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In this paper, a new scheme based on integral observer approach is designed for a class of chaotic systems to achieve synchronization. Unlike the proportional observer approach, the proposed scheme is demonstrated to be effective under a noisy environment in the transmission channel. Based on the Lyapunov stability theory, a sufficient condition for synchronization is derived in the form of a Lyapunov inequality. This Lyapunov inequality is further transformed into a linear matrix inequality (LMI) form by using the Schur theorem and some matrix operation techniques, which can be easily solved by the LMI toolboxes for the design of suitable control gains. It is demonstrated with the Murali-Lakshmanan-Chua system that a better noise suppression and a faster convergence speed can be achieved for chaos synchronization by using this integral observer scheme, as compared with the traditional proportional observer approach. Index Terms--Chaos synchronization, disturbance, integral observer, linear matrix inequality, Lyapunov stability theory.

Abstract: It is proved that a linear time-invariant system with internal point delays is asymptotically hyperstable independent of the delays if an associate delay-free system is asymptotically hyperstable and the [...]

The dynamics of cortical cognitive maps developed by self-organization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point. Index Terms--Flow invariance, global exponential stability, multitime scale neural network.

Two integral versions of input to state stability are considered: integral input-to-state stability (iISS) and integral input-to-integral-state stability (iIiSS). We present sufficient conditions that guarantee that if a controller achieves semiglobal practical iISS (respectively, iIiSS) of an approximate discrete-time model of a nonlinear sampled-data system, then the same controller achieves semiglobal practical iISS (respectively, iIiSS) of the exact discrete-time model by reducing the sampling period. Recent results on numerical methods for systems with measurable disturbances can be used to generate approximate models that we consider. Results are presented for arbitrary dynamic controllers that can be discontinuous in general. Index Terms--Disturbances, integral input-to-state stability (iISS), Lyapunov, nonlinear, sampled-data.

Exact nonlinear differential equations of boost pulsewidth modulation converters (either with hard-switching or soft-switching, in continuous and discontinuous conduction mode) are obtained, without using average approximations. An extended system of equations also contains the differential equation of the integrated error between the reference and the output voltage. A positive-definite Lyapunov function is defined, including in its expression a nonlinear integral of type Lurie. Sufficient conditions for the Lyapunov stability of the system are determined. Consequently, constraints on the feedback circuit parameters that assure the large-signal stability of the system are formulated. These constraints relate the parameters of the feedback circuit with the values of the power stage elements, load, switching frequency, steady-state duty-cycle. For different values of the steady-state duty-cycle, graphical boundaries which assure the large-signal stability for all the variation of the load are designed. Simulations and experiments confirm the stability of the design. Index Terms--Boost regulator, large-signal stability, Lyapunov stability.

In this note, we revisit the Lyapunov theory for singular systems. There are basically two well-known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of the work by Lewis. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clarified. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of Takaba et al. Index Terms--Lyapunov equation, singular systems.

This note addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this note looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation. Index Terms--Hybrid systems, static output feedback, switched Lyapunov function, switched systems.

A sufficient condition for checking the robust stability of a polytope of polynomial matrices is proposed in this note. A simple feasibility test performed in a convex set of linear-matrix inequalities defined at the vertices of the polytope yields sufficient conditions for the robust stability of the entire domain, Both continuous-time (left haft-plane) and discrete-time stability (unit disk) are investigated. Numerical comparisons with quadratic stability results and with another method recently appeared in the literature show that the conditions proposed provide, in general, less conservative results. Index Terms--Linear-matrix inequalities (LMIs), parameter dependent Lyapunov functions, polynomial matrices, robust stability, uncertain polynomials.

The problem of point-to-point control design for differentially steered nonholonomic mobile robots is considered in this paper. The control variables are derived using Lyapunov's stability technique and are piecewise continuous. The proposed control law guarantees the exponential stability of the closed-loop system and ensures the convergence of the position and the orientation of the robot to their desired fixed values. Keywords: Mobile robot; Nonholonomic constraint; Point-to-point motion; Stabilization and control; Piecewise continuous control; Backstepping technique; Lyapunov stability method; Exponential stability

We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class KL function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems. Index Terms--Input-to-state dynamical stability (ISDS), Lyapunov function, nonlinear stability margin, small gain theorem.

