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In this paper, we analyze the behavior of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee. We prove that after a large number of iterations, if the stochastic approximation process enters the domain of attraction of an attracting set, it gets locked into the attracting set with high probability. We demonstrate that the above-mentioned result is an effective instrument for analyzing stochastic approximation schemes in the absence of a stability guarantee, by using it to obtain an alternate criterion for convergence in the presence of a locally attracting set for the mean field and by using it to show that a feedback mechanism, which involves resetting the iterates at regular time intervals, stabilizes the scheme when the mean field possesses a globally attracting set, thereby guaranteeing convergence. The results in this paper build on the works of Borkar, Andrieu et al., and Chen et al., by allowing for the presence of set-valued drift functions. [ABSTRACT FROM AUTHOR]

In this note, it is shown that the results claimed in the paper —as well as the examples presented there—are incorrect. [ABSTRACT FROM AUTHOR]

In this paper, the locally fixed-time and globally fixed-time $\mathcal {H}_{\infty }$ control problems for the port-controlled Hamiltonian (PCH) systems are investigated via the interconnection and damping assignment passivity-based control (IDA-PBC) technique. Compared with finite-time stabilization, where the convergence time of the closed-loop system's states relies on the initial values, the settling time of fixed-time stabilization can be adjusted to achieve desired equilibrium point regardless of initial conditions. The concepts of fixed-time $\mathcal {H}_{\infty }$ control, fixed-time stability region (or region of attraction), and fixed-time stability boundary are presented in this paper, and the criterions of globally fixed-time attractivity of a prespecified locally fixed-time stability region are obtained. Combining the locally fixed-time stability of an equilibrium point and the globally fixed-time attractivity of a prespecified fixed-time stability region, the globally fixed-time $\mathcal {H}_{\infty }$ control problem of PCH system is effectively solved. Two novel control laws are designed to deal with the globally fixed-time $\mathcal {H}_{\infty }$ control problem, and the conservativeness in estimating the settling time is also briefly discussed. An illustrative example shows that the theoretical results obtained in this paper work very well in the fixed-time $\mathcal {H}_{\infty }$ control design for PCH systems. [ABSTRACT FROM AUTHOR]

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied primarily on the conditions of homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems by deriving convergence results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results shows that if the undelayed system has a convergent trajectory that is unbounded in all components as $t\rightarrow-\infty$, then the system is globally asymptotically stable for arbitrary bounded time-varying delays. This follows from a more general result derived in the paper that allows to quantify delay-independent regions of attraction, which can be used to prove global asymptotic stability for various classes of systems. These also recover various known delay-independent stability results that are discussed within the paper. [ABSTRACT FROM PUBLISHER]

Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at times $0, \tau, 2\tau, \ldots$ , so that the controlled hybrid SDE becomes asymptotically stable? It has been proved that this is possible if the drift and diffusion coefficients of the given hybrid SDE satisfy the linear growth condition. However, many hybrid SDEs in the real world do not satisfy this condition (namely, they are highly nonlinear) and there is no answer to the question, yet if the given SDE is highly nonlinear. The aim of this paper is to tackle the stabilization problem for a class of highly nonlinear hybrid SDEs. Under some reasonable conditions on the drift and diffusion coefficients, we show how to design the feedback control function and give an explicit bound on $\tau$ (the time duration between two consecutive state observations), whence the new theory established in this paper is implementable. [ABSTRACT FROM AUTHOR]

