integral versions of iss for sampled data nonlinear systems via their approximate discrete time models 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 ap proximate 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

discrete time homogeneity robustness and approximation in this paper we study robustness properties of discrete time homogeneous systems we also analyse stability and robustness properties of a more general class of nonlin ear discrete time systems that can be approximated by homogeneous ones we apply the results to the investigation of stability properties of discretized continuous time systems

discrete time certainty equivalence output feedback allowing discontinuous control laws including those from model predictive control we present certainty equivalence output feedback results for discrete time nonlinear systems that employ possibly discontinuous control laws in the feedback loop coupling assumptions of nominal robustness with uniform observability or detectability assumptions we assert nominally robust stability for output feedback closed loops we further show that model predictive control mpc can be used to generate a feedback control law that is robustly globally asymptotically stabilizing when used in a certainty equivalence output feedback closed loop allowing for discontinuous feedback control laws is important for systems employing mpc since the method can and sometimes necessarily does result in discontinuous control laws

lyapunov theory for 2 d nonlinear roesser models application to asymptotic and exponential stability this technical note deals with a general class of discrete 2 d possibly nonlinear systems based on the roesser model we first motivate the introduction of lyapunov type definitions of asymptotic and exponential stability this will allow us to introduce and discuss several particularities that cannot be found in 1 d systems once this background has been carefully designed we develop different lyapunov theorems in order to check asymptotic and exponential stability of nonlinear 2 d systems finally we propose the first converse lyapunov theorem in the case of exponential stability

on uniform asymptotic stability of time varying parameterized discrete time cascades recently a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been proposed in the literature in this paper we develop novel tools that can be used within this framework and that are useful for tracking problems in particular results for stability analysis of parameterized time varying discrete time cascaded systems are given this class of models arises naturally when one uses an approximate discrete time model to design a stabilizing or tracking controller for a sampled data plant while some of our results parallel their continuous time counterparts the stability properties that are considered the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete time systems and are technically different from those in the continuous time context a result on constructing strict lyapunov functions from nonstrict ones that is of independent interest is also presented we illustrate the utility of our results in the case study of the tracking control of a mobile robot this application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete time parameterized systems

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

on the rational derivation of a hierarchy of dynamic equations for a homogeneous isotropic elastic plate abstract flexural equations of motion for a homogeneous isotropic elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components all but the lowest order expansion functions are eliminated with the help of the three dimensional equations of motion and are plugged into the boundary conditions eliminating between these an equation is obtained for the mean plane vertical displacement which also includes arbitrary loading on the plate surface this equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact rayleigh lamb dispersion relation to all orders various truncations of the equation are discussed and are compared numerically with each other the exact three dimensional solution and mindlin s plate equation both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons the theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode

asymptotic and exponential stability of nonlinear two dimensional continuous discrete roesser models abstract sufficient conditions guaranteeing lyapunov stability asymptotic stability and exponential stability of nonlinear two dimensional continuous discrete systems are proposed special attention is paid to neutrally stable systems such as some two dimensional system descriptions of vehicle platoons which may be stable or asymptotically stable but never exponentially stable our conditions for lyapunov stability and asymptotic stability only require the corresponding two dimensional lyapunov function to have a negative semidefinite divergence they are thus suitable for the analysis of non exponential versions of 2d stability examples are given to illustrate the results

an approximate green s function for a locally excited fluid loaded thin elastic plate in the classic treatment of the line driven fluid loaded thin elastic plate a branch cut integral typically needs to be evaluated this branch cut arises due to a square root operator in the spectral form of the acoustic impedance in a previous paper j acoust soc am 110 3018 2001 diperna and feit developed a methodology complex layer analysis cla to approximate this impedance the resulting approximation was in the form of a rational function although this was not explicitly stated in this paper a rational function approximation rfa to the acoustic impedance is derived the advantage of the rfa as compared to the cla approach is that a smaller number of terms are required the accuracy of the rfa is examined both in the fourier transform domain and the spatial domain the rfa is then used to obtain a differential relationship between the pressure and velocity on the surface of the plate finally using the rfa in conjunction with the equation of motion of the plate an approximate expression for the green s function for a line driven plate is obtained in terms of a sum of propagating and evanescent waves comparisons of these results with the numerical inversion of the exact integral show reasonable agreement

on uniform boundedness of parameterized discrete time systems with decaying inputs applications to cascades a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been established recently within this framework naturally arises the need to investigate stability properties of parameterized discrete time systems further results that guarantee appropriate stability of the parameterized family of discrete time systems that is used within this framework have been also established for systems with cascaded structure a fundamental condition that is required in this framework is uniform boundedness of solutions of the cascade however this is difficult to check in general in this paper we provide a range of sufficient conditions for uniform boundedness that are easier to check these results further contribute to the toolbox for controller design of sampled data nonlinear systems via their approximate discrete time models

