operations research and economics, where energy-efficient and profit-maximizing decisions must be made without knowing the future. Topics in the text include the following: - Queue stability theory - Backpressure, max-weight, and virtual queue methods - Primal-dual methods for non-convex stochastic utility maximization - Universal scheduling theory for arbitrary sample paths - Approximate and randomized scheduling theory - Optimization of renewal systems and Markov decision systems Detailed examples and numerous problem set questions are provided to reinforce the main concepts. Table of Contents: Introduction / Introduction to Queues / Dynamic Scheduling Example / Optimizing Time Averages / Optimizing Functions of Time Averages / Approximate Scheduling / Optimization of Renewal Systems / Conclusions

1,781 citations


Journal ArticleDOI
TL;DR: The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented for both fixed and switching interaction topologies, which describe the information exchange between the multi-agent systems.

1,252 citations


Journal ArticleDOI
TL;DR: The decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases.
Abstract: Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. With the definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.

1,200 citations


Journal ArticleDOI
TL;DR: Novel time-dependent Lyapunov functionals in the framework of the input delay approach are introduced, which essentially improve the existing results and can guarantee the stability under the sampling which may be greater than the analytical upper bound on the constant delay that preserves the stability.

982 citations


Journal ArticleDOI
TL;DR: A newly developed NCS model including all these network phenomena is provided, including communication constraints, to provide an explicit construction of a continuum of Lyapunov functions that guarantee stability of the NCS in the presence of communication constraints.
Abstract: There are many communication imperfections in networked control systems (NCS) such as varying transmission delays, varying sampling/transmission intervals, packet loss, communication constraints and quantization effects. Most of the available literature on NCS focuses on only some of these aspects, while ignoring the others. In this paper we present a general framework that incorporates communication constraints, varying transmission intervals and varying delays. Based on a newly developed NCS model including all these network phenomena, we will provide an explicit construction of a continuum of Lyapunov functions. Based on this continuum of Lyapunov functions we will derive bounds on the maximally allowable transmission interval (MATI) and the maximally allowable delay (MAD) that guarantee stability of the NCS in the presence of communication constraints. The developed theory includes recently improved results for delay-free NCS as a special case. After considering stability, we also study semi-global practical stability (under weaker conditions) and performance of the NCS in terms of Lp gains from disturbance inputs to controlled outputs. The developed results lead to tradeoff curves between MATI, MAD and performance gains that depend on the used protocol. These tradeoff curves provide quantitative information that supports the network designer when selecting appropriate networks and protocols guaranteeing stability and a desirable level of performance, while being robust to specified variations in delays and transmission intervals. The complete design procedure will be illustrated using a benchmark example.

827 citations


Journal ArticleDOI
TL;DR: A barrier Lyapunov function (BLF) is introduced to address two open and challenging problems in the neuro-control area: for any initial compact set, how to determine a priori the compact superset on which NN approximation is valid; and how to ensure that the arguments of the unknown functions remain within the specified compact supersets.
Abstract: In this brief, adaptive neural control is presented for a class of output feedback nonlinear systems in the presence of unknown functions. The unknown functions are handled via on-line neural network (NN) control using only output measurements. A barrier Lyapunov function (BLF) is introduced to address two open and challenging problems in the neuro-control area: 1) for any initial compact set, how to determine a priori the compact superset, on which NN approximation is valid; and 2) how to ensure that the arguments of the unknown functions remain within the specified compact superset. By ensuring boundedness of the BLF, we actively constrain the argument of the unknown functions to remain within a compact superset such that the NN approximation conditions hold. The semiglobal boundedness of all closed-loop signals is ensured, and the tracking error converges to a neighborhood of zero. Simulation results demonstrate the effectiveness of the proposed approach.

818 citations


Journal ArticleDOI
TL;DR: It is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed-loop system.
Abstract: In this paper, robust adaptive neural network (NN) control is investigated for a general class of uncertain multiple-input-multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities. For nonsymmetric input nonlinearities of saturation and deadzone, variable structure control (VSC) in combination with backstepping and Lyapunov synthesis is proposed for adaptive NN control design with guaranteed stability. In the proposed adaptive NN control, the usual assumption on nonsingularity of NN approximation for unknown control coefficient matrices and boundary assumption between NN approximation error and control input have been eliminated. Command filters are presented to implement physical constraints on the virtual control laws, then the tedious analytic computations of time derivatives of virtual control laws are canceled. It is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed-loop system. Finally, simulation results are presented to illustrate the effectiveness of the proposed adaptive NN control.