This paper addresses the stability issue of a class of delayed switched cascade nonlinear systems consisting of separate subsystems and coupling terms between them. Some global and local asymptotic stability sufficient conditions are proposed, drawing stability conclusion of the overall cascade system from those of separate systems. These results essentially rely on the following observation: For a general delayed switched nonlinear system being asymptotically stable, the trajectories of the perturbed system asymptotically approach zero if so does the perturbation. This observation is one of the main results in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of disturbance attenuation with internal stability for nonlinear, input-affine systems via measurement feedback. The solution to the above-mentioned problem has been provided, three decades ago, in terms of the solution to a system of coupled nonlinear, first-order partial differential equations (PDEs). As a consequence, despite the rather elegant characterisation of the solution, the presence of PDEs renders the control design synthesis almost infeasible in practice. Therefore, to circumvent such a computational bottle-neck, in this paper we provide a novel characterisation of the exact solution to the problem that does not hinge upon the explicit computation of the solution to any PDE. The result is achieved by considering the immersion of the nonlinear dynamics into an extended system for which locally positive definite functions solving the required PDEs may be directly provided in closed form by relying only on the solutions to Riccati-like, state-dependent, algebraic matrix equations. [ABSTRACT FROM AUTHOR]

In a discrete time nonhomogeneous Markov chain (TNHMC), the states spaces, transition probabilities, and reward functions at different times may be different. In this paper, with the confluencity previously introduced, we show that the states of a TNHMC can be classified into the branching states and a number of classes of confluent states (versus the transient and recurrent states in the time homogeneous case). The optimization of average reward in TNHMC's consisting of a single confluent class (uni-chain) have been addressed in a previous paper by the author. In this paper, we show that with confluencity and the state classification and under some bound conditions, we can obtain the necessary and sufficient conditions for optimal policies of the average reward of TNHMCs consisting of multiple confluent classes (multi-chains). Just like in the uni-chain TNHMC case, the sufficient condition does not need to hold in any “zero frequently visited” time sequence. This “under-selectivity” makes the problem not amenable to dynamic programming. A direct comparison based approach is used to prove the results. The results enhance our understanding of state classification and performance optimization with the notion of confluencity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of energy-to-peak stochastic stability (EPSS) of two-dimensional (2-D) Roesser systems in the presence of state time-varying delays and multiplicative noises. First, a scheme that ensures a 2-D stochastic time-delay system is stochastically stable with an attenuation performance is proposed. The scheme presented in this paper can be regarded as an extension of the Lyapunov–Krasovskii functional method for 2-D stochastic time-delay systems, focusing on the EPSS problem. The proposed scheme is then utilized to derive delay-dependent EPSS conditions in terms of tractable linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the derived stability conditions. [ABSTRACT FROM AUTHOR]

This paper investigates a formation control problem in which a group of kinematic unicycles is made to converge to a desired formation with parallel heading angles and come to a stop. A control law is presented, which solves this problem for almost all initial conditions in any given compact set. The proposed control law is local and distributed, meaning that each unicycle is only required to sense its relative displacement measured in its own body frame, and the relative heading angle with respect to each of its neighbors. No communication between the unicycles is required. The sensing graph is assumed to be connected, undirected, and time invariant. The idea used to solve the above-mentioned formation control problem is to rigidly attach to the body frame of each unicycle an appropriate fixed offset vector. Stabilizing the desired formation amounts to achieving consensus of the endpoints of the offset vectors, and simultaneously synchronizing the unicycles’ heading angles. A control law achieving this goal is constructed by combining a bounded translational consensus controller with an attitude synchronizer. As a special case, the proposed solution solves the full unicycle synchronization problem, in which the unicycle positions are made to converge to each other, while the unicycle headings are made to align. [ABSTRACT FROM AUTHOR]

