Showing papers on "Lyapunov function published in 2005"
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13 Jul 2005
TL;DR: This paper presents SVD-Krylov Methods and Case Studies, a monograph on model reduction using Krylov methods for linear dynamical systems, and some examples of such reduction schemes.
Abstract: Preface Part I. Introduction: 1. Introduction 2. Motivating examples Part II. Preliminaries: 3. Tools from matrix theory 4. Linear dynamical systems, Part 1 5. Linear dynamical systems, Part 2 6. Sylvester and Lyapunov equations Part III. SVD-based Approximation Methods: 7. Balancing and balanced approximations 8. Hankel-norm approximation 9. Special topics in SVD-based approximation methods Part IV. Krylov-based Approximation Methods: 10. Eigenvalue computations 11. Model reduction using Krylov methods Part V. SVD-Krylov Methods and Case Studies: 12. SVD-Krylov methods 13. Case studies 14. Epilogue 15. Problems Bibliography Index.
2,893 citations
TL;DR: It is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
Abstract: We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
2,828 citations
TL;DR: It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS) and the result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.
1,045 citations
TL;DR: It is proved that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Holder continuous state feedback.
982 citations
TL;DR: Several sufficient conditions are derived for the existence, uniqueness, and GRS of equilibria for interval neural networks with time delays by use of a new Lyapunov function and matrix inequality.
Abstract: In this paper, two related problems, global asymptotic stability (GAS) and global robust stability (GRS) of neural networks with time delays, are studied. First, GAS of delayed neural networks is discussed based on Lyapunov method and linear matrix inequality. New criteria are given to ascertain the GAS of delayed neural networks. In the designs and applications of neural networks, it is necessary to consider the deviation effects of bounded perturbations of network parameters. In this case, a delayed neural network must be formulated as a interval neural network model. Several sufficient conditions are derived for the existence, uniqueness, and GRS of equilibria for interval neural networks with time delays by use of a new Lyapunov function and matrix inequality. These results are less restrictive than those given in the earlier references.
498 citations
TL;DR: Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established and a new hybrid impulsive and switching control strategy for nonlinear systems is developed.
Abstract: In this note, a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated. Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established. In addition, a new hybrid impulsive and switching control strategy for nonlinear systems is developed. A typical example, the unified chaotic system, is given to illustrate the theoretical results.
401 citations
TL;DR: The adaptive control laws proposed in this paper are optimal with respect to a family of cost functionals by the inverse optimality approach, without solving the associated Hamilton-Jacobi-Isaacs partial differential equation directly.
Abstract: The attitude tracking control problem of a rigid spacecraft with external disturbances and an uncertain inertia matrix is addressed using the adaptive control method. The adaptive control laws proposed in this paper are optimal with respect to a family of cost functionals. This is achieved by the inverse optimality approach, without solving the associated Hamilton-Jacobi-Isaacs partial differential (HJIPD) equation directly. The design of the optimal adaptive controllers is separated into two stages by means of integrator backstepping, and a control Lyapunov argument is constructed to show that the inverse optimal adaptive controllers achieve H/sub /spl infin// disturbance attenuation with respect to external disturbances and global asymptotic convergence of tracking errors to zero for disturbances with bounded energy. The convergence of adaptive parameters is also analyzed in terms of invariant manifold. Numerical simulations illustrate the performance of the proposed control algorithms.
383 citations
TL;DR: The main idea is to design a Lyapunov-based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed-loop system.
Abstract: In this work, a predictive control framework is proposed for the constrained stabilization of switched nonlinear systems that transit between their constituent modes at prescribed switching times. The main idea is to design a Lyapunov-based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed-loop system. This is achieved as follows: For each constituent mode, a Lyapunov-based model predictive controller (MPC) is designed, and an analytic bounded controller, using the same Lyapunov function, is used to explicitly characterize a set of initial conditions for which the MPC, irrespective of the controller parameters, is guaranteed to be feasible, and hence stabilizing. Then, constraints are incorporated in the MPC design which, upon satisfaction, ensure that: 1) the state of the closed-loop system, at the time of the transition, resides in the stability region of the mode that the system is switched into, and 2) the Lyapunov function for each mode is nonincreasing wherever the mode is reactivated, thereby guaranteeing stability. The proposed control method is demonstrated through application to a chemical process example.
358 citations
13 Mar 2005
TL;DR: In this paper, a nonsmooth optimization technique is proposed to solve H∞ synthesis problems under additional structural constraints on the controller, which avoids the use of Lyapunov variables and therefore leads to moderate size optimization programs even for very large systems.
