further results on least squares based adaptive minimum variance control based on the recently established results on self tuning regulators originally proposed by astrom and wittenmark this paper presents various novel and extended results on least squares based adaptive minimum variance control for linear stochastic systems these results establish self optimality self tuning property and the best possible convergence rate of the control law in a variety of situations of interest

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

the weak painleve property as a criterion for the integrability of dynamical systems we investigate the validity of the weak painleve property as an integrability criterion we present an example of a time dependent hamiltonian system which possesses a weak painleve type expansion while presenting a chaotic behavior however this system presents also critical fixed singularities the importance of the latter as far as integrability is concerned is discussed here

adaptive control of parametric nonlinear autoregressive models via a new martingale approach the purpose of this note is to investigate the stability and the optimality of the adaptive tracking for a wide class of parametric nonlinear autoregressive models via a new martingale approach several asymptotic results for the standard least squares estimator of the unknown model parameter such as a central limit theorem a law of iterated logarithm and strong laws of large numbers are also provided

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

central limit theorem and law of iterated logarithm for least squares algorithms in adaptive tracking in autoregressive adaptive tracking we prove that the least squares and the weighted least squares algorithms possess the same asymptotic properties sharing the same central limit theorem and the same law of iterated logarithm we also obtain the same asymptotic behavior and show the limitations of these results in the autoregressive with moving average framework

stochastic adaptive prediction and model reference control guo and chen 1991 have recently shown how to establish the self optimality and mean square stability of a self tuning regulator the idea allows us to proceed with the development of a more comprehensive theory of stochastic adaptive filtering control and identification in adaptive filtering we examine both indirect and noninterlaced direct schemes for prediction using both least squares and gradient parameter estimation algorithms in addition to analyzing similar direct adaptive control algorithms we propose new generalized certainty equivalence adaptive model reference control laws with simultaneous disturbance rejection we also establish that the parameters converge to the null space of a certain matrix from this one may deduce the convergence of several adaptive controllers

almost sure rate of convergence of the parameter estimates in stochastic approximation algorithm this paper presents novel results on the almost sure convergence of the parameter estimates in stochastic approximation algorithm it is proved that this rate has the same order as the best one established for the least squares algorithm although we consider the case of self tuning controllers the presented results can easily be extended to some other adaptive processes

on the one and two dimensional toda lattices and the painleve property the toda lattice and the two dimensional toda lattice 2 dtl are shown to possess a type of painleve property that is based on the use of separate singular manifolds for each dependent variable the isospectral problem for the 2 dtl found by both mikhailov and by fordy and gibbons can be simply and logically derived from this analysis some remarks are made about the connection between our work and independent work of kametaka and airhault on the relationship between the toda lattice and the second painleve transcendent

nonparametric estimation and adaptive control of functional autoregressive models this paper deals with nonparametric estimation and adaptive control of nonlinear systems of the form x n 1 f x n u n xi n 1 n in bkn where the state x n is observed f is an unknown function and the control un is chosen in order to track a given reference trajectory we estimate the function f using a nonparametric estimator and study two adaptive control laws built from this nonparametric estimator and derived from the self tuning control the first one can be used for open loop stable systems and requires an additional exciting noise the second one needs some a priori knowledge on function f but allows us to control open loop unstable systems we establish some general results on the nonparametric estimator of f like the uniform almost sure convergence over dilating sets and then prove that both adaptive control laws are asymptotically optimal in quadratic mean in addition we give a strongly consistent estimator of the covariance matrix of the unobservable white noise xi n

self convergence of weighted least squares with applications to stochastic adaptive control a recursive least squares algorithm with slowly decreasing weights for linear stochastic systems is found to have self convergence property i e it converges to a certain random vector almost surely irrespective of the control law design such algorithms enjoy almost the same nice asymptotic properties as the standard least squares this universal convergence result combined with a method of random regularization then easily can be applied to construct a self convergent and uniformly controllable estimated model and thus may enable us to form a general framework for adaptive control of possibly nonminimum phase autoregressive moving average with exogenous input armax systems as an application we give a simple solution to the well known stochastic adaptive pole placement and linear quadratic gaussian lqg control problems in the paper

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

indirect adaptive pole placement control of mimo stochastic systems self tuning results in this paper we consider indirect adaptive pole placement control appc of linear multivariable stochastic systems instead of the canonical representation often used in the literature we propose using a non minimal but otherwise uniquely identifiable pseudo canonical parameterization that is more suitable for multivariable armax model identification to identify the plant we use the weighted extended least squares wels algorithm a least squares method with slowly decreasing weights which was introduced in bercu 1995 the pole placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable we prove that with a reasonable amount of prior information the resulting appc scheme is globally stabilizing and asymptotically self tuning regardless of the degree of persistency of external excitation these results represent the most complete study of stochastic multivariable appc systems to this date

