discrete boundary conditions for problems with interface a discrete boundary condition on an artificial boundary is obtained for some second order elliptic boundary value problems with interface the problem is reduced to a problem without singularities and the standard finite element method is used to solve this problem some numerical examples are given

on the convergence of least squares estimates in white noise the problem of convergence of least squares ls estimates in a stochastic linear regression model with white noise is considered it is well known that if the parameter estimates are known to converge the convergence analysis for many adaptive systems can be rendered considerably less arduous for an important case where the regression vector is a measurable function of the observations and the noise is gaussian it has been shown by using a bayesian embedding argument that the ls estimates converge almost surely for almost all true parameters in the parameter space except for a zero measure set however nothing can be said about a particular given system which is usually the objective it has long been conjectured that such a bad zero measure set in the parameter space does not actually exist a conclusive answer to this important question is provided and it is shown that the set can indeed exist this then shows that to provide conclusive convergence results for stochastic adaptive systems it is necessary to resort to a sample pathwise analysis instead of the bayesian embedding approach

convergence rate for rls based direct self tuning minimum variance regulation of armax minimum phase plants an upper bound for the error convergence rate of recursive least squares rls based direct self tuning minimum variance dstmv regulation of minimum phase multiple time delay autoregressive moving average with exogenous input armax plant is derived the rls algorithm used is k interlaced with k being the plant time delay the bound is derived for known fixed b sub 0 by extending a previously proposed methodology the bound provides a joint explanation of dstmv regulation stability and parameter estimate convergence the paper demonstrates that self tuning is based on convergence properties of rls as well as on the excitation quality of plant white noise which generates via controller feedback the plant input

error and stability analysis of boundary methods for elliptic problems with interfaces in boundary methods piecewise particular solutions are employed to solve a given elliptic equation within subdomains of some region of interest a boundary approximation is then obtained by satisfying the interior and exterior boundary conditions in a least squares sense we examine convergence derive error norm bounds for approximate solutions and conduct a stability analysis of the associated algebraic problem the aim of this analysis is to help choosing good partitions of subdomains

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

the numerical solutions of interface problems by infinite element method in this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems furthermore approximations of stress intensity factors are given the infinite element method may be considered as a certain scheme of mesh refinement but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently and that an approximate solution which has a singularity at the singular point can be also obtained

parameter convergence and uniqueness in nonminimal parameterizations for multivariable adaptive control the issue of parameter convergence in multivariable adaptive control is addressed in a general framework parameter convergence is guaranteed if a certain design identity has a unique solution and if the inputs satisfy persistency of excitation conditions the uniqueness of the solution of the design identity can be obtained in general by using parameterizations that although nonminimal are structured so as to guarantee uniqueness this concept is illustrated with a direct adaptive pole placement algorithm which is modified to guarantee uniqueness and it is shown how the results can be used to establish stability and convergence properties of the algorithm

asymptotically efficient parameter estimation using quantized output observations this paper studies identification of systems in which only quantized output observations are available an identification algorithm for system gains is introduced that employs empirical measures from multiple sensor thresholds and optimizes their convex combinations strong convergence of the algorithm is first derived the algorithm is then extended to a scenario of system identification with communication constraints in which the sensor output is transmitted through a noisy communication channel and observed after transmission the main results of this paper demonstrate that these algorithms achieve the cramer rao lower bounds asymptotically and hence are asymptotically efficient algorithms furthermore under some mild regularity conditions these optimal algorithms achieve error bounds that approach optimal error bounds of linear sensors when the number of thresholds becomes large these results are further extended to finite impulse response and rational transfer function models when the inputs are designed to be periodic and full rank

the method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities babuska and oh have introduced a new approach called the method of auxiliary mapping mam to deal with elliptic boundary value problems with singularities they showed that for the laplace equation with corner singularities in the context of the p version of the finite element method mam yielded an exponential rate of convergence at virtually no extra cost in this paper those results are extended by showing the exponential convergence of mam for homogeneous laplace equations with boundary singularities and helmholtz equations with both corner and boundary singularities in addition a convergence result is developed for mam as applied to the h p version of the finite element method to clarify the power of mam a series of benchmark runs are made for three examples using our implementation mapfem comparisons are made with two h version finite element codes pltmg6 and fesop both of which use adaptive meshes the finite difference code ellpack and the recent singular element code isbfm the examples include the well known motz problem and both homogeneous and nonhomogeneous helmholtz equations

