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A method is proposed for classifying estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation. A natural class of automorphisms of this algebra is examined and a systematic method of generating equivalent problems is developed. Finally, a new class of nonlinear filtering problems with essentially nonlinear filtering equations are presented.
The formation of nonlinear filtering problems are discussed along with approaches to solving nonlinear filtering problems. An approach to the design and evaluation of the performance of suboptimal filters is presented in terms of upper and lower bounds.
The problem of optimal control of linear discrete-time stochastic dynamical system with unknown and, possibly, stochastically varying parameters is considered on the basis of noisy measurements. It is desired to minimize the expected value of a quadratic cost functional. Since the simultaneous estimation of the state and plant parameters is a nonlinear filtering problem, the extended Kalman filter algorithm is used. Several qualitative and asymptotic properties of the open loop feedback optimal control and the enforced separation scheme are discussed. Simulation results via Monte Carlo method show that, in terms of the performance measure, for stable systems the open loop feedback optimal control system is slightly better than the enforced separation scheme, while for unstable systems the latter scheme is far better.
Generalization of superposition principle for linear systems applied to classification of nonlinear systems and solution of nonlinear filtering problems
This paper is concerned with the parameter adaptive least squares estimation of random processes. The main result is a general representation theorem for the conditional expectation of a random variable on a product probability space. Using this theorem along with the general likelihood ratio expression, the least squares estimate of the process is found in terms of the parameter conditioned estimates. The stochastic differential for the a posteriori probability and the stochastic differential equation for the a posteriori density are found by using simple stochastic calculus on the representations obtained. The results are specialized to the case when the parameter has a discrete distribution. The results can be used to construct an implementable recursive estimator for certain types of nonlinear filtering problems. This is illustrated by some simple examples.
Approximations to optimal nonlinear filters for problems involving signal and noise processes
In the investigation, the filtering problem is considered in the continuous time domain. The postulated simple suboptimal nonlinear filter structure closely parallels the structure of the Kalman-Bucy optimal linear filter algorithm. Two filter performance evaluation methods are developed based on the Kolmogorov equations for the transition density of Markov processes. The expansions in the approximations for the nonlinear system and observation functions are in effect carried out up to second-order terms in both methods. The description of the filter's performance is sought in terms of second-order statistics in both methods.
Sigma-point filters have received a lot of attention in recent years as a better alternative to extended Kalman filters for highly nonlinear problems. In this paper, we compare the performance of the additive divided difference sigma-point filter to the extended Kalman filter when applied to orbit determination of a realistic operational scenario based on the Interstellar Boundary Explorer mission. For the scenario studied, both filters provided equivalent results. The performance of each is discussed in detail.
This paper discusses the Kalman filter used in the navigation software of the Space Shuttle. The form of the filter will be discussed: the standard Kalman filter versus the square-root version of this filter. Both of these filters have severe nonlinearity problems. Two successful solutions of the nonlinearity problem are presented. A real-time program must have a good method of editing bad data. The data editing scheme is discussed. Those elements of the Kalman filter state vector which are random variables will be discussed.
The two step filter is applied to process intersatellite radar measurements to determine the motion of one satellite relative to another in close elliptical orbits. This filter breaks a nonlinear estimation problem into two state vectors. The 'first step' state is chosen so as to have a linear measurement equation. This is nonlinearly related to the 'second step' state which describes the dynamics. Two different forms are used. In one, the first step state is the second step state vector augmented by the measurement equation. In the other, the first step and second step state vectors are of equal dimension. The two step filter is compared against an iterated extended Kalman filter and a Kalman filter using a change of variables. Analytical differences between the two step estimator and these conventional filters are highlighted. Special concerns for initializing the first step state covariance matrix and handling the possibility of numerically rank deficient covariance matrices are addressed. Numerical simulations are performed which show that the Two Step estimator produces a lower estimation bias under two circumstances; large apriori initial error; and small dimension observation vectors which require a longer arc of measurements to generate observability of the state.
One approach to the problem of simplifying complex nonlinear filtering algorithms is through using stratified probability approximations where the continuous probability density functions of certain random variables are represented by discrete mass approximations. This technique is developed in this paper and used to simplify the filtering algorithms developed for the optimum receiver for signals corrupted by both additive and multiplicative noise.