670 citations


Journal ArticleDOI
TL;DR: In this paper, a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems is presented. But the approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results.

649 citations


Journal ArticleDOI
TL;DR: A Lyapunov technique is presented for designing a robust adaptive synchronization control protocol for distributed systems having non-identical unknown nonlinear dynamics, and for a target dynamics to be tracked that is also nonlinear and unknown.

603 citations


Journal ArticleDOI
TL;DR: A new type of augmented Lyapunov functional is proposed which contains some triple-integral terms and some new stability criteria are derived in terms of linear matrix inequalities without introducing any free-weighting matrices.

549 citations


Journal ArticleDOI
TL;DR: It is shown using Lyapunov theory that the position, orientation, and velocity tracking errors, the virtual control and observer estimation errors, and the NN weight estimation errors for each NN are all semiglobally uniformly ultimately bounded (SGUUB) in the presence of bounded disturbances and NN functional reconstruction errors while simultaneously relaxing the separation principle.
Abstract: In this paper, a new nonlinear controller for a quadrotor unmanned aerial vehicle (UAV) is proposed using neural networks (NNs) and output feedback. The assumption on the availability of UAV dynamics is not always practical, especially in an outdoor environment. Therefore, in this work, an NN is introduced to learn the complete dynamics of the UAV online, including uncertain nonlinear terms like aerodynamic friction and blade flapping. Although a quadrotor UAV is underactuated, a novel NN virtual control input scheme is proposed which allows all six degrees of freedom (DOF) of the UAV to be controlled using only four control inputs. Furthermore, an NN observer is introduced to estimate the translational and angular velocities of the UAV, and an output feedback control law is developed in which only the position and the attitude of the UAV are considered measurable. It is shown using Lyapunov theory that the position, orientation, and velocity tracking errors, the virtual control and observer estimation errors, and the NN weight estimation errors for each NN are all semiglobally uniformly ultimately bounded (SGUUB) in the presence of bounded disturbances and NN functional reconstruction errors while simultaneously relaxing the separation principle. The effectiveness of proposed output feedback control scheme is then demonstrated in the presence of unknown nonlinear dynamics and disturbances, and simulation results are included to demonstrate the theoretical conjecture.

Journal ArticleDOI
TL;DR: A feedback motion-planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories and proves the property of probabilistic coverage.
Abstract: Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth non-linear systems. Here we present a feedback motion-planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories. The region of attraction of this non-linear feedback policy “probabilistically covers” the entire controllable subset of state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic non-linear feedback design algorithm on simple non-linear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm.

Journal ArticleDOI
TL;DR: Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping to solve the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings.

Journal ArticleDOI
TL;DR: The proposed controller is continuous and successfully overcomes the problem of computing the control law when the approximation model becomes uncontrollable and a switching robust control Lyapunov function (RCLF)-based adaptive, state feedback controller is designed.
Abstract: We consider the tracking problem of unknown, robustly stabilizable, multi-input multi-output (MIMO), affine in the control, nonlinear systems with guaranteed prescribed performance. By prescribed performance we mean that the tracking error converges to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, exhibiting maximum overshoot as well as undershoot less than some sufficiently small preassigned constants. Utilizing an output error transformation, we obtain a transformed system whose robust stabilization is proven necessary and sufficient to achieve prescribed performance guarantees for the output tracking error of the original system, provided that initially the transformed system is well defined. Consequently, a switching robust control Lyapunov function (RCLF)-based adaptive, state feedback controller is designed, to solve the stated problem. The proposed controller is continuous and successfully overcomes the problem of computing the control law when the approximation model becomes uncontrollable. Simulations illustrate the approach.

Journal ArticleDOI
01 Dec 2010
TL;DR: The URED is based on a STA modification and includes high-degree terms providing finite-time, and exact convergence to the derivative of the input signal, with a convergence time that is bounded by some constant independent of the initial conditions of the differentiation error.
Abstract: The differentiators based on the Super-Twisting Algorithm (STA) yield finite-time and theoretically exact convergence to the derivative of the input signal, whenever this derivative is Lipschitz. However, the convergence time grows unboundedly when the initial conditions of the differentiation error grow. In this technical note a Uniform Robust Exact Differentiator (URED) is introduced. The URED is based on a STA modification and includes high-degree terms providing finite-time, and exact convergence to the derivative of the input signal, with a convergence time that is bounded by some constant independent of the initial conditions of the differentiation error. Strong Lyapunov functions are used to prove the convergence of the URED.

