New exact nonlinear filters with large Lie algebras

The ideas previously used (Stochastics. Vol. 5, pp. 65–92, 1981) to construct some finite-dimensional nonlinear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.In engineering language, the new filters are obtained from the old by smoothing the output through an n-pole linear system before adding observation noise in the usual way; this adds 2n to the Lie algebra dimension. In the simplest case (drift = tanh x, OBSERVATION = x) we put x through a one-pole described by a new variable ζ, and observe ζ + noise instead of x + noise; the new Lie algebra has additional generators ζ and ∂/∂ξ besides the four from the oscillator algebra, to give dimension 6.The filter, which gives a recursive construction of the conditional density, can be ‘derived’ by any of three (here equivalent methods: (i) direct integration of the Kallianpur-Striebel formula as a Gaussian integral; (ii) solution of a parabolic PDE with quadratic potential; and (iii) the Wei-Norman procedure.     

Finite dimensional filters with non-linear drift IX Construction of finite dimensional estimation algebras of non-maximal rank

The idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It has proven to be an invaluable tool in the study of nonlinear filtering problem. In 1983, Brockett proposed to classify all finite-dimensional estimation algebras. In this paper, we give the construction of finite-dimensional estimation algebras of non-maximal rank. These non-maximal rank finite-dimensional estimation algebras play an important role in Brockett's classification problem.     

On necessary conditions for the existence of finite-dimensional filters in discrete time

Under some regularity assumptions we show that, if a finite-dimensional filter in discrete time exists, then the observation, prediction, and filtering distributions are all of exponential class. Our results, motivated by analogous results in the field of statistics, hold for arbitrary (finite) dimensions of the state and observation spaces as well as of the filter itself.     

On a new class of finite dimensional estimation algebras

We introduce a new class of solvable finite dimensional estimation algebras that contains both the linear and the Bene cases. The condition for finite dimensionality is expressed in terms of two matrices. They play an important role in understanding the structure of estimation algebras.     

On some finite dimensional nonlinear filters for certain diffusions observed in correlated noise

An extension of the one dimensional Bene nonlinear filter to the case of correlated process and observation noises is presented. Conditions are derived so that the optimal nonlinear filter is finite dimensional.     

A note on the memory length of optimal nonlinear filters

Consider the one-dimensional filtering problem with state diffusion coefficient ^γ and observation noise coefficient ^1−γ, where is fixed. In the regime →0, we provide an example of state dynamics and a corresponding asymptotically optimal filter whose memory length is short in a strong sense. This stands in contrast with the behavior of the correlation memory length, proposed in [7], which stays bounded away from zero as →0.     

Finite-dimensional filters with nonlinear drift,VI: Linear structure of Ω

Ever since the concept of estimation algebra was first introduced by Brockett and Mitter independently, it has been playing a crucial role in the investigation of finite-dimensional nonlinear filters. Researchers have classified all finite-dimensional estimation algebras of maximal rank with state space less than or equal to three. In this paper we study the structure of quadratic forms in a finite-dimensional estimation algebra. In particular, we prove that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂f _j /∂x _i −∂f _i /∂x _j )matrix, wheref denotes the drift term, is a linear matrix in the sense that all the entries in Ω are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank. This research was supported by Army Research Office Grants DAAH 04-93-0006 and DAAH 04-1-0530.     

A state prediction scheme for discrete time nonlinear dynamic systems

A state prediction scheme is proposed for discrete time nonlinear dynamic systems with non-Gaussian disturbance and observation noises. This scheme is based upon quantization, multiple hypothesis testing, and dynamic programming. Dynamic models of the proposed scheme are as general as dynamic models of particle predictors, whereas the nonlinear models of the extended Kalman (EK) predictor are linear with respect to the disturbance and observation noises. The performance of the proposed scheme is compared with both the EK predictor and sampling importance resampling (SIR) particle predictor. Monte Carlo simulations have shown that the performances of the proposed scheme, EK predictor, and SIR particle predictor are all model-dependent, that is, one performs better than the others for a given example. Some examples, for which the proposed scheme performs better than the others do, are also given in the paper.     