This paper is concerned with stability analysis and synthesis for discrete-time linear systems with stochastic dynamics. Equivalence is first proved for three stability notions under some key assumptions on the randomness behind the systems. In particular, we use the assumption that the stochastic process determining the system dynamics is independent and identically distributed with respect to the discrete time. Then, a Lyapunov inequality condition is derived for stability in a necessary and sufficient sense. Although our Lyapunov inequality will involve decision variables contained in the expectation operation, an idea is provided to solve it as a standard linear matrix inequality; the idea also plays an important role in state feedback synthesis based on the Lyapunov inequality. Motivating numerical examples are further discussed as an application of our approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability (gain and phase) margin of networked dynamical systems, e.g., vehicles in formation, each of which has access to the state of its neighbors and subsequently uses a state feedback gain $F$ for a certain global objective such as attitude synchronization. Here, the network topology is directed and described by a generalized Laplacian matrix $L$. An individual dynamical system can adopt its own state feedback control law such as a linear-quadratic-regulator controller for an ample stability margin, but it may lose the stability margin to a great extent when the same control strategy utilizing relative state information is used after being interconnected with other dynamical systems. This paper reveals and elaborates upon the following four facts: First, the stability margin after interconnection is quantified via the minimum singular value of a frequency-dependent matrix made up of $F$ and $L$; Second, the stability margin of a networked dynamical system having a pole at the origin is at most the inverse of the zero-eigenvalue sensitivity of $L$; Third, there exists an upper bound of the stability margin that has a computational merit, and asymptotically converges to the exact margin with respect to network size, probability of link existence, and control gain in a random network setting; and finally, $L$ can be designed to maximize the stability margin. Numerical examples are provided to demonstrate the elaboration. [ABSTRACT FROM AUTHOR]

This paper is concerned with the energy shaping of 1-D linear boundary controlled port-Hamiltonian systems. The energy-Casimir method is first proposed to deal with power preserving systems. It is shown how to use finite dimensional dynamic boundary controllers and closed-loop structural invariants to partially shape the closed-loop energy function and how such controller finally reduces to a state feedback. When dissipative port-Hamiltonian systems are considered, the Casimir functions do not exist anymore (dissipation obstacle) and the immersion (via a dynamic controller)/reduction (through invariants) method cannot be applied. The main contribution of this paper is to show how to use the same ideas and state functions to shape the closed-loop energy function of dissipative systems through direct state feedback i.e. without relying on a dynamic controller and a reduction step. In both cases, the existence of solution and the asymptotic stability (by additional damping injection) of the closed-loop system are proven. The general theory and achievable closed-loop performances are illustrated with the help of a concluding example, the boundary stabilization of a longitudinal beam vibrations. [ABSTRACT FROM AUTHOR]

In this paper, we establish results similar in nature to the invariance principle for a class of stochastic hybrid systems modeled by set-valued mappings. In particular, we characterize the set to which bounded random solutions converge in terms of a new concept called weak total recurrence. A refinement of the result under the existence of a non-increasing on average Lyapunov-like function is also established. A comparison between weak total recurrence and the frequently studied invariance concepts are presented. Finally, application of the main results to establish weak sufficient conditions for certifying stochastic stability properties like uniform global asymptotic stability in probability for compact sets and uniform global recurrence for open, bounded sets are discussed. [ABSTRACT FROM PUBLISHER]

This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min–max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush–Kuhn–Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption. [ABSTRACT FROM AUTHOR]

This paper presents reduction theorems for stability, attractivity, and asymptotic stability of compact subsets of the state space of a hybrid dynamical system. Given two closed sets $\Gamma _1 \subset \Gamma _2 \subset \mathbb {R}^n$ , with $\Gamma _1$ compact, the theorems presented in this paper give conditions under which a qualitative property of $\Gamma _1$ that holds relative to $\Gamma _2$ (stability, attractivity, or asymptotic stability) can be guaranteed to also hold relative to the state space of the hybrid system. As a consequence of these results, sufficient conditions are presented for the stability of compact sets in cascade-connected hybrid systems. We also present a result for hybrid systems with outputs that converge to zero along solutions. If such a system enjoys a detectability property with respect to a set $\Gamma _1$ , then $\Gamma _1$ is globally attractive. The theory of this paper is used to develop a hybrid estimator for the period of oscillation of a sinusoidal signal. [ABSTRACT FROM AUTHOR]

