on boundedness of solutions of state periodic systems a multivariable cell structure approach many dynamical systems are periodic with respect to several state variables this periodicity typically leads to the coexistence of multiple invariant solutions equilibria or limit cycles as a consequence while there are many classical techniques for analysis of boundedness and stability of such systems most of these only permit to establish local properties motivated by this a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented the proposed method is inspired by the cell structure approach developed by leonov and noldus and characterized by two main advances first the conventional cell structure framework is extended to the case of dynamics which are periodic with respect to multiple states second by introducing the notion of a leonov function the usual definiteness requirements of standard lyapunov functions are relaxed to sign indefinite functions furthermore it is shown that under mild additional conditions the existence of a leonov function also ensures input to state stability i e robustness with respect to exogenous perturbations the performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system

almost iss property for feedback connected systems abstract small gain conditions used in analysis of feedback interconnections are contraction conditions which imply certain stability properties such conditions are applied to a finite or infinite interval in this paper we consider the case when a small gain condition is applied to several disjunct intervals and use the density propagation condition in the gaps between these intervals to derive global stability properties for an interconnection this extends and improves recent results from stein shiromoto et al 2015

almost global stabilization of fully actuated rigid bodies this paper addresses the problem of stabilizing a fully actuated rigid body the problem is formulated by considering the natural configuration space for rigid bodies the special euclidean group se 3 the proposed solution consists of a landmark based controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point as such the equilibrium point is asymptotically stable and only a nowhere dense set of measure zero lies outside its region of attraction the controller uses velocity measurements and the position coordinates of a collection of landmarks fixed in the environment as an additional feature the control law is designed so as to verify prescribed bounds on the actuation

differentiability of palmer s linearization theorem and converse result for density functions we study differentiability properties in a particular case of the palmer s linearization theorem which states the existence of a homeomorphism h between the solutions of a linear ode system having exponential dichotomy and a quasilinear system indeed if the linear system is uniformly asymptotically stable sufficient conditions ensuring that h is a c2 preserving orientation diffeomorphism are given as an application we generalize a converse result of density functions for a nonlinear system in the nonautonomous case

brief paper vision based control for rigid body stabilization this paper addresses the problem of stabilizing to a desired equilibrium point an eye in hand system which consists of a single camera mounted on a rigid body free to move on se 3 it is assumed that there is a collection of landmarks fixed in the environment and that the image coordinates of those landmarks are provided to the system by an on board ccd camera the proposed method addresses not only the problem of stabilization but also that of maintaining feature visibility along the system s trajectory the resulting solution consists of a feedback control law based on the current and desired image coordinates and reconstructed attitude and depth ratio information which guarantees that i the desired equilibrium point is an almost global attractor ii a set of necessary conditions for feature visibility holds throughout the system s trajectories and iii the image of a predefined feature point is kept inside the camera s field of view

global convergence of oja s pca learning algorithm with a non zero approaching adaptive learning rate a non zero approaching adaptive learning rate is proposed to guarantee the global convergence of oja s principal component analysis pca learning algorithm most of the existing adaptive learning rates for oja s pca learning algorithm are required to approach zero as the learning step increases however this is not practical in many applications due to the computational round off limitations and tracking requirements the proposed adaptive learning rate overcomes this shortcoming the learning rate converges to a positive constant thus it increases the evolution rate as the learning step increases this is different from learning rates which approach zero which slow the convergence considerably and increasingly with time rigorous mathematical proofs for global convergence of oja s algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time ddt system extensive simulations are carried out to illustrate and verify the theory derived simulation results show that this adaptive learning rate is more suitable for oja s pca algorithm to be used in an online learning situation

