robust and adaptive design of numerical optimization based extremum seeking control we consider the employment of numerical optimization and state regulation to solve the extremum seeking control esc problem which does not assume the time scale separation between the plant dynamics and the extremum seeking loop extremum seeking is realized via a state regulator that drives the state traveling along a convergent set point sequence generated by a numerical optimization algorithm in this paper we propose a novel design of an asymptotic state regulator via output tracking for state feedback linearizable systems where we trade off finite time state regulation to obtain flexibility in designing a robust extremum seeking controller existing techniques such as nonlinear damping and nonlinear adaptive control are then used to deal with input disturbance and unmodeled plant dynamics simulation examples illustrate the effectiveness of the basic and robust extremum seeking schemes and some design guidelines are provided for engineering applications

formation control of mobile agents based on inter agent distance dynamics we propose a novel formation control strategy based on inter agent distances for single integrator modeled agents in the plane attempting to directly control the inter agent distances we derive a control law from the distance dynamics the proposed control law achieves the local asymptotic stability of infinitesimally rigid formations triangular infinitesimally rigid formations are globally asymptotically stable under the proposed control law with all squared distance errors exponentially and monotonically converging to zero as an extension of existing results the stability analysis in this paper reveals that any control laws related with the gradient law by multiplication of a positive matrix ensure the local asymptotic stability of infinitesimally rigid formations

high order multivariable transfer function curve fitting algorithms sparse matrix methods and experimental results abstract this paper develops a computational approach to multivariable frequency domain curve fitting based on two norm minimization the algorithm is specifically tailored to the identification of complex systems having a large number of parameters and includes a sparse matrix method for reducing computation and memory requirements on large problems the algorithm is also well suited for identification of lightly damped systems such as flexible structures the overall approach is successfully demonstrated on a high order multivariable flexible structure experiment requiring the estimation of 780 parameters over a 100 hz band width

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

newton based stochastic extremum seeking in this paper we introduce a newton based approach to stochastic extremum seeking and prove local stability of newton based stochastic extremum seeking algorithm in the sense of both almost sure convergence and convergence in probability the convergence of the newton algorithm is proved to be independent of the hessian matrix and can be arbitrarily assigned which is an advantage over the standard gradient based stochastic extremum seeking simulation shows the effectiveness and advantage of the proposed algorithm over gradient based stochastic extremum seeking

bounded extremum seeking with discontinuous dithers the analysis of discontinuous extremum seeking es controllers e g those applicable to digital systems has historically been more complicated than that of continuous controllers we establish a simple and general extension of a recently developed bounded form of es to a general class of oscillatory functions including functions discontinuous with respect to time such as triangle or square waves with dead time we establish our main results by combining a novel idea for oscillatory control with an extension of functional analytic techniques originally utilized by kurzweil jarnik sussmann and liu in the late 80s and early 90s and recently studied by durr et al we demonstrate the value of the result with an application to inverter switching control

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

non parametric nonlinear system identification a data driven orthogonal basis function approach in this paper a data driven orthogonal basis function approach is proposed for non parametric fir nonlinear system identification the basis functions are not fixed a priori and match the structure of the unknown system automatically this eliminates the problem of blindly choosing the basis functions without a priori structural information further based on the proposed basis functions approaches are proposed for model order determination and regressor selection along with their theoretical justifications

stability and robustness for saddle point dynamics through monotone mappings abstract motivated by recent interest in saddle point dynamics where given a convex in x and concave in y function trajectories follow the steepest descent in x and the steepest ascent in y and where convergence of trajectories to saddle points is desired this note revisits the maximal monotone mapping approach to saddle point dynamics mildly improves one convergence result and proposes new results on the robustness of pointwise asymptotic stability of the set of saddle points the results apply to nonsmooth convex concave functions and constraints and as a special case to projected saddle point dynamics

on the stability of gradient polynomial systems at infinity abstract we first define the notion of the infimum at infinity of a polynomial function and the notion of stability at infinity near the fiber of the gradient descent system then we prove that the gradient descent system is stable at infinity near the fiber of the infimum value at infinity

