estimation of the domain of attraction for discrete time linear impulsive control systems with input saturation abstract this paper discusses the domain of attraction and stability analysis for discrete time linear impulsive systems with input saturation the first goal is to give a stable condition for the system the second is to present a strategy to find an inner estimate of the domain of attraction via establishing lyapunov functions and using other mathematical methods finally we present several examples to illustrate the validity of the conditions we have derived

a piecewise approximation approach to nonlinear systems stability and region of attraction this paper discusses piecewise bilinear models and recurrent fuzzy systems as particular classes of dynamic fuzzy systems which can be dealt with in the same framework due to their structural similarity as universal approximators they are capable of representing any continuous nonlinear system dynamics with arbitrary accuracy by means of a rule base first basic definitions of both systems are revisited here and unifying canonical forms are given then a stability criterion is derived for the system classes which is based on approximating piecewise quadratic lyapunov functions and formulated in terms of linear matrix inequalities the main contribution is the demonstration of the stability analysis and estimation of region of attraction by means of these models outperforming methods reported in the literature

an analysis and design method for discrete time linear systems under nested saturation this paper considers a discrete time linear system under nested saturation nested saturation arises for example in systems with actuators subject to both magnitude and rate saturation a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains is then formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design

nonlinear system stability boundary approximation by polytopes in state space an algorithm that estimates the attracting basin boundary for a stable equilibrium point of a nonlinear system has been developed this boundary is an important characterization of the global properties of a nonlinear dynamic system the algorithm begins with a convex polytope that is entirely within the region of attraction and that surrounds the stable equilibrium point the basin boundary of this stable equilibrium point is then approximated by a possibly non convex polytope whose vertices are generated by integrating vertices of the initial polytope backwards in time an accuracy test is applied on facets of the latter polytope if any facet fails the test then the approximation is refined by adding more vertices and facets to both the initial convex polytope and the final polytope three examples are presented to demonstrate the method examples include a third order pid control system a fourth order balancing acrobot system and a fourth order lateral directional aircraft system for the pid pos

global stability analysis of fuzzy controllers using cell mapping methods linear and nonlinear stability analysis of a nonlinear fuzzy logic controller is considered the linear analysis is carried out by lyapunov s first method i e by determination of the eigenvalues of the linearized system the nonlinear global stability analysis is executed by a combination of two methods simple and interpolated cell mapping examples of controllers with a complex dynamic behavior are presented these include controllers with multiple solutions such as i stable equilibrium point and stable limit circle ii five unstable equilibrium points and three stable limit circles and iii examples where the linear analysis is completely insufficient and only nonlinear investigation provides desirable results the structure of domains of attraction of stable solutions is determined comparison of the results obtained by the cell mapping method with those provided by direct numerical integration are presented

convex computation of the region of attraction of polynomial control systems we address the long standing problem of computing the region of attraction roa of a target set e g a neighborhood of an equilibrium point of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints we show that the roa can be computed by solving an infinite dimensional convex linear programming lp problem over the space of measures in turn this problem can be solved approximately via a classical converging hierarchy of convex finite dimensional linear matrix inequalities lmis our approach is genuinely primal in the sense that convexity of the problem of computing the roa is an outcome of optimizing directly over system trajectories the dual infinite dimensional lp on nonnegative continuous functions approximated by polynomial sum of squares allows us to generate a hierarchy of semialgebraic outer approximations of the roa at the price of solving a sequence of lmi problems with asymptotically vanishing conservatism this sharply contrasts with the existing literature which follows an exclusively dual lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative lmi conditions the approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description the approach is demonstrated on several numerical examples

robust sequential detection in distributed sensor networks we consider the problem of sequential binary hypothesis testing with a distributed sensor network in a non gaussian noise environment to this end we present a general formulation of the consensus innovations sequential probability ratio test mathcal ci sprt furthermore we introduce two different concepts for robustifying the mathcal ci sprt and propose four different algorithms namely the least favorable density mathcal ci sprt the median mathcal ci sprt the m mathcal ci sprt and the myriad mathcal ci sprt subsequently we analyze their suitability for different binary hypothesis tests before verifying and evaluating their performance in a shift in mean and a shift in variance scenario for different network connectivities and amounts of noise contamination