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

stability of two dimensional linear systems with singularities on the stability boundary using lmis this paper gives results on stability and asymptotic stability of two dimensional systems using linear matrix inequalities lmis despite a long history of research in this area systems with singularities on the stability boundary ssb have received limited attention because they cannot produce a sign definite solution to the required lmi however 2d systems describing some classes of models of vehicle platoons generically involve an ssb therefore commonly used definitions for asymptotic stability and strict lmi conditions are not suitable to discuss the stability of these systems it is shown that the existence of a negative semidefinite solution together with simple additional conditions is sufficient to guarantee asymptotic stability thus the stability conditions discussed here can be used to study a wider range of dynamical systems including systems with singularities on the stability boundary ssb which cannot be exponentially stable a unified framework is used to analyse continuous continuous continuous discrete and discrete discrete systems simultaneously

stability analysis of 2 d digital filters with saturation arithmetic an lmi approach an improved lmi based criterion for the nonexistence of overflow oscillations in two dimensional 2 d digital filters described by the roesser model employing saturation arithmetic is presented the criterion makes use of the structural properties as prevailing in the system under consideration of the saturation nonlinearities in a greater detail than the usual sector restriction 0 1

discrete time stability with perturbations application to model predictive control exponential stability and asymptotic perturbed stability results are derived for nonlinear discrete time systems some related results are available in the literature but proofs for the discrete time case are difficult to find our motivation for obtaining these results was to determine conditions under which discrete time nonlinear model predictive control is stabilizing in the face of perturbations this problem has particular relevance when output rather than state feedback is used a nonlinear state estimator is then required to reconstruct the state and it is necessary to identify the class of reconstruction errors for which the cascaded estimator controller assembly is stabilizing

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

a solvable lie algebra condition for stability of linear multidimensional systems this note analyzes exponential stability of a class of linear discrete multidimensional systems using a multidimensional comparison principle for estimating the system componentwise exponential convergence and a solvable lie algebra condition a sufficient condition for exponential stability of linear multidimensional systems is presented the stability condition can be easily examined by computing the system matrices in finite steps this is demonstrated by an example

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

approximate dynamic boundary conditions for a thin piezoelectric layer abstract approximate dynamic boundary conditions of different orders are derived for the case of a thin piezoelectric coating layer bonded to an elastic material the approximate boundary conditions are derived using series expansions of the elastic displacements and the electric potential in the thickness coordinate of the layer all the expansion functions are then eliminated with the aid of the equations of motion and boundary interface conditions of the layer this results in boundary conditions on the elastic material that may be truncated to different orders in the thickness of the layer to obtain approximate boundary conditions the approximate boundary conditions may be used as a replacement for the piezoelectric layer and thus simplify the analysis significantly numerical examples show that the approximate boundary conditions give good results for low frequencies and or thin piezoelectric layers

dynamic cylindrical shell equations by power series expansions the dynamics of an infinite circular cylindrical shell is considered the derivation process is based on power series expansions of the displacement components in the radial direction by using these expansions in the three dimensional equations of motions a set of recursion relations is identified expressing higher displacement coefficients in terms of lower order ones the new approximate shell equations are hereby obtained from the boundary conditions resulting in a set of six partial differential equations these equations are believed to be asymptotically correct and it is in principle possible to go to any order equations of order up to and including the square of the thickness h2 are presented explicitly this set of six equations are reduced to five as well as three equations and compared to classical theories dispersion curves together with the eigenfrequencies for a 2d case are calculated using exact classical and expansion theories it is shown that the approximate equations containing order h2 are in general as good as or better than the established theory of the same order