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

extremum seeking with bounded update rates a b s t r a c t in this work we present a form of extremum seeking es in which the unknown function being mini mized enters the system s dynamics as the argument of a cosine or sine term thereby guaranteeing known bounds on update rates and control efforts we present general n dimensional optimization and stabi lization results as well as 2d vehicle control with bounded velocity and control efforts for application to autonomous vehicles tracking a source in a gps denied environment with unknown orientation this es approach allows for smooth heading angle actuation with constant velocity and in application to a unicycle type vehicle results in control ability as if the vehicle is fully actuated our stability analysis is made possible by the classic results of kurzweil jarnik sussmann and liu regarding systems with highly oscillatory terms in our stability analysis we combine the averaging results with a semiglobal practical stability result under small parametric perturbations developed by moreau and aeyels published by elsevier b v

lipschitz stability for stochastic programs with complete recourse this paper investigates the stability of optimal solution sets to stochastic programs with complete recourse where the underlying probability measure is understood as a parameter varying in some space of probability measures in math programming 67 1994 pp 99 108 sha piro proved lipschitz upper semicontinuity of the solution set mapping inspired by this result we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions for a subclass of recourse models we show that the solution set mapping is hausdorff lipschitz continuous with respect to the subgradient distance moreover the subgradient distance is estimated above by the kolmogorov smirnov distance of certain distribution functions related to the recourse model the lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations finally the lipschitz continuity result is used to derive some new results on the asymptotic behavior ofoptimal solutions when the probability measure underlying the recourse model is estimated via empirical measures law of iterated logarithm large deviation estimate estimate for asymptotic distribution

conjugate variables in continuous maximum entropy inference for a continuous maximum entropy distribution obtained from an arbitrary number of simultaneous constraints we derive a general relation connecting the lagrange multipliers and the expectation values of certain particularly constructed functions of the states of the system from this relation an estimator for a given lagrange multiplier can be constructed from derivatives of the corresponding constraining function these estimators sometimes lead to the determination of the lagrange multipliers by way of solving a linear system and in general they provide another tool to widen the applicability of jaynes s formalism this general relation especially well suited for computer simulation techniques also provides some insight into the interpretation of the hypervirial relations known in statistical mechanics and the recently derived microcanonical dynamical temperature we illustrate the usefulness of these new relations with several applications in statistics

lie bracket approximation of extremum seeking systems extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map it often requires advanced system theoretic tools to understand the qualitative behavior of extremum seeking systems in this paper a novel interpretation of extremum seeking is introduced we show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain lie brackets of the vector fields of the extremum seeking system it turns out that the lie bracket system directly reveals the optimizing behavior of the extremum seeking system furthermore we establish a theoretical foundation and prove that uniform asymptotic stability of the lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system we use the established results in order to prove local and semi global practical uniform asymptotic stability of the extrema of a certain map for multi agent extremum seeking systems

a numerical approach to optimal coherent quantum lqg controller design using gradient descent abstract this paper is concerned with coherent quantum linear quadratic gaussian cqlqg control the problem is to find a stabilizing measurement free quantum controller for a quantum plant so as to minimize a mean square cost for the fully quantum closed loop system the plant and controller are open quantum systems interconnected through bosonic quantum fields in comparison with the observation actuation structure of classical controllers coherent quantum feedback is less invasive to the quantum dynamics the plant and controller variables satisfy the canonical commutation relations ccrs of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations qsdes in order to correspond to such oscillators these qsdes must satisfy physical realizability pr conditions in the form of quadratic constraints on the state space matrices reflecting the ccr preservation in time the symmetry of the problem is taken into account by introducing equivalence classes of coherent quantum controllers generated by symplectic similarity transformations we discuss a modified gradient flow which is concerned with norm balanced realizations of controllers a line search gradient descent algorithm with adaptive stepsize selection is proposed for the numerical solution of the cqlqg control problem the algorithm finds a local minimum of the lqg cost over the parameters of the hamiltonian and coupling operators of a stabilizing coherent quantum controller thus taking the pr constraints into account a convergence analysis of the algorithm is presented numerical examples of designing locally optimal cqlqg controllers are provided in order to demonstrate the algorithm performance