minimax robust decentralized detection decentralized detection problems are studied where the sensor distributions are not specified completely the sensor distributions are assumed to belong to known uncertainty classes it is shown for a broad class of such problems that a set of least favorable distributions exists for minimax robust testing between the hypotheses it is hence established that the corresponding minimax robust tests are solutions to simple decentralized detection problems for which the sensor distributions are specified to be the least favorable distributions

convergence analysis of nonlinear dynamical systems by nested lyapunov functions a method for estimating the domain of attraction of an asymptotically stable equilibrium point of a nonlinear dynamical system and for deriving an upper bound on the time of convergence in the estimated domain is presented it is based on a set of lyapunov functions defined on nested regions in the state space the estimated domain obtained as the union of a subset of these regions is based on a local lyapunov like condition for the convergence of the solution in each region to its inner boundary a bound on the time of convergence within the estimated domain is given by the sum of the local bounds this concept is implemented using a class of regions whose boundaries are described by fourier series

dynamic lyapunov functions lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems the existence of lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse lyapunov theorems nevertheless the actual computation of the analytic expression of the function may be difficult herein we propose an approach to avoid the computation of an explicit solution of the lyapunov partial differential inequality introducing the concept of dynamic lyapunov function these functions allow to study stability properties of equilibrium points similarly to standard lyapunov functions in the former however a positive definite function is combined with a dynamical system that render dynamic lyapunov functions easier to construct than lyapunov functions moreover families of standard lyapunov functions can be obtained from the knowledge of a dynamic lyapunov function by rendering invariant a desired submanifold of the extended state space the invariance condition is given in terms of a system of partial differential equations similar to the lyapunov pde differently from the latter however in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right hand side of the equation several applications and examples conclude the paper

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

a numerical method for computing domains of attraction for dynamical systems the backward mapping approach for computation of global domains of attraction of asymptotically stable non critical equilibrium points of dynamical systems is presented a basis for the proposed approach is an extension of lyapunov s direct method due to lasalle and lefschetz an iterative process that converges to the global domain of attraction of an asymptotically stable equilibrium point is formulated the method applies to both continuous time and discrete time multidimensional systems it is shown that the backward mapping approach proposed by c s hsu for spiral equilibrium points of second order discrete time systems is a particular case of the algorithm presented here the proposed method can be used for autonomous systems as well as for systems with periodic coefficients when applied to discrete time formulation of dynamical systems the method can be used to determine the regions of stability of periodic solutions the paper concludes with a number of illustrative examples that demonstrate the usefulness of the proposed approach

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

rational lyapunov functions for estimating and controlling the robust domain of attraction this paper addresses the estimation and control of the robust domain of attraction rda of equilibrium points through rational lyapunov functions lfs and sum of squares sos techniques specifically continuous time uncertain polynomial systems are considered where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set the estimation problem consists of computing the largest estimate of the rda lerda provided by a given rational lf the control problem consists of computing a polynomial static output controller of given degree for maximizing such a lerda in particular the paper shows that the computation of the best lower bound of the lerda for chosen degrees of the sos polynomials which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities bmis can be reformulated as a quasi convex optimization problem under some conditions moreover the paper provides a necessary and sufficient condition for establishing tightness of this lower bound lastly the paper discusses the search for optimal rational lfs using the proposed strategy

a generalization of zubov s method to perturbed systems a generalization of zubov s theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point for the construction it is necessary to consider solutions in the viscosity sense as a consequence maximal robust lyapunov functions can be characterized as viscosity solutions

stability regions of nonlinear dynamical systems a constructive methodology a constructive methodology for estimating stability regions of general nonlinear dynamical systems is developed the constructive methodology starts with a given lyapunov function either a global or a local lyapunov function and yields a sequence of lyapunov functions which are then used to estimate the stability region the resulting sequence of estimated stability regions is shown to be a strictly monotonic increasing sequence and yet each of them remains inside the entire stability region the significance of this methodology includes its ability to reduce significantly the conservativeness in estimating the stability regions its computational efficiency its adaptability and its sound theoretical basis

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

singular perturbation margin and generalised gain margin for linear time invariant systems in this paper we propose a singular perturbation margin spm and a generalised gain margin ggm as stability metrics for single input single output siso linear time invariant lti systems from the view of singular perturbations and regular perturbations which have bijective correspondences with the classical phase margin pm and the gain margin gm respectively both of the numerical and analytical time domain spm and ggm assessment methods are provided and relationships between the singular perturbation parameter pm of the perturbed system pm and spm of the nominal system and the monotonically increasing phase of the fast system are also revealed these results make it possible to assess the pm of the nominal system in the time domain for siso lti systems using the spm with a standardised testing system called pm gauge as demonstrated by examples the concepts of spm and ggm can be used as metrics of stability margins for linear time varying systems and nonlinear systems

