on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

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

fast extremum seeking for wiener hammerstein plants this paper considers an observer based extremum seeking es scheme acting on a wiener hammerstein wh plant the es scheme utilises a high frequency sinusoidal dither in order to achieve fast minimisation of the plant s static nonlinearity unlike the prevailing analyses of fast es schemes acting on wh plants the presented result is semi global and applicable to plants with fairly general unknown nonlinearities design of the es scheme requires knowledge of the relative orders of both the input and output dynamics of the plant two different tunings of the es parameters are considered one which forces the es dynamics to be fast and one which is less restrictive it is shown that approximate knowledge of the phase of the plant s input dynamics is required if the es scheme is to use the less restrictive tuning for both tunings it is shown how the es scheme may be designed in order to simultaneously achieve arbitrarily fast and accurate minimisation of the static nonlinearity from an arbitrarily large set of initial conditions moreover the result shows that a wh plant with unstable input dynamics can in fact be stabilised by the es scheme

integration over matrix spaces with unique invariant measures we present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of wick contractions the method gives exact results for monomials of low order for higher order monomials it leads to an error of order 1 n where n is the dimension of the matrix and where is independent of the degree of the monomial we give a lower bound on the integer and show how can be increased systematically the method is particularly suited for symbolic computer calculation explicit results are given for o n u n and for the circular orthogonal ensemble

extremum seeking based perfect adaptive tracking of non pe references despite nonvanishing variance of perturbation abstract this paper presents a novel algorithm for adaptive stabilization of unstable discrete time systems with unknown control directions in a departure from commonly used recursive least squares and gradient type parameter estimators the proposed algorithm is based on extremum seeking es method the perturbation signal is a martingale difference sequence m d s with a non decaying bounded from below variance in spite of a non vanishing perturbation somewhat surprisingly it is shown that globally almost surely a s the tracking error converges to zero input and output signals are uniformly bounded and the parameter estimates are convergent sequences

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

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

uniform asymptotic solutions of boundary and eigenvalue problems for a second order ordinary differential equation with singular and turning points abstract a uniform asymptotic solution to the boundary value problem u x 2 r x u 2 g x u z i u z i i i 1 2 and x i z1 z2 is developed by langer s method r x can have poles at the end points of i the case in which r x e u x where e is an unknown eigenvalue and u has poles in i is also discussed the asymptotic nature of the formal solution to the boundary value problem is investigated and specific examples are solved

stochastic averaging in continuous time and its applications to extremum seeking we investigate stochastic averaging theory in continuous time for locally lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms first we establish a general stochastic averaging principle and some related stability theorems for a class of continuous time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process then we propose a continuous time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals we analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems

using persistent excitation with fixed energy to stabilize adaptivecontrollers and obtain hard bounds for the parameter estimation error two important instability problems in certainty equivalence adaptive control are solved by external excitation the first instability is parameter drift along an unstable manifold when the excitation level is not high enough the second instability is numerical and due to a division with zero in the adaptive law global methods based on excitation have been developed to solve this problem but the energy of the excitation has been tuned on line the main contribution of the current paper is in showing that the estimator is stabilized when we apply excitation with fixed and finite energy the level of excitation should be sufficiently high relative to the magnitudes of the external disturbances and the unmodeled dynamics the approach can be generalized to more complex adaptive laws this together with the fact that we obtain hard bounds for the parameter estimation error opens up for the possibility of designing robust controllers that are adaptive

adaptive control via extremum seeking global stabilization and consistency of parameter estimates we propose an extremum seeking es based algorithm for reference tracking and stabilization of unstable discrete time systems with unknown control directions the probing signal is a martingale difference sequence m d s with a vanishing variance it is proved that almost surely a s the input and output signals are uniformly bounded and the tracking error converges to zero it is also shown that a s the parameter estimates converge toward their true values this shows that the proposed algorithm can also perform identification of open loop unstable systems with unknown control directions

uniform approximations for the natural modes and frequencies of spindle shaped acoustical resonators we wish to obtain uniform approxiations for the natural modes and frequencies of some elongated acoustic bodies or spindles these are bodies generated by plain curves of the type y h a a dxb also the end points are pointed finally we remark that when s o r ex r x is a body of infinite length with a nonzero cross section such a body is generated by a curve of the type y x i o kl x xx 1 2