Journal ArticleDOI
TL;DR: Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task, and through a direct stability domain characterization, three LMI stability analysis conditions are proposed.
Abstract: After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task. If the fractional order @n is such that 0<@n<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.

Journal ArticleDOI
TL;DR: In this paper, it is shown that under standard assumptions ensuring incremental stability of a switched system, it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori.
Abstract: Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e., existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost dc-dc converter.

Journal ArticleDOI
TL;DR: Krasovskii's method to find Lyapunov functions, and recently obtained extensions of the LaSalle invariance principle for hybrid systems are used to obtain stability proofs of primal-dual laws in different scenarios, and applications to cross-layer network optimization are exhibited.

Journal ArticleDOI
TL;DR: In this article, the global dynamics of SIR models with distributed delay and with discrete delay were studied and the endemic equilibrium was shown to be globally asymptotically stable whenever it exists.
Abstract: SIR models with distributed delay and with discrete delay are studied. The global dynamics are fully determined for R 0 > 1 by using a Lyapunov functional. For each model it is shown that the endemic equilibrium is globally asymptotically stable whenever it exists.

Journal ArticleDOI
TL;DR: In this paper, the authors consider interconnections of nonlinear subsystems in the input-to-state stability (ISS) framework, where a gain matrix is used to encode the mutual dependencies of the systems in the network.
Abstract: We consider interconnections of $n$ nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS; the cases of summation, maximization, and separation with respect to external gains are obtained as corollaries.

Book
15 Dec 2010
TL;DR: The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term as mentioned in this paper, and applies to infinite-dimensional as well as to finite-dimensional dynamical systems.
Abstract: The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows. This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called ""average Lyapunov functions"". Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat.