a robust adaptive minimum variance controller this paper addresses the twin questions of performance and robustness of an adaptive controller for single input single output linear stochastic systems the authors present an adaptive controller that has the following properties 1 attaining optimal regulation and tracking in the ideal minimum phase known upper bound on system order known sign and lower bound for the leading coefficient b 0 positive real condition on noise case and self tuning in a cesaro sense to a minimum variance regulator in the case of pure regulation 2 providing mean square stability when the system is of minimum phase with known upper bound on order but not necessarily satisfying a positive real condition on the noise 3 providing mean square stability when the system is in a graph topological neighborhood of computable size of an ideal plant as in 1 and the statistical properties of the disturbance are violated 4 continuing to stabilize the system when the adaptation gain is prevented from vanishing

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

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

high order finite difference schemes for elliptic problems with intersecting interfaces in this work we derive high order second and fourth of convergence but with first and third order local truncation error respectively compact finite difference schemes for elliptic equations with intersecting interfaces the approach uses the differential equation and the jump interface relations as additional identities which can be differentiated to eliminate higher order local truncation errors numerical experiments are carried out to demonstrate the high order accuracy and to show that our method is effective to sharp contrast in the diffusion coefficients of the problems

theoretical and numerical analysis on a thermo elastic system with discontinuities a second order accurate numerical scheme is proposed for a thermo elastic system which models a bar made of two distinct materials the physical parameters involved may be discontinuous across the joint of the two materials where there might be also singular heat and or force sources the solution components the temperature and the displacement may change rapidly across the joint by transforming the system into a different one time marching schemes can be used for the new system which is well posed the immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources the proposed numerical method can fit both explicit and implicit formulation for the implicit version a stable and fast prediction correction scheme is also developed convergence analysis shows that our method is second order accurate at all grid points in spite of the discontinuities across the interface numerical experiments are performed to support the theoretical analysis in this paper

on consistency of bayes estimates in a certainty equivalence adaptive system the authors analyze a control problem with data generated by the linear regression model where intercept and slope coefficients are unknown they propose a certainty equivalence control rule based on bayes estimates of the intercept and slope coefficients it is shown that the control rule converges to the optimal control rule which requires complete knowledge of intercept and slope coefficients furthermore under the proposed control rule if the total control cost tends to infinity they show that the bayes estimates for slope and intercept are consistent

the simplified hybrid combined methods for laplace s equation with singularities abstract in li and liang 1983 the simplified hybrid combined method is presented for combining the ritz galerkin method and the finite element method in this paper we will apply this method to solve singularity problems of laplace s equation error bounds and stability analyses will be provided while taking into account the integration approximation along the coupling boundary a significant coupling relation between the ritz galerkin and the finite element method has been found for the laplace equation with singularities an optimal rate of convergence has also been achieved numerical experiments have been carried out for solving the benchmark problem motz s problem to verify the theoretical results

why you cannot transform your way out of trouble for small counts summary while data transformation is a common strategy to satisfy linear modeling assumptions a theoretical result is used to show that transformation cannot reasonably be expected to stabilize variances for small counts under broad assumptions as counts get smaller it is shown that the variance becomes proportional to the mean under monotonic transformations g that satisfy g 0 0 excepting a few pathological cases a suggested rule of thumb is that if many predicted counts are less than one then data transformation cannot reasonably be expected to stabilize variances even for a well chosen transformation this result has clear implications for the analysis of counts as often implemented in the applied sciences but particularly for multivariate analysis in ecology multivariate discrete data are often collected in ecology typically with a large proportion of zeros and it is currently widespread to use methods of analysis that do not account for differences in variance across observations nor across responses simulations demonstrate that failure to account for the mean variance relationship can have particularly severe consequences in this context and also in the univariate context if the sampling design is unbalanced

brief paper stochastic adaptive control using a modified least squares algorithm recent papers on stochastic adaptive control have established global convergence for algorithms using a stochastic approximation iteration however to date global convergence has not been established for algorithms incorporating a least squares iteration this paper establishes global convergence for a slightly modified least squares stochastic adaptive control algorithm it is shown that with probability one the algorithm will ensure that the system inputs and outputs are sample mean square bounded and the mean square output tracking error achieves its global minimum possible value for linear feedback control

