the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

a polynomial approach to robust estimation of uncertain discrete time systems the robust estimation problem for uncertain discrete time systems is treated in this paper considering parametric uncertainties in the polynomial framework the estimator is the minimiser of a cost function written in a epsilon contaminated form uncertainties are not formally modelled it is shown that in the present context the proposed approach can be connected to stochastic modelling of uncertainties the optimal robust estimator is obtained by computing a spectral factorisation and by solving a single diophantine equation an example shows the efficiency of the proposed method

solution of the h sub infinity optimal linear filtering problem for discrete time systems the solution of l sub 2 minimum variance and h sub infinity estimation problems is considered using a polynomial systems approach the results for the l sub 2 filtering problem which corresponds with wiener or kalman filtering prediction are first presented in polynomial matrix form attention then turns to the solution of the h sub infinity estimation problem for scalar systems numerous examples are presented to illustrate the computational procedures the two types of estimator are appropriate to very different estimation problems and the new h sub infinity devices should be valuable in certain application areas

a fundamental mean square convergence theorem for sdes with locally lipschitz coefficients and its applications a version of the fundamental mean square convergence theorem is proved for stochastic differential equations sdes in which coefficients are allowed to grow polynomially at infinity and which satisfy a one sided lipschitz condition the theorem is illustrated on a number of particular numerical methods including a special balanced scheme and fully implicit methods the proposed special balanced scheme is explicit and its mean square order of convergence is 1 2 some numerical tests are presented

exact schrodinger wavefunctions of n coupled time dependent harmonic oscillators for n coupled time dependent harmonic oscillators with any couplings coordinates and momentum we have derived the primary invariant and the generalized lewis and riesenfeld invariant represented in terms of the classical solution matrix of hamilton s oscillator equation using the lewis and riesenfeld invariant method and exploiting the properties of the symplectic matrix we have obtained the exact schrodinger wavefunctions for the oscillators

feedforward control is dual to deconvolution a duality is demonstrated between optimal feedforward control and optimal deconvolution or input estimation these two problems are normally discussed separately in the literature but have close similarities duality between them can be demonstrated if and only if one uses general problem formulations with frequency shaped weighting in the criteria from one of the problems the dual problem can then be obtained immediately from the block diagram by reversing the directions of arrows interchanging summation points and node points and transposing all transfer function matrices this result applies for continuous and discrete time problems as well as for minimization of j g for any transfer function norms for which gt g a derivation of a polynomial solution to the frequency weighted discrete time mimo lqg feedforward control problem illustrates the use of the duality

a novel derivation methodology for polynomial lq controller design a simple derivation method for optimization of linear quadratic lq controllers is presented in the discrete time polynomial systems framework a control law variation regarded as a potential feedforward from the innovations is used orthogonality is evaluated in the frequency domain by collectively cancelling unstable poles by zeros the suggested method summarized as a three step scheme is exemplified on a disturbance measurement feedforward and an output feedback problem it is a simple more direct alternative to the completing the squares approach

an ellipsoid algorithm for probabilistic robust controller design abstract in this paper a new iterative approach to probabilistic robust controller design is presented which is applicable to any robust controller filter design problem that can be represented as an lmi feasibility problem recently a probabilistic subgradient iteration algorithm was proposed for solving lmis it transforms the initial feasibility problem to an equivalent convex optimization problem which is subsequently solved by means of an iterative algorithm while this algorithm always converges to a feasible solution in a finite number of iterations it requires that the radius of a non empty ball contained into the solution set is known a priori this rather restrictive assumption is released in this paper while retaining the convergence property given an initial ellipsoid that contains the solution set the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes all containing the solution set at each iteration a random uncertainty sample is generated with a specified probability density which parameterizes an lmi for this lmi the next minimum volume ellipsoid that contains the solution set is computed an upper bound on the maximum number of possible correction steps that can be performed by the algorithm before finding a feasible solution is derived a method for finding an initial ellipsoid containing the solution set which is necessary for initialization of the optimization is also given the proposed approach is illustrated on a real life diesel actuator benchmark model with real parametric uncertainty for which a h 2 robust state feedback controller is designed

optimal differentiation based on stochastic signal models the problem of estimating the time derivative of a signal from sampled measurements is addressed the measurements may be corrupted by colored noise a key idea is to use stochastic models of the signal to be differentiated and of the measurement noise two approaches are suggested the first is based on a continuous time stochastic process as a model of the signal the second uses a discrete time arma model of the signal and a discrete time approximation of the derivative operator digital differentiators are presented in a shift operator polynomial form they minimize the mean square estimation error and are calculated from a linear polynomial equation and a polynomial spectral factorization the three obstacles to perfect differentiation namely a finite smoothing lag measurement noise and aliasing effects due to sampling are discussed

symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies the symmetry algebra of the m dimensional quantum anisotropic harmonic oscillator hamiltonian h with commensurate frequencies is shown to be u m each eigenspace of h carries a single irreducible representation of u m however distinct eigenspaces can yield equivalent u m representations the dynamical algebra is the non compact symplectic algebra sp m r for m 3 the anisotropic hamiltonian is relevant to superdeformed nuclei

noether s theorem and accidental degeneracy it is shown that the determination of the symmetry lie algebra of a quantum system with accidental degeneracy can be obtained in principle by using noether s theorem the procedure is illustrated by considering a quantum system which represents a generalization of the degeneracies present in an anisotropic two dimensional harmonic oscillator thus besides the u 2 algebra which gives the fundamental degeneracy of a two dimensional oscillator the studied system can have an infinite set of states with the same energy characterized by an u 1 1 lie algebra

schrodinger cat states of a non stationary generalized oscillator the two mode even and odd coherent schrodinger cat states are introduced for the generalized stationary and non stationary harmonic oscillator the expected values of positions and momenta and their dispersion matrices are calculated for the schrodinger cat states the quadrature squeezing and correlation phenomenon are studied and the wigner and husimi functions are constructed explicitly for these states the photon statistics of the two mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function which is expressed in terms of hermite polynomials the photon number means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon

convergence and stability of the backward euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients abstract this paper firstly investigates convergence of the backward euler method for stochastic differential equations sdes driven by brownian motion and compound poisson process different from existing works we do not impose a globally lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super linear growth it is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on sdes allowing for possibly super linearly growing drift diffusion and jump coefficients an exact order 1 2 is recovered when further differentiability assumption is put on the coefficients furthermore the considered method is able to inherit the mean square stability of a wider class of levy noise driven sdes for all stepsizes these results are finally supported by some numerical experiments

the projected explicit ito taylor methods for stochastic differential equations under locally lipschitz conditions and polynomial growth conditions abstract although the numerical methods to stochastic differential equations with the coefficients of locally lipschitz and polynomial growth have been discussed commonly by some authors there are few works on the high strong order numerical methods in this paper the mean square convergence of the general projected explicit ito taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition suitable local lipschitz conditions and polynomial growth conditions our analysis follows the idea of stochastic c stability and stochastic b consistency after giving the selection strategy of optimal parameters we prove that the projected explicit ito taylor methods owning optimal parameters share the same mean square convergence orders with non projected ones finally two numerical experiments are presented to show the effectiveness of theoretical results

on explicit tamed milstein type scheme for stochastic differential equation with markovian switching abstract we propose a new tamed milstein type scheme for stochastic differential equation with markovian switching when drift coefficient is assumed to grow super linearly the strong rate of convergence is shown to be equal to 1 0 under mild regularity e g once differentiability requirements on drift and diffusion coefficients novel techniques are developed to tackle two fold difficulties arising due to jumps of the markov chain and the reduction of regularity requirements on the coefficients

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

noether s theorem and time dependent quantum invariants the time dependent integrals of motion linear in position and momentum operators of a quantum system are extracted from noether s theorem prescription by means of special time dependent variations of coordinates for the stationary case of the generalized two dimensional harmonic oscillator the time independent integrals of motion are shown to correspond to special bragg type symmetry properties a detailed study for the nonstationary case of this quantum system is presented the linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength they are also obtained from noether s theorem the general treatment for a multi dimensional quadratic system is indicated and it is shown that the time dependent variations that give rise to the linear invariants as conserved quantities satisfy the corresponding classical homogeneous equations of motion for the coordinates

robust wiener filtering with non parametric spectral uncertainty in this article robust least squares filtering problems are considered for non parametric multivariate spectral uncertainty defined by the so called spectral band and generalised moment constraints its major aim is to provide a basis for computing approximate solutions to worst case wiener filtering optimisation problems involving causal filters and multivariate signals it hinges upon associating upper and lower bounds on the minimum worst case performance achievable with causal filters with linear cost linear matrix inequality lc lmi constraint optimisation problems on the basis of a lagrangean duality formulation for the worst case least squares performance of a given filter upper bounds on it are obtained as the optimal values of lc lmi problems then for linearly parameterised classes of filter transfer functions a causal filter which optimises such an upper bound on worst case performance can also be obtained from an lc lmi optimisation problem to estimate the amount of conservatism incur