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

exponential stability of 2 d systems exponential stability of 2 d systems is defined and characterized the definition of exponential stability works also with unbounded data as for the case of 1 d systems we show in particular that exponential stability and asymptotic stability are equivalent properties

on timoshenko reissner type theories of plates and shells abstract linear tr theories timoshenko reissner theories of isotropic plates and shells are discussed these theories take into account the transversal shear deformation and rotation inertia the main subject under consideration is the construction of these theories by the asymptotic method and the related error estimates for static and dynamic problems in the dynamic case a method is suggested for the extension of the range of applicability of the tr theory

relaxed logarithmic barrier function based model predictive control of linear systems in this paper we investigate the use of relaxed logarithmic barrier functions in the context of linear model predictive control we present results that allow to guarantee asymptotic stability of the corresponding closed loop system and discuss further properties like performance and constraint satisfaction in dependence of the underlying relaxation the proposed stabilizing mpc schemes are not necessarily based on an explicit terminal set or state constraint and allow to characterize the stabilizing control input sequence as the minimizer of a globally defined continuously differentiable and strongly convex function the results are illustrated by means of a numerical example

approximate boundary conditions for a fluid loaded elastic plate approximate boundary conditions for an infinite elastic layer immersed in a fluid are derived by using series expansions in the thickness coordinate of the plate fields the displacements fields are eliminated adopting the three dimensional equations of motion the sums and differences of the boundary pressure fields and their normal derivatives are related through a set of approximate boundary conditions one symmetric and one antisymmetric these equations involve powers in the layer thickness together with partial derivatives with respect to time as well as the spatial variables in the plate plane the approximate boundary conditions can be truncated to an arbitrary order and explicit relations are presented including terms of order five comparisons are made with effective boundary conditions using classical plate theories the numerical examples involve reflection and transmission of plane waves incident on the plate at different angles as well as the pressure fields due to a line force three fluid loading cases are studied modest heavy and light loadings the results using truncated approximate boundary conditions are compared to exact and classical plate solutions the examples show that the accuracies of the power series approximations of order three and higher are very good in the frequency interval considered

gronwall inequality for hybrid systems the gronwall inequality a well known and useful result both for continuous time and discrete time signals is extended to hybrid signals namely those that combine continuous time and discrete time an application of the result to establishing a bounded energy bounded state property for hybrid systems with inputs is provided

asymptotic characterizations of input output to state stability for discrete time systems abstract we establish the equivalence between global detectability and output to state stability for difference inclusions with outputs and we present equivalent asymptotic characterizations of input output to state stability for discrete time nonlinear systems these new stability characterizations for discrete time systems parallel what have been developed for continuous time systems in angeli et al uniform global asymptotic stability of differential inclusions j dynamical control systems 10 2004 391 412 and angeli et al seperation principles for input output and integral input to state stability siam j control optim 43 2004 256 276

robust stability and performance analysis of 2d mixed continuous discrete time systems with uncertainty this paper investigates 2d mixed continuous discrete time systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set it is shown that a nonconservative linear matrix inequality lmi condition for ensuring robust stability can be obtained by introducing complex lyapunov functions depending polynomially on the uncertain vector and a frequency moreover it is shown that nonconservative lmi conditions for establishing upper bounds of the robust h and h 2 norms can be obtained by introducing analogous lyapunov functions depending rationally on the frequency some numerical examples illustrate the proposed methodology

two dimensional digital filters without overflow oscillations and instability due to finite word length new criteria for the absence of finite word length effects such as overflow oscillations and instability in two dimensional digital filters are presented the criteria are formulated using the state space representations and are based on results concerning the 2d lyapunov equation several examples illustrate the theoretical results

stability of nonlinear parameter varying digital 2 d systems a general state space model for nonlinear parameter varying digital two dimensional 2 d systems is proposed sufficient conditions for system stability are given presented theorem can be considered as a generalization of the lyapunov stability theorem for one dimensional nonlinear systems given results can be useful in stability analysis for systems described by 2 d models

structural stability asymptotic stability and exponential stability for linear multidimensional systems the good the bad and the ugly abstractin this paper we investigate three concepts of stability for linear two dimensional systems the good structural stability an algebraic property linked to the location of the roots of a certain characteristic polynomial the bad asymptotic stability roughly the trajectory converges to the equilibrium point and the ugly exponential stability the rate of convergence is at least exponential more precisely we show that for a usual set of boundary conditions taken along the positive semi axes structural stability and exponential stability are equivalent notions for this particular set of boundary conditions we further prove that structural stability implies asymptotic stability but a counterexample shows that asymptotic stability does not imply structural stability which is a major difference compared to the one dimensional case this also highlights the importance of the boundary conditions when one works with multidimensional systems