This paper presents a small-gain result for large-scale interconnected systems composed of hybrid subsystems. It is assumed that the subsystems are input-to-output practically stable (IOpS) with offset and unboundedness observability (UO), and the interconnected system can be proved to be IOpS with offset and UO as long as the proposed small-gain condition is satisfied. If, moreover, the assumption of IOpS on the subsystems is reduced to input-to-output stability (IOS), then the interconnected system is IOS. [ABSTRACT FROM AUTHOR]

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, which are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr {L}_2$ -gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables. [ABSTRACT FROM AUTHOR]

We establish a new condition for global consensus of continuous-time multiagent systems with undirected switching interaction graphs. The condition is strictly weaker than the persistency of excitation, one recently shown in , to be necessary and sufficient for global exponential consensus. The price paid for the relaxation of the excitation requirement is twofold; on one hand, we establish only global asymptotic consensus without the “exponential” qualifier. On the other hand, we prove that our condition is not necessary. We also prove in this paper that an additional assumption is needed in to extend the consensus result to the case of negatively weighted edges. [ABSTRACT FROM AUTHOR]

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. [ABSTRACT FROM AUTHOR]

Pre-orders and equivalence relations between processes, like simulation and bisimulation, have played a central role in the minimization and abstraction based verification and analysis of discrete-state systems for modal and temporal properties. In this paper, we investigate the pre-orders and equivalence relations on hybrid systems which preserve stability. We first show that stability with respect to reference trajectories is not preserved by either the traditional notion of bisimulation or the more recently proposed stronger notions with additional continuity constraints. We introduce the concept of uniformly continuous simulation and bisimulation—namely, simulation and bisimulation with some additional uniform continuity conditions on the relation—that can be used to reason about stability of trajectories. Finally, we show that uniformly continuous simulations and bisimulations are widely prevalent, by recasting many classical results on proving stability of dynamical and hybrid systems as establishing the existence of a simple, obviously stable system that (bi)-simulates the given system through uniformly continuous (bi)-simulations. We also discuss briefly a new abstraction method for stability analysis which is based on the foundations developed in the paper. [ABSTRACT FROM PUBLISHER]

The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and nonsmooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the control protocols designed in this paper. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to provide an analysis of gradient descent ( $\text{GD}$ ) algorithms with gradient errors that do not necessarily vanish, asymptotically. In particular, sufficient conditions are presented for both stability (almost sure boundedness of the iterates) and convergence of $\text{GD}$ with bounded (possibly) nondiminishing gradient errors. In addition to ensuring stability, such an algorithm is shown to converge to a small neighborhood of the minimum set, which depends on the gradient errors. It is worth noting that the main result of this paper can be used to show that $\text{GD}$ with asymptotically vanishing errors indeed converges to the minimum set. The results presented herein are not only more general when compared to previous results, but our analysis of $\text{GD}$ with errors is new to the literature to the best of our knowledge. Our work extends the contributions of Mangasarian and Solodov, Bertsekas and Tsitsiklis, and Tadić and Doucet. Using our framework, a simple yet effective implementation of $\text{GD}$ using simultaneous perturbation stochastic approximations, with constant sensitivity parameters, is presented. Another important improvement over many previous results is that there are no “additional” restrictions imposed on the step sizes. In machine learning applications where step sizes are related to learning rates, our assumptions, unlike those of other papers, do not affect these learning rates. Finally, we present experimental results to validate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider the class of input-affine MIMO nonlinear systems which are assumed to be invertible and input-output linearizable. This class includes, as a trivial special case, the class of MIMO systems possessing a well-defined vector relative degree. It is shown that if a system in this class is strongly minimum phase, in a sense specified in the paper, it is globally stabilizable via partial-state feedback or semiglobally stabilizable via (dynamic) output feedback. The result substantially broadens the class of nonlinear MIMO systems for which global/semiglobal stabilization via limited state information is known to be possible. [ABSTRACT FROM PUBLISHER]

