Approximate finite-dimensional filtering for polynomial states over polynomial observations

In this article, the mean-square filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, this article deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third-order sensor filtering problem for a quadratic state, assuming a conditionally Gaussian initial condition for the extended third-order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.     

Root-mean-square filtering of the state of polynomial stochastic systems with multiplicative noise

Some results obtained by the present author in the field of designing the finitedimensional root-mean-square filters for stochastic systems with polynomial equations of state and multiplicative noise from the linear observations were overviewed. A procedure to derive the finite-dimensional system of approximate filtering equations for a polynomial arbitrary-order equation of state was presented. The closed system of filtering equations for the root-mean-square estimate and covariance matrix error was deduced explicitly for special cases of linear and quadratic coefficients of drift and diffusion in the equation of state. For linear stochastic systems with unknown parameters, the problem of joint root-mean-square state filtering and identification of the parameters from linear observations was considered in the Appendix.     

Optimal filtering for linear state delay systems

In this note, the optimal filtering problem for linear systems with state delay over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the error variance consists of a set of equations, whose number is specified by the ratio between the current filtering horizon and the delay value in the state equation and increases as the filtering horizon tends to infinity. In the example, performance of the designed optimal filter for linear systems with state delay is verified against the best Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time-delay systems.     

Discrete-time filtering for nonlinear polynomial systems over linear observations

This paper designs a discrete-time filter for nonlinear polynomial systems driven by additive white Gaussian noises over linear observations. The solution is obtained by computing the time-update and measurement-update equations for the state estimate and the error covariance matrix. A closed form of this filter is obtained by expressing the conditional expectations of polynomial terms as functions of the estimate and the error covariance. As a particular case, a third-degree polynomial is considered to obtain the finite-dimensional filtering equations. Numerical simulations are performed for a third-degree polynomial system and an induction motor model. Performance of the designed filter is compared with the extended Kalman one to verify its effectiveness.     

Optimal filtering for linear systems with state and observation delays

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Discrete-time optimal control for stochastic nonlinear polynomial systems

This paper presents a solution to the discrete-time optimal control problem for stochastic nonlinear polynomial systems over linear observations and a quadratic criterion. The solution is obtained in two steps: the optimal control algorithm is developed for nonlinear polynomial systems by considering complete information when generating a control law. Then, the state estimate equations for discrete-time stochastic nonlinear polynomial system over linear observations are employed. The closed-form solution is finally obtained substituting the state estimates into the obtained control law. The designed optimal control algorithm can be applied to both distributed and lumped systems. To show effectiveness of the proposed controller, an illustrative example is presented for a second degree polynomial system. The obtained results are compared to the optimal control for the linearized system.     

Optimal linear filtering over observations with multiple delays

In this paper, the optimal filtering problem for a linear system over observations with multiple delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and its variance. As a result, the optimal filtering equations similar to the traditional Kalman–Bucy ones are obtained in the form dual to the Smith predictor, commonly used for robust control design in time‐delay systems. In the example, the obtained optimal filter over observations with multiple delays is verified for a sample system and compared with the best Kalman–Bucy filter available for delayed measurements. Copyright © 2004 John Wiley & Sons, Ltd.     

Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

This paper addresses the mean-square finite-dimensional filtering problem for polynomial system states with both, Gaussian and Poisson, white noises over linear observations. A constructive procedure is established to design the mean-square filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial system. The theoretical result is complemented with an illustrative example verifying performance of the designed filter.     

Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances

This article presents the optimal quadratic-Gaussian controller for uncertain stochastic polynomial systems with unknown coefficients and matched deterministic disturbances over linear observations and a quadratic criterion. The optimal closed-form controller equations are obtained through the separation principle, whose applicability to the considered problem is substantiated. As intermediate results, this article gives closed-form solutions of the optimal regulator, controller and identifier problems for stochastic polynomial systems with linear control input and a quadratic criterion. The original problem for uncertain stochastic polynomial systems with matched deterministic disturbances is solved using the integral sliding mode algorithm. Performance of the obtained optimal controller is verified in the illustrative example against the conventional quadratic-Gaussian controller that is optimal for stochastic polynomial systems with known parameters and without deterministic disturbances. Simulation graphs demonstrating overall performance and computational accuracy of the designed optimal controller are included.     