blow up for nonlinear dissipative wave equations in rn abstract we consider the cauchy problem u t t u u t m 1 u t u p 1 u t x 0 r n u 0 x u 0 x u t 0 x v 0 x for 1 m p p n n 2 for n 3 we prove that for any given numbers 0 0 there exist infinitely many data u 0 v 0 in the energy space such that the initial energy e 0 the gradient norm u 0 2 and the solution of the above cauchy problem blows up in finite time

dynamic equations for a fully anisotropic elastic plate a hierarchy of dynamic plate equations is derived for a fully anisotropic elastic plate using power series expansions in the thickness coordinate for the displacement components recursion relations are obtained among the expansion functions adopting these in the boundary conditions on the plate surfaces and along the edges a set of dynamic equations with pertinent edge boundary conditions are derived on implicit form these can be truncated to any order and are believed to be asymptotically correct for the special case of an orthotropic plate explicit plate equations are presented and compared analytically and numerically to other approximate theories given in the literature these results show that the present theory capture the plate behavior accurately concerning dispersion curves eigenfrequencies as well as stress and displacement distributions

cal h _ infty and cal h _ 2 norms of 2 d mixed continuous discrete time systems via rationally dependent complex lyapunov functions this paper addresses the problem of determining the cal h infty and cal h 2 norms of 2 d mixed continuous discrete time systems the first contribution is to propose a novel approach based on the use of complex lyapunov functions with even rational parametric dependence which searches for upper bounds on the sought norms via linear matrix inequalities lmis the second contribution is to show that the upper bounds provided are nonconservative by using lyapunov functions in the chosen class with sufficiently large degree the third contribution is to provide conditions for establishing the tightness of the upper bounds the fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient lmi condition for establishing positive semidefiniteness of even hermitian matrix polynomials this result is also exploited to derive an improved necessary and sufficient lmi condition for establishing exponential stability of 2 d mixed continuous discrete time systems some numerical examples illustrate the proposed approach it is worth remarking that nonconservative lmi methods for determining the cal h infty and cal h 2 norms of 2 d mixed continuous discrete time systems have not been proposed yet in the literature

a sufficient condition on the exponential stability of two dimensional 2 d shift variant systems in this paper we consider a class of two dimensional 2 d shift variant systems based on the bound on the deviation of parameters a sufficient condition is derived which ensures exponential stability with different decaying rates along the two directions an example is presented to illustrate the result

dynamic equations for fluid loaded porous plates using approximate boundary conditions systematically derived equations for fluid loaded thin poroelastic layers are presented for time harmonic conditions the layer is modeled according to biot theory for both open and closed pores series expansion techniques in the thickness variable are used resulting in separate symmetric and antisymmetric plate equations these equations which are believed to be asymptotically correct are expressed in terms of approximate boundary conditions and can be truncated to arbitrary order analytical and numerical results are presented and compared to the exact three dimensional theory and a flexural plate theory numerical comparisons are made for two material configurations and two thicknesses the results show that the presented theory predicts the plate behavior accurately

stability and absence of overflow oscillations for 2 d discrete time systems it is shown that a 2 d linear time invariant discrete time system could be bounded input bounded output bibo unstable even if the 2 d system which is implemented by finite word length is free of overflow oscillations with zero input according to a previous criterion for the absence of overflow oscillations in 2 d discrete time systems an improved criterion is presented to ensure both stability properties

input output to state stability for discrete time systems this paper presents lyapunov characterizations of uniform output to state stability and uniform input output to state stability ioss with respect to disturbances for discrete time dt nonlinear systems we show the equivalence of the following three properties for dt systems uniform ioss existence of a smooth lyapunov function for uniform ioss and existence of a state norm estimator this equivalence result is a dt counterpart of krichman et al 2001 input output to state stability siam journal on control and optimization 39 1874 1928 theorem 2 4 and a generalization of jiang and wang 2002 a converse lyapunov theorem for discrete time systems with disturbances systems and control letters 45 49 58 theorem 1 1 and jiand and wang 2001 input to state stability for discrete time nonlinear systems automatica 37 857 869 theorem 1 email protected 4