This paper addresses the problem of local stabilization of nonlinear polynomial control systems subject to time-varying input delay and polytopic parameter uncertainty. A linear matrix inequality approach based on the Lyapunov–Krasovskii theory is proposed for designing a nonlinear polynomial state feedback controller ensuring the robust local uniform asymptotic stability of the system origin along with an estimate of its region of attraction. Two convex optimization procedures are presented to compute a stabilizing controller ensuring either a maximized set of admissible initial states for given upper bounds on the delay and its variation rate or a maximized lower bound on the maximum admissible input delay considering a given set of admissible initial states. Numerical examples demonstrate the potentials of the proposed stabilization approach. [ABSTRACT FROM AUTHOR]

In this paper, we provide a general strategy based on Lyapunov functionals to analyze global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping. To do so, we use the fact that for any linear infinite-dimensional system that is globally exponentially stable, there exists a Lyapunov functional. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term that compensates the nonlinearity. Our results are then applied to the linearized Korteweg–de Vries equation and some wave equations. [ABSTRACT FROM AUTHOR]

This paper employs detectability ideas to decide uniform global asymptotic stability (UGAS) of the trivial solution for a class of switched nonlinear time-varying systems when the trivial solution is uniformly globally stable. Using the notion of limiting behaviors of the state, output, and switching signals, the concept of a limiting zeroing-output solution is introduced. This leads to a definition of weak zero-state detectability (WZSD) that can be used to check UGAS, (uniformly for a set of switched signals). En route to establish this, a number of new stability results are derived. For example, under appropriate conditions, it is feasible to decide UGAS even when the switching signal does not satisfy an averaged dwell-time condition. It is also shown that WZSD of the original switched system can be verified by detectability conditions of much simpler auxiliary systems. Moreover, UGAS can be guaranteed without requiring that in each allowable system (without switching), the trivial solution is attractive. The effectiveness of the proposed concept is illustrated by a few examples including a switched semi-quasi-Z-source inverter. [ABSTRACT FROM AUTHOR]

It is proved in this paper that the existence of a delay-dependent suitable Lyapunov function is a necessary and sufficient condition for a discrete-time fully nonlinear time-delay system, with given delays digraph, to be globally asymptotically stable. The same result is provided for the input-to-state stability. The less is the number of edges in the delays digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. The case of arbitrary time-varying time delays, with no constraints as long as bounded, is covered as a special case. [ABSTRACT FROM AUTHOR]

In this paper, we present a method to robustify asymptotically stable nonlinear systems by adding an integral action that rejects unknown additive disturbances. The proposed approach uses a port-Hamiltonian (pH) representation of the open-loop dynamics, which, relying on the asymptotic stability property, is guaranteed to exist. The integral action controller preserves the pH structure, and, by adding a suitable cross term between the plant and the controller states to the closed-loop energy function, it avoids the unnatural coordinate transformation used in the past. The controller is shown to be robust against some common types of modeling uncertainty, including unknown friction dynamics in mechanical systems. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for assessing the stability of feedback interconnections where one element is a static slope-restricted nonlinearity and the other element is a linear system. The approach is based on the use of Zames–Falb multipliers where the dynamic portion of the multiplier is chosen as an externally positive noncausal transfer function. By restricting attention to a subset of these multipliers, a set of pure linear matrix inequality conditions is obtained which requires no initial parameterization by the user. A useful byproduct of using externally positive systems is that the results are applicable to nonodd slope restricted nonlinearities, which is not the case for all classes of Zames–Falb multipliers. [ABSTRACT FROM AUTHOR]