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

asymptotic semiclassical equivalence for schrodinger equations with singular potentials and for related systems of two first order equations the asymptotic equivalence of systems of two ordinary first order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points the set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed by persistently using the determinant properties of the transformation matrices the number of integration operations at each step of this procedure is halved compared with the algorithms known before the theory is specialized to the case of time independent one dimensional schrodinger equations with singular potentials some generalizations to multichannel schrodinger equations are also presented

stochastic adaptive stabilization via extremum seeking in case of unknown control directions this technical note presents a novel extremum seeking es based algorithm for global stabilization of unstable first order linear discrete time ldt stochastic systems with unknown control directions the probing signal is a martingale difference sequence with a vanishing variance the controller parameter estimate converges almost surely a s to the optimizing value the sample mean square value of the output converges a s to the minimal variance

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

uniform asymptotic solutions of second order linear ordinary differential equations with singular points i formal theory abstract a formal uniform asymptotic solution of the differential equation d 2 u dz 2 2 r z u 0 z d for large when d contains any number and order of poles and turning points is constructed the development is a generalization of langer s method an asymptotic relation between the wronskian of any pair of linearly independent solutions of the differential equation and the wronskian of the pair of corresponding solutions of the comparison equation is derived a formal uniform asymptotic solution of a system of two first order ordinary differential equation is also obtained

stochastic averaging in discrete time and its applications to extremum seeking we investigate stochastic averaging theory for locally lipschitz discrete time nonlinear systems with stochastic perturbation and its applications to convergence analysis of discrete time stochastic extremum seeking algorithms firstly by defining two average systems one is continuous time the other is discrete time we develop discrete time stochastic averaging theorem for locally lipschitz nonlinear systems with stochastic perturbation our results only need some simple and applicable conditions which are easy to verify and remove a significant restriction present in existing results global lipschitzness of the nonlinear vector field secondly we provide a discrete time stochastic extremum seeking algorithm for a static map in which measurement noise is considered and an ergodic discrete time stochastic process is used as the excitation signal finally for discrete time nonlinear dynamical systems in which the output equilibrium map has an extremum we present a discrete time stochastic extremum seeking scheme and with a singular perturbation reduction we prove the stability of the reduced system compared with classical stochastic approximation methods while the convergence that we prove is in a weaker sense the conditions of the algorithm are easy to verify and no requirements e g boundedness are imposed on the algorithm itself

uniform asymptotic solutions of second order linear ordinary differential equations with singular points ii some expansions abstract in this paper the general expansion produced in the companion paper j math anal appl 133 1988 is specialized to obtain expansions in terms of some special functions

level repulsion in constrained gaussian random matrix ensembles introducing sets of constraints we define new classes of random matrix ensembles the constrained gaussian unitary cgue and the deformed gaussian unitary dgue ensembles the latter interpolate between the gue and the cgue we derive a sufficient condition for gue type level repulsion to persist in the presence of constraints for special classes of constraints we extend this approach to the orthogonal and to the symplectic ensembles a generalized fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones we find that in the dgues level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints

spectral fluctuation properties of constrained unitary ensembles of gaussian distributed random matrices we investigate the spectral fluctuation properties of constrained ensembles of random matrices defined by the condition that a number nq of matrix elements vanish identically that condition is imposed in unitarily invariant form in the limit of large matrix dimension we show that as long as nq is smaller than a critical value at which the quadratic level repulsion of the gaussian unitary ensemble of random matrices may be destroyed all spectral fluctuation measures have the same form as for the gaussian unitary ensemble

review of the k body embedded ensembles of gaussian random matrices the embedded ensembles were introduced by mon and french 1975 ann phys ny 95 90 as physically more plausible stochastic models of many body systems governed by one and two body interactions than provided by standard random matrix theory we review several approaches aimed at determining the spectral density the spectral fluctuation properties and the ergodic properties of these ensembles moments methods numerical simulations the replica trick the eigenvector decomposition of the matrix of second moments and supersymmetry the binary correlation approximation and the study of correlations between matrix elements

remarks on adaptive stabilization of first order non linear systems abstract adaptive stabilization of first order non linear systems of the form x x f x ug x are studied for the case where g x 0 one necessary and two sufficient conditions for adaptive stabilization are given for the case of unknown sign of the gain a sufficient condition is givem simulation results are presented