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

stability criteria with nonlinear eigenvalues for diagonalizable nonlinear systems stability of the linear system especially the diagonalizable system can be verified by computing the signs of the real parts of its eigenvalues in this paper we generalize this result to the diagonalizable nonlinear system in terms of nonlinear eigenvalues according to the main results the unique equilibrium point of the diagonalizable system is globally asymptotically stable if a specific set of nonlinear eigenvalues is not positive at each point in the state space

finding inverse systems from coordinates abstract let i be a homogeneous ideal in r k x 0 x n such that r i is an artinian gorenstein ring a famous theorem of macaulay says that in this instance i is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial f a major question related to this result is to be able to describe f in terms of the ideal i in this note we give a partial answer to this question by analyzing the case when i is the artinian reduction of the ideal of a reduced arithmetically gorenstein zero dimensional scheme p n we obtain f from the coordinates of the points of

a dual to lyapunov s stability theorem lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations here we introduce a theorem which can be viewed as a dual to lyapunov s result from existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero the scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories if the stationary density is bounded everywhere except at a singularity in the origin then almost all trajectories tend towards the origin the weaker notion of stability allows for applications also in situations where lyapunov s theorem cannot be used moreover the new criterion has a striking convexity property related to control synthesis

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a lyapunov density criterion for almost everywhere stability of a class of lipschitz continuous and almost everywhere c 1 nonlinear systems this paper is concerned with the analysis of almost everywhere stability aes of a class of nonlinear systems via lyapunov densities while many of the previous results on lyapunov densities assume that the vector field is continuously differentiable this paper deals with nonlinear systems that can have points of non differentiability a key to prove aes is monotonicity of a measure defined with a lyapunov density to deduce that the set of non convergent initial values has zero lebesgue measure the monotonicity condition of a measure involves a change of variables in integrals of the lyapunov density over a set such a change of variables along trajectories is not available for vector fields that are not globally continuously differentiable even though they are smooth at almost all points of the state space in this paper an alternative inequality is presented that proves monotonicity of a measure along trajectories of a class of nonlinear systems that are lipschitz continuous and almost everywhere c

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

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

actuator and sensor placement in linear advection pde with building system application abstract we study the problem of actuator and sensor placement in a linear advection partial differential equation pde the problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants we propose a gramian based approach to the problem of actuator and sensor placement the special structure of the advection pde is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator the explicit formula for the gramian as a function of actuator and sensor location is used to provide test criteria for the suitability of a given sensor and actuator location furthermore the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field in particular the almost everywhere stability property of the advective vector field is shown to play a crucial role in deciding the location of actuators and sensors simulation results are performed to support the main results of this paper

almost global stochastic stability we develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point with respect to the lebesgue measure is asymptotically attracted to the origin with unit probability the method can be viewed as a dual to lyapunov s second method for stochastic differential equations and extends the deterministic result of a rantzer syst control lett 42 2001 pp 161 168 the result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization the main technical tool is the theory of stochastic flows of diffeomorphisms

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

on the inflation method in adaptive noise subspace estimator it is shown that yang and kaveh s 1987 1988 inflation method is not necessary to accomplish an adaptive noise subspace estimator without the inflation method an alternative noise subspace estimator involving less computation and simpler parallel implementation is developed we also prove that if the initial noise subspace basis consists of a set of mutually orthonormal vectors the proposed estimator has a better performance than that with the inflation method both in the mean sense and in the mean square sense computer simulations are provided to substantiate the superiority of the estimator without the inflation method

uncertainty propagation in dynamical systems dynamical propagation of parametric and initial condition uncertainty is studied the notion of input measure of an observable is defined and its propagation to output measure of the observable is studied by means of transfer operators uncertainty of these measures is defined in terms of their cumulative probability distributions comparison with alternative uncertainty metrics such as variance and entropy is pursued the developed formalism is illustrated through an analysis of the effect of pitchfork bifurcation on uncertainty finally the implications of these concepts in the design of nonlinear systems are discussed

optimal motion planning using navigation measure abstractwe introduce navigation measure as a new tool to solve the motion planning problem in the presence of static obstacles existence of navigation measure guarantees collision free convergence at the final destination set beginning with almost every initial condition with respect to the lebesgue measure navigation measure can be viewed as a dual to the navigation function while the navigation function has its minimum at the final destination set and peaks at the obstacle set navigation measure takes the maximum value at the destination set and is zero at the obstacle set a linear programming formalism is proposed for the construction of navigation measure set oriented numerical methods are utilised to obtain finite dimensional approximation of this navigation measure application of the proposed navigation measure based theoretical and computational framework is demonstrated for a motion planning problem in a complex fluid flow