online learning of feasible strategies in unknown environments define an environment as a set of convex constraint functions that vary arbitrarily over time and consider a cost function that is also convex and arbitrarily varying agents that operate in this environment intend to select actions that are feasible for all times while minimizing the cost s time average such action is said optimal and can be computed offline if the cost and the environment are known a priori an online policy is one that depends causally on the cost and the environment to compare online policies to the optimal offline action define the fit of a trajectory as a vector that integrates the constraint violations over time and its regret as the cost difference with the optimal action accumulated over time fit measures the extent to which an online policy succeeds in learning feasible actions while regret measures its success in learning optimal actions this paper proposes the use of online policies computed from a saddle point controller which are shown to have fit and regret that are either bounded or grow at a sublinear rate these properties provide an indication that the controller finds trajectories that are feasible and optimal in a relaxed sense concepts are illustrated throughout with the problem of a shepherd that wants to stay close to all sheep in a herd numerical experiments show that the saddle point controller allows the shepherd to do so

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

singularly perturbed lie bracket approximation we consider the interconnection of two dynamical systems where one has an input affine vector field by employing a singular perturbation and a lie bracket analysis technique we show how the trajectories can be approximated by two decoupled systems from this trajectory approximation result and the stability properties of the decoupled systems we derive stability properties of the overall system

on a class of generating vector fields for the extremum seeking problem lie bracket approximation and stability properties abstract in this paper we describe a broad class of control functions for extremum seeking problems we show that it unifies and generalizes existing extremum seeking strategies which are based on lie bracket approximations and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks the second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems it provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability in contrast to many known results we also prove asymptotic and exponential stability in the sense of lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields

observer based perturbation extremum seeking control with input constraints for direct contact membrane distillation process abstractan observer based perturbation extremum seeking control is proposed for a direct contact membrane distillation dcmd process the process is described with a dynamic model that is based on a 2d advection diffusion equation model which has pump flow rates as process inputs the objective of the controller is to optimise the trade off between the permeate mass flux and the energy consumption by the pumps inside the process cases of single and multiple control inputs are considered through the use of only the feed pump flow rate or both the feed and the permeate pump flow rates a nonlinear lyapunov based observer is designed to provide an estimation for the temperature distribution all over the designated domain of the dcmd process moreover control inputs are constrained with an anti windup technique to be within feasible and physical ranges performance of the proposed structure is analysed and simulations based on real dcmd process parameters for each control input are provided

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

extremum seeking on submanifolds in the euclidian space extremum seeking is a powerful control method to steer a dynamical system to an extremum of a partially unknown function in this paper we introduce extremum seeking systems on submanifolds in the euclidian space using a trajectory approximation technique based on lie brackets we prove that uniform asymptotic stability of the so called lie bracket system on the manifold implies practical uniform asymptotic stability of the corresponding extremum seeking system on the manifold we illustrate the approach with an example of extremum seeking on a torus

batch to batch finite horizon lq control for unknown discrete time linear systems via stochastic extremum seeking we employ our recent discrete time stochastic averaging theorems and stochastic extremum seeking to iteratively batch to batch optimize open loop control sequences for unknown but reachable discrete time linear systems with a scalar input and without known system dimension for a cost that is quadratic in the measurable output and the input first for a multivariable gradient based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open loop control sequence second to remove the convergence rate s dependence on the hessian matrix of the cost function which is unknown since the system s model the system matrices a b c is unknown we develop a multivariable discrete time newton based stochastic extremum seeking method design the newton based algorithm for the iteration of the input sequence and prove local exponential convergence to the optimal input sequence finally two simulation examples are given to illustrate the effectiveness of the two methods