In this note, the problem of learning unknown functions in a class of cascaded nonlinear systems will be studied. The functions to be learned are those functions that are imbedded in the system dynamics and are of known period of time. In addition to the unknown periodic time functions, nonlinear uncertainties bounded by known functions of the state are also admissible. The objective of the note is to find an iterative learning control under which the class of nonlinear systems are globally stabilized (in the sense of being uniform bounded), their outputs are asymptotically convergent, and a combination of the time functions contained in system dynamics are asymptotically learned. To this end, a new type of differential-difference learning law is utilized to generate the proposed learning control that yields both asymptotic stability of the system output and asymptotic convergence of the learning error. The design is carried out by applying the Lyapunov direct method and the backward recursive design method. Index Terms--Learning control, Lyapunov design, periodic function, stability, uncertain system.

The sensitivity of coupled continuous-time Lyapunov equations is analyzed through three types of norms of the inverse Lyapunov-like operator. This analysis leads to bounds on the relative perturbation of the solution. The results extend the bounds obtained for standard Lyapunov equation in [6]. Index Terms--Jump linear systems, Lyapunov equation, stability.

This note presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the system can be established if a piecewise Lyapunov function can be constructed and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software. Index Terms--Discrete-time systems, linear matrix inequality (LMI), piecewise linear systems, stability.

This paper is devoted to stability analysis of linear systems with state delay and input saturation. The domain of attraction resulting from an a priori designed state feedback law is analyzed using Lyapunov-Razumikhin and Lyapunov-Krasovskii functional approach. Both delay-independent and delay-dependent estimation of the domain of attraction are presented using the linear matrix inequality technique. The problem of designing linear state feedback laws such that the domain of attraction is enlarged is formulated and solved as an optimization problem with LMI constraints. Numerical examples are used to demonstrate the effectiveness of the proposed design technique. Index Terms--Actuator saturation, domain of attraction, linear matrix inequality, time-delay.

In this paper, we stabilize the parabolic equilibrium profile in a two-dimensional (2-D) channel flow using actuators and sensors only at the wall. The control of channel flow was previously considered by Speyer and coworkers, and Bewley and coworkers, who derived feedback laws based on linear optimal control, and implemented by wall-normal actuation. With an objective to achieve global Lyapunov stabilization, we arrive at a feedback law using tangential actuation (using teamed pairs of synthetic jets or rotating disks) and only local measurements of wall shear stress, allowing to embed the feedback in micro-electromechanical systems (MEMS) hardware, without need for wiring. This feedback is shown to guarantee global stability in at least [H.sup.2] norm, which by Sobolev's embedding theorem implies continuity in space and time of both the flow field and the control (as well as their convergence to the desired steady state). The theoretical results are limited to low values of Reynolds number, however, we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher. Index Terms--Boundary feedback, Lyapunov stability, Navier--Stokes equations, tangential velocity actuation, two-dimensional (2-D) channel flow.

This paper establishes a global convergent algorithm for discrete-time model reference adaptive systems (MRAS's), an extension of the classical decreasing gain algorithm. Using this approach, the process, in adaptive control, may contain zeros outside the unit circle if the external reference signal is a solution of a linear homogeneous difference equation. Index Terms: Adaptive control, identification, Lyapunov theorem, stability theorem

The authors consider multirate digital control systems which consist of an interconnection of a continuous-time nonlinear plant (described by ordinary differential equations), a digital controller (described by ordinary difference equations) which has quantizers (but is otherwise linear), along with the required interface elements (A/D and D/A converters). The input to the digital controller consists of the multirate sampled output of the plant. In the present paper, the authors show that when quantizer nonlinearities are neglected, then under reasonable conditions (which exclude the critical cases), the stability properties (in the Lyapunov sense) of the trivial solution of a nonlinear multirate digital control system can be deduced from the stability properties of the trivial solution of its linearization. For such systems we also present a result concerning the existence and construction of stabilizing multirate-output digital controllers. In the present paper, the authors also show that the solutions of a multirate digital feedback control system with nonlinear plant and quantizers are uniformly ultimately bounded if the trivial solution of the corresponding linear system consisting of the linearization of the plant and with the quantization removed from the digital controller is asymptotically stable. We also provide a result which compares the response of multirate digital control systems with nonlinear plant and quantizers in the controller with the response of the corresponding nonlinear multirate digital control systems without quantizers in the digital controller. Index Terms - Lagrange stability, Lyapunov stability, multirate digital control systems.

Stochastic averaging equations have been derived for integrable and nonresonant Hamiltonian system with multi-degree-of-freedom and subjected to light damping and real noise excitations. Khasminskii's method is then used with the averaged equations expression for obtaining the largest Lyapunov exponents of the square root of the Hamiltonian. The stochastic stability and bifurcation of the original systems can then be determined approximately.

Diagonal solutions of a Lyapunov inequality for companion matrices are studied. Such solutions are required if states of a discrete-time linear system are computed with a finite-precision arithmetic. Index Terms - Companion matrix, critical exponent, diagonal stability, discrete Lyapunov equation.

This paper presents a hyperstability theorem for a class of hybrid dynamic systems composed of coupled differential and difference equations subject to (possibly) time-varying nonlinearities satisfying a Popov-type inequality. The nonlinear controller generates the plant input at all times from its sampled values by defining an extended discrete system. The hyperstability results are obtained from this discrete system of special type whose state consists of the sampled continuous substate and the digital substate of the given hybrid system. Some corollaries and related physical interpretations are also given. Index Terms - Digital systems, hybrid systems, hyperstability.

In this paper, we study the global stabilization, by means of smooth state feedback, of partially linear composite systems. We show how to compute the stabilizing feedback thanks to a weak Lyapunov function for a nonlinear subsystem instead of a strict one. Index Terms - Feedback, global stabilization, Lyapunov function, nonlinear systems.

The parameterized Lyapunov bounding technique of Haddad and Bernstein is extended to include an [H.sub.[infinity]]-disturbance attenuation constraint. The results presented in this paper provide a framework for designing fixed-order (i.e., full- and reduced-order) controllers that guarantee robust [H.sub.2] and [H.sub.[infinity]] performance in the presence of structured constant real parameter variations in the state space model. Index Terms - [H.sub.2]/[H.sub.[infinity]] design, real parameter uncertainty, parameter-dependent Lyapunov functions.

We introduce a new method to investigate stability of work-conserving policies in multiclass queueing networks. The method decomposes feasible trajectories and uses linear programming to test stability. We show that this linear program is a necessary and sufficient condition for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class of (piecewise linear convex) Lyapunov functions. Using linear programming duality, we show that for two-station systems the Lyapunov function approach is equivalent to ours and therefore characterizes stability exactly.

In this paper, some globally asymptotical stability criteria for the equilibrium states of a general class of discrete-time dynamic neural networks with continuous states are presented using a diagonal Lyapunov function approach. The neural networks are assumed to have the asymmetrical weight matrices throughout the paper. The resulting criteria are described by the diagonal stability of some matrices associated with the network parameters. Some novel stability conditions represented by either the existence of the positive diagonal solutions of the Lyapunov equations or some inequalities are given. Using the equivalence between the diagonal stability and the Schur stability for a nonnegative matrix, some simplified global stability conditions are also presented. Finally, some examples are provided for demonstrating the effectiveness of the global stability conditions presented.

A token passing ring can be described as a system of M queues with one server that rotates around the queues sequentially. Georgiadis and Szpankowski [4, pp. 7-33] considered rings where the token (server) performs x [conjunction] [l.sub.p] services on queue j, where x is the size of queue j upon arrival of the token, and [l.sub.j] is a fixed limit of service for queue j. The token then spends some random time switching to the next queue. For j = 1, . . ., M, arrivals to queue j are Poisson with rate [[Lambda].sub.j], and service times have mean [s.sub.j] and are independent of the arrival and switchover processes. The purpose of this paper is to give an alternate and simpler proof of the stability conditions given in [4] using Lyapunov functions. An additional assumption is made about the second moments of the service and switchover times being finite.

We note that several of the results on Schur-stability of interval matrices can be viewed as special cases of results obtained in a more general framework, and some of the latter were previously reported in the literature.

The stability and instability of fluid models are analyzed in consideration of Dai's (1995) recently introduced theorem stating that the stability of a scheduling policy depends on the stability of the corresponding fluid model. The study uses the piecewise linear Lyapunov functions introduced by Botvitch and Zamyatin (1992). The stability of First-Buffer-First-Served and Last-Buffer-First-Served disciplines is determined in all reentrant lines. The stability of the Lu and Kumar's four buffer network and of Kelly-type networks is also established.

This paper describes an approach to the stability analysis of two-dimensional (2-D) digital filters that are modeled in the Fornasini-Marchesini state space using a class of generalized 2-D Lyapunov equations. The generalization was made based on the constant 2-D Lyapunov equation proposed recently by Hinamoto. It is shown that the use of the generalize Lyapunov equations narrows the gap between 'sufficiency' and 'necessity' for a state-space digital filter to be stable, which occurs in Hinamoto's Lyapunov theorem. Feasible methods for finding numerical solutions of the generalized 2-D Lyapunov equation are also proposed. An example is included to illustrate the main results of the paper.

An algorithm and its intended application to large-scale system decentralized controller designs are presented in this paper. Index Terms - Decentralized control, large-scale systems.

Sufficient conditions for the global asymptotic stability of the equilibrium [x.sub.e] = 0 of discrete-time dynamical systems which have saturation nonlinearities on part of the states are established. We utilize a class of positive definite and radially unbounded Lyapunov functions in establishing our results. When using quadratic form Lyapunov functions, our results involve necessary and sufficient conditions under which positive definite matrices can be used to generate Lyapunov functions for the systems considered herein.

In this letter, we develop a new adjustment rule for a perceptron with a saturating nonlinearity that ensures perfect classification when the input patterns are linearly separable. The proof is based on the Lyapunov stability formalism, is widely used in deterministic process identification, and is rather straightforward. It should therefore be of pedagogical interest.

A stability criterion for uncertain systems with time-varying delay is derived via the Lyapunov functional approach. By checking the Hamiltonian matrix and solving an algebraic Riccati equation, a new bound on allowable stability preserving nonlinear perturbations is presented. The result obtained here is shown to be less conservative than others reported in the literature. Index Terms--Robust stability, time-delay systems, uncertain systems.

A one-dimensional iterative chaotic map with infinite collapses within symmetrical region [-1, 0) [union] (0, +1] is proposed. The stability of fixed points and that around the singular point are analyzed. Higher Lyapunov exponents of proposed map show stronger chaotic characteristics than some iterative and continuous chaotic models usually used. There exist inverse bifurcation phenomena and special main periodic windows at certain positions shown in the bifurcation diagram, which can explain the generation mechanism of chaos. The chaotic model with good properties can be generated if choosing the parameter of the map properly. Stronger inner pseudorandom characteristics can also be observed through [chi square] test on the supposition of even distribution. This chaotic model may have many advantages in practical use. Index Terms--Bifurcation, iterative chaotic map with infinite collapses, Lyapunov exponents, main periodic windows, stability, [chi square] test.

In this note, we consider production-distribution systems with buffer and capacity constraints. For such systems, we assume that the model is not known exactly. More precisely, the entries of the matrix representing the system structure may be affine functions of some uncertain time-varying parameters that take values within assigned bounds. We give stabilizability conditions that can be checked, in principle, by solving a min-max problem on the surface of the state-space (buffer level space) unit ball. Then, we consider a special case in which each uncertain parameter affects a single column of the system matrix and is independent of all the other ones. In this case, we propose a mixed integer linear programming algorithm to check the stabilizability conditions and we provide a stabilizing control in an explicit form. Index Terms--Constrained control, inventory systems, Lyapunov functions, min-max control, mixed integer linear programming, uncertain systems.

In this note, we analyze the discrete behavior to identify all kinds of cycles of hybrid nonlinear systems and then study the continuous behavior along each kind of cycle. Based on these analysis, we construct some continuous functions to bound Lyapunov functions along all subsystems and identify a subsequence of time points where the Lyapunov functions are nonincreasing. We use these results to derive some new sufficient conditions for the robust stability of a class of hybrid nonlinear systems with polytopic uncertainties. These conditions do not require the Lyapunov functions to be nonincreasing along each subsystem nor the whole sequence of the switchings. Furthermore, they do not require the knowledge of continuous trajectory. Index Terms--Hybrid nonlinear systems, robust stability.

This work evolved from an endeavor to derive stability criteria and Poincare normal forms for nonlinear systems associated with a nonsemisimple zero (in Part I) or a pair of imaginary eigenvalues (in Part II). The stability criteria are given in terms of the noninteracting vector restoring and restraining forces, which are motivated from the Lienard equation for nonlinear mass-damper-spring system models. Lyapunov functions are constructed explicitly to fulfill the La Salle invariant principle for local or global stability assertion. It turned out that the Lyapunov functions thus constructed apply to a wide variety of linear stability scenarios. By introducing the notions of restoring and restraining forces, how the Lyapunov functions, the stability criteria and the system dynamics interplay are also exhibited. Two distinct classes of nonlinearities which we refered to as being arithmetical and transcendental, emerged. In some sense, such systems carry nonlinear lags coexisting with the linear lead. In particular, a characteristic of the nonlinear dynamics, a staircase structure, is discovered. Further extension is also made to incorporate nondestabilizing perturbation, which bears important bifurcational implications. Index Terms--Lyapunov function/stability, nonsemisimple zero mode, Poincare normal forms.

This paper studies nonlinear systems associated with or linked to a nonsemisimple (NSS) pair of imaginary eigenvalues. Stability criteria and Poincare normal forms (PNF's) are derived via explicit construction of Lyapunov functions. It is a counterpart and continuation of Part I, which considered the case with an NSS zero eigenvalue. While originating from stability analysis for a nonsemisimple critical mode, the development extends to co-existing (semi-) simple critical modes and to stable modes, as in Part I. The present NSS imaginary pairs (NSSIP) case is shown to possess certain characteristics common to NSS zero (NSSZ), such as the nonlinear lag and the staircase structure, but retains its own identities as well. These include the less surprising, the submodular variable-pair circles which emerge from the units of the potential generator and the more remarkable stricter specification in the dynamics, which is possibly attributable to the inherited conjugacy. An elementary partial differential equation traversing the development shows potential for nontrivial generalization. Index Terms--Lyapunov function/stability, nonsemisimple imaginary pair mode, Poincare normal forms.

This paper comments on the results of a recent paper. We note that Theorems 3.1 and 4.1 are incorrectly stated, i.e., they are not valid for the non-[Omega]-transformable regions. The results cannot cover the ride quality region listed in Table 1 since it is a non-[Omega]-transformable region.

In this paper, some novel criteria for the global robust stability of a class of interval Hopfield neural networks with constant delays are given. Based on several new Lyapunov functionals, delay-independent criteria are provided to guarantee the global robust stability of such systems. For conventional Hopfield neural networks with constant delays, some new criteria for their global asymptotic stability are also easily obtained. All the results obtained are generalizations of some recent results reported in the literature for neural networks with constant delays. Numerical examples are also given to show the correctness of our analysis. Index Terms--Interval Hopfield neural networks, Lyapunov functionals, robust stability, time delays.

This note presents a new approach to deriving stability conditions for continuous-time linear systems interconnected with a saturation. The method presented here can be extended to handle a deadzone, or in general, nonlinearities in the form of piecewise linear functions. By representing the saturation as a constrained optimization problem, the necessary (Kuhn-Tucker) conditions for optimality are used to derive linear and quadratic constraints which characterize the saturation. After selecting a candidate Lyapunov function, we pose the question of whether the Lyapunov function is decreasing along trajectories of the system as an implication between the necessary conditions derived from the saturation optimization, and the time derivative of the Lyapunov function. This leads to stability conditions in terms of linear matrix inequalities (LMIs), which are obtained by an application of the S-procedure to the implication. An example is provided where the proposed technique is compared and contrasted with previous analysis methods. Index Terms--Kuhn-Tucker conditions, linear system, Lyapunov function, S-procedure, saturation.

The authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields. They allow switching between members of the family of vector fields to be completely random. The focus of this paper is to prove a converse Lyapunov theorem for this class of systems. Index Terms - Hybrid systems, nonlinear, polysystems, stability, switching.

In this paper, the authors give a necessary and sufficient condition for globally stabilizing a nonlinear system, robustly with respect to unstructured uncertainties [Mathematical Expression Omitted] (u,x,t), norm-bounded for each fixed x and u. This condition requires one to find a smooth, proper, and positive definite solution V (x) of a suitable partial differential inequality depending only on the system data. A procedure, based on the knowledge of V(x), is outlined for constructing almost smooth robustly stabilizing controllers. Our approach, based on Lyapunov functions, generalizes previous results for linear uncertain systems and establishes a precise connection between robust stabilization, on one hand, and [H.sub.[infinity]]-control sector conditions and input-to-state stabilization on the other. Index Terms - Lyapunov functions, robust stabilization, unstructured uncertainties.