attractors of nonlinear dynamical systems with a weakly monotone measure abstract this paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure it is assumed that the measure of a set of points in the phase space is increasing along the flow we prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions for a system of autonomous ordinary differential equations we present attractivity conditions in terms of the divergence with a density function unlike previous results in the literature our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure as an example the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions

neural network learning algorithms for tracking minor subspace in high dimensional data stream a novel random gradient based algorithm is developed for online tracking the minor component mc associated with the smallest eigenvalue of the autocorrelation matrix of the input vector sequence the five available learning algorithms for tracking one mc are extended to those for tracking multiple mcs or the minor subspace ms in order to overcome the dynamical divergence properties of some available random gradient based algorithms we propose a modification of the oja type algorithms called ojam which can work satisfactorily the averaging differential equation and the energy function associated with the ojam are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set the corresponding energy or lyapunov functions exhibit a unique global minimum attained if and only if its state matrices span the ms of the autocorrelation matrix of a vector data stream the other stationary points are saddle unstable points the globally convergence of ojam is also studied the ojam provides an efficient online learning for tracking the ms it can track an orthonormal basis of the ms while the other five available algorithms cannot track any orthonormal basis of the ms the performances of the relative algorithms are shown via computer simulations

a negative bendixson like criterion for a class of hybrid systems a condition which ensures the absence of periodic orbits for nonsmooth dynamical systems is presented the condition is a higher dimensional generalization of bendixson s criterion applicable to differential inclusions that are useful in the description of hybrid systems the main argument is based on contraction analysis of the d measured volume along the system trajectories a connection to methods for estimating the hausdorff dimension is emphasized for a class of hybrid systems described by a linear system with relay feedback the conditions are presented in the form of linear matrix inequalities a simple but illustrative example is analyzed

development and analysis of a neural network approach to pisarenko s harmonic retrieval method pisarenko s harmonic retrieval phr method is perhaps the first eigenstructure based spectral estimation technique the basic step in this method is the computation of eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix of the underlying data the authors recast a known constrained minimization formulation for obtaining this eigenvector into the neural network nn framework using the penalty function approach they develop an appropriate energy function for the nn this nn is of feedback type with the neurons having sigmoidal activation function analysis of the proposed approach shows that the required eigenvector is a minimizer with a given norm of this energy function further all its minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of nonlinear differential equations which define the network dynamics have been derived results of computer simulations are presented to support their analysis

robust stability analysis using lyapunov density we state and prove a robustness result for the concept of almost everywhere uniform stability of an invariant attractor for a nonlinear autonomous differential equation we also show that the concept of almost everywhere uniform stability rectifies the drawback of vanishing density on stable manifolds of hyperbolic equilibrium points which exists for a weaker concept of almost everywhere stability

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

a remark on an example by teel hespanha with applications to cascaded systems the properties of a system proposed by teel and hespanha which is globally exponentially stable but with state that can be driven to infinity by an arbitrarily small exponentially decaying disturbance are discussed in detail these are used to propose a family of systems with a similar property and to argue that unstable behavior may be nongeneric and not detected by means of simulations finally sufficient conditions for the existence of unbounded trajectories in cascaded systems are given

robustness of delayed multistable systems with application to droop controlled inverter based microgrids motivated by the problem of phase locking in droop controlled inverter based microgrids with delays the recently developed theory of input to state stability iss for multistable systems is extended to the case of multistable systems with delayed dynamics sufficient conditions for iss of delayed systems are presented using lyapunov razumikhin functions it is shown that iss multistable systems are robust with respect to delays in a feedback the derived theory is applied to two examples first the iss property is established for the model of a nonlinear pendulum and delay dependent robustness conditions are derived second it is shown that under certain assumptions the problem of phase locking analysis in droop controlled inverter based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum for this case corresponding delay dependent conditions for asymptotic phase locking are given