the structure and design of realizable decision feedback equalizers for iir channels with colored noise a simple algorithm for optimizing decision feedback equalizers dfes by minimizing the mean square error mse is presented a complex baseband channel and correct past decisions are assumed the dispersive channel may have infinite impulse response and the noise may be colored consideration is given to optimal realizable stable and finite lag smoothing forward and feedback filters in discrete time they are parameterized as recursive filters in the special case of transmission channels with finite impulse response and autoregressive noise the minimum mse can be attained with transversal feedback and forward filters in general the forward part should include a noise whitening filter the inverse noise model the finite realizations of the filters are calculated using a polynomial equation approach to the linear quadratic optimization problem the equalizer is optimized essentially by solving a system of linear equations ax b where a contains transfer function coefficients from the channel and noise model no calculation of correlations is required with this method a simple expression for the minimal mse is presented the dfe is compared to mse optimal linear recursive equalizers expressions for the equalizer in the limiting case of infinite smoothing lags are also discussed

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

wiener design of adaptation algorithms with time invariant gains a design method is presented that extends least mean squared lms adaptation of time varying parameters by including general linear time invariant filters that operate on the instantaneous gradient vector the aim is to track time varying parameters of linear regression models in situations where the regressors are stationary or have slowly time varying properties the adaptation law is optimized with respect to the steady state parameter error covariance matrix for time variations modeled as vector arima processes the design method systematically uses prior information about time varying parameters to provide filtering prediction or fixed lag smoothing estimates for arbitrary lags the method is based on a transformation of the adaptation problem into a wiener filter design problem the filter works in open loop for slow parameter variations whereas a time varying closed loop has to be considered for fast variations in the latter case the filter design is performed iteratively the general form of the solution at each iteration is obtained by a bilateral diophantine polynomial matrix equation and a spectral factorization for white gradient noise the diophantine equation has a closed form solution further structural constraints result in very simple design equations under certain model assumptions the wiener designed adaptation laws reduce to lms adaptation compared with kalman estimators the channel tracking performance becomes nearly the same in mobile radio applications whereas the complexity is in general much lower

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

paper efficient algorithm for matrix spectral factorization an algorithm is presented for the spectral factorization of polynomial or rational matrices arising in optimal control and filtering theory as well as in network theory there are two versions of the algorithm one applicable to continuous time problems the other to discrete time ones both versions are based on newton s method feature quadratic convergence and provide a significant improvement in efficiency over the existing methods

robust kalman filtering for uncertain discrete time systems with probabilistic parameters bounded within a polytope we consider the problem of finite horizon discrete time kalman filtering for systems with parametric uncertainties specifically we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density the condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over bound are derived

realizable mimo decision feedback equalizers structure and design we present and discuss the structure and design of optimum multivariable decision feedback equalizers dfes the equalizers are derived under the constraint of realizability that is causal and stable filters and finite decision delay the design is based on a known dispersive discrete time multivariable channel model with infinite impulse response the additive noise is described by a multivariate arma model equations for obtaining minimum mean square error mmse and zero forcing dfes are derived under the assumption of correct past decisions the design of a realizable mmse dfe requires the solution of a linear system of equations in the model parameters no spectral factorization is required we derive novel necessary and sufficient conditions for the existence of zero forcing dfes and point out the significance of these conditions for the design of multiuser detectors an optimal mmse dfe will contain iir filters in general simulations indicate that the performance improvement obtained with an iir dfe is reduced more than for a suboptimal fir dfe when error propagation is taken into account since the use of iir feedback filters tends to worsen the error propagation

adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations abstract we consider the use of adaptive timestepping to allow a strong explicit euler maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non negative non globally lipschitz coefficients solutions of such equations may display a tendency towards explosive growth countered by a sufficiently intense and nonlinear diffusion we construct an adaptive timestepping strategy which closely reproduces the almost sure a s asymptotic stability and instability of the equilibrium and which can ensure the positivity of solutions with arbitrarily high probability our analysis adapts the derivation of a discrete form of the ito formula from appleby et al 2009 in order to deal with the lack of independence of the wiener increments introduced by the adaptivity of the mesh we also use results on the convergence of certain martingales and semi martingales which influence the construction of our adaptive timestepping scheme in a way proposed by liu mao 2017

robust kalman filtering for uncertain discrete time systems this paper is concerned with the problem of a kalman filter design for uncertain discrete time systems the system under consideration is subjected to time varying norm bounded parameter uncertainty in both the state and output matrices the problem addressed is the design of a linear filter such that the variance of the filtering error is guaranteed to be within a certain bound for all admissible uncertainties furthermore the guaranteed cost can be optimized by appropriately searching a scaling design parameter

robust kalman filtering for uncertain systems abstract this paper studies the problem of kalman filter design for uncertain systems the system under consideration is subjected to time varying norm bounded parameter uncertainties in both the state and measurement matrices the problem we address is the design of a state estimator such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties a riccati equation approach is proposed to solve the above problem furthermore a suboptimal covariance upper bound can be computed by a convex optimization