Abstract: We develop nonsmooth optimization techniques to solve H∞ synthesis problems under additional structural constraints on the controller. Our approach avoids the use of Lyapunov variables and therefore leads to moderate size optimization programs even for very large systems. The proposed framework is very versatile and can accommodate a number of challenging design problems including static, fixed-order, fixed-structure, decentralized control, design of PID controllers and simultaneous design and stabilization problems. Our algorithmic strategy uses generalized gradients and bundling techniques suited for the H∞-norm and other nonsmooth performance criteria. Convergence to a critical point from an arbitrary starting point is proved (full version) and numerical tests are included to validate our methods.
334 citations
TL;DR: In this article, a command filtered backstepping approach is presented that uses adaptive function approximation to control UAVs using three feedback loops, including an inner loop that generates surface position commands.
Abstract: A command filtered backstepping approach is presented that uses adaptive function approximation to control unmanned air vehicles. The controller is designed using three feedback loops. The command inputs to the airspeed and flight-path angle controller are x c , γ c , V c and the bounded first derivatives of these signals. That loop generates comand inputs μ c , α c for a wind-axis angle loop. The sideslip angle command β c is always zero. The wind-axis angle loop generates rate commands P c , Q c , R c for an inner loop that generates surface position commands. The control approach includes adaptive approximation of the aerodynamic force and moment coefficient functions. The approach maintains the stability (in the sense of Lyapunov) of the adaptive function approximation process in the presence of magnitude, rate, and bandwidth limitations on the intermediate states and the surfaces.
330 citations
TL;DR: This paper revisits the problem of mixed H/sub 2//H/sub /spl infin// filtering for polytopic discrete-time systems and makes full use of the parameter-dependent stability idea, which results in a much less conservative filter design method.
Abstract: This paper revisits the problem of mixed H/sub 2//H/sub /spl infin// filtering for polytopic discrete-time systems. Differing from previous results in the quadratic framework, the filter design makes full use of the parameter-dependent stability idea: Not only is the filter dependent of the parameters (which are assumed to reside in a polytope and be measurable online), but in addition, the Lyapunov matrices are different for the entire polytope domain, as well as for different channels with respect to the mixed performances. These ideas are realized by introducing additional slack variables to the well-established performance conditions and by employing new bounding techniques, which results in a much less conservative filter design method. A numerical example is presented to illustrate the effectiveness and advantage of the developed filter design method.
TL;DR: The switching logic tracks the evolution of the state estimates generated by the observers and orchestrates switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system.
TL;DR: In this paper, a modification based on Lyapunov stability theory to design a controller is proposed in order to overcome the limitation of nonlinear control for making two identical chaotic systems or two different chaotic systems be synchronized.
Abstract: Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2eTe. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.
TL;DR: It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals, and it is demonstrated that under the constant population size assumption, these models exhibit asymptotically stable steady states.
Abstract: In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds.
TL;DR: In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability of neutral systems and two methods with or without decoupling the LyAPunov matrices and system matrices are developed and shown to be equivalent to each other.
Abstract: In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability of neutral systems Two methods with or without decoupling the Lyapunov matrices and system matrices are developed and shown to be equivalent to each other The resulting delay-dependent stability criteria are less conservative than the existing ones owing to the augmented Lyapunov functional and the introduction of free-weighting matrices The delay-independent criteria are obtained as an easy corollary Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods Copyright © 2005 John Wiley & Sons, Ltd
TL;DR: Stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model predictive control algorithms that do not require the terminal cost to be a local control Lyapunov function are presented.
Abstract: We present stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model predictive control (MPC) algorithms that do not require the terminal cost to be a local control Lyapunov function. The two key assumptions we make are that the value function is bounded by a K/sub /spl infin// function of a state measure related to the distance of the state to the target set and that this measure is detectable from the stage cost. We show that these assumptions are sufficient to guarantee closed-loop asymptotic stability that is semiglobal and practical in the horizon length and robust to small perturbations. If the assumptions hold with linear (or locally linear) K/sub /spl infin// functions, then the stability will be global (or semiglobal) for long enough horizon lengths. In the global case, we give an explicit formula for a sufficiently long horizon length. We relate the upper bound assumption to exponential and asymptotic controllability. Using terminal and stage costs that are controllable to zero with respect to a state measure, we can guarantee the required upper bound, but we also require that the state measure be detectable from the stage cost to ensure stability. While such costs and state measures may not be easy to construct in general, we explore a class of systems, called homogeneous systems, for which it is straightforward to choose them. In fact, we show for homogeneous systems that the associated K/sub /spl infin// functions are linear, thereby guaranteeing global asymptotic stability. We discuss two examples found elsewhere in the MPC literature, including the discrete-time nonholonomic integrator, to demonstrate our methods. For these systems, we give a new result: They can be globally asymptotically stabilized by a finite-horizon MPC algorithm that has guaranteed robustness. We also show that stable linear systems with control constraints can be globally exponentially stabilized using finite-horizon MPC without requiring the terminal cost to be a global control Lyapunov function.