optimal stabilization using lyapunov measures numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note set theoretic notion of almost everywhere stability introduced by the lyapunov measure weaker than conventional lyapunov function based stabilization methods is used for optimal stabilization the linear perron frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program set oriented numerical methods are used to obtain the finite dimensional approximation of the linear program we provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program the approach is demonstrated on stabilization of period two orbit in a controlled standard map

a minor subspace analysis algorithm this paper proposes a learning algorithm which extracts adaptively the minor subspace spanned by the eigenvectors corresponding to the smallest eigenvalues of the autocorrelation matrix of an input signal we show both analytically and by simulation results that the weight vectors provided by the proposed algorithm are guaranteed to converge to the minor subspace of the input signal for wider applications we also present the complex valued version of the proposed minor subspace analysis algorithm

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

density functions for navigation function based systems in this paper we present a scheme for constructing density functions for systems that are almost globally asymptotically stable i e systems for which all trajectories converge to an equilibrium except for a set of measure zero using navigation functions nfs although recently proven converse theorems guarantee the existence of density functions for such systems such results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task we show that for a specific class of dynamical systems that are defined based on an nf a density function can be easily derived from the system s underlying nf

coupled principal component analysis a framework for a class of coupled principal component learning rules is presented in coupled rules eigenvectors and eigenvalues of a covariance matrix are simultaneously estimated in coupled equations coupled rules can mitigate the stability speed problem affecting noncoupled learning rules since the convergence speed in all eigendirections of the jacobian becomes widely independent of the eigenvalues of the covariance matrix a number of coupled learning rule systems for principal component analysis two of them new is derived by applying newton s method to an information criterion the relations to other systems of this class the adaptive learning algorithm ala the robust recursive least squares algorithm rrlsa and a rule with explicit renormalization of the weight vector length are established

stability in the almost everywhere sense a linear transfer operator approach abstract the problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework the infinitesimal generator of a linear transfer operator perron frobenius is used to provide stability conditions of an autonomous ordinary differential equation it is shown that almost everywhere uniform stability of a nonlinear differential equation is equivalent to the existence of a non negative solution for a steady state advection type linear partial differential equation we refer to this non negative solution verifying almost everywhere global stability as lyapunov density a numerical method using finite element techniques is used for the computation of lyapunov density

convex computation of the maximum controlled invariant set for polynomial control systems we characterize the maximum controlled invariant mci set for discrete as well as continuous time nonlinear dynamical systems as the solution of an infinite dimensional linear programming problem for systems with polynomial dynamics and compact semialgebraic state and control constraints we describe a hierarchy of finite dimensional linear matrix inequality lmi relaxations whose optimal values converge to the volume of the mci set dual to these lmi relaxations are sum of squares sos problems providing a converging sequence of outer approximations to the mci set the approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description a number of numerical examples illustrate the approach

noncommutative plurisubharmonic polynomials part i global assumptions abstract we consider symmetric polynomials p in the noncommutative nc free variables x 1 x 2 x g we define the nc complex hessian of p as the second directional derivative replacing x t by y q x x t h h t 2 p s t x t h y s k t s 0 y x t k h t we call an nc symmetric polynomial nc plurisubharmonic nc plush if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n n matrices for every size n i e q x x t h h t 0 for all x h r n n g for every n 1 in this paper we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables i e an nc symmetric polynomial p is nc plush if and only if it has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic in this paper we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the frobenius theorem a subsequent paper j m greene preprint 6 proves that if the nc complex hessian q of p takes positive semidefinite values on an nc open set then q takes positive semidefinite values on all tuples x h thus p has the form in eq 0 1 the proof in j m greene preprint 6 draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions

contraction theory on riemannian manifolds abstract contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another in this work we present the fundamental results of contraction theory in an intrinsic coordinate free setting with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties we provide coordinate free proofs of the main results for autonomous vector fields and clarify the assumptions under which the results hold we state and prove several interesting corollaries to the main result study cascade and feedback interconnections of contracting systems study some simple examples and highlight how contraction theory has arisen independently in other scientific disciplines we conclude by illustrating the developed theory for the case of gradient dynamics