a frequency domain iterative identification algorithm using general orthonormal basis functions in this paper a two step frequency domain identification algorithm is proposed in which the identified model is parameterized in terms of general orthonormal basis functions furthermore an update scheme is presented where the poles of the basis functions are adjusted iteratively and the convergence properties of this scheme with respect to unknown but bounded noise are studied the algorithm is applied to experimental frequency response data obtained from a compact disk player

emergence of tsallis statistics as a consequence of invariance abstract for non equilibrium systems in a steady state we present two necessary and sufficient conditions for the emergence of q canonical ensembles also known as tsallis statistics these conditions are invariance requirements over the definition of subsystem and environment and over the joint rescaling of temperature and energy our approach is complementary to the notions of tsallis non extensive statistics and superstatistics

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

convergence and convergence rate of the balanced realization truncations for infinite dimensional discrete time systems abstract in this paper we study approximation via balanced realizations for a large class of infinite dimensional discrete time linear systems we give properties of the truncated system and prove that the approximation is l 2 convergent in the case of nuclear systems we prove convergence in nuclear norm and give an estimation of the l convergence rate

a dual purpose principal and minor component flow abstract principal component flows are flows converging to the eigenvectors associated with the largest eigenvalues of a given symmetric matrix similarly minor component flows converge to the eigenvectors associated with the smallest eigenvalues traditional flows required the matrix to be positive definite and moreover finding well behaved minor component flows appeared to be harder and unrelated to the principal component case this paper derives a flow which can be used to extract either the principal or the minor components and which does not require the matrix to be positive definite the flow is shown to be a generalisation of the oja brockett flow

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

temperature is not an observable in superstatistics abstract superstatistics beck and cohen 2003 is a formalism that attempts to explain the presence of distributions other than the boltzmann gibbs distributions in nature typically power law behavior for systems out of equilibrium such as fluids under turbulence plasmas and gravitational systems superstatistics postulates that those systems are found in a superposition of canonical ensembles at different temperatures and sometimes the physical interpretation is one of local thermal equilibrium in the sense of an inhomogeneous temperature distribution in different regions of space or instants of time here we show that in order for superstatistics to be internally consistent it is impossible to define a phase space function or microscopic observable b p q corresponding one to one to the local value of 1 k b t thus unlike energy which is defined by a phase space function h p q the hamiltonian temperature is not a microscopic observable an important consequence of our proof is that in superstatistics the identification of temperature with the kinetic energy is limited to the expectation of and cannot be used to measure the different temperatures in local thermal equilibrium or its fluctuations

stochastic source seeking with forward and angular velocity regulation abstract this paper studies a stochastic extremum seeking method to steer a nonholonomic vehicle to the unknown source of a spatial field in a plane the key challenge lies in the lack of vehicle s position information and the distribution of the scalar field different from the existing stochastic strategy that keeps the forward velocity constant and controls only the angular velocity we design a stochastic extremum seeking controller to regulate both forward and angular velocities simultaneously in this work thus the vehicle decelerates near the source and stays within a small area as if it comes to a full stop which solves the overshoot problem in the constant forward velocity case we use the stochastic averaging theory to prove the local exponential convergence both almost surely and in probability to a small neighborhood near the source for elliptical level sets finally simulations are included to illustrate the theoretical results

extremum seeking for static maps with delays in this paper we address the design and analysis of multi variable extremum seeking for static maps subject to arbitrarily long time delays both gradient and newton based methods are considered multi input systems with different time delays in each individual input channel as well as output delays are dealt with the phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation based averaging based estimate of the hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point even in the presence of delays the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the time delay in particular a new backstepping like transformation is introduced to design the predictor for the gradient based extremum seeking scheme with multiple and distinct input delays the proposed newton based extremum seeking approach removes the dependence of the convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable as in the literature free of delays a source seeking example illustrates the performance of the proposed delay compensated extremum seeking schemes