This paper considers a scalar continuous-time linear time-invariant system, whose feedback signal is transmitted through a communication network. Such a network has only finite bit rate and suffers from transmission delay which is characterized by both lower and upper delay bounds. The concerned system implements event-triggering strategies, i.e., only when certain events are triggered, the system samples and transmits feedback signals. This paper first derives some lower bounds on the feedback bit rate required to achieve the input-to-state stability under arbitrary event-triggering strategies. Then this paper proposes some constructive methods to design the event-triggering strategy and the controller, and can achieve the input-to-state stability at a bit rate being arbitrarily close to these obtained lower bit rate bounds. Moreover, this paper proves that the stabilizing bit rates under the proposed event-triggering strategies can be strictly lower than the stabilizing bit rate required by any time-triggering strategy. This bit rate superiority comes from the fact that under event triggering, the state information can be freely extracted from the receive time instants of feedback packets without consuming any bit rate. Simulations are done to demonstrate the bit rate superiority of the proposed event-triggering strategies. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of global finite-time stabilization for a class of uncertain nonlinear systems with event-triggered inputs. The existing event-based design methods can only partially compensate for the effects of the event-triggered errors and cannot completely counteract them to achieve finite-time control. For this reason, a new method about event triggering mechanism and event-triggered controller codesign is presented based on the idea of backstepping design and the sign function technique. It is proved that the event-triggered control system is the Zeno-free and the newly proposed control strategy ensures the global finite-time stability of the closed-loop systems via Lyapunov analyses and finite-time stability theory, which improves the existing results of only boundedness or asymptotic stability. Finally, two examples are performed to demonstrate the validity of the proposed strategy. [ABSTRACT FROM AUTHOR]

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and, hence, such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function, which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones. [ABSTRACT FROM AUTHOR]

This paper develops a new partial-state feedback model reference adaptive control (MRAC) scheme, which has full capability to deal with plant uncertainties for output tracking and desired flexibility to combine the advantages of full-state feedback MRAC and output feedback MRAC. For partial-state feedback MRAC, plant-model matching is achievable as with full-state feedback control, while the controller structure enjoys less complexity as compared with an output feedback MRAC design. Adaptive partial-state feedback control designs are developed for relative-degree-one plants and for general plants. Both adaptive control designs ensure closed-loop system stability and asymptotic output tracking. New results are presented for plant-model matching, error model, adaptive law, and stability analysis. New features of partial-state feedback MRAC are addressed, including its design flexibility. Simulation studies are conducted whose results verify the effectiveness of partial-state feedback MRAC. [ABSTRACT FROM AUTHOR]

This paper investigates state feedback stabilization problem of a class of stochastic cascade nonlinear time-delay systems with stochastic inverse dynamics. By characterizing unmeasured stochastic inverse dynamics with stochastic input-to-state stability condition, without imposing any growth condition on time-delay system nonlinearities, a delay-independent, dynamic state feedback controller is constructed. It is shown that the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the second-order practical consensus of homogeneous sampled-data multiagent systems (MASs). To do this, a new nonlinear emulation strategy based on homogeneity is developed. It is then applied to MASs under synchronously variable sampling. Finally, a comparison with the classical linear strategy is provided in the case of MASs under synchronously periodic sampling. [ABSTRACT FROM AUTHOR]

This paper considers a class of reinforcement-based learning (namely, perturbed learning automata) and provides a stochastic-stability analysis in repeatedly played, positive-utility, finite strategic-form games. Prior work in this class of learning dynamics primarily analyzes asymptotic convergence through stochastic approximations, where convergence can be associated with the limit points of an ordinary-differential equation (ODE). However, analyzing global convergence through an ODE-approximation requires the existence of a Lyapunov or a potential function, which naturally restricts the analysis to a fine class of games. To overcome these limitations, this paper introduces an alternative framework for analyzing asymptotic convergence that is based upon an explicit characterization of the invariant probability measure of the induced Markov chain. We further provide a methodology for computing the invariant probability measure in positive-utility games, together with an illustration in the context of coordination games. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of asymptotically stabilizing the splay state configuration of a network of identical pulse coupled oscillators through the design of the their phase response function. The network of pulse coupled oscillators is modeled as a hybrid system. The design of the phase response function is performed to achieve almost global asymptotic stability of a set, wherein oscillators’ phases are evenly distributed on the unit circle. To establish such a result, a novel Lyapunov function is proposed. Robustness with respect to frequency perturbation is assessed. Finally, the effectiveness of the proposed methodology is shown in an example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce continuous-time linear compartmental switched systems and investigate their stability and stabilizability properties. By their nature, these systems are always stable. Necessary and sufficient conditions for asymptotic stability for arbitrary switching functions, and sufficient conditions for asymptotic stability under certain dwell-time conditions on the switching functions are proposed. Finally, stabilizability is thoroughly investigated and proved to be equivalent to the existence of a Hurwitz convex combination of the subsystem matrices, a condition that, for positive switched systems, is only sufficient for stabilizability. [ABSTRACT FROM AUTHOR]