inverse systems of zero dimensional schemes in p n abstract the authors construct the global macaulay inverse system l z for a zero dimensional subscheme z of projective n space p n over an algebraically closed field k from the local inverse systems of the irreducible components of z they show that when z is locally gorenstein a generic element f of degree d apolar to z determines z if d is larger than an invariant z as a consequence of this globalization they show that a natural upper bound for the hilbert function of gorenstein artin quotients of the coordinate ring of z is achieved for large socle degree they also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms under suitable hypotheses the main tools are elementary but delicate they involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates

nonlinear control synthesis by convex optimization a stability criterion for nonlinear systems recently derived by the third author can be viewed as a dual to lyapunov s second theorem the criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories toward the equilibrium the new criterion has a remarkable convexity property which in this note is used for controller synthesis via convex optimization recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

a nonlinear position and attitude observer on se 3 using landmark measurements this paper addresses the problem of position and attitude estimation based on landmark readings and velocity measurements a derivation of a nonlinear observer on se 3 is presented using a lyapunov function conveniently expressed as a function of the difference between the estimated and the measured landmark coordinates the resulting feedback laws are explicit functions of the landmark measurements and velocity readings exploiting the sensor information directly in the observer the proposed observer yields almost global asymptotic stabilization of the position and attitude errors and exponential convergence in any closed ball inside the region of attraction also it is shown that the asymptotic convergence of the estimation error trajectories is shaped by the landmark geometry and observer design parameters the problem of non ideal velocity readings is also considered and the observer is augmented to compensate for bias in the angular and linear velocity measurements the resulting position attitude and bias estimation errors are shown to converge exponentially fast to the desired equilibrium points for bounded initial estimation errors simulation results are presented to illustrate the stability and convergence properties of the observer

on the stability and stabilization of quaternion equilibria of rigid bodies we study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives depending on how the system dynamics are assumed to evolve in the first case we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction in the second case we relax the assumption that the sense of rotation is constant finally a third scenario is considered in which hybrid switching control is used to choose the direction in which to spin that is both equilibria are continuously considered with regard to less energy consumption it is showed that each of three scenarios must be treated in a different theoretical setting a comparative study in simulations is also provided

combination of lyapunov and density functions for stability of rotational motion lyapunov methods and density functions provide dual characterizations of the solutions of a nonlinear dynamic system this work exploits the idea of combining both techniques to yield stability results that are valid for almost all the solutions of the system based on the combination of lyapunov and density functions analysis methods are proposed for the derivation of almost input to state stability and of almost global stability in nonlinear systems the techniques are illustrated for an inertial attitude observer where angular velocity readings are corrupted by non idealities

performance analysis of adaptive eigenanalysis algorithms we present a rigorous analysis of several popular forms of short memory adaptive eigenanalysis algorithms using a stochastic averaging method a first order analysis shows that the algorithms do not have any equilibrium points despite published claims to the contrary through averaging analysis we show that they hover around an appropriate eigenvector a second order analysis is also given without the gaussian noise assumption and our results greatly outperform an earlier approximation in the literature the second order analysis has been of much interest in the offline study but in the dynamic adaptive case is uncommon

optimisation and asymptotic stability abstractthe problems of unconstrained optimisation and establishing asymptotic stability have much in common understanding the analogy between these two sheds light on their interconnection and may lead to a number of new results for instance in this paper we provide estimates of the rate of convergence when analysing asymptotic stability of differential equations rather than just ascertain the very fact of stability also standard methods for the design of lyapunov functions e g those having the meaning of the full energy of the system turn out to be unsatisfactory from this point of view and have to be modified these claims are exemplified in the paper by considering the heavy ball method for function minimisation in parallel with the problem of asymptotic stability for the synchronous motor equation

robust synchronization for multistable systems in this note we study a robust synchronization problem for multistable systems evolving on manifolds within an input to state stability iss framework based on a recent generalization of the classical iss theory to multistable systems a robust synchronization protocol is designed with respect to a compact invariant set of the unperturbed system the invariant set is assumed to admit a decomposition without cycles that is with neither homoclinic nor heteroclinic orbits numerical simulation examples illustrate our theoretical results

