Manoeuvring target tracking with coordinated-turn motion using stochastic non-linear filter

Allowing for perturbations in speed and turn rate of a target moving in a coordinated turn obeys a non-linear stochastic differential equation. Existing algorithms for coordinated turn tracking avoid this problem by ignoring perturbations in the continuous time model and adding process noise only after discretisation. The dynamic model used here adds small perturbations, modelled as independent Brownian motion processes, to the speed and turn rate. The target state is to be recursively estimated from noisy discrete-time measurements of the target's range and bearing. In particular, this paper examines the effect of the perturbations in speed and turn rate on the coordinated turn motion of the aircraft, and subsequently the stochastic algorithm is developed by deriving the evolutions of conditional means and variances for estimating the state of the aircraft. By linearizing the stochastic differential equations about the mean of the state vector using first-order approximation, the mean trajectory of the resulting first-order approximated stochastic differential model does not preserve the perturbation effect felt by the moving target; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the perturbed model is examined on the basis of the second-order approximations of the system non-linearity. The theory of the non-linear filter of this paper is developed using the Kolmogorov forward equation ‘between the observation’ and a functional difference equation for the conditional probability density ‘at the observation’. The effectiveness of the second-order non-linear filter is examined on the basis of its ability to preserve perturbation effect felt by the aircraft. The Kolmogorov forward equation, however, is not appropriate for numerical simulations, since it is the equation for the evolution of the conditional probability density. Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions, velocities and turn rate of the manoeuvring aircraft. Even these equations are not appropriate for the numerical simulations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher-order moments. Hence we consider the approximations to these moment evolution equations. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.     

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.     

Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering†

Using Bayes' theorem the conditional mean of the posterior probability density function is estimated via Monte Carlo techniques. Multi-stage, non-linear filtering requires the solution of high dimensional integrals. The new feature of the approach presented is that a combination of analytical and numerical methods yields a variance reduction which can also be interpreted as an accuracy improvement of approximate non-linear filter equations. Theorems are derived to prove zero sampling variance for the linear Gaussian case and experimental results indicate that the proposed estimators are feasible in non-linear situations.     

A second-order method for state estimation of non-linear dynamical systems†

The paper is concerned with further elaboration of the concept of statistical approximation of a given nonlinear function and its application to the problem of state estimation of a non-linear noisy dynamical system from noise-corrupted observations. It is shown that under the assumption that the conditional probability density of the state variable is gaussian, it is possible to approximate a non-linear function of the state by a polynomial of arbitrary order. Using the second-order approximation, an algorithm is then developed for the problem of state estimation. Results obtained from this algorithm are compared with those obtained from a second-order algorithm based on Taylor's series expansion of the non-linear function.     

Limited memory optimal filtering

Linear and nonlinear optimal filters with limited memory length are developed. The filter output is the conditional probability density function and, in the linear Gaussian case, is the conditional mean and covariance matrix where the conditioning is only on a fixed amount of most recent data. This is related to maximum-likelihood least-squares estimation. These filters have application in problems where standard filters diverge due to dynamical model errors. This is demonstrated via numerical simulations.     

Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements

This paper presents the derivation of the dynamical equations of a second-order filter which estimates the states of a non-linear plant on the basis of discrete noisy measurements. The filter equations contain terms involving the second-order partial derivatives of the plant and output equations. Simulation results are presented which yield a comparison of the performance of the first-versus the second-order filter when applied to a nonlinear third-order system. The results indicate that the inclusion of second-order terms can markedly improve the filter performance.     

Performance assessment,diagnosis, and optimal selection of non-linear state filters

Non-linear state filters of different approximations and capabilities allow for real-time estimation of unmeasured states in non-linear stochastic processes. It is well known that the performance of non-linear filters depends on the underlying numerical and statistical approximations used in their design. Despite the theoretical and practical interest in evaluating the performance of non-linear filtering methods, it remains one of the most complex problems in the area of state estimation. We propose the use of posterior Cramér–Rao lower bound (PCRLB) or mean square error (MSE) inequality as a filtering performance benchmark. Using the PCRLB inequality, we develop assessment and diagnosis tools for monitoring and evaluating the performance of non-linear filters. Using the PCRLB inequality-based performance assessment tool, an optimal non-linear filter switching strategy is proposed for state estimation in general non-linear systems. The non-linear filter switching strategy is an optimal performance strategy, which maintains high filtering performance under all operating conditions. The complex, high dimensional integrals involved in the computation of the PCRLB inequality-based non-linear filter assessment and diagnosis tools are approximated using sequential Monte-Carlo (SMC) methods. The utility and efficacy of the developed tools are illustrated through a numerical example.     