Polynomial filtering of discrete-time stochastic linear systemswith multiplicative state noise

The problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied. For such models, a technique of state augmentation is used, leading to the definition of a general polynomial filter. The theory is developed for time-varying systems with nonstationary and non-Gaussian noises. Moreover, the steady-state polynomial filter for stationary systems is also studied. Numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques     

Discrete-time controller for stochastic nonlinear polynomial systems with Poisson noises

This paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.     

Optimal filtering for linear systems with state and multiple observation delays

This article solves the optimal filtering problem for linear systems with state and multiple observation delays. The optimal estimate equation similar to the traditional Kalman–Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed optimal filter is compared to the traditional Kalman–Bucy filter.     

Solution in the z-domain to the discrete-time stationary Kalman filtering problem for systems with perfect measurements

The discrete-time stationary Kalman filtering problem is solved in thez-domain for general right invertible linear systems whose measurements are all noise free. A simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases. It provides an optimal solution also for systems with uniform and nonuniform time delays. The minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle, finite and infinite, and the excess of the number of inputs over the number of measurements are investigated.     

Local linearization filters for non-linear continuous-discrete state space models with multiplicative noise

In this paper, the local linearization method for the approximate computation of the prediction and filtering estimates of continuous-discrete state space models is extended to the general case of non-linear non-autonomous models with multiplicative noise. The approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a local linearization of both the non-linear state equation and the non-linear measurement equation. In addition, the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained, and an algorithm for their numerical computation is also given. The performance of the LL filters is illustrated by mean of numerical experiments.     

Quantised polynomial filtering for nonlinear systems with missing measurements

This paper is concerned with the polynomial filtering problem for a class of nonlinear systems with quantisations and missing measurements. The nonlinear functions are approximated with polynomials of a chosen degree and the approximation errors are described as low-order polynomial terms with norm-bounded coefficients. The transmitted outputs are quantised by a logarithmic quantiser and are also subject to randomly missing measurements governed by a Bernoulli distributed sequence taking values on 0 or 1. Dedicated efforts are made to derive an upper bound of the filtering error covariance in the simultaneous presence of the polynomial approximation errors, the quantisations as well as the missing measurements at each time instant. Such an upper bound is then minimised through designing a suitable filter gain by solving a set of matrix equations. The filter design algorithm is recursive and therefore applicable for online computation. An illustrative example is exploited to show the effectiveness of the proposed algorithm.     

Linear quadratic regulation for discrete-time systems with state delays and multiplicative noise

In this paper, the linear quadratic regulation problem for discrete-time systems with state delays and multiplicative noise is considered. The necessary and sufficient condition for the problem admitting a unique solution is given. Under this condition, the optimal feedback control and the optimal cost are presented via a set of coupled difference equations. Our approach is based on the maximum principle. The key technique is to establish relations between the costate and the state.     

Design and analysis of discrete-time robust Kalman filters

In this paper, the problem of finite and infinite horizon robust Kalman filtering for uncertain discrete-time systems is studied. The system under consideration is subject to time-varying norm-bounded parameter uncertainty in both the state and output matrices. The problem addressed is the design of linear filters having an error variance with an optimized guaranteed upper bound for any allowed uncertainty. A novel technique is developed for robust filter design. This technique gives necessary and sufficient conditions to the design of robust quadratic filters over finite and infinite horizon in terms of a pair of parameterized Riccati equations. Feasibility and convergence properties of the robust quadratic filters are also analyzed.     

Kalman filtering in extended noise environments

This note introduces an extended environment for Kalman filtering that considers also the presence of additive noise on input observations in order to solve the problem of optimal (minimal variance) estimation of noise-corrupted input and output sequences. This environment includes as subcases both errors-in-variables filtering (optimal estimate of inputs and outputs from noisy observations) and traditional Kalman filtering (optimal estimate of state and output in presence of state and output noise). A Monte Carlo simulation shows that the performance of this extended filtering technique leads to the expected minimal variance estimates.     

线性广义系统的最优和鲁棒满阶平滑器

对于线性离散随机广义系统,利用增广状态方法将平滑器问题转化为增广状态的滤波器问题.基于极大似然线性估计准则,提出了最优的满阶平滑器,其中增广状态滤波器的误差方差阵满足广义Riccati方程.当线性离散广义系统的过程噪声和观测噪声的方差不确定时,基于极大极小鲁棒设计原理和最优满阶平滑算法,得到了鲁棒满阶平滑器.应用动态误差方差分析方法证明了其鲁棒性,即鲁棒平滑误差方差阵存在一个上界方差矩阵.数值仿真例子验证了其有效性和正确性.