Morphological representation of nonlinear filters

Morphological operators provide very efficient algorithms for signal (image) processing. The efficiency of morphological operators has been captured by using them as approximations of nonlinear operators in numerous applications (e.g., image restoration). Our approach to the approximation of nonlinear operators is the construction of morphological bounds on them. We present a general theory on the morphological bounds on nonlinear operators, propose conditions for the existence of these bounds, and derive several fundamental morphological bounds.We also derive morphological bounds on iterations of nonlinear operators, which are superior to the original nonlinear operator in some applications. Because obtaining the results of the convergence of iterations of a nonlinear operator is often particularly desirable, we provide morphological bounds on the convergence of such iterations, and propose conditions for their convergence based on morphological properties. Finally, we propose several criteria for the morphological characterization of roots of nonlinear operators.This work was supported by U.S. Office of Naval Research award N00014-91-J-1725.     

伪效应代数中的模糊滤子与模糊理想

在伪效应代数中提出了模糊滤子和模糊理想的概念,讨论了它们的性质;引入并研究了强模糊滤子和强模糊理想,得到了一些好的结论。     

Advanced point-mass method for nonlinear state estimation

State estimation of discrete-time nonlinear non-Gaussian stochastic systems by point-mass approach, which is based on discretization of state space by a regular grid and numerical solution of Bayesian recursive relations, is treated. The stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator. Boundary-based grid design, thrifty convolution, and multigrid design with grid splitting and merging are proposed. The main advantages of these techniques are nonnegligible support delimitation, time-saving computation of convolution, and effective processing of multimodal probability density functions, respectively. The techniques are involved into the basic point-mass approach and a new general-purpose, more sophisticated point-mass algorithm is designed. Computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

Sonar-based robot navigation using nonlinear robust observers

This paper addresses the sonar-based navigation of mobile robots. The extended Kalman filtering (EKF) technique is considered, but from a deterministic, nonstochastic, point of view. For this problem, new results are presented on the robustness of the nonlinear observation scheme. The original feature is that the region-of-convergence question is posed in its complete nonlinear framework, that is, considering the dynamics not only of the estimation error ζ(t), but also of the covariance matrix P(t). In this way the approach followed makes less conservative the treatment and improves the convergence analysis. The proposed ideas were tested successfully on simulation experiments of a mobile platform.     

Nonlinear observers comprising high-gain observers and extended Kalman filters

A full order observer is designed for a class of nonlinear systems that can potentially admit unstable zero dynamics. The structure of the observer is composed of an Extended High Gain Observer (EHGO), for the estimation of the derivatives of the output, augmented with an Extended Kalman Filter (EKF) for the estimation of the states of the internal dynamics. The EHGO is also utilized to estimate a signal that is used as a virtual output to an auxiliary system comprised of the internal dynamics. In the special case of the system being linear in the states of the internal dynamics, we achieve semi-global asymptotic convergence of the estimation error. We demonstrate the efficacy of the observer in two examples; namely, a synchronous generator connected to an infinite bus and a Translating Oscillator with a Rotating Actuator (TORA) system.     

A separation theorem for nonlinear systems

In this paper we have obtained a nonlinear separation result for controlled stochastic systems. The result is based on a sequential technique, introduced by the second author, which has been applied with significant success for nonlinear deterministic systems.     

Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems

We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system (ε x_k+1 =φ(x_k), y_k = h(x_k)+η(x_k)ν_k, K=0, 1,…. This system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations.The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (ε) and of the minimal filter.     

On designing filters for uncertain sampled-data nonlinear systems

This paper is concerned with the problem of nonlinear filtering for sampled-data systems with nonlinear time-varying parameter uncertainty. The aim is to design a digital filter such that the ratio between the energy of the estimation errors and the energy of the exogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. A nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty is provided. Based on this nonlinear bounded real lemma, the robust filtering problem is solved in terms of both continuous and discrete Hamilton–Jacobi equations.