A geometric homogeneity is introduced for evolution equations in a Banach space. Scalability property of solutions of homogeneous evolution equations is proven. Some qualitative characteristics of stability of trivial solution are also provided. In particular, finite-time stability of homogeneous evolution equations is studied. Theoretical results are illustrated on important classes of partial differential equations. [ABSTRACT FROM PUBLISHER]

Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not completely understood. The goal of this paper is to develop a stability theory for Zeno hybrid systems that parallels classical Lyapunov theory; that is, we present Lyapunov-like sufficient conditions for Zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler Zeno hybrid systems defined on the first quadrant of the plane. These conditions are applied to Lagrangian hybrid systems, which model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior. Finally, the results are applied to robotic bipedal walking. [ABSTRACT FROM AUTHOR]

It is known that state-dependent, multi-step Lyapunov bounds lead to greatly simplified verification theorems for stability for large classes of Markov chain models. This is one component of the “fluid model” approach to stability of stochastic networks. In this paper we extend the general theory to randomized multi-step Lyapunov theory to obtain criteria for stability and steady-state performance bounds, such as finite moments. These results are applied to a remote stabilization problem, in which a controller receives measurements from an erasure channel with limited capacity. Based on the general results in the paper it is shown that stability of the closed loop system is assured provided that the channel capacity is greater than the logarithm of the unstable eigenvalue, plus an additional correction term. The existence of a finite second moment in steady-state is established under additional conditions. [ABSTRACT FROM AUTHOR]

This paper introduces an \cal H-representation method to express an n^2\times \,1 vector \overrightarrow X as \overrightarrow X=H\widetilde X. Based on the introduced \cal H-representation approach, several topics are extensively discussed, including the generalized Lyapunov equations (GLEs) arising from stochastic control, stochastic observability, generalized \cal D-stability and \cal D-stabilization, weak stability, and stabilization. A necessary and sufficient condition for the existence and uniqueness of the symmetric and skew-symmetric solutions of GLEs is presented, respectively. Moreover, the solution structure of GLEs is also clarified. Through the \cal H-representation method, several necessary and sufficient conditions are also obtained for stochastic observability, generalized \cal D-stability and \cal D-stabilization, weak stability, and stabilization. [ABSTRACT FROM AUTHOR]

Classical methods for estimating additive faults are based on state-space models, e.g., moving horizon estimation (MHE) and unknown input observers (UIOs). This paper contributes new direct design methods from closed-loop I/O data for systems with stable inversion, which do not require building a state-space model by first principles, nor require identifying it. Inspired by subspace identification, we use the input and output (I/O) relationship of a plant in a Vector ARX (VARX) form to parameterize least-squares (LS) problems for estimating faults. We prove that with the order of the VARX descriptions tending to infinity, the fault estimates are unbiased. Under lower relative degrees, we prove that our new methods are equivalent to system-inversion-based estimation for both LTI and LTV systems. We will show more general unbiased estimation conditions for higher relative degrees. These require that the underlying inverted system from faults to outputs is stable. Algorithms of identifying unbiased fault estimation filters from data will be developed in this paper based on single LS. Moreover, covariance of the fault estimates can also be extracted from data. [ABSTRACT FROM AUTHOR]