BookDOI
29 Nov 2010
TL;DR: In this article, the authors present an approach for stability analysis of nonlinear dynamical systems based on energy function theory and quantization of the stability region of the dynamical system.
Abstract: Preface. Acknowledgments. 1. Introduction and Overview. 1.1 Introduction. 1.2 Trends of Operating Environment. 1.3 Online TSA. 1.4 Need for New Tools. 1.5 Direct Methods: Limitations and Challenges. 1.6 Purposes of This Book. 2. System Modeling and Stability Problems. 2.1 Introduction. 2.2 Power System Stability Problem. 2.3 Model Structures and Parameters. 2.4 Measurement-Based Modeling. 2.5 Power System Stability Problems. 2.6 Approaches for Stability Analysis. 2.7 Concluding Remarks. 3. Lyapunov Stability and Stability Regions of Nonlinear Dynamical Systems. 3.1 Introduction. 3.2 Equilibrium Points and Lyapunov Stability. 3.3 Lyapunov Function Theory. 3.4 Stable and Unstable Manifolds. 3.5 Stability Regions. 3.6 Local Characterizations of Stability Boundary. 3.7 Global Characterization of Stability Boundary. 3.8 Algorithm to Determine the Stability Boundary. 3.9 Conclusion. 4. Quasi-Stability Regions: Analysis and Characterization. 4.1 Introduction. 4.2 Quasi-Stability Region. 4.3 Characterization of Quasi-Stability Regions. 4.4 Conclusions. 5. Energy Function Theory and Direct Methods. 5.1 Introduction. 5.2 Energy Functions. 5.3 Energy Function Theory. 5.4 Estimating Stability Region Using Energy Functions. 5.5 Optimal Schemes for Estimating Stability Regions. 5.6 Quasi-Stability Region and Energy Function. 5.7 Conclusion. 6. Constructing Analytical Energy Functions for Transient Stability Models. 6.1 Introduction. 6.2 Energy Functions for Lossless Network-Reduction Models. 6.3 Energy Functions for Lossless Structure-Preserving Models. 6.4 Nonexistence of Energy Functions for Lossy Models. 6.5 Existence of Local Energy Functions. 6.6 Concluding Remarks. 7. Construction of Numerical Energy Functions for Lossy Transient Stability Models. 7.1 Introduction. 7.2 A Two-Step Procedure. 7.3 First Integral-Based Procedure. 7.4 Ill-Conditioned Numerical Problems. 7.5 Numerical Evaluations of Approximation Schemes. 7.6 Multistep Trapezoidal Scheme. 7.7 On the Corrected Numerical Energy Functions. 7.8 Concluding Remarks. 8. Direct Methods for Stability Analysis: An Introduction. 8.1 Introduction. 8.2 A Simple System. 8.3 Closest UEP Method. 8.4 Controlling UEP Method. 8.5 PEBS Method. 8.6 Concluding Remarks. 9. Foundation of the Closest UEP Method. 9.1 Introduction. 9.2 A Structure-Preserving Model. 9.3 Closest UEP. 9.4 Characterization of the Closest UEP. 9.5 Closest UEP Method. 9.6 Improved Closest UEP Method. 9.7 Robustness of the Closest UEP. 9.8 Numerical Studies. 9.9 Conclusions. 10. Foundations of the Potential Energy Boundary Surface Method. 10.1 Introduction. 10.2 Procedure of the PEBS Method. 10.3 Original Model and Artifi cial Model. 10.4 Generalized Gradient Systems. 10.5 A Class of Second-Order Dynamical Systems. 10.6 Relation between the Original Model and the Artifi cial Model. 10.7 Analysis of the PEBS Method. 10.8 Concluding Remarks. 11. Controlling UEP Method: Theory. 11.1 Introduction. 11.2 The Controlling UEP. 11.3 Existence and Uniqueness. 11.4 The Controlling UEP Method. 11.5 Analysis of the Controlling UEP Method. 11.6 Numerical Examples. 11.7 Dynamic and Geometric Characterizations. 11.8 Concluding Remarks. 12. Controlling UEP Method: Computations. 12.1 Introduction. 12.2 Computational Challenges. 12.3 Constrained Nonlinear Equations for Equilibrium Points. 12.4 Numerical Techniques for Computing Equilibrium Points. 12.5 Convergence Regions of Equilibrium Points. 12.6 Conceptual Methods for Computing the Controlling UEP. 12.7 Numerical Studies. 12.8 Concluding Remarks. 13. Foundations of Controlling UEP Methods for Network-Preserving Transient Stability Models. 13.1 Introduction. 13.2 System Models. 13.3 Stability Regions. 13.4 Singular Perturbation Approach. 13.5 Energy Functions for Network-Preserving Models. 13.6 Controlling UEP for DAE Systems. 13.7 Controlling UEP Method for DAE Systems. 13.8 Numerical Studies. 13.9 Concluding Remarks. 14. Network-Reduction BCU Method and Its Theoretical Foundation. 14.1 Introduction. 14.2 Reduced-State System. 14.3 Analytical Results. 14.4 Static and Dynamic Relationships. 14.5 Dynamic Property (D3). 14.6 A Conceptual Network-Reduction BCU Method. 14.7 Concluding Remarks. 15. Numerical Network-Reduction BCU Method. 15.1 Introduction. 15.2 Computing Exit Points. 15.3 Stability-Boundary-Following Procedure. 15.4 A Safeguard Scheme. 15.5 Illustrative Examples. 15.6 Numerical Illustrations. 15.7 IEEE Test System. 15.8 Concluding Remarks. 16. Network-Preserving BCU Method and Its Theoretical Foundation. 16.1 Introduction. 16.2 Reduced-State Model. 16.3 Static and Dynamic Properties. 16.4 Analytical Results. 16.5 Overall Static and Dynamic Relationships. 16.6 Dynamic Property (D3). 16.7 Conceptual Network-Preserving BCU Method. 16.8 Concluding Remarks. 17. Numerical Network-Preserving BCU Method. 17.1 Introduction. 17.2 Computational Considerations. 17.3 Numerical Scheme to Detect Exit Points. 17.4 Computing the MGP. 17.5 Computation of Equilibrium Points. 17.6 Numerical Examples. 17.7 Large Test Systems. 17.8 Concluding Remarks. 18. Numerical Studies of BCU Methods from Stability Boundary Perspectives. 18.1 Introduction. 18.2 Stability Boundary of Network-Reduction Models. 18.3 Network-Preserving Model. 18.4 One Dynamic Property of the Controlling UEP. 18.5 Concluding Remarks. 19. Study of the Transversality Conditions of the BCU Method. 19.1 Introduction. 19.2 A Parametric Study. 19.3 Analytical Investigation of the Boundary Property. 19.4 The Two-Machine Infi nite Bus (TMIB) System. 19.5 Numerical Studies. 19.6 Concluding Remarks. 20. The BCU-Exit Point Method. 20.1 Introduction. 20.2 Boundary Property. 20.3 Computation of the BCU-Exit Point. 20.4 BCU-Exit Point and Critical Energy. 20.5 BCU-Exit Point Method. 20.6 Concluding Remarks. 21. Group Properties of Contingencies in Power Systems. 21.1 Introduction. 21.2 Groups of Coherent Contingencies. 21.3 Identifi cation of a Group of Coherent Contingencies. 21.4 Static Group Properties. 21.5 Dynamic Group Properties. 21.6 Concluding Remarks. 22. Group-Based BCU-Exit Method. 22.1 Introduction. 22.2 Group-Based Verifi cation Scheme. 22.3 Linear and Nonlinear Relationships. 22.4 Group-Based BCU-Exit Point Method. 22.5 Numerical Studies. 22.6 Concluding Remarks. 23. Group-Based BCU-CUEP Methods. 23.1 Introduction. 23.2 Exact Method for Computing the Controlling UEP. 23.3 Group-Based BCU-CUEP Method. 23.4 Numerical Studies. 23.5 Concluding Remarks. 24. Group-Based BCU Method. 24.1 Introduction. 24.2 Group-Based BCU Method for Accurate Critical Energy. 24.3 Group-Based BCU Method for CUEPs. 24.4 Numerical Studies. 24.5 Concluding Remarks. 25. Perspectives and Future Directions. 25.1 Current Developments. 25.2 Online Dynamic Contingency Screening. 25.3 Further Improvements. 25.4 Phasor Measurement Unit (PMU)-Assisted Online ATC Determination. 25.5 Emerging Applications. 25.6 Concluding Remarks. Appendix. A1.1 Mathematical Preliminaries. A1.2 Proofs of Theorems in Chapter 9. A1.3 Proofs of Theorems in Chapter 10. Bibliography. Index.