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

analysis and design for discrete time linear systems subject to actuator saturation we present a method to estimate the domain of attraction for a discrete time linear system under a saturated linear feedback a simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant moreover the condition can be expressed as linear matrix inequalities lmis in terms of all the varying parameters and hence can easily be used for controller synthesis the following surprising result is revealed for systems with single input suppose that an ellipsoid is made invariant with a linear feedback then it is invariant under the saturated linear feedback if and only if there exists a saturated nonlinear feedback which makes the ellipsoid invariant finally the set invariance condition is extended to determine invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements

control of rational systems using linear fractional representations and linear matrix inequalities abstract every system of the form x f x u y g x u where f and g are rational functions of the state x and linear functions of the input u possesses a linear fractional representation lfr in this lfr the system is viewed as an lti system connected with a diagonal feedback element linear in the state we devise an algorithm for computing lfrs based on this construction we give sufficient conditions for various properties to hold for the open loop system these include checking whether a given polytope is stable finding a lower bound on the decay rate of trajectories initiating in this polytope computing an upper bound on the l2 gain etc all these conditions are obtained by analyzing the properties of a differential inclusion related to the lfr and given as convex optimization problems over linear matrix inequalities lmis we show how to use this approach for static state feedback synthesis we then generalize the results to dynamic output feedback synthesis in the case when f and g are linear in every state coordinate that is not measured extensions towards a class of nonrational and uncertain nonliner systems are discussed

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

brief paper estimating the domain of attraction for non polynomial systems via lmi optimizations this paper proposes a strategy for estimating the domain of attraction da for non polynomial systems via lyapunov functions lfs the idea consists of converting the non polynomial optimization arising for a chosen lf in a polynomial one which can be solved via lmi optimizations this is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst case remainders in truncated taylor expansions moreover a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen lf and another condition is provided for establishing whether such an estimate has been found the proposed strategy can readily be exploited with variable lfs in order to search for optimal estimates lastly it is worth remarking that no other method is available to estimate the da for non polynomial systems via lmis

on permutations convex hulls and normal operators abstract a spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite dimensional space this is used to prove the following if a and b are normal operators on an n dimensional complex hilbert space h with eigenvalues given by 1 n and 1 n respectively and if a b is also normal then a b max s n diag k k for any unitarily invariant norm on l h

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

minimax robust hypothesis testing minimax robust hypothesis testing is studied for the cases where the collected data samples are corrupted by outliers and are mismodeled due to modeling errors for the former case huber s clipped likelihood ratio test is introduced and analyzed for the latter case first a robust hypothesis testing scheme based on the kullback leibler divergence is designed this approach generalizes a previous work by levy second dabak and johnson s asymptotically robust test is introduced and other possible designs based on f divergences are investigated all proposed and analyzed robust tests are extended to fixed sample size and sequential probability ratio tests simulations are provided to exemplify and evaluate the theoretical derivations

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

a class of nonlinear time delay systems and related optimal problems abstract the method of synthesis for a class of time delay nonlinear systems affine in control is considered the proposed approach is based on the solution of a special optimal control problem the integrand of the optimized functional is chosen in such a way that the bellman equation has a desired solution the nonlinear system design is reduced to the examination of the integrand of the optimized functional to extend the domain of asymptotic stability of the nonlinear system a sequence of the lyapunov krasovskii functionals is used the whole system becomes the system with variable structure the proposed approach gives a systematic framework for an almost sliding mode control

second order bounds on the domain of attraction and the rate of convergence of nonlinear dynamical systems and neural networks we present a method for analyzing the convergence properties of nonlinear dynamical systems yielding second order bounds on the domain of attraction of an asymptotically stable equilibrium point and on the time of convergence in the estimated domain we show that under certain conditions on the system there exists an analytic solution to the corresponding optimization problem the method is applied in analyzing the dynamics of a neural network model

an analysis and design method for linear systems under nested saturation this paper considers a linear system under nested saturation nested saturation arises for example when the actuator is subject to magnitude and rate saturation simultaneously a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design