extremum seeking control with an adaptive dither signal we present an output feedback law that steers a fully actuated control affine system with general drift vector field to a minimum of the output function the proposed control law is based on a suitable approximation of descent directions by lie brackets in the high frequency limit a novelty of our approach is an adaptive choice of the frequency parameter in this way the problem of determining a sufficiently large frequency parameter becomes obsolete the adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift the proposed control law does not only lead to convergence into a neighborhood of the set of minima but leads to exact convergence for the case of an output function with a global minimum and no other stationary points we prove global convergence

representation and identification of non parametric nonlinear systems of short term memory and low degree of interaction in this paper we propose a new representation which is particularly useful for a class of non parametric nonlinear systems that have short term memory and low degree of interaction advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically furthermore results regarding structural estimation based on the analysis of variance and on full scale identification are also provided

rational basis functions for robust identification from frequency and time domain measurements this paper investigates the use of general bases with fixed poles for the purposes of robust estimation these bases which generalise the common fir laguerre and two parameter kautz ones are shown to be fundamental in the disc algebra provided a very mild condition on the choice of poles is satisfied it is also shown that by using a min max criterion these bases lead to robust estimators for which error bounds in different norms can be explicitly quantified the key idea facilitating this analysis is to re parameterise the chosen model structures into a new one with equivalent fixed poles but for which the basis functions are orthonormal in h 2 d

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

subspace based identification of infinite dimensional multivariable systems from frequency response data a new identification algorithm which identifies low complexity models of infinite dimensional systems from equidistant frequency response data is presented the new algorithm is a combination of the fourier transform technique with the recent subspace techniques given noisefree data finite dimensional systems are exactly retrieved by the algorithm when noise is present it is shown that identified models strongly converge to the balanced truncation of the identified system if the measurement errors are covariance bounded several conditions are derived on consistency illustrating the trade offs in the selection of certain parameters of the algorithm two examples are presented which clearly illustrate the good performance of the algorithm

extremum seeking for limit cycle minimization in many physical problems equilibrium stabilization is not possible and the controlled system is in a limit cycle if the size of the limit cycle depends on some of the control parameters then a reasonable objective would be to tune this parameter to minimize the size of the limit cycle we propose a method for achieving this this method is an extension of our earlier result kristic and wang 2000 on extremum seeking for equilibria we illustrate the method with a van der pol oscillator example and present analysis for it using averaging and singular perturbations

a stochastic approximation framework for a class of randomized optimization algorithms we study a class of random sampling based algorithms for solving general non differentiable optimization problems these are iterative approaches that are based on sampling from and updating an underlying distribution function over the set of feasible solutions in particular we propose a novel and systematic framework to investigate the convergence and asymptotic convergence rates of these algorithms by exploiting their connections to the well known stochastic approximation sa method such an sa framework unifies our understanding of these randomized algorithms and provides new insight into their design and implementation issues our preliminary numerical experiments indicate that new implementations of these algorithms based on the proposed framework may lead to improved performance over existing procedures

on the stable equilibrium points of gradient systems this paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f in particular it is shown that under the assumption that f is real analytic local minimality is necessary and sufficient for stability under the weaker assumption that f is indefinitely continuously differentiable local minimality is neither necessary nor sufficient for stability

extremum seeking for dynamic maps using lie brackets and singular perturbations abstract we introduce a framework for the analysis of extremum seeking systems for dynamic maps based on lie bracket approximations and singular perturbation theory using the introduced framework we provide an extremum seeking regulator for a unicycle and prove its stability properties

practical stability and stabilization presents a practical stability result for dynamical systems depending on a small parameter this result is applied to a practical stability analysis of fast time varying systems studied in averaging theory and of highly oscillatory systems studied by sussmann and liu 1999 furthermore the problem of practically stabilizing control affine systems with drift is discussed

brief paper parameter convergence in adaptive extremum seeking control this paper addresses the problem of parameter convergence in adaptive extremum seeking control design an alternate version of the popular persistence of excitation condition is proposed for a class of nonlinear systems with parametric uncertainties the condition is translated to an asymptotic sufficient richness condition on the reference set point since the desired optimal set point is not known a priori in this type of problem the proposed method includes a technique for generating perturbation signal that satisfies this condition in closed loop this demonstrates its superiority in terms of parameter convergence the method guarantees parameter convergence with minimal but sufficient level of perturbation the effectiveness of the proposed method is illustrated with a simulation example