Journal ArticleDOI
TL;DR: A discrete-time model for networked control systems (NCSs) that incorporates all network phenomena: time-varying sampling intervals, packet dropouts and time- varying delays that may be both smaller and larger than the sampling interval is presented.

Journal ArticleDOI
TL;DR: This brief investigates the problem of robust sampled-data H ∞ control for active vehicle suspension systems by using an input delay approach and a quarter-car model is considered.
Abstract: This brief investigates the problem of robust sampled-data H ∞ control for active vehicle suspension systems. By using an input delay approach, the active vehicle suspension system with sampling measurements is transformed into a continuous-time system with a delay in the state. The transformed system contains non-differentiable time-varying state delay and polytopic parameter uncertainties. A Lyapunov functional approach is employed to establish the H ∞ performance, and the controller design is cast into a convex optimization problem with linear matrix inequality (LMI) constraints. A quarter-car model is considered in this brief and the effectiveness of the proposed approach is illustrated by a realistic design example.

Journal ArticleDOI
01 Dec 2010
TL;DR: A robust consensus tracking problem for a class of second-order multi-agent systems has been addressed in the presence of disturbances and unmodeled dynamics using algebraic graph theory, Lyapunov-based analysis, and an invariance-like theorem.
Abstract: In this paper, a robust consensus tracking problem for a class of second-order multi-agent systems has been addressed in the presence of disturbances and unmodeled dynamics. The desired trajectory to be tracked is only provided to a small group of team members. An identifier is developed to estimate the unknown disturbances and unmodeled dynamics. A continuous consensus tracking controller is developed based on this identifier to achieve asymptotic consensus tracking using the local information obtained from neighboring agents. A sufficient condition is derived to ensure consensus tracking and asymptotic stability of the closed-loop system using algebraic graph theory, Lyapunov-based analysis, and an invariance-like theorem. Numerical simulations are provided to demonstrate the effectiveness of the developed robust consensus controller.

Journal ArticleDOI
TL;DR: Robust adaptive-fuzzy-tracking control of a class of uncertain multi-input/multi-output nonlinear systems with coupled interconnections is considered and it is shown via Lyapunov theory that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded.
Abstract: Robust adaptive-fuzzy-tracking control of a class of uncertain multi-input/multi-output nonlinear systems with coupled interconnections is considered in this paper. Takagi-Sugeno (T-S) fuzzy systems are used to approximate the unknown system functions. A novel adaptive-control scheme is developed on the basis of the so-called ?dynamic-surface control? and ?minimal-learning parameters? techniques. The proposed scheme has following two key features. First, the number of parameters updated online for each subsystem is reduced to one, and both problems of ?curse of dimension? for high-dimensional systems and ?explosion of complexity? inherent in the conventional backstepping methods are circumvented. Second, the potential controller-singularity problem in some of the existing adaptive-control schemes with feedback-linearization techniques is overcome. It is shown via Lyapunov theory that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded. Finally, simulation results via two examples are presented to demonstrate the effectiveness and advantages of the proposed scheme.