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

injectivity and almost global asymptotic stability of hurwitz vector fields abstract we present in dimension n 3 a survey of examples to the jacobian conjecture the weak markus yamabe conjecture furthermore we show and construct new examples of vector fields where the origin is almost globally asymptotic stable by using the novel concept of density functions introduced by rantzer

analysis of positive invariance and almost regional attraction via density functions with converse results this note is concerned with analysis of positive invariance of nonlinear systems and convergence of trajectories in a region of the state space via density functions if there exists a density function that is positive inside a set containing an equilibrium point and tends to zero as approaches to the boundary the set is positively invariant and almost all of the trajectories starting from there converge to the equilibrium converse results are also provided to prove the existence of such density functions

characterizations of integral input to state stability for systems with multiple invariant sets we generalize the theory of input to state stability iss and of its characterizations by means of lyapunov dissipation inequalities to the study of systems admitting invariant sets which are not necessarily stable in the sense of lyapunov but admit a suitable hierarchical decomposition it is the latter which allows to greatly extend the class of systems to which iss theory can be applied allowing in a unified treatement to deal with oscillators in euclidean coordinates almost globally asymptotically stable systems on manifolds systems with multiple equilibria in bbr n just to name a few

some applications of hausdorff dimension inequalities for ordinary differential equations upper bounds are obtained for the hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable from these are derived sufficient conditions for dissipative analytic n dimensional periodic differential equations to have only a finite number of periodic solutions for autonomous equations the same conditions ensure that each bounded semi orbit converges to a critical point these results yield some information about the lorenz equation and the forced duffing equation

controller design and value function approximation for nonlinear dynamical systems this work considers the infinite time discounted optimal control problem for continuous time input affine polynomial dynamical systems subject to polynomial state and box input constraints we propose a sequence of sum of squares sos approximations of this problem obtained by first lifting the original problem into the space of measures with continuous densities and then restricting these densities to polynomials these approximations are tightenings rather than relaxations of the original problem and provide a sequence of rational controllers with value functions associated to these controllers converging under some technical assumptions to the value function of the original problem in addition we describe a method to obtain polynomial approximations from above and from below to the value function of the extracted rational controllers and a method to obtain approximations from below to the optimal value function of the original problem thereby obtaining a sequence of asymptotically optimal rational controllers with explicit estimates of suboptimality numerical examples demonstrate the approach

adaptive multiple minor directions extraction in parallel using a pca neural network a principal component analysis pca neural network is developed for online extraction of the multiple minor directions of an input signal the neural network can extract the multiple minor directions in parallel by computing the principal directions of the transformed input signal so that the stability speed problem of directly computing the minor directions can be avoided to a certain extent on the other hand the learning algorithms for updating the net weights use constant learning rates this overcomes the shortcoming of the learning rates approaching zero in addition the proposed algorithms are globally convergent so that it is very simple to choose the initial values of the learning parameters this paper presents the convergence analysis of the proposed algorithms by studying the corresponding deterministic discrete time ddt equations rigorous mathematical proof is given to prove the global convergence the theoretical results are further confirmed via simulations

a linear algebra method to decompose forms whose length is lower than the number of variables into weighted sum of squares abstractin this paper an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables

the apolar bilinear form in geometric modeling some recent methods of computer aided geometric design are related to the apolar bilinear form an inner product on the space of homogeneous multivariate polynomials of a fixed degree already known in 19th century invariant theory using a generalized version of this inner product we derive in a straightforward way some of the recent results in cagd like marsden s identity the expression for the de boor fix functionals and recursion schemes for the computation of b patches and their derivatives