weighted estimation and tracking for branching processes with immigration for branching processes with immigration we propose an approach which allows us to consistently estimate the means m spl lambda and the variances spl sigma sup 2 b sup 2 of the offspring and immigration distributions respectively generally statistical results for branching processes are established under the well known trichotomy m 1 for example no parameters of the immigration distribution can be consistently estimated if m 1 the purpose of the paper is to obtain through the introduction of a suitable adaptive control strongly consistent estimators for all the parameters m spl lambda spl sigma sup 2 and b sup 2 without any restriction on the range of m central limit theorems and laws of iterated logarithm are also provided

convergence and logarithm laws of self tuning regulators abstract this paper starts with a survey of some recent results on least squares ls and ls based self tuning regulators str several long standing problems concerning the basic properties of ls based str such as stability optimality consistency and the best convergence rate are solved within a unified framework some previously related results are also subsumed and extended various new techniques for analysing ls based adaptive tracking systems are presented which may also be useful for analysing other adaptive control problems

on critical stability of discrete time adaptive nonlinear control in this paper we examine the global stability and instability problems for a class of discrete time adaptive nonlinear stochastic control the systems to be controlled may exhibit chaotic behavior and are assumed to be linear in unknown parameters but nonlinear in output dynamics which are characterized by a nonlinear function say f x it is found and proved that in the scalar parameter case there is a critical stability phenomenon for least squares ls based adaptive control systems to be specific let the growth rate of f x be f x o spl par x spl par sup 6 with b spl ges 0 then it is found that b 4 is a critical value for global stability i e the closed loop adaptive system is globally stable if b 4 and is unstable in general if b spl ges 4 as a consequence we find an interesting phenomenon that the linear case does not have for some ls based certainty equivalence adaptive controls even if the ls parameter estimates are strongly consistent the closed loop systems may still be unstable this paper also indicates that adaptive nonlinear stochastic control that is designed based on e g taylor expansion or weierstrass approximation for nonlinear models may not be feasible in general

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

adaptive tracking control of linear systems with binary valued observations and periodic target this technical note studies the adaptive control for linear systems with set valued observations to track a given periodic target based on the system parameters accessorial parameters with the same order as that of the tracking targets are introduced and estimated considering the system parameters are unknown and set valued observations can supply only limited information each time a two scale adaptive control algorithm is designed each control input is designed at the large time scale and lasts for a holding time small scale during which the parameter estimation algorithm is constructed from the estimate of accessorial parameters the control signal is updated at the large time scale by the certainty equivalence principle as the holding time goes to infinity the algorithm can be proved to be asymptotically efficient in a certain sense meanwhile the adaptive tracking algorithm is shown to be asymptotically optimal a numerical example is given to demonstrate the effectiveness of the algorithms and the main results obtained

brief paper a new concept of strong controllability via the schur complement for arx models in adaptive tracking we propose a new concept of strong controllability related to the schur complement of a suitable limiting matrix this new notion allows us to extend the previous convergence results associated with multidimensional arx models in stochastic adaptive tracking on the one hand we carry out a sharp analysis of the almost sure convergence for both least squares and weighted least squares algorithms on the other hand we also provide a central limit theorem and a law of iterated logarithm for these two algorithms

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

on the discrete adjoints of adaptive time stepping algorithms we investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation ad by differentiating the time step controller and the error estimator of the original algorithm reverse mode ad generates spurious adjoint derivatives of the time steps the resulting discrete adjoint models become inconsistent with the adjoint ode and yield incorrect derivatives to regain consistency one has to cancel out the contributions of the non physical derivatives in the discrete adjoint model we demonstrate that the discrete adjoint models of one step explicit adaptive algorithms such as the runge kutta schemes can be made consistent with their continuous counterparts using simple code modifications furthermore we extend the analysis to cover second order adjoint models derived through an extra forward mode differentiation of the discrete adjoint code several numerical examples support the mathematical derivations

backlund transformation and the painleve property when a differential equation possesses the painleve property it is possible for specific equations to define a backlund transformation by truncating an expansion about the singular manifold at the constant level term from the backlund transformation it is then possible to derive the lax pair modified equations and miura transformations associated with the completely integrable system under consideration in this paper completely integrable systems are considered for which backlund transformations as defined above may not be directly defined these systems are of two classes the first class consists of equations of toda lattice type e g sine gordon bullough dodd equations we find that these equations can be realized as the minus one equation of sequences of integrable systems although the backlund transformation may or may not exist for the minus one equation it is shown for specific sequences that the backlund transformation does exist for the positive equations of the sequence this in turn allows the derivation of lax pairs and the recursion operation for the entire sequence the second class of equations consists of sequences of harry dym type these equations have branch point singularities and thus do not directly possess the painleve property yet by a process similar to the uniformization of algebraic curves their solutions may be parametrically represented by meromorphic functions for specific systems this is shown to provide a natural extension of the painleve property

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

kernel density estimation and goodness of fit test in adaptive tracking we investigate the asymptotic properties of a recursive kernel density estimator of the driven noise of multivariate armax models in adaptive tracking we provide an almost sure pointwise and uniform strong law of large numbers as well as a pointwise and multivariate central limit theorem we also carry out a goodness of fit test together with some simulation experiments