brief paper consistency in least squares estimation in a previous paper the convenience of using martingale theory in the analysis of bayesian least squares estimation was demonstrated however certain restrictions had to be imposed on either the feedback structure or on the initial values for the estimation in the present paper these restrictions are removed and necessary and sufficient conditions for strong consistency in a bayesian sense are given for the gaussian white noise case without any assumptions on closed loop stability or on the feedback structure in the open loop case the poles are shown to be consistently estimated almost everywhere and in the closed loop case certain choices of control law are shown to assure consistency finally adaptive control laws are treated and implicit self tuning regulators are shown to converge to the desired control laws

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

the zero divisor problem of multivariable stochastic adaptive control abstract stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time in order that the analysis of these algorithms is valid zero divisions must be events of probability zero this property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to lebesgue measure this result is deduced from the following general result established in this paper a non constant rational function of a finite set of random variables x 1 x n is absolutely continuous with respect to lebesgue measure if the joint distribution function of x 1 x n has this property

finite time identification in unstable linear systems identification of the parameters of stable linear dynamical systems is a well studied problem in the literature both in the low and high dimensionalsettings however there are hardly any results for the unstable case especially regarding finite time bounds for this setting classical results on least squares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue unstable linear systems arise in key real applications in control theory econometrics and finance this study establishes finite time bounds for the identification error of the least squares estimates for a fairly large class of heavy tailed noise distributions and transition matrices of such systems the results relate the time length samples required for estimation to a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution to establish them appropriate concentration inequalities for random matrices and for sequences of martingale differences are leveraged

self tuning trackers we examine the problem of obtaining adaptive control laws which tune themselves to control laws minimizing the variance of the tracking error between the output of the linear armax system and a specified reference trajectory if the reference trajectory is sufficiently rich of order greater than or equal to the sum of the degrees of the control and noise polynomials in the armax system then an adaptive controller is exhibited for which the parameter estimates are strongly consistent for the linear model following problem where the trajectory to be tracked is generated as the output of a linear system it is enough for the order of sufficient richness to be greater than the degree of the noise polynomial alone further if the order of sufficient richness is even smaller as is often the case then a lower dimensional adaptive controller which does not attempt to estimate all the coefficients of the noise polynomial is self tuning

on the stability and convergence of self tuning control virtual equivalent system approach this article presents a unified understanding and judgement of the stability and convergence of a general self tuning control stc system which consists of arbitrary control strategy arbitrary parameter estimation algorithm and a deterministic stochastic linear time invariant lti plant the necessary conditions required for the global stability and convergence of a general stc system are relaxed i e the convergence of parameter estimates is removed for both deterministic and stochastic stc schemes to reach this goal virtual equivalent system ves concept and methodology is adopted with the help of ves the original nonlinear dominant nonlinear in structure problem is converted to a linear dominant linear in structure problem the results developed in this article show that stc systems are stable and convergent for the abundance of control strategies and parameter estimation algorithms which will provide great flexibility in the applications of stc

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

a nonconforming combination for solving elliptic problems with interfaces abstract nonconforming combinations are provided for solving interface problems of elliptic equations in these approaches the ritz galerkin method with particular solutions is used for the part of a solution domain where there are interface singular points and the conventional finite element method is used for the rest of the solution domain in addition admissible functions chosen are constrained to be continuous only at the element nodes on the common boundary of the subdomains error bounds are derived in the sobolev norms and numerical experiments are given for solving a model interface problem of the equation u u 0 moreover a significant coupling relation l 1 o ln h is found for interface problems by using the nonconforming combinations where l 1 is the total number of particular solutions used in the ritz galerkin method and h is the maximal boundary length of triangular elements in the finite element method

tuning of gaussian stochastic control systems a closed loop system consisting of a control system and an adaptive controller is called tuning for a specified control objective if the real system and the ideal system defined below achieve the same value for the control objective the real system is the system consisting of the unknown control system in closed loop with the adaptive controller in which the parameters of the adaptive controller have been determined by identification under feedback or in closed loop the ideal system is the system consisting of the unknown control system in closed loop with a controller in which the controller has been synthesized with knowledge of the unknown control system and such that the closed loop system satisfies the control objective both the gaussian stochastic control system with full observations and with partial observations are considered the approach to the problem is based on stochastic realization theory for gaussian systems the control objectives of minimum variance control and pole placement are also given necessary conditions for tuning are discussed