minimum seeking for clfs universal semiglobally stabilizing feedback under unknown control directions employing extremum seeking es for seeking minima of control lyapunov function clf candidates we develop 1 the first systematic design of es controllers for unstable plants 2 a simple non model based universal feedback law that emulates in an average sense the l g v controllers for stabilization with inverse optimality and 3 a new strategy for stabilization of systems with unknown control directions as an alternative to nussbaum gain controllers that lack exponential stability lack transient performance guarantees and lack robustness to changes in the control direction the stability analysis that underlies our designs is inspired by an analysis approach synthesized in a recent work by durr stankovic and johansson which combines a lie bracket averaging result of gurvits and li with a semiglobal practical stability result under small parametric perturbations by moreau and aeyels

discrete time system modelling in with orthonormal basis functions abstract in this paper model sets for linear time invariant systems spanned by fixed pole orthonormal bases are investigated the obtained model sets are shown to be complete in l p t 1 the lebesque spaces of functions on the unit circle t and in c t the space of periodic continuous functions on t the l p norm error bounds for estimating systems in l p t by the partial sums of the fourier series formed by the orthonormal functions are computed for the case 1 some inequalities on the mean growth of the fourier series are also derived these results have application in estimation and model reduction

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

extremum seeking with sliding mode gradient estimation and asymptotic regulation for a class of nonlinear systems this paper presents a model based extremum seeking approach for a class of single input single output nonlinear systems with the analytic form of the performance function unknown a priori we focus on a practically implementable design with robustness to model uncertainties and disturbances a discrete time sliding mode gradient estimator is developed for estimating the gradient of the performance profile based on the estimate a variable structure output feedback regulator is proposed to enforce the system states toward the optimal trajectory we analyze convergence conditions of the switching system toward a neighborhood of the optimal trajectory and establish an ultimate bound on the size of the neighborhood the robustness of the proposed controller is discussed with respect to measurement noise

a framework for extremum seeking control of systems with parameter uncertainties traditionally the design of extremum seeking algorithm treats the system as essentially a black box which for many applications means disregarding known information about the model structure in contrast to this approach there have been recent examples where a known plant structure with uncertain parameters has been used in the online optimization of plant operation however the results for these approaches have been restricted to specific classes of plants and optimization algorithms this paper seeks to provide general results and a framework for the design of extremum seekers applied to systems with parameter uncertainties general conditions for an optimization method and a parameter estimator are presented so that their combination guarantees convergence of the extremum seeker for both static and dynamic plants tuning guidelines for the closed loop scheme are also presented the generality and flexibility of the proposed framework is demonstrated through a number of parameter estimators and optimization algorithms that can be combined to obtain extremum seeking examples of anti lock braking and model reference adaptive control are used to illustrate the effectiveness of the proposed framework

proof of the gradient conjecture of r thom let x t be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x t we prove the gradient conjecture of r thom which states that the secants of x t a tx0 have a limit actually we show a stronger statement the radial projection of x t from x0 onto the unit sphere has flnite length

dynamical analysis of neural networks of subgradient system in this technical note we consider a class of neural network which is a generalization of neural network models considered in the optimization context under some mild assumptions this neural network can be translated into a negative subgradient dynamical system at first we study the existence and uniqueness of solution of this neural network then by nonsmooth ojasiewicz inequality we prove the convergence of the trajectories of this neural network in the end a constrained minimization problem is studied which can be associated with this neural network it is proved that the local constrained minimum of the cost function coincides with the stable equilibria point of this neural network