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

optimal self tuning filtering prediction and smoothing for discrete multivariable processes an algorithm is proposed for self tuning optimal fixed lag smoothing or filtering for linear discrete time multivariable processes a z transfer function solution to the discrete multivariable estimation problem is first presented this solution involves spectral factorization of polynomial matrices and assumes knowledge of the process parameters and the noise statistics the assumption is then made that the signal generating process and noise statistics are unknown the problem is reformulated so that the model is in an innovations signal form and implicit self tuning estimation algorithms are proposed the parameters of the innovation model of the process can be estimated using an extended kalman filter or alternatively extended recursive least squares these estimated parameters are used directly in the calculation of the predicted smoothed or filtered estimates the approach is an attempt to generalize the work of hagander and wittenmark

the linear steklov method for sdes with non globally lipschitz coefficients we present an explicit numerical method for solving stochastic differential equations with non globally lipschitz coefficients a linear version of the steklov average under a split step formulation supports our new solver the linear steklov method converges strongly with a standard one half order also we present numerical evidence that the explicit linear steklov reproduces almost surely stability solutions with high accuracy for diverse application models even for stochastic differential systems with super linear diffusion coefficients

robust filtering and prediction for linear systems with uncertain dynamics a game theoretic approach we examine the existence and behavior of game theoretic solutions for robust linear filters and predictors our basic uncertainty class includes m th order time varying discrete time systems with uncertain dynamics uncertain initial state covariance and uncertain nonstationary input and observation noise covariance our results include recursive kalman filter predictor realizations for the resulting robust procedures our approach is based on saddle point theory we emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique in this paper we highlight the role such distributions with finite support play in these decision models in particular we demonstrate that in these decision models the least favorable prior distribution is always discrete

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

heisenberg picture approach to the invariants and the exact quantum motions for coupled parametric oscillators for n coupled generalized time dependent oscillators primary invariants and a generalized invariant are found in terms of classical solutions exact quantum motions satisfying the heisenberg equation of motion are also found for number states and coherent states of the generalized invariant the uncertainties in positions and momenta are obtained

the ermakov lewis invariants for coupled linear oscillators we consider n coupled linear oscillators with time dependent coefficients an exact complex amplitude real phase decomposition of the oscillatory motion is constructed this decomposition is further used to derive n exact constants of motion which generalize the so called ermakov lewis invariant of a single oscillator in the floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing floquet solutions

quantifying the error in estimated transfer functions with application to model order selection previous results on estimating errors or error bounds on identified transfer functions have relied on prior assumptions about the noise and the unmodeled dynamics this prior information took the form of parameterized bounding functions or parameterized probability density functions in the time or frequency domain with known parameters it is shown that the parameters that quantify this prior information can themselves be estimated from the data using a maximum likelihood technique this significantly reduces the prior information required to estimate transfer function error bounds the authors illustrate the usefulness of the method with a number of simulation examples how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and under modeling is shown another simulation study compares the method to akaike s well known fpe and aic criteria

on tamed euler approximations of sdes driven by levy noise with applications to delay equations we extend the taming techniques for explicit euler approximations of stochastic differential equations driven by levy noise with superlinearly growing drift coefficients strong convergence results are presented for the case of locally lipschitz coefficients moreover rate of convergence results are obtained in agreement with classical literature when the local lipschitz continuity assumptions are replaced by global assumptions and in addition the drift coefficients satisfy polynomial lipschitz continuity finally we further extend these techniques to the case of delay equations

weiner and kalman filters for systems with random parameters a linear stationary optimal filtering problem is considered in which the plant dynamics and noise covariances are incompletely known unknown plant parameters in the plant model such as gains and time constants are treated as random variables with specified means and variances generalized wiener and kalman bucy filters are derived on the basis of transfer function matrix or state space representations of the plant respectively an application of the generalized filter to the linear quadratic optimal control of plants with unknown disturbances is also described and a certainty equivalence principle is shown to apply

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

note on invariants for uncoupled ermakov systems several invariants for an ermakov system consisting of a parametric oscillator and a milne oscillator in one dimension are derived this system appears in amplitude phase analysis of parametric oscillator solutions first and second order invariants for such a system are derived using the wronskian relations of a fundamental pair of parametric oscillator solutions in this way wronskian invariants of the parametric oscillator system alone imply ermakov lewis type invariants for the combined ermakov system as well as invariants for the milne oscillator alone