TL;DR: The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricIAL forms and the Lyapunov matrix equation.
Abstract: In this note, robust stability of state-space models with respect to real parametric uncertainty is considered. Specifically, a new class of parameter-dependent quadratic Lyapunov functions for establishing stability of a polytope of matrices is introduced, i.e., the homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs). The choice of this class, which contains parameter-dependent quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form, is motivated by the property that a polytope of matrices is stable if and only there exists an HPD-QLF. The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricial forms and the Lyapunov matrix equation. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.
TL;DR: A direct adaptive state-feedback controller is proposed for highly nonlinear systems and employs a neural network with flexible structure, i.e., an online variation of the number of neurons that approximates and adaptively cancels an unknown plant nonlinearity.
Abstract: A direct adaptive state-feedback controller is proposed for highly nonlinear systems. We consider uncertain or ill-defined nonaffine nonlinear systems and employ a neural network (NN) with flexible structure, i.e., an online variation of the number of neurons. The NN approximates and adaptively cancels an unknown plant nonlinearity. A control law and adaptive laws for the weights in the hidden layer and output layer of the NN are established so that the whole closed-loop system is stable in the sense of Lyapunov. Moreover, the tracking error is guaranteed to be uniformly asymptotically stable (UAS) rather than uniformly ultimately bounded (UUB) with the aid of an additional robustifying control term. The proposed control algorithm is relatively simple and requires no restrictive conditions on the design constants for the stability. The efficiency of the proposed scheme is shown through the simulation of a simple nonaffine nonlinear system.
TL;DR: In this paper, robust adaptive control for a class of parametric-strict-feedback nonlinear systems with unknown time delays is presented, and a systematic backstepping design method is proposed to guarantee global uniform ultimate boundedness of all the signals.
01 Jun 2005
TL;DR: The aim is to design a mode-independent fuzzy controller such that the closed-loop Markovian jump fuzzy system (MJFS) is robustly stochastically stable and derived for the uncertain MJFS in terms of linear matrix inequalities (LMIs).
Abstract: This paper is concerned with the robust-stabilization problem of uncertain Markovian jump nonlinear systems (MJNSs) without mode observations via a fuzzy-control approach. The Takagi and Sugeno (T-S) fuzzy model is employed to represent a nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. The aim is to design a mode-independent fuzzy controller such that the closed-loop Markovian jump fuzzy system (MJFS) is robustly stochastically stable. Based on a stochastic Lyapunov function, a robust-stabilization condition using a mode-independent fuzzy controller is derived for the uncertain MJFS in terms of linear matrix inequalities (LMIs). A new improved LMI formulation is used to alleviate the interrelation between the stochastic Lyapunov matrix and the system matrices containing controller variables in the derivation process. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method.
TL;DR: It is proved that the constructed controller can render the closed-loop system asymptotically stable and based on Lyapunov stability theory, it is shown that the designed observer and controller are independent of the time delays.
Abstract: In this note, the problem of robust output feedback control for a class of nonlinear time delayed systems is considered. The systems considered are in strict-feedback form. State observer is first designed, then based on the observed states the controller is designed via backstepping method. Both the designed observer and controller are independent of the time delays. Based on Lyapunov stability theory, we prove that the constructed controller can render the closed-loop system asymptotically stable. Simulation results further verify the effectiveness of the proposed approach.
TL;DR: An observer-based direct adaptive fuzzy-neural control scheme is presented for nonaffine nonlinear systems in the presence of unknown structure of nonlinearities and based on strictly-positive-real (SPR) Lyapunov theory, the stability of the closed-loop system can be verified.
Abstract: In this paper, an observer-based direct adaptive fuzzy-neural control scheme is presented for nonaffine nonlinear systems in the presence of unknown structure of nonlinearities. A direct adaptive fuzzy-neural controller and a class of generalized nonlinear systems, which are called nonaffine nonlinear systems, are instead of the indirect one and affine nonlinear systems given by Leu et al. By using implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the fuzzy-neural controller are derived for the nonaffine nonlinear systems. Based on strictly-positive-real (SPR) Lyapunov theory, the stability of the closed-loop system can be verified. Moreover, the overall adaptive scheme guarantees that all signals involved are bounded and the output of the closed-loop system will asymptotically track the desired output trajectory. To demonstrate the effectiveness of the proposed method, simulation results are illustrated in this paper.