quasi newton methods based on ordinary differential equation approach for unconstrained nonlinear optimization in this paper we propose a hybrid ode based quasi newton qn method for unconstrained optimization problems which combines the idea of low order implicit runge kutta rk techniques for gradient systems with the qn type updates of the jacobian matrix such as the symmetric rank one sr1 update the main idea of this approach is to associate a qn matrix to approximate numerically the jacobian matrix in the gradient system fundamentally this is a gradient system based on the first order optimality conditions of the optimization problem to further extend the methods for solving large scale problems a feature incorporated to the proposed methods is that a limited memory setting for matrix vector multiplications is used thus avoiding the computational and storage issues when computing jacobian information under suitable assumptions global convergence of the proposed method is proved practical insights on the effectiveness of these approaches on a set of test functions are given by a numerical comparison with that of the limited memory bfgs algorithm l bfgs and conjugate gradient algorithm cg

a sequential homotopy method for mathematical programming problems we propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract hilbert spaces under the guignard constraint qualification the method is equivalent to performing projected backward euler timestepping on a projected gradient antigradient flow of the augmented lagrangian the projected backward euler equations can be interpreted as the necessary optimality conditions of a primal dual proximal regularization of the original problem the regularized problems are always feasible satisfy a strong constraint qualification guaranteeing uniqueness of lagrange multipliers yield unique primal solutions provided that the stepsize is sufficiently small and can be solved by a continuation in the stepsize we show that equilibria of the projected gradient antigradient flow and critical points of the optimization problem are identical provide sufficient conditions for the existence of global flow solutions and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient antigradient flow practically eradicating convergence to saddle points and maxima the sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework we demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth newton approach for the regularized subproblems

hamiltonian formalism and path entropy maximization maximization of the path information entropy is a clear prescription for constructing models in non equilibrium statistical mechanics here it is shown that following this prescription under the assumption of arbitrary instantaneous constraints on position and velocity a lagrangian emerges which determines the most probable trajectory deviations from the probability maximum can be consistently described as slices in time by a hamiltonian according to a nonlinear langevin equation and its associated fokker planck equation the connections unveiled between the maximization of path entropy and the langevin fokker planck equations imply that missing information about the phase space coordinate never decreases in time a purely information theoretical version of the second law of thermodynamics all of these results are independent of any physical assumptions and thus valid for any generalized coordinate as a function of time or any other parameter this reinforces the view that the second law is a fundamental property of plausible inference

finite time distributed linear equation solver for solutions with minimum l_1 norm this paper proposes a continuous time distributed algorithm for multiagent networks to achieve a solution with the minimum l 1 norm to underdetermined linear equations the proposed algorithm comes from a combination of the filippov set valued map with the projection consensus flow given the underlying network is undirected and fixed it is shown that the proposed algorithm drives all agents individual states to converge in finite time to a common value which is the minimum l 1 norm solution

orthonormal basis functions in time and frequency domain hambo transform theory the class of finite impulse response fir laguerre and kautz functions can be generalized to a family of rational orthonormal basis functions for the hardy space h2 of stable linear dynamical systems these basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals as required in e g system identification system approximation and adaptive filtering in this paper the basis functions are derived from a transfer function perspective as well as in a state space setting it is shown how this approach leads to alternative series expansions of systems and signals in time and frequency domain the generalized basis functions induce signal and system transforms hambo transforms which have proved to be useful analysis tools in various modelling problems these transforms are analyzed in detail in this paper and a large number of their properties are derived principally it is shown how minimal state space realizations of the system transform can be obtained from minimal state space realizations of the original system and vice versa

a perturbation based proportional integral extremum seeking control approach this paper proposes a proportional integral extremum seeking control technique the technique is a generalization of the standard perturbation based techniques which provides fast transient performance of the closed loop system to the optimum equilibrium of a measured cost function under the assumption that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown it is shown that for a class of minimum phase nonlinear system with a cost function of relative degree one the closed loop esc can practically stabilize the unknown equilibrium optimum the stabilization result provides an alternative approach to output feedback control of nonlinear systems the effectiveness of the proposed approach is demonstrated using simulation examples

instability of a non isolated equilibrium in this paper non isolated equilibria of conservative mechanical systems are studied under mild non degeneracy conditions such equilibria are shown to be unstable in the sense of liapunov

on the estimation of the gradient lines of a density and the consistency of the mean shift algorithm we consider the problem of estimating the gradient lines of a density which can be used to cluster points sampled from that density for example via the mean shift algorithm of fukunaga and hostetler 1975 we prove general convergence bounds that we then specialize to kernel density estimation