on kalman filtering for conditionally gaussian systems with random matrices abstract we consider linear stochastic systems with additive white gaussian noise with the added generality that the system matrices are random and adapted to the observation process the main result of this paper is that in order for the standard kalman filter to generate the conditional mean and conditional covariance of the conditionally gaussian distributed state it is sufficient for the random matrices to be finite with probability one at each time instant this generalizes the best previous results available to date to our knowledge which require the more stringent hypothesis that the entries of the random matrices should possess finite second moments at each time instant a significant application of the results of this paper is to the problem of recursive identification of the unknown parameters of a controlled linear stochastic system in such problems the observation matrix is typically generated by complicated nonlinear feedback as for example in adaptive control and the finiteness of the second moments is difficult if not impossible to establish while the finiteness with probability one has been established in many applications

the direct method of lines for the numerical solutions of interface problem abstract in this paper a direct method of lines is proposed for solving the interface problem on a polygonal domain numerically a suitable transformation of coordinates is introduced then the interface problem is reduced to a variational differential problem on a semi infinite strip in the new variables and furthermore the variational differentional problem is discretized respect with variable thus we obtain the semi discrete approximation of the interface problem finally the optimal error estimate of the semi discrete approximation is given and the numerical examples show that the direct method of lines is feasible and very effective for solving the interface problem numerically

adaptive control with the stochastic approximation algorithm geometry and convergence new geometric properties possessed by the sequence of parameter estimates are exhibited which yield valuable insight into the behavior of the stochastic approximation based algorithm as it is used in minimum variance adaptive control in particular these geometric properties together with certain probabilistic arguments prove that if the system does not have a reduced order minimum variance controller then the parameter estimates converge to a random multiple of the true parameter an explicit expression for the limiting parameter estimate is also available with strictly positive probability the limiting parameter estimate is not the true parameter and in some cases differs from the true parameter with probability one if the system possesses reduced order minimum variance controllers then convergence to a minimum variance controller in a cesaro sense is shown the geometry of the limit set is described sufficient conditions are also given for some of these results to hold for parameter estimation schemes other than stochastic approximation

convergence of adaptive control schemes using least squares parameter estimates the stability convergence asymptotic optimality and self tuning properties of stochastic adaptive control schemes based on least squares estimates of the unknown parameters are examined it is assumed that the additive noise is i i d and gaussian and that the true system is of minimum phase the bayesian embedding technique is used to show that the recursive least squares parameter estimates converge in general the normal equations of least squares are used to establish that all stable control law designs used in a certainty equivalent i e indirect procedure generally yield a stable adaptive control system four results are given to characterize the limiting behavior precisely a certainty equivalent self tuning regulator is shown to yield strongly consistent parameter estimates when the delay is strictly greater than one even without any excitation in the reference trajectory

adaptive lqg control of input output systems a cost biased approach in this paper we consider linear systems in input output form and introduce a new adaptive linear quadratic gaussian lqg control scheme which is shown to be self optimizing the identification algorithm incorporates a cost biasing term which favors the parameters with smaller lqg optimal cost and a second term that aims at moderating the time variability of the estimate the corresponding closed loop scheme is proven to be stable and to achieve an asymptotic lqg cost equal to the one obtained under complete knowledge of the true system self optimization the results of this paper extend in a nontrivial way previous results established along the cost biased approach in other settings

a nonconforming combined method for solving laplace s boundary value problems with singularities for solving laplace s boundary value problems with singularities a nonconforming combined approach of the ritz galerkin method and the finite element method is presented in this approach singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain in addition the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods this method is nonconforming however the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used

the problem of pole zero cancellation in transfer function identification and application to adaptive stabilization the asymptotic controllability of the identified system is a central problem in adaptive control if controllability is ascertained the analysis of even complex adaptive controllers based on multistep performance indices is drastically simplified in this paper we study the controllability issue in connection with the recursive least squares rls algorithm we show that standard rls does not generally provide models that are controllable however a variant of this method that preserves all the basic properties of the standard rls and also guarantees asymptotic controllability is introduced the algorithm can be safely used in any adaptive control system provided that the control law is able to stabilize known invariant plants