This work explores Lyapunov characterizations of the input-output-to-state stability (oss) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators", and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.

We show that, like continuous-time systems, zeroinput locally asymptotically stable hybrid systems are locally input-to-state-stable (LISS). We demonstrate by examples that, unlike continuous-time systems, zero-input locally exponentially stable hybrid systems may not be LISS with linear gain, inputto-state stable (ISS) hybrid systems may not admit any ISS Lyapunov function, and nonuniform ISS hybrid systems may not be (uniformly) ISS. We then provide a strengthened ISS condition as an equivalence to the existence of an ISS Lyapunov function for hybrid systems. This strengthened condition reduces to standard ISS for continuous-time and discrete-time systems. Finally under some other assumptions we establish the equivalence among ISS, several asymptotic characterizations of ISS, and the existence of an ISS Lyapunov function for hybrid systems.

Abstract A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.

The paper presents a methodology for analyzing the stability of formations of interconnected vehicles that are based on leader-follower relations. The methodology exploits input-to-state stability properties of basic leader-follower interconnections and builds on the propagation of these properties throughout the network to establish global stability bounds. This is formalized using the notion of (formation ISS),aweakerform of stability than string or mesh stability, which relates leader input(s) to formation state errors. In this paper we focus on cyclic interconnections of vehicles and show how the ISS framework can be extended to include these structures. This is the first such result for cyclic graphs that represent formations based on leader-follower controllers.

INTRODUCTION A central question in control theory is how to formulate, for general non-linear systems, notions of robustness and stability with respect to exogenous input disturbances. The linear case is by now very well understood, and, at least in a finite-dimensional set-up, most `reasonable' definitions of `inputto -state' or `input--output' stability (provided in this last case that additional reachability and observability assumptions are met) boil down to local asymptotic stability, viz. to the classical condition on the systems poles lying in the complex open left half-plane. However, for non-linear systems the range of possibilities is much broader, and the goal of coming up with an effective classification for many different behaviours that might be labeled as `stable' together with methods which would allow to establish relationships between such stability notions has attracted a substantial research effort within the past years. In this respect, input to state stability (i

(Communicated by Sergei Pilyugin) Abstract. This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. The main result determines global asymptotic stability of the network from the diagonal stability of a dissipativity matrix which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the secant criterion for cyclic networks presented in [1], and extends it to a general interconnection structure represented by a graph. The new stability test is illustrated on a mitogen-activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. The next problem addressed is the robustness of stability in the presence of diffusion terms. A compartmental model is used to represent the localization of the reactions, and conditions are presented under which stability is preserved despite the diffusion terms between the compartments.

This paper presents the role of vector relative degree in the formulation of stationarity conditions of optimal control problems for affine control systems. After translating the dynamics into a normal form, we study the Hamiltonian structure. Stationarity conditions are rewritten with a limited number of variables. The approach is demonstrated on two and three inputs systems, then, we prove a formal result in the general case. A mechanical system example serves as illustration.

This paper presents a Lyapunov-type proof of the Input-to-State Stable(ISS) small-gain theorem. The proof given in this paper demonstrates how to construct a Lyapunov function explicitly, which contrasts with existing proofs based on input-output analysis. The Lyapunov functions are constructed by selecting state-dependent scaling functions properly. The construction of Lyapunov functions motivates the author to formulate a new criterion for stability and performance of interconnected nonlinear systems. Furthermore, an ISS small-gain condition with nonlinearly-scaled supply rates is obtained naturally in order to reduce the conservatism arising in the application of the ISS small-gain condition.

Abstract—This paper reports a stable online adaptive identification technique for the identification of finite-dimensional dynamical models of dynamically positioned underwater robotic vehicles. Proofs for the identifier’s global stability, and for the input-tostate stability of this class of plants are reported. A direct comparison of the adaptive identification method to a conventional, offline, least-squares method is reported. Using experimental data obtained with the Johns Hopkins University Remotely Operated underwater robotic Vehicle (JHUROV), both methods are employed to identify decoupled, single-degree-of-freedom dynamical plant models. Performance of the resulting identified dynamical plant models is quantitatively compared to the experimentally observed motion of the actual vehicle. Index Terms—Adaptive estimation, dynamics, least-squares methods, robot dynamics, underwater vehicles.

We study the recently introduced notion of outputinput stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. This paper develops the theory of output-input stability in the multi-input, multi-output setting. We show that output-input stability is a combination of two system properties, one related to detectability and the other to left-invertibility. For systems affine in controls, we derive a necessary and sufficient condition for output-input stability, which relies on a global version of the nonlinear structure algorithm. This condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output-input stable systems.