Kalman-Bucy Filter Research Articles

Block matrices with L-block-banded inverse: inversion algorithms

Block-banded matrices generalize banded matrices. We study the properties of positive definite full matrices P whose inverses A are L-block-banded. We show that, for such matrices, the blocks in the L-block band of P completely determine P; namely, all blocks of P outside its L-block band are computed from the blocks in the L-block band of P. We derive fast inversion algorithms for P and its inverse A that, when compared to direct inversion, are faster by two orders of magnitude of the linear dimension of the constituent blocks. We apply these inversion algorithms to successfully develop fast approximations to Kalman-Bucy filters in applications with high dimensional states where the direct inversion of the covariance matrix is computationally unfeasible.

Discretization and Simulation of the Zakai Equation

This paper is concerned with numerical approximations for the stochastic partial differential Zakai equation of nonlinear filtering problems. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyze the error caused by an Euler‐type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order $\sqrt{\delta}$ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. Formally, this consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from a correlated Kalman–Bucy filter.

Information and Entropy Flow in the Kalman?Bucy Filter

We investigate the information theoretic properties of Kalman–Bucy filters in continuous time, developing notions of information supply, storage and dissipation. Introducing a concept of energy, we develop a physical analogy in which the unobserved signal describes a statistical mechanical system interacting with a heat bath. The abstract ‘universe’ comprising the signal and the heat bath obeys a non-increase law of entropy; however, with the introduction of partial observations, this law can be violated. The Kalman–Bucy filter behaves like a Maxwellian demon in this analogy, returning signal energy to the heat bath without causing entropy increase. This is made possible by the steady supply of new information. In a second analogy the signal and filter interact, setting up a stationary non-equilibrium state, in which energy flows between the heat bath, the signal and the filter without causing any overall entropy increase. We introduce a rate of interactive entropy flow that isolates the statistical mechanics of this flow from marginal effects. Both analogies provide quantitative examples of Landauer’s Principle.

OPTIMAL LINEAR FILTERING 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, the resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is then 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 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.

Comparative Study of Sliding Mode, Optimal and Extended Kalman-Bucy Filters Performance for Quadratic Stochastic Systems

This paper presents a filter for nonlinear quadratic stochastic systems over linear observations, based on the sliding mode technique. Performance of the designed filter is compared to those of the optimal filter for quadratic systems and an extended Kalman-Bucy filter. Numerical simulation results are obtained and graphically presented. Specifics of the suggested approach are discussed.

Optimal Filtering for Nonlinear Polynomial Systems Over Linear Observations With Time Delay

In this paper, the optimal filtering problem for nonlinear systems over linear observations with time delay is treated proceeding from the general expression for the stochastic !to differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining a closed system of the filtering equations for a polynomial state over linear observations with delay is then established, which yields the explicit closed form of the filtering equations in the particular case of a bilinear system state. In the example, performance of the designed optimal filter for bilinear states over linear observations with delay is verified against the best filter available for bilinear states over linear observations without delays and the conventional extended Kalman-Bucy filter.

A solution to the linear estimation problem with correlated signal and noise

A solution to linear estimation problems involving correlated signal and noise is provided by using the approximate Karhunen–Loève expansion. The proposed solution takes the form of a suboptimum estimator that converges to the desired optimum estimate as the length of the series representation goes to infinity and it can be expressed as a function of the solution to a stochastic differential equation which is similar to a Kalman–Bucy filter. The main advantage of this suboptimum solution is that it can be applied without introducing structural assumptions regarding the properties of the correlation functions (stationarity, state-space structure, etc.).

Optimal linear filtering over observations with multiple delays

AbstractIn 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.

Fast Implementations of the Kalman–Bucy Filter for Satellite Data Assimilation

We present practical data assimilation algorithms based on the Kalman-Bucy filter (KBf) for combining satellite altimetry data with the nonlinear ocean circulation models. Data assimilation in such applications is computationally challenging because of the large dimensions of the state fields. Compared with the direct KBf, our KBf implementations provide computational savings of two orders of the magnitude of the linear dimension of the state field. We run twin experiments by interfacing our data assimilation algorithms with the NLOM, a nonlinear ocean circulation model developed at the Naval Research Laboratory.

On precise integration method

The numerical integration for the ordinary differential equations is extremely important for applications. So far, almost all the numerical integration methods apply the finite difference approximation even for a time-invariant system. The precise integration method (PIM) solves the time step integration for the time-invariant system first. For such a problem, precise integration gives a highly precise numerical result, which approaches the full computer precision. After the integration of time-invariant system is solved, various approximate methods can be applied to other problems such as time-variant or nonlinear time integration. The numerical integration of two-point boundary value problems (TPBVP) is also very important in applications. Such as wave propagation, optimal control, structural mechanics, electro-magnetic wave guide problems, etc. The PIM can also be applied to solve the TPBVP. The TPBVP induced initial value problems such as the Lyapunov differential equation, the Riccati differential equation and the Kalman–Bucy filter equation, etc. can also be solved along the same way. In this paper, the essence of precise integration will be explained.

Optimal filtering for linear systems with multiple delays in observations

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.

Wind Estimation on a Lightweight Vertical-Takeoff-and-Landing Uninhabited Vehicle

Wind-velocity measurement on a rotary-wing aircraft is a difficult task because of the flow induced by the rotors. The purpose of this paper is to develop a method to estimate the wind velocity components from the measurement of the state variables of a rotorcraft in the moving atmosphere. The algorithm presented is in the framework of the output error method. The wind-velocity components were estimated using a novel variational formulation. The method uses airframe and rotor models that calculate the aerodynamic and thrust coefficients by means of an artificial neural-network technique. To validate the method, the results are compared to wind-velocity estimates from a Kalman-Bucy filter

Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs

In this note, we develop a real-time and accurate solution for nonlinear filtering problems based on the Gaussian distribution. Specifically, we present an explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation of the Yau filtering system, which includes the linear filtering system and exact filtering system. The solution is given in terms of a solution of a system of ordinary differential equations. In particular, our method can be implemented in hardware. The complexity of our algorithms is the same as those of Kalman-Bucy filters in the case of linear filtering systems.