On the iterated forms of Kalman filters using statistical linearization

The extended Kalman filter (EKF) is a suboptimal estimator of the conditional mean and covariance for nonlinear state estimation. It is based on first order Taylor series approximation of nonlinear state functions. The unscented Kalman filter (UKF) and the ensemble Kalman filter (EnKF) are suboptimal estimators that are termed as Jacobian free because they do not require the existence of the Jacobian of the nonlinearity. The iterated form of EKF is an estimator of the conditional mode that employs an approximate Newton–Raphson iterative scheme to solve the maximization of the conditional probability density function. In this paper, the iterated forms of UKF and EnKF are presented that perform Newton–Raphson iteration without explicitly differentiating the nonlinear functions. The use of statistical linearization in iterated UKF and EnKF is a nondifferentiable optimization method when the measurement function is nonsmooth or discontinuous. All three iterated forms can be shown to be conditional mean estimators after the first iteration. A simple numerical example involving continuous and discontinuous measurment functions is included to evaluate the performance of the algorithms for the estimation of conditional mean, covariance and mode. A batch reactor simulation is shown for estimating both the states and unknown parameters.     

On the state estimation for non-linear dynamic systems †

The object of this paper is to present an approximate technique for state estimation of non-linear dynamical systems under noisy observations. The conditional cumulant is introduced, by which the conditional probability density can be characterized. A set of dynamical equations satisfied by conditional cumulants is derived, and an approximate method is proposed for computing the cumulants. The relation of the cumulant method to the stochastic linearization technique is also discussed. Finally the state estimation problem for linear stochaatic system with state-dependent disturbance is solved to illustrate the use of the Gaussian approximation.     

Non-linear filtering of counting processes driven by Ornstein-Uhlenbeck processes

The intensity of the doubly stochastic Poisson process (DSPP) considered in this paper is a linear function of a first-order Gauss-Markov process x ₁, (Ornstein-Uhlenbeck process).

By observing a DSPP realization and by analysing the conditional characteristic function of x ₁, we intend to find a non-linear recursive filter that gives an estimation of the intensity.

The expression of the centred conditional moments up to any order is established recursively, and a practical numerical algorithm is developed on the basis of a suboptimal non-linear filter. Consideration of the centred odd moments is also justified. The results of the numerical simulations are presented and enable a comparison to be made between the behaviour of the suboptimal non-linear filter and that of an adapted linear filter. The number of centred conditional moments to be retained in the formulation of the suboptimal non-linear filter is discussed.

Finally, numerical simulation results are given and commented on.     

Recursive Filtering for Systems with Small but Non-negligible Non-linearities†

Linear estimation theory has been applied extensively to non-linear systems by assuming that perturbations from a reference solution can be described by linear equutious. As long as the second-order (and higher) terms in the perturbation, equations me negligible, linear estimation techniques have been found to yield satisfactory response. Many examples have been encountered in which the linear theory is not satisfactory, however, and it is to this situation that attention is directed here. Time diserete systems in which the second-order effects are small but non-negligible are considered. Recursion relations for the conditional mean and covarianoo are developed. While these relations yield approximations to the true values of these moments, they are superior to the approximations provided by applying linear theory to a non-linear system. Some results for a simple system are presented in which the response from linear and non-linear filters is compared.     

Series PID controller tuning based on the SIMC rule and signal filtering

The SIMC (Simple control) rule, proposed by Skogestad, is ineffective for a class of processes with oscillatory dynamics and processes defined by transfer functions obtained as a result of ideal decoupling of multiple-input multiple-output systems. For this class of stable processes it is proposed to apply a higher-order filtering to the open-loop process step response and to approximate the filtered step response with stable SOPDT models. These models are used to obtain a high performance/robustness tradeoff by the ideal series PID controllers, tuned by the SIMC rule, with the higher-order filter in the feedback loop. Parallel PID controllers, with higher-order noise filters, tuned by applying exact process frequency response and optimization under constraints on the robustness and sensitivity to measurement noise, are used to demonstrate merits of the proposed simple design and tuning of the series PID controller. Experimental results on a mechanical laboratory plant are presented in Appendix.     