TL;DR: A proportional-derivative control with on-line gravity compensation is proposed for regulation tasks of robot manipulators with elastic joints and it is proved via a Lyapunov argument that the control law globally asymptotically stabilizes the desired robot configuration.
TL;DR: It is proved that the optimal fusion estimators under three weighted fusion rules are locally optimal, but are globally suboptimal.
TL;DR: A Lyapunov-based approach for trajectory tracking of the Schrodinger equation, when the reference trajectory is an eigenstate, a modification based on adiabatic invariance is proposed, illustrating the simplicity and also the interest for trajectory generation.
TL;DR: This work considers the full 3n-dimensional model of the n-generator system with lossy transmission lines and loads and proves the existence of a nonlinear static state feedback law for the generator excitation field that ensures asymptotic stability of the operating point with a well-defined estimate of the domain of attraction provided by a bona fide Lyapunov function.
Abstract: We provide a solution to the long-standing problem of transient stabilization of multimachine power systems with nonnegligible transfer conductances. More specifically, we consider the full 3n-dimensional model of the n-generator system with lossy transmission lines and loads and prove the existence of a nonlinear static state feedback law for the generator excitation field that ensures asymptotic stability of the operating point with a well-defined estimate of the domain of attraction provided by a bona fide Lyapunov function. To design the control law we apply the recently introduced interconnection and damping assignment passivity-based control methodology that endows the closed-loop system with a port-controlled Hamiltonian structure with desired total energy function. The latter consists of terms akin to kinetic and potential energies, thus has a clear physical interpretation. Our derivations underscore the deleterious effects of resistive elements which, as is well known, hamper the assignment of simple "gradient" energy functions and compel us to include nonstandard cross terms. A key step in the construction is the modification of the energy transfer between the electrical and the mechanical parts of the system which is obtained via the introduction of state-modulated interconnections that play the role of multipliers in classical passivity theory.
TL;DR: In this paper, a new construction of Lyapunov-krasovskii functionals (LKFs) is applied to a discrete-time system with time-varying delays from given intervals.
Abstract: Robust stability and the guaranteed cost control problem are considered for discrete-time systems with time-varying delays from given intervals. A new construction of Lyapunov–Krasovskii functionals (LKFs), which has been recently introduced in the continuous-time case, is applied. To a nominal LKF, which is appropriate to the system with nominal delays, terms are added that correspond to the system with the perturbed delays and that vanish when the delay perturbations approach zero. The nominal LKF is chosen in the form of the descriptor type and is applied either to the original or to the augmented system. The delay-independent result is derived via the Razumikhin approach. Guaranteed cost state-feedback control is designed. The advantage of the new tests is demonstrated via illustrative examples.
TL;DR: Five different initial conditions are studied, disclose the inherent relationship between each initial condition and corresponding learning convergence (or boundedness) property, and the iterative learning control method under consideration is based on Lyapunov theory.
Abstract: Initial conditions, or initial resetting conditions, play a fundamental role in all kinds of iterative learning control methods. In this note, we study five different initial conditions, disclose the inherent relationship between each initial condition and corresponding learning convergence (or boundedness) property. The iterative learning control method under consideration is based on Lyapunov theory, which is suitable for plants with time-varying parametric uncertainties and local Lipschitz nonlinearities.
TL;DR: A delay-dependent stability criterion is derived and a set of model-based fuzzy controllers is synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay large-scale system.
Abstract: To overcome the effect of modeling errors between nonlinear multiple time-delay subsystems and Takagi-Sugeno (T-S) fuzzy models with multiple time delays, a robustness design of fuzzy control is proposed in This work. In terms of Lyapunov's direct method, a delay-dependent stability criterion is hence derived to guarantee the asymptotic stability of nonlinear multiple time-delay large-scale systems. Based on this criterion and the decentralized control scheme, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay large-scale system. Finally, a numerical example with simulations is given to demonstrate the concepts discussed throughout This work.
TL;DR: In this paper, the effects of time-delay effects on the master-slave synchronization scheme are studied by a time delay feedback control technique, and several new delay-independent and delay-dependent sufficient conditions are presented for master/slave synchronization of Lur'e systems based upon Lyapunov method and linear matrix inequalities (LMI) approaches.
Abstract: In this paper time-delay effects on the master–slave synchronization scheme are studied by a time-delay feedback control technique. Several new delay-independent and delay-dependent sufficient conditions are presented for master–slave synchronization of Lur’e systems based upon Lyapunov method and linear matrix inequalities (LMI’s) approaches. These new synchronization criteria are easily verifiable and offer some fairly adjustable real parameters, which are of important significance in the design and applications of such chaos synchronization systems, and the proposed results improve and generalize the earlier works.