Kalman-Bucy Filter Research Articles

Linear and non-linear filters for linear, but not gaussian processes†

Suppose {Xt} is a zero mean second order stationary process satisfying the difference equation Xt= θ Xt–1, where {ut} is a sequence of i.i.d. random variables. Further, we assume that we observe the process yt = Xt + vt, where {Vt} is also a sequence of i.i.d. random variables. The object is to obtain a recursive filter for Xt given {Yt}. If the process {ut} and {vt} are gaussian, then the recursive filter is linear, and this is the well known Kalman-Bucy filter. Otherwise, the optimal filters are non-linear. In this paper, we define two non-linear filters and compare their performance with respect to the linear filter.

Recursive input-output and state-space solutions for continuous-time linear estimation problems

A general linear least-squares estimation problem is considered. It is shown how the optimal filters for filtering and smoothing can be recursively and efficiently calculated under certain structural assumptions about the covariance functions involved. This structure is related to an index known as the displacement rank, which is a measure of non-Toeplitzness of a covariance kernel. When a state space type structure is added, it is shown how the Chandrasekhar equations for determining the gain of the Kalman-Bucy filter can be derived directly from the covariance function information; thus we are able to imbed this class of state-space problems into a general input-output framework.

New classes of finite-dimensional nonlinear filters

We derive sufficient conditions for the estimation algebras of two special classes of nonlinear estimation systems to be finite dimensional. By carrying out the Brockett—Wei—Norman program new classes of finite-dimensional filters are obtained. These systems can be regarded respectively as extensions of the Kalman—Bucy filters and the Benes filters.

Estimation and control for linear, partially observable systems with non-gaussian initial distribution

The nonlinear filtering problem of estimating the state of a linear stochastic system from noisy observations is solved for a broad class of probability distributions of the initial state. It is shown that the conditional density of the present state, given the past observations, is a mixture of Gaussian distributions, and is parametrically determined by two sets of sufficient statistics which satisfy stochastic DEs; this result leads to a generalization of the Kalman–Bucy filter to a structure with a conditional mean vector, and additional sufficient statistics that obey nonlinear equations, and determine a generalized (random) Kalman gain. The theory is used to solve explicitly a control problem with quadratic running and terminal costs, and bounded controls.

Duality of the discrete-time optimal filter and regulator problems

The duality exhibited between the discrete-time Kalman-Bucy filter with correlated process and measurement noise and the discrete-time optimal regulator is investigated.

On the Kalman-Bucy filter with specified input

In a recent paper, Vathsal suggested that a new configuration had been obtained for linear filtering problems, which was distinctly different from the Kalman-Bucy filter. It is shown that this in fact is merely a special case of the filter with a specified input.

Synthesis of Feedback-Feedforward Controller by LQ Regulator Theory and its Application to Boiler Control

Methods to synthesize a feedforward control and a reduced order feedback compensator which is expected to be low sensitive to parameter variations are presented under the condition that the optimal state feedback regulator is already synthesized. The feedforward is designed so as to minimize the value of the performance index to the step change of measurable disturbance. The compensator is the mode reduced one of the Kalman-Bucy filter type optimal observer. A method to let poles of the optimal observer are almost the same as those of the optimal regulator is presented. A method to choose the poles to be reduced to synthesize the reduced order compensator is also presented. The methods presented apply to the design of boiler controller and verify the effectiveness by simulation study.

Filtering of partitioned large scale hydrological systems / Le filtrage d'un système hydrologique de grande dimension, divisé en sous-bassins

ABSTRACT Recent work in on-line hydrological forecasting has utilized the Kalman-Bucy filter for state (discharge) forecasting. One drawback of using these algorithms has been restrictions related to the size of the state vector due to computation and computer limitations; resulting in limiting hydrological applications to small headwater catchments. Forecasts of headwater catchments are of limited use to agencies such as the US National Weather Service which routinely have catchments with between 10 and 100 forecast points. This paper sets forth the methodology that allows forecasting of large systems by partitioning the system into subsystems where the filtering of subsystems is performed in parallel. The interactions between the partitioned subsystems are accounted for by supplementing the noise processes. An added institutional advantage to the partitioning approach is the ability for forecasting agencies to set up local forecast centres that are coordinated through a regional centre. This is especial...