Error analysis of solutions of second-order boundary value problems obtained with residual correction method

Based on the maximum principle of differential equations and with the aid of asymptotic iteration technique, this paper tries to establish monotonic relation of second-order obstacle boundary value problems with their approximate solutions to eventually obtain the upper and lower approximate solutions of the exact solution. To obtain numerical solutions, the cubic spline approximation method is applied to discretize equations, and then according to the ‘residual correction method’ proposed in this paper, residual correction values are added into discretized grid points to translate once complex inequalities’ constraint mathematical programming problems into simple equational iteration problems. The numerical results also show that such method has the characteristic of correcting residual values to symmetrical values for such problems, as a result, the mean approximate solutions obtained even with a considerably small quantity of grid points still quite approximate the exact solution. Furthermore, the error range of approximate solutions can be identified very easily by using the obtained upper and lower approximate solutions, even if the exact solution is unknown.     

模糊加权中值滤波器

本文介绍了一种模糊加权中值滤波器，该滤波器由模糊布尔函数和滤波加权确定。本文用S型函数逼近模糊布尔函数。此外，用模糊理论领域中使用的S型函数逼近所滤波的加权。模糊加权中值滤波器只由4个参数确定。所提出的滤波在均方误差准则下能够由最小均方算法导出。图像复原的实验结果表明，本文介绍的模糊加权中值滤波方法既能去除脉冲噪声和平滑高斯噪声，又能同时有效地保持边缘和图像细节，漠糊加权中值滤波器明显优于加权中值滤波器，也优于Wiener滤波器。     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstract second-order infinite-dimensional dynamical systems is considered. It is proved using the frequency-domain method, that approximate controllability of second-order system can be verified by the approximate controllability conditions for the corresponding simplified first-order system. General results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends earlier results on approximate controllability of second-order abstract dynamical systems.     

Adaptive neural network output feedback control of stochastic nonlinear systems with dynamical uncertainties

In this paper, a robust adaptive neural network (NN) backstepping output feedback control approach is proposed for a class of uncertain stochastic nonlinear systems with unknown nonlinear functions, unmodeled dynamics, dynamical uncertainties and without requiring the measurements of the states. The NNs are used to approximate the unknown nonlinear functions, and a filter observer is designed for estimating the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function is incorporated into the backstepping recursive design technique, and a new robust adaptive NN output feedback control approach is constructed. It is mathematically proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing design parameters appropriately. The simulation example and comparison results further justify the effectiveness of the proposed approach.     

On the simulation of continuous non-linear filters A comparative study of second- and third-order approximations

Among several second-order approximations to the filter of a non-linear system, it is found that the extended Kalman filter is generally the most versatile. The second-order likelihood filter, also known ns the Detchmendy—Sridhar filter is inferior to the ahove and at the same time involves more computation. In the special ease when analytical expressions For the gaussian expectation integrals of the non-linearities can be found, the extended Kalman filter can be further improved by using stochastic linear approximations as suggested by Sunahara. The third-order likelihood filter derived in this paper is more accurate than the above, but calls for considerable storage space and computing time.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

A survey of feedback particle filter and related controlled interacting particle systems (CIPS)

In this survey, we describe controlled interacting particle systems (CIPS) to approximate the solution of the optimal filtering and the optimal control problems. Part I of the survey is focussed on the feedback particle filter (FPF) algorithm, its derivation based on optimal transportation theory, and its relationship to the ensemble Kalman filter (EnKF) and the conventional sequential importance sampling–resampling (SIR) particle filters. The central numerical problem of FPF—to approximate the solution of the Poisson equation—is described together with the main solution approaches. An analytical and numerical comparison with the SIR particle filter is given to illustrate the advantages of the CIPS approach. Part II of the survey is focussed on adapting these algorithms for the problem of reinforcement learning. The survey includes several remarks that describe extensions as well as open problems in this subject.     

A modified extended Kalman filter for linear discrete-time systems with unknown parameters

This technical communique presents a modified extended Kalman filter for estimating the states and unknown parameters in discrete-time, multi-input multi-output linear systems. The hyperstability of the filter is guaranteed by introducing a compensator into the estimation mechanism. It is proved that the estimates for the states and unknown parameters converge to the exact values if some conditions are assumed to the estimation mechanism. A numerical example shows that the proposed filter is much more effective than the extended Kalman filter in the estimation of unknown parameters.