Journal ArticleDOI
TL;DR: By constructing a novel Lyapunov-Krasovskii functional, and using some new approaches and techniques, several novel sufficient conditions are obtained to ensure the exponential stability of the trivial solution in the mean square.
Abstract: This paper is concerned with the problem of exponential stability for a class of Markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays and known or unknown parameters. The jumping parameters are determined by a continuous-time, discrete-state Markov chain, and the mixed time delays under consideration comprise both time-varying delays and continuously distributed delays. To the best of the authors' knowledge, till now, the exponential stability problem for this class of generalized neural networks has not yet been solved since continuously distributed delays are considered in this paper. The main objective of this paper is to fill this gap. By constructing a novel Lyapunov-Krasovskii functional, and using some new approaches and techniques, several novel sufficient conditions are obtained to ensure the exponential stability of the trivial solution in the mean square. The results presented in this paper generalize and improve many known results. Finally, two numerical examples and their simulations are given to show the effectiveness of the theoretical results.

Journal ArticleDOI
TL;DR: In this paper, an infection-age model of disease transmission was studied, where both the infectiousness and the removal rate may depend on the infection age, and the system was described using integrated semigroups.
Abstract: We study an infection-age model of disease transmission, where both the infectiousness and the removal rate may depend on the infection age. In order to study persistence, the system is described using integrated semigroups. If the basic reproduction number R 0 1, a Lyapunov functional is used to show that the unique endemic equilibrium is globally stable amongst solutions for which disease transmission occurs.

Journal ArticleDOI
TL;DR: The globally exponential stabilization problem is investigated for a general class of stochastic systems with both Markovian jumping parameters and mixed time-delays and it is shown that the desired state feedback controller can be characterized explicitly in terms of the solution to a set of LMIs.
Abstract: In this technical note, the globally exponential stabilization problem is investigated for a general class of stochastic systems with both Markovian jumping parameters and mixed time-delays. The mixed mode-dependent time-delays consist of both discrete and distributed delays. We aim to design a memoryless state feedback controller such that the closed-loop system is stochastically exponentially stable in the mean square sense. First, by introducing a new Lyapunov-Krasovskii functional that accounts for the mode-dependent mixed delays, stochastic analysis is conducted in order to derive a criterion for the exponential stabilizability problem. Then, a variation of such a criterion is developed to facilitate the controller design by using the linear matrix inequality (LMI) approach. Finally, it is shown that the desired state feedback controller can be characterized explicitly in terms of the solution to a set of LMIs. Numerical simulation is carried out to demonstrate the effectiveness of the proposed methods.

Book
27 Oct 2010
TL;DR: A range of methods and tools to design observers for nonlinear systems represented by a special type of a dynamic nonlinear model -- the Takagi--Sugeno (TS) fuzzy model are provided.
Abstract: Many problems in decision making, monitoring, fault detection, and control require the knowledge of state variables and time-varying parameters that are not directly measured by sensors. In such situations, observers, or estimators, can be employed that use the measured input and output signals along with a dynamic model of the system in order to estimate the unknown states or parameters. An essential requirement in designing an observer is to guarantee the convergence of the estimates to the true values or at least to a small neighborhood around the true values. However, for nonlinear, large-scale, or time-varying systems, the design and tuning of an observer is generally complicated and involves large computational costs. This book provides a range of methods and tools to design observers for nonlinear systems represented by a special type of a dynamic nonlinear model -- the Takagi--Sugeno (TS) fuzzy model. The TS model is a convex combination of affine linear models, which facilitates its stability analysis and observer design by using effective algorithms based on Lyapunov functions and linear matrix inequalities. Takagi--Sugeno models are known to be universal approximators and, in addition, a broad class of nonlinear systems can be exactly represented as a TS system. Three particular structures of large-scale TS models are considered: cascaded systems, distributed systems, and systems affected by unknown disturbances. The reader will find in-depth theoretic analysis accompanied by illustrative examples and simulations of real-world systems. Stability analysis of TS fuzzy systems is addressed in detail. The intended audience are graduate students and researchers both from academia and industry. For newcomers to the field, the book provides a concise introduction dynamic TS fuzzy models along with two methods to construct TS models for a given nonlinear system