Zero-miss-distance guidance law based on line-of-sight rate measurement only

This paper presents a high performance, simple and robust guidance method which utilizes line-of-sight (LOS) rate measurement only to yield zero-miss-distance against highly maneuvering targets. The novel guidance law adopts the basic framework of proportional navigation guidance, yet instead of using an acceleration command which is proportional to the measured LOS rate, the acceleration command is applied proportionally to an equivalent LOS rate. The equivalent LOS rate is a linear combination of the measured LOS rate and higher-order LOS rate derivatives, which are estimated from the noisy LOS rate measurement using a Kalman–Bucy filter. A considerable part of this paper is devoted to a comprehensive simulation study of the new guidance law. Deterministic simulations and Monte Carlo analyses show that excellent performance is obtained against highly maneuvering targets.

Model Predictive Control of Structures under Earthquakes using Acceleration Feedback

This paper presents a general formulation of the model predictive control (MPC) scheme with special reference to acceleration feedback in structural control under earthquakes. The MPC scheme is based on a prediction model of the system response to obtain the control action by minimizing an objective function. Optimization objectives include minimization of the difference between the predicted and desired response trajectories, and of the control effort subject to certain constraints. The effectiveness of MPC has been demonstrated to be equivalent to the optimal control. In this study, the prediction model is formulated using a feedback loop containing acceleration measurements from various locations in the structure. The state observer utilizes the Kalman-Bucy filter to estimate the states of the system from the acceleration feedback. Examples of single-story and three-story buildings equipped with control devices are used to demonstrate the effectiveness of the MPC scheme based on acceleration feedback. Both buildings are analyzed using an active tendon control device and an active mass damper (AMD). A two-story building with an AMD is used to experimentally validate the numerical control scheme. The results demonstrate the effectiveness of the MPC scheme using acceleration feedback. The acceleration feedback framework developed in this paper should serve as a building block for future extensions of MPC in capturing and benefiting from the attractive features of MPC, i.e., computational expediency, real-time applications, intrinsic compensation for time delays, and treatment of constraints, for implementation in civil structures.

KALMAN-BUCY FILTERING FOR SINGULAR STOCHASTIC DIFFERENTIAL SYSTEMS

This work investigates the problem of state estimation for singular stochastic differential systems. A Kalman-Bucy-like filter is proposed, based on a suitable decomposition of the descriptor vector into two components. The first one is expressed as a function of the observation, and therefore does not need to be estimated, while the second component is described by a regular linear stochastic system and can be estimated by a Kalman-Bucy filter. Numerical simulations are presented on the case of a stochastic system with an unknown input, modeled as a singular system.

Extension of the Kalman–Bucy Filter to Elementary Linear Systems with Fractional Brownian Noises

We investigate the optimal filtering problem in the simplest Gaussian linear system driven by fractional Brownian motions. At first we extend to this setting the Kalman–Bucy filtering equations which are well-known in the specific case of usual Brownian motions. Closed form Volterra type integral equations are derived both for the mean of the optimal filter and the variance of the filtering error. Then the asymptotic stability of the filter is analyzed. It is shown that the variance of the filtering error converges to a finite limit as the observation time tends to infinity.

On Filtering Problems Over Ito-Volterra and Delayed Observations

In this paper, the Kalman-Bucy filter is designed for an Ito-Volterra process over Ito-Volterra observations that cannot be reduced to the case of a differential observation equation. The Kalman-Bucy filter is then designed for an Ito-Volterra process over discontinuous Ito-Volterra observations. Based on the obtained results the optimal filtering equations over observations with delays are obtained for an Ito-Volterra process, proceeding from discontinuous observations of the Ito-Volterra type. It is shown that the designed filter can be significantly simplified for a dynamic system state governed by a differential equation. The filtering problems are considered over observations with multiple delays and a set of delays of the continuum power.

Adaptive friction compensation using extended Kalman–Bucy filter friction estimation

An extended Kalman–Bucy filter (EKBF)-based friction compensation method is presented and validated. The method relies on an accurate model of system rigid-body dynamics and measured motion, rather than a structured nonlinear friction model, to estimate external friction torque. The estimate is used in a traditional friction compensator to cancel friction effects. The EKBF compensator is compared with other model-based and non-model-based friction compensation strategies through position tracking experiments. Results show that when motion is dominated by static and Stribeck friction, non-model-based friction estimation and compensation using the EKBF consistently provides equal or superior performance over model-based adaptive friction compensation methods.

Etch profile control of high-aspect ratio deep submicrometer α-Si gate etch

The well-acknowledged etch profile drift problem in chip production was investigated with a more accurate meads of measuring actual etch thickness to monitor and correct this drift. Using a high-aspect ratio, 0.1-/spl mu/m /spl alpha/-Si gate structure, the investigation was specifically focused on the control of transition timing in the critical interval from main etch (ME) to over etch (OE). This required reliable endpoint detection of /spl alpha/-Si which was achieved through the development of a method employing a Kalman-Bucy filter with a real-time spectroscopic ellipsometer (RTSE). The robustness of our endpoint detection technique was tested and demonstrated under the actual physical and chemical disturbance environments of the etching process. Application of this endpoint detection technique to the etch of a 0.1-/spl mu/m patterned /spl alpha/-Si gate also achieved a significant improvement on the etch profile repeatability.