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

convergence properties of gradient descent noise reduction gradient descent noise reduction is a technique that attempts to recover the true signal or trajectory from noisy observations of a non linear dynamical system for which the dynamics are known this paper provides the first rigorous proof that the algorithm will recover the original trajectory for a broad class of dynamical systems under certain conditions the proof is obtained using ideas from linearisation theory since the first introduction of the algorithm it has been recognised that the algorithm can fail to recover the true trajectory and it has been suggested that this is a practical or numerical limitation that is a consequence of near tangencies between stable and unstable manifolds this paper demonstrates through numerical experiments and details of the proof that the situation is worse than expected in that near tangencies impose essential limitations on noise reduction not just practical or numerical limitations that is gradient descent noise reduction will sometimes fail to recover the true trajectory even with unlimited perfect computation on the other hand the numerical experiments suggest that the gradient descent noise reduction algorithm will always recover a trajectory that is entirely consistent with the evidence provided by the observations that is it attains the best that can be achieved given the observations it is argued that near tangencies will therefore impose the same limitations on any noise reduction algorithm

the multimode procrustes problem in this paper we consider a generalization of the well known procrustes problem relevant to principal component analysis of multidimensional data arrays this multimode procrustes problem is a complex constrained minimization problem which involves the simultaneous least squares fitting of several matrices we propose two solutions of the problem the projected gradient approach which leads to solving ordinary differential equations on matrix manifolds and differential geometric approach for optimization on products of matrix manifolds a numerical example concerning the three mode procrustes illustrates the developed algorithms

continuous dynamical systems that realize discrete optimization on the hypercube we study the problem of finding a local minimum of a multilinear function e over the discrete set 0 1 n the search is achieved by a gradient like system in 0 1 n with cost function e under mild restrictions on the metric the stable attractors of the gradient like system are shown to produce solutions of the problem even when they are not in the vicinity of the discrete set 0 1 n moreover the gradient like system connects with interior point methods for linear programming and with the analog neural network studied by vidyasagar vid95 in the same context ams subject classifications 34d20 90c51 key words gradient like systems multilinear functions 0 1 programming interior point methods lyapunov stability

general orthonormal bases for robust identification in h_ infty in this paper the problem of system identification in h infty with general orthonormal basis functions is investigated a two stage algorithm is shown to be robust provided that the number of basis elements as a function of the amount of data does not increase faster than o n frac 2 5 worst case identification error bounds in the h infty norm are derived the algorithm also works on nonuniformly spaced frequency response measurements an example is provided to illustrate the application of the algorithm

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

a gradient flow approach to decentralized output feedback optimal control this paper visits the quadratic optimal control problem of decentralised control systems via static output feedback a gradient flow approach is introduced as a tool to compute the optimal output feedback gain several nice properties are revealed concerning the convergence of the gain matrix along the trajectory of an ordinary differential equation obtained from the gradient of objective cost i e the objective cost is decreasing along this trajectory if the equilibrium points are isolated the convergence can be guaranteed a simulation example is given to illustrate the effectiveness of this approach

synchronization with partial state coupling on so n this paper studies autonomous synchronization of k agent orientations in so n by using links that each convey a priori different partial state information between two agents the latter compare the action of their states on a common link dependent vector of mathbb r n such setting appears when each pair of agents compares the expression in their respective body frames of a common external direction force field heading of moving object etc to synchronize their states agents must combine this information over time or throughout the network a gradient coupling law for synchronization is proposed extensive convergence analysis of the coupled agents is provided both for fixed and time varying reference vectors the results are compared to a similar setting with states in mathbb r n and agents coupled through scalar product of their state with a reference vector