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

a note on strong approximation of sdes with smooth coefficients that have at most linearly growing derivatives abstract in the literature on strong approximation of sdes polynomial error rate results for numerical schemes are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition in the present article we show that the latter condition is far from being sufficient for polynomial error rates we construct an sde with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can at best be approximated with a logarithmic rate whatever numerical method based on observations of the driving brownian motion at finitely many fixed times is used most interestingly it furthermore turns out that using a method that properly adjusts the number of evaluations of the driving brownian motion to its actual path the latter sde can in fact be approximated with polynomial rate 1 in terms of the average number of evaluations that are used to the best of our knowledge this is only the second example in the literature of an sde for which there exist adaptive methods that perform superior to non adaptive ones with respect to the convergence rate

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

robust wiener filters for random signals in correlated noise corresp minimax robust wiener filtering is considered for the case in which the signal and noise spectral density matrix is not completely specified results are obtained for spectral density matrix classes which are defined by upper and lower bounds on the components of the matrix these results form an extension of earlier results on robust wiener filtering for the case of uncorrelated signals and noise

convergence rate and stability of the truncated euler maruyama method for stochastic differential equations abstract recently mao 2015 developed a new explicit method called the truncated euler maruyama em method for the nonlinear sde and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition in his another follow up paper mao 2016 he discussed the rates of l q convergence of the truncated em method for q 2 and showed that the order of l q convergence can be arbitrarily close to q 2 under some additional conditions however there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated em method would be inapplicable the key aim of this paper is to establish the convergence rate without these restrictions the other aim is to study the stability of the truncated em method the advantages of our new results will be highlighted by the comparisons with the results in mao 2015 2016 as well as others on the tamed em and implicit methods

cautious h2 optimal control using uncertain markov parameters identified in closed loop abstract in this paper we present a new cautious h 2 optimal design approach based on the noisy and biased markov parameters identified from a finite number of input and output samples in a closed loop plant this approach not only links closed loop subspace identification with optimal control but also directly evaluates parametric uncertainties on the identified markov parameters neither a state space model nor its stochastic uncertainty has to be realized the effects of the parametric uncertainties on the output predictor and a quadratic cost function are explicitly analyzed an h 2 optimal control problem is formulated as a min max problem of the expectation of the cost function with respect to the stochastic noise in the identified parameters analytic solution to this problem is derived in a closed form which avoids computing the empirical mean of the quadratic cost as required by randomized algorithms an extension of the cautious design to the solution associated with an arbitrary probability is also proposed the solutions hence lead to easily implementable control laws robust to the uncertainties in the identified markov parameters from a closed loop plant

polynomial systems approach to optimal linear filtering and prediction the solution of the optimal linear estimation problem is considered using a polynomial matrix description for the discrete system the filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and new insights into the disturbance rejection properties of the filter are obtained the plant or signal model can be stable or unstable and allowance is made for both control and disturbance input subsystems and white and coloured measurement noise or an output disturbance subsystem the model structure was determined by the needs of several industrial filtering problems the polynomial form of filter may easily be included in a self tuning algorithm and a simple adaptive estimator is described

h sub 2 optimal linear robust sampled data filtering design using polynomial approach a new frequency domain approach to robust multi input multi output mimo linear filter design for sampled data systems is presented the system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set the optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties the design problem is cast in the context of h sub 2 via the polynomial matrix representation of systems with norm bounded unstructured uncertainties the sampled data mix of continuous and discrete time systems is handled by means of a lifting technique however it does not increase the dimensionality or alter the computational cost of the solution the setup adopted allows dealing with several filtering problems a simple deconvolution example illustrates the procedure

on stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions in a recent article jentzen et al 2016 commun math sci 14 1477 1500 doi 10 4310 cms 2016 v14 n6 a1 it has been established that for every arbitrarily slow convergence speed and every nat

mean square convergence of explicit two step methods for highly nonlinear stochastic differential equations abstract in this paper we propose the projected two step euler maruyama method and the projected two step milstein method for highly nonlinear stochastic differential equations under a global monotonicity condition we first examine the strong convergence in mean square sense for these two explicit schemes based on the notions of stochastic stability and b consistency for two step methods we prove that the convergence rates of the projected two step euler maruyama method and the projected two step milstein method are 1 2 and 1 respectively in particular our results can be applied to equations with super linearly growing drift and diffusion coefficients finally we numerically verify the optimal mean square convergence orders of these two schemes by a series of examples