Signal and noise problem as solution to nonlinear stochastic differential equations
Theory of linear estimation and applicability to problems of smoothing, filtering, extrapolation, and nonlinear estimation
Model-based prognostics approaches use domain knowledge about a system and its failure modes through the use of physics-based models. Model-based prognosis is generally divided into two sequential problems: a joint state-parameter estimation problem, in which, using the model, the health of a system or component is determined based on the observations; and a prediction problem, in which, using the model, the stateparameter distribution is simulated forward in time to compute end of life and remaining useful life. The first problem is typically solved through the use of a state observer, or filter. The choice of filter depends on the assumptions that may be made about the system, and on the desired algorithm performance. In this paper, we review three separate filters for the solution to the first problem: the Daum filter, an exact nonlinear filter; the unscented Kalman filter, which approximates nonlinearities through the use of a deterministic sampling method known as the unscented transform; and the particle filter, which approximates the state distribution using a finite set of discrete, weighted samples, called particles. Using a centrifugal pump as a case study, we conduct a number of simulation-based experiments investigating the performance of the different algorithms as applied to prognostics.
When dealing with imperfect data and general models of dynamic systems, the best estimate is always sought in the presence of uncertainty or unknown parameters. In many cases, as the first attempt, the Extended Kalman filter (EKF) provides sufficient solutions to handling issues arising from nonlinear and non-Gaussian estimation problems. But these issues may lead unacceptable performance and even divergence. In order to accurately capture the nonlinearities of most real-world dynamic systems, advanced filtering methods have been created to reduce filter divergence while enhancing performance. Approaches, such as Gaussian sum filtering, grid based Bayesian methods and particle filters are well-known examples of advanced methods used to represent and recursively reproduce an approximation to the state probability density function (pdf). Some of these filtering methods were conceptually developed years before their widespread uses were realized. Advanced nonlinear filtering methods currently benefit from the computing advancements in computational speeds, memory, and parallel processing. Grid based methods, multiple-model approaches and Gaussian sum filtering are numerical solutions that take advantage of different state coordinates or multiple-model methods that reduced the amount of approximations used. Choosing an efficient grid is very difficult for multi-dimensional state spaces, and oftentimes expensive computations must be done at each point. For the original Gaussian sum filter, a weighted sum of Gaussian density functions approximates the pdf but suffers at the update step for the individual component weight selections. In order to improve upon the original Gaussian sum filter, Ref. [2] introduces a weight update approach at the filter propagation stage instead of the measurement update stage. This weight update is performed by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. By adaptively updating each component weight during the nonlinear propagation stage an approximation of the true pdf can be successfully reconstructed. Particle filtering (PF) methods have gained popularity recently for solving nonlinear estimation problems due to their straightforward approach and the processing capabilities mentioned above. The basic concept behind PF is to represent any pdf as a set of random samples. As the number of samples increases, they will theoretically converge to the exact, equivalent representation of the desired pdf. When the estimated qth moment is needed, the samples are used for its construction allowing further analysis of the pdf characteristics. However, filter performance deteriorates as the dimension of the state vector increases. To overcome this problem Ref. [5] applies a marginalization technique for PF methods, decreasing complexity of the system to one linear and another nonlinear state estimation problem. The marginalization theory was originally developed by Rao and Blackwell independently. According to Ref. [6] it improves any given estimator under every convex loss function. The improvement comes from calculating a conditional expected value, often involving integrating out a supportive statistic. In other words, Rao-Blackwellization allows for smaller but separate computations to be carried out while reaching the main objective of the estimator. In the case of improving an estimator's variance, any supporting statistic can be removed and its variance determined. Next, any other information that dependents on the supporting statistic is found along with its respective variance. A new approach is developed here by utilizing the strengths of the adaptive Gaussian sum propagation in Ref. [2] and a marginalization approach used for PF methods found in Ref. [7]. In the following sections a modified filtering approach is presented based on a special state-space model within nonlinear systems to reduce the dimensionality of the optimization problem in Ref. [2]. First, the adaptive Gaussian sum propagation is explained and then the new marginalized adaptive Gaussian sum propagation is derived. Finally, an example simulation is presented.
The application of parameter estimation to highly unstable aircraft is discussed. Included are a discussion of the problems in applying the output error method to such aircraft and demonstrates that the filter error method eliminates these problems. The paper shows that the maximum likelihood estimator with no process noise does not reduce to the output error method when the system is unstable. It also proposes and demonstrates an ad hoc method that is similar in form to the filter error method, but applicable to nonlinear problems. Flight data from the X-29 forward-swept-wing demonstrator is used to illustrate the problems and methods discussed.
This paper discusses the application of parameter estimation to highly unstable aircraft. It includes a discussion of the problems in applying the output error method to such aircraft and demonstrates that the filter error method eliminates these problems. The paper shows that the maximum likelihood estimator with no process noise does not reduce to the output error method when the system is unstable. It also proposes and demonstrates an ad hoc method that is similar in form to the filter error method, but applicable to nonlinear problems. Flight data from the X-29 forward-swept-wing demonstrator is used to illustrate the problems and methods discussed.
Finite dimensional approximations of optimal nonlinear filters for class of large-noise, nonlinear, continuous-time problems