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

output feedback control for stabilization on se 3 this paper addresses the problem of stabilizing systems that evolve on se 3 the proposed solution consists of an output feedback controller that guarantees almost global asymptotic stability of the desired equilibrium point in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a nowhere dense set of measure zero the output vector is formed by the position coordinates of a collection of landmarks fixed in the environment the resulting closed loop system exhibits the following properties i the position and orientation subsystems are decoupled ii the position error is globally exponentially stable and iii the orientation error is almost globally exponentially stable results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

a small gain theorem for motone systems with multivalued input state characteristics we provide a small gain theorem for feedback interconnections of monotone input output systems with multi valued input state characteristics this extends a small gain theorem of angeli and sontag for monotone systems with singleton valued characteristics we prove our theorem using thieme s convergence theory for asymptotically autonomous systems we also provide an illustrative example

homogeneous polynomial solutions to constant coefficient pde s suppose p is a homogeneous polynomial over a field k let p d be the differential operator defined by replacing each occurrence of xj in p by xj defined formally in case k is not a subset of c the classical example is p x1 xn j xj for which p d is the laplacian in this paper we solve the equation p d q 0 for homogeneous polynomials q over k under the restriction that k be an algebraically closed field of characteristic 0 this restriction is mild considering that the natural field is c our main theorem and its proof are the natural extensions of work by sylvester clifford rosanes gundelfinger cartan maass and helgason the novelties in the presentation are the generalization of the base field and the removal of some restrictions on p the proofs require only undergraduate mathematics and hilbert s nullstellensatz a fringe benefit of the proof of the main theorem is an expansion theorem for homogeneous polynomials over r and c this theorem is trivial for linear forms goes back at least to gauss for p x1 x 2 2 x 2 3 and has also been studied by maass and helgason

sensitivity to cumulative perturbations for a class of piecewise constant hybrid systems we consider a class of continuous time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces and which includes the fluid level dynamics of the max weight scheduling policy as a special case we study the effect of an external disturbance perturbation on the state trajectory and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation

inverse optimal stabilization of a rigid spacecraft the authors present an approach for constructing optimal feedback control laws for regulation of a rotating rigid spacecraft they employ the inverse optimal control approach which circumvents the task of solving a hamilton jacobi equation and results in a controller optimal with respect to a meaningful cost functional the inverse optimality approach requires the knowledge of a control lyapunov function and a stabilizing control law of a particular form for the spacecraft problem they are both constructed using the method of integrator backstepping the authors give a characterization of nonlinear stability margins achieved with the inverse optimal control law

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

on necessary conditions for almost global stability in the year 2000 the sufficiency of the existence of density functions for the almost global stability of a system was proved in this note we prove the necessity of the existence of density functions for the particular case of global asymptotical stability

convex duality and nonlinear optimal control problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals in this way methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems the result is a necessary and sufficient condition of optimality that generalizes well known sufficient conditions referred to as verification theorems in dynamic programming as a byproduct we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the hamilton jacobi equation it is a striking illustration of the wide range of problems to which convex analysis and in particular convex duality is applicable the approach applied to parametric problems in the calculus of variations was pioneered by l c young lectures on the calculus of variations and optimal control theory w b saunders philadelphia pa 1969 as recent work has shown however it is equally fruitful when applied in optimal control thi

lyapunov redesign of analog phase lock loops the second method of lyapunov is proposed as a method for analyzing the stability and tracking of the nonlinear phase lock loop model it is shown that a fairly regular procedure known as lyapunov design which is quite well known in the area nonlinear control theory is applicable to solving this class of problems the ability of loops designed using these techniques to track a phase step is also proven