This paper considers the input-to-state stability (ISS) of time-varying switched systems with time delays, where the upper bound estimation for the operator of Lyapunov function (UBEOL) is assumed to be time varying and mode dependent. The ISS and integral ISS are investigated for time-varying switched systems with time delays by using the Lyapunov–Razumikhin and comparison theorem methods. Since the coefficient in the UBEOL is time varying and takes a positive/negative value, the subsystems consist of both ISS and non-ISS subsystems, simultaneously. It is shown that our presented results have wider applications than some existing works. Two examples, including one of the consensus for time-varying multiagent systems with cooperative and competitive protocols, are presented to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper introduces the internally positive representation (IPR) of linear coupled differential-difference systems with time-varying delays. Then, a stability analysis is conducted on this class of systems: Relying on results established under positivity assumptions, the IPR technique leads to a novel sufficient condition for the delay-independent asymptotic stability of coupled differential-difference systems that holds for not necessarily positive systems. As a consequence, an analogous stability condition for neutral-type equations with time-varying delays is also obtained. The results are discussed and favorably compared to similar conditions proposed in the literature, and finally validated by means of numerical examples. [ABSTRACT FROM AUTHOR]

Stability analysis of nonlinear singular systems, arising from differential algebraic equations, is addressed in this paper, exploiting the fact that it belongs to the class of Positvstellensatz problems. By means of the direct Lyapunov method and a variety of convex descriptor forms, sufficient conditions in terms of linear matrix inequalities are developed. Issues concerning proper initialization and simulation are also considered. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

This paper deals with the stabilization of the feasible equilibrium point of a special class of nonlinear quadratic systems known as Lotka–Volterra (LV) systems, in the presence of interval uncertainty via a fixed linear state feedback (FLSF). It is well known that for a linear time-invariant system with interval uncertainty, stability at the vertices of a polytope implies stability in the interior of the whole polytope. It has been shown that this idea can be extended to examine the stability of LV systems in the presence of interval uncertainty, recently. In fact, it has been proved that in spite of nonlinear nature of the system, stability at a special vertex guarantees stability at all other vertices, which makes it easier to check the stability of the whole uncertain system. A subtle use of this fact in stabilization of interval LV systems via FLSF leads to a sufficient condition in terms of linear matrix inequalities. Also, the proposed approach is extended to a wider class of LV systems with time-varying and state-dependent system matrices. The efficiency of the proposed scheme is shown through numerical examples and simulations. [ABSTRACT FROM AUTHOR]

This paper studies event-based control of nonlinear systems with state quantization. Two configurations of the event-based quantized controller are considered: Quantization after sampling and quantization before sampling. The considered quantizer is assumed to have a finite quantization range. With input-to-state stability (ISS) tools, new event-triggering, and dynamic quantization mechanisms are designed to deal with the interaction of the quantizer and the sampler. For both of the configurations, infinitely fast sampling, and in particular the Zeno phenomenon, is avoided. And the system state asymptotically converges to the origin, if a growth condition is satisfied by the ISS gain of the controlled system. [ABSTRACT FROM AUTHOR]

This paper considers the problem of asynchronous $\mathcal H_{\infty }$ control for two-dimensional (2-D) Markov jump systems. The underlying system is described based upon the Roesser model. Especially, the hidden Markov model is employed when dealing with the asynchronization between a controlled system and a controller, and the relation between them is constructed through a conditional probability matrix. Based on the Lyapunov function technique, the asymptotic mean square stability and $\mathcal H_{\infty }$ noise attenuation performance are investigated for the closed-loop 2-D system. Moreover, the controller gain can be obtained by solving a convex optimization problem. An example is presented to show the effectiveness and potential of the proposed new design technique. [ABSTRACT FROM AUTHOR]