getting robustness against unstructured uncertainty a tube based mpc approach this note extends tube based model predictive control to the control of nonlinear systems with unmodeled dynamics the problem of obtaining robustness against unstructured uncertainty is converted into the easier problem of achieving robustness against an additional bounded disturbance while satisfying an additional output constraint the output constraint depends on the maximum l gain of the unstructured dynamics

control systems analysis for the fornasini marchesini 2d systems model progress after four decades the development of a control systems theory for multidimensional linear systems i e systems with more than one independent variable is an area where a very significant volume of research is based

strong iss implies strong iiss for time varying impulsive systems abstract for time invariant nonimpulsive systems it is already well known that the input to state stability iss property is strictly stronger than integral input to state stability iiss very recently we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics iss implies iiss also for time varying systems in this paper we show that this implication remains true for impulsive systems provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems

input to state stable finite horizon mpc for neutrally stable linear discrete time systems with input constraints abstract mpc or model predictive control is representative of control methods which are able to handle inequality constraints closed loop stability can therefore be ensured only locally in the presence of constraints of this type however if the system is neutrally stable and if the constraints are imposed only on the input global asymptotic stability can be obtained until recently use of infinite horizons was thought to be inevitable in this case a globally stabilizing finite horizon mpc has lately been suggested for neutrally stable continuous time systems using a non quadratic terminal cost which consists of cubic as well as quadratic functions of the state the idea originates from the so called small gain control where the global stability is proven using a non quadratic lyapunov function the newly developed finite horizon mpc employs the same form of lyapunov function as the terminal cost thereby leading to global asymptotic stability a discrete time version of this finite horizon mpc is presented here furthermore it is proved that the closed loop system resulting from the proposed mpc is iss input to state stable provided that the external disturbance is sufficiently small the proposed mpc algorithm is also coded using an sqp sequential quadratic programming algorithm and simulation results are given to show the effectiveness of the method

qualitative equivalences of iss and lp gain stability properties for discrete time nonlinear systems input to state stability iss and inputoutput l2 gain are two important robust stability concepts in analyzing the interconnections of nonlinear dynamical systems in this paper we demonstrate several qualitative equivalences between different iss related properties and lp gain properties for discrete time systems via nonlinear changes of coordinates prior to that certain summation to summation estimates are shown to be equivalent to the standard definitions of iss and integral iss iiss which lead to sufficient lyapunov function conditions to verify lp gain properties combined with known results on the equivalence of 0 input global asymptotic stability and iiss and a superposition principle we subsequently outline interesting implicative relationships between various discrete time robust stability properties

brief paper supervisory control using a new control relevant switching this paper presents a new supervisory control scheme which is based on a control relevant switching logic unlike most of the existing switching methods considering only estimator performance the proposed scheme takes both estimator and controller performance into account as an index to the controller performance an iiss integral input to state stability lyapunov function is employed it is ensured that the lyapunov function satisfies a certain inequality this lyapunov based switching is then coupled to the state dependent dwell time switching developed recently and the state of the uncertain plant is shown to converge asymptotically the proposed supervisory control scheme is applied to an input constrained neurally stable linear plant

the first darboux problem for wave equations with a nonlinear positive source term abstract we consider the first darboux problem for nonlinear wave equations with positive power nonlinearity source term depending on the power of nonlinearity we investigate the problem on a global existence and blow up of solutions of the first darboux problem the question of local solvability of the problem is also considered

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

wave propagation of functionally graded layers treated by recursion relations and effective boundary conditions wave propagation through a layer of a material that is inhomogeneous in the thickness direction typically a functionally graded material fgm is investigated the material parameters and the displacement components are expanded in power series in the thickness coordinate leading to recursion relations among the displacement expansion functions these can be used directly in a numerical scheme as a means to get good field representations when applying boundary conditions and this can be done even if the layer is not thin this gives a schema that is much more efficient than the approach of subdividing the layer into many sublayers with constant material properties for thin layers for which the material parameters do not depend on the layer thickness the recursion relations can be used to eliminate all but the lowest order expansion functions employing the boundary conditions this leads to a set of effective boundary conditions relating the displacements and stresses on the two sides of the layer thus completely replacing the layer by these effective boundary conditions numerical examples illustrate the convergence properties of the scheme for fg layers and the influence of different variations of the material parameters in the fg layer