the mca exin neuron for the minor component analysis the minor component analysis mca deals with the recovery of the eigenvector associated to the smallest eigenvalue of the autocorrelation matrix of the input data and is a very important tool for signal processing and data analysis it is almost exclusively solved by linear neurons this paper presents a linear neuron endowed with a novel learning law called mca exinn and analyzes its features the neural literature about mca is very poor in the sense that both a little theoretical basis is given almost always focusing on the ode asymptotic approximation and only experiments on toy problems at most four dimensional problems are presented without any numerical analysis this work addresses these problems and lays sound theoretical foundations for the neural mca theory in particular it classifies the mca neurons according to the riemannian metric and justifies from the analysis of the degeneracy of the error cost the different behavior in approaching convergence the cost landscape is studied and used as a basis for the analysis of the asymptotic behavior all the phases of the dynamics of the mca algorithms are investigated in detail and together with the numerical analysis lead to the identification of three possible kinds of divergence here called sudden dynamic and numerical the importance of the choice of low initial conditions is also explained a lot of importance is given to the experimental part where simulations on high dimensional problems are presented and analyzed the orthogonal regression or total least squares tls technique is also presented together with a real world application on the identification of the parameters of an electrical machine it can be concluded that mca exin is the best mca neuron in terms of stability no finite time divergence speed and accuracy

global lyapunov analysis of multistable nonlinear systems a new type of stability is introduced and its equivalent lyapunov characterization is presented the problem of global stability for the compact set composed of all invariant solutions of a nonlinear system several equilibriums for instance is studied such a problem statement allows us to analyze global stability properties for multistable systems it is shown that several well known multistable systems satisfy this new stability property

an almost global notion of input to state stability a new definition of almost global input to state stability for systems on differentiable manifolds is discussed some of its properties are investigated and specific examples are shown which can be treated by means of dual lyapunov techniques

a region dependent gain condition for asymptotic stability a sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem roughly speaking to apply this theorem it is required that the gains composition is continuous increasing and upper bounded by the identity function in this work an alternative sufficient condition is presented for the case in which this criterion fails due to either lack of continuity or the bound of the composed gain is larger than the identity function more precisely the local resp non local asymptotic stability of the origin resp global attractivity of a compact set is ensured by a region dependent small gain condition under an additional condition that implies convergence of solutions for almost all initial conditions in a suitable domain the almost global asymptotic stability of the origin is ensured two examples illustrate and motivate this approach

algorithms for accelerated convergence of adaptive pca we derive and discuss adaptive algorithms for principal component analysis pca that are shown to converge faster than the traditional pca algorithms due to oja and karhunen 1985 sanger 1989 and xu 1993 it is well known that traditional pca algorithms that are derived by using gradient descent on an objective function are slow to converge furthermore the convergence of these algorithms depends on appropriate choices of the gain sequences since online applications demand faster convergence and an automatic selection of gains we present new adaptive algorithms to solve these problems we first present an unconstrained objective function which can be minimized to obtain the principal components we derive adaptive algorithms from this objective function by using 1 gradient descent 2 steepest descent 3 conjugate direction and 4 newton raphson methods although gradient descent produces xu s lmser algorithm the steepest descent conjugate direction and newton raphson methods produce new adaptive algorithms for pca we also provide a discussion on the landscape of the objective function and present a global convergence proof of the adaptive gradient descent pca algorithm using stochastic approximation theory extensive experiments with stationary and nonstationary multidimensional gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods we also compare the steepest descent adaptive algorithm with state of the art methods on stationary and nonstationary sequences

global stability analysis using the eigenfunctions of the koopman operator we propose a novel operator theoretic framework to study global stability of nonlinear systems based on the spectral properties of the so called koopman operator our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems the main results establish the necessary and sufficient relationship between the existence of specific eigenfunctions of the koopman operator and the global stability property of hyperbolic fixed points and limit cycles these results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space

almost global attraction in planar systems abstract in this work we present a result relating the recent ideas of almost global stability and density functions with the classical poincare bendixson theory for planar systems

convex programs for temporal verification of nonlinear dynamical systems a methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently conditions that must be satisfied by a barrier certificate can be formulated as a convex program and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region the dual of this problem i e the reachability problem concerns proving the existence of a trajectory starting from the initial set that reaches another given set using insights from the linear programming duality appearing in the discrete shortest path problem we show in this paper that reachability of continuous systems can also be verified through convex programming several convex programs for verifying safety and reachability as well as other temporal properties such as eventuality avoidance and their combinations are formulated some examples are provided to illustrate the application of the proposed methods finally we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates