characterizations of detectability notions in terms of discontinuous dissipation functions we consider a new lyapunov type characterization of detectability for non linear systems without controls in terms of lower semicontinuous not necessarily smooth or even continuous dissipation functions and prove its equivalence to the gasmo global asymptotic stability modulo outputs and uoss uniform output to state stability properties studied in previous work the result is then extended to provide a construction of a discontinuous dissipation function characterization of the ioss input to state stability property for systems with controls this paper complements a recent result on smooth lyapunov characterizations of ioss the utility of non smooth lyapunov characterizations is illustrated by application to a well known transistor network example

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

output to state stability for hybrid systems output to state stability oss is a dual notion of input to state stability for dynamical systems this paper presents lyapunov and asymptotic characterizations of oss for hybrid dynamical systems emphasizing that a globally detectable i e nonuniformly oss hybrid system admits a smooth oss lyapunov function

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss 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 ioss 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

on the overlap dynamics of multi state neural networks with a finite number of patterns neural networks with multi state neurons are studied in the case of low loading for symmetric couplings satisfying a certain positivity condition a lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns furthermore an algorithm is derived for zero temperature to determine all the fixed points as an illustration the three state model is worked out explicitly for hebbian couplings for finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric the stability properties are discussed in detail for the three state model for asymmetric couplings limit cycle behaviour is shown to be possible

singular perturbations and input to state stability this paper establishes a type of total stability for the input to state stability property with respect to singular perturbations in particular if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input to state stable with respect to disturbances then these properties continue to hold up to an arbitrarily small offset for initial conditions disturbances and their derivatives in an arbitrarily large compact set as long as the singular perturbation parameter is sufficiently small

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

global almost disturbance decoupling with stability for non minimum phase single input single output nonlinear systems the so called problem of almost disturbance decoupling with internal stability addps is the following one given a system and an arbitrarily small number 0 find a feedback law yielding a closed loop system which is stable and in which the gain in the l2 sense between the exogenous input and the regulated output is less than or equal to the complete solution of this problem has been known since a long time in the case of linear systems in the case of nonlinear systems the only global results available so far in the literature were about siso systems having an asymptotically stable zero dynamics in this paper a new set of results are presented dealing with nonlinear siso systems having a possibly unstable zero dynamics which include the general class of linear siso systems as a special case

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

input to state stable nonlinear filtering for a class of continuous time delayed nonlinear systems this paper investigates the input to state stable iss nonlinear filtering problem for a class of continuous time delayed nonlinear systems with external disturbance a new delay dependent nonlinear iss filter is established through available measurements to estimate the states of delayed nonlinear systems such that the filtering error system is both exponentially and input to state stable for any bounded external disturbance the design of the nonlinear iss filter for these nonlinear systems is achieved by solving a linear matrix inequality lmi which can be easily facilitated by using standard numerical packages an illustrative example is given to demonstrate the effectiveness of the proposed filter

bistable biological systems a characterization through local compact input to state stability many biological systems have the capacity to operate in two distinct modes in a stable manner typically the system can switch from one stable mode to the other in response to a specific external input mathematically these bistable systems are usually described by models that exhibit at least two distinct stable steady states on the other hand to capture biological variability it seems more natural to associate to each stable mode of operation an appropriate invariant set in the state space rather than a single fixed point a general formulation is proposed in this paper which allows freedom in the form of kinetic interactions and is suitable for establishing conditions on the existence of one or more disjoint forward invariant sets for the given system stability with respect to each set is studied in terms of a local notion of input to state stability with respect to compact sets two well known systems that exhibit bistability are analyzed in this framework the lac operon and an apoptosis network for the first example the question of designing an input that drives the system to switch between modes is also considered

on a periodic predator prey system with time delays abstract this paper deals with discuss the global existence of positive periodic solutions for a predator prey system with time delays based on the theory of coincidence degree

forward completeness unboundedness observability and their lyapunov characterizations a finite dimensional continuous time system is forward complete if solutions exist globally for positive time this paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities moreover a version of this fact is also proved for systems with inputs and a generalization is also provided for systems with outputs and a notion unboundedness observability of relative completeness we apply these results to obtain a bound on reachable states in terms of energy like estimates of inputs

construction of lyapunov functions for interconnected parabolic systems an iiss approach this paper is devoted to two issues the first one is to provide lyapunov based tools to establish integral input to state stability iiss and input to state stability iss for some classes of nonlinear parabolic equations the other one is to provide a stability criterion for interconnections of iiss parabolic systems the results addressing the former problem allow us to overcome obstacles arising in tackling the latter one the results for the latter problem are a small gain condition and a formula of lyapunov functions which can be constructed for interconnections whenever the small gain condition holds it is demonstrated that for interconnections of partial differential equations the choice of a right state and input spaces is crucial in particular for iiss subsystems which are not iss as illustrative examples stability of two highly nonlinear reaction diffusion systems is established by the proposed small gain criterion

output to state stability and detectability of nonlinear systems the notion of input to state stability iss has proved to be useful in nonlinear systems analysis this paper discusses a dual notion output to state stability oss a characterization is provided in terms of a dissipation inequality involving storage lyapunov functions combining iss and oss there results the notion of input output to state stability ioss which is also studied and related to the notion of detectability the existence of observers and output injection

conditions for abstract nonlinear regulation this paper studies the general problem of the existence of output dynamic stabilizers for a control system the controller is not assumed to have any special structure necessary and sufficient conditions are given in terms of new notions which generalize the usual ones of stabilizability and detectability in the particular cases of analytic and bilinear systems these conditions are considerably simplified

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

on state observers for nonlinear systems abstract for linear systems it is known that there exists a state observer if and only if the system is detectable or in other words asymptotically stabilizable by output injection in this paper it is shown that asymptotic stabilizability by output injections is neither necessary nor sufficient for the construction of a nonlinear observer which may not have an exponential error decay rate the concept of uniform observers is introduced and necessary conditions and sufficient conditions are given for the construction of uniform observers

bounded input bounded output stabilization of nonlinear systems using state detectors abstract this paper is devoted to the study of relations between lyapunov stabilization and bounded input bounded output bibo stabilization of nonlinear systems when the state is not available for measurement and various state detectors are employed for implementing feedback control it is found that with some constraints imposed on the detectors lyapunov stabilization implies bibo stabilization a feedback control scheme for achieving bibo stability is presented

certainty equivalence implies detectability abstract it is shown that any stabilizing certainty equivalence control used within an adaptive control system causes the familiar interconnection of a controlled process and associated output estimator to be detectable through the estimator s output error e p for every frozen value of the index or parameter vector p upon which both the estimator and controller dynamics depend the fact that certainty equivalence implies detectability has been known for some time this has been shown to be so whenever the process model is linear and the controller and estimator models are also linear for every frozen value of p in this paper use is made of recently introduced concepts of input to state stability and detectability for nonlinear systems to prove that the same implication is valid in a more general nonlinear setting

a class of globally stabilizing controllers for nonlinear systems abstract a class of globally stabilizing controllers for nonlinear systems are parametrized as fractional transformations on some input to state stable parameters

continuous control lyapunov functions for asymptotically controllable time varying systems this paper shows that for time varying systems global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control lyapunov function with respect to the set

separation principles for input output and integral input to state stability we present new characterizations of input output to state stability this is a notion of detectability formulated in the iss input to state stability framework equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals these results provide a set of separation principles for input output to state stability characterizations of the property as conjunctions of weaker stability notions when applied to the notion of integral iss these characterizations yield analogous results

global stabilization by output feedback examples and counterexamples abstract we show by means of examples that global complete observability and global stabilizability by state feedback are not sufficient to guarantee global stabilizability by dynamic output feedback we show that a main obstruction is related to unboundedness unobservability this is that some unmeasured state components may escape in finite time whereas the measurements remain bounded

sontag s input to state stability condition and global stabilization using state detection this paper deals with the global stabilization problem for nonlinear system using state detection sufficient algebraic conditions for stabilization and detectability concerning particular classes of systems are included these conditions are based to a version of a lyapunov like condition proposed by sontag which guarantees input to state stability

intrinsic robustness of global asymptotic stability abstract equivalence is shown for discrete time systems between global asymptotic stability and the so called integral input to state stability the latter is a notion of robust stability with respect to exogenous disturbances which informally translates into the statement no matter what is the initial condition if the energy of the inputs is small then the state must eventually be small

nonlinear output feedback tracking with almost disturbance decoupling addresses the design of global output feedback controls for a class of uncertain nonlinear single input single output systems which are globally transformable into an observable minimum phase system whose nonlinearities depend on the output only they may be affected by unknown time varying disturbances or parameter variations entering linearly in the state equations the proposed dynamic controller is of order spl rho 1 spl rho is the relative degree of the given system and ensures that the closed loop system enjoys for any initial condition and for any smooth bounded output reference signal with bounded time derivatives up to order spl rho the following properties input to state stability with respect to disturbance inputs and almost disturbance decoupling i e the influence of disturbances both on the l sub 2 and on the l sub spl infin norm of the output tracking error is arbitrarily attenuated by increasing a positive scalar control parameter when disturbances are zero the reference signal is exponentially tracked by the output and the equilibrium point corresponding to the reference signal is globally asymptotically stable

remarks regarding the gap between continuous lipschitz and differentiable storage functions for dissipation inequalities appearing in h control abstract this paper deals with the regularity of solutions of the hamilton jacobi inequality which arises in h control it shows by explicit counterexamples that there are gaps between existence of continuous and locally lipschitz positive definite and proper solutions and between lipschitz and continuously differentiable ones on the other hand it is shown that it is always possible to smooth out solutions provided that an infinitesimal increase in gain is allowed