Comment on "Observer Stabilization of Singularly Perturbed Systems"

Comparison with Optimal Controller It is interesting to note that the vertical channel is optimal in a larger sense. In the vertical channel, the control variables were constrained to be the simple functions of the measurement variable given in Eq. (3). Can a more complicated compensation connecting the measurement and control variables provide smaller mean-square errors? It is well known that optimal control is provided by feedback compensation that is the cascade combination of the KalmanBucy filter and the state-feedback optimal regulator. The filter was given in Eq. (11). The regulator is

Kalman-bucy and minimax filtering

It is shown that the Kalman-Bucy filter is also a minimax filter.

Estimation of states in a polymerization reactor

LUENBERGER observers and KALMAN-BUCY filters have been applied to estimate non-measurable states in a styrene polymerization reactor. The states of the nonlinear reactor model are: the temperature; the concentrations of the monomer and the initiator; and some characteristic numbers representing the chain length distribution of the produced polymer. The temperature and the refractive index of the reaction medium are used as measuring variables. Applying non-linear estimation techniques, it is possible to reconstruct the unknown state vector from these measurements. Simulation results are given for different cases. It is described how one may consider the Gel-Permeation-Chromatographic measurement of the distribution in the estimation.

Estimation of States in a Polymerization Reactor

LUENBERGER observers and KALMAN-BUCY filters have been applied to estimate non-measurable states in a styrene polymerization reactor. The states of the nonlinear reactor model are: the temperature; the concentrations of the monomer and the initiator; and some characteristic numbers representing the chain length distribution of the produced polymer. The temperature and the refractive index of the reaction medium are used as measuring variables. Applying non-linear estimation techniques, it is possible to reconstruct the unknown state vector from these measurements. Simulation results are given for different cases. It is described how one may consider the Gel-Permeation-Chromatographic measurement of the distribution in the estimation.

Stochastic distributed systems with point observations and boundary control: an abstract theory

There already exists a fairly complete theory for the problems of estimation and stochastic optimal control for linear distributed parameter systems, with Gaussian or non Gaussian noise disturbance. In [8] and [12] generalizations of the familiar finite dimensional results of the Kalman-Bucy filter and the separation principle are obtained using an abstract input-output Hilbert space representation for the system. However, in [8] and [12] all the input operators are assumed to be bounded and so it does not cover the important practical cases of control and noise on submanifolds of the spatial domain or point observations. Here we introduce unbounded system operators in the abstract input-output Hilbert space representation and thus extend all the results of [8] and [12] to allow for point observations and noise and control on submanifolds including the boundary. The theory is illustrated by several examples.

A stochastic realization approach to the smoothing problem

The purpose of this paper is to develop a theory of smoothing for finite dimensional linear stochastic systems in the context of stochastic realization theory. The basic idea is to embed the given stochastic system in a class of similar systems all having the same output process and the same Kalman-Bucy filter. This class has a lattice structure with a smallest and a largest element; these two elements completely determine the smoothing estimates. This approach enables us to obtain stochastic interpretations of many important smoothing formulas and to explain the relationship between them.

Design of single linear functional filters

This paper treats the simultaneous design problem of a single linear functional filter (SLFF) and a feedback gain for the so-called single-input linear-quadratic-gaussian (LQG) problem. The SLFF is a linear unbiased filter which estimates a scalar-valued linear function of the system state, and is generally of lower order than the Kalman-Bucy filter. A canonical form, the minimal order and the optimal design procedure of an SLFF are obtained for the steady-state case. Finally, the results are compared with the Kalman-Bucy filter by an example.

On the application of modern control theory to improving the fidelity of an underwater projector

The paper presents a technique for controlling the radiated sound pressure from an electroceramic projector in water. This technique involves the implementation of a state variable feedback tracking algorithm. The state variables of the system are estimated with a Kalman–Bucy filter from the measured radiated pressure. The state variables are then fed back to effect an appropriate compensation of the system, which results in a least-square minimization of the difference between the measured and desired pressure signatures. The algorithm has been tested with an actual projector. The results show that the measured pressure tracks the desired pressure reasonably well.

A direct approach to jump detection in linear time-invariant systems with application to power system perturbation detection

A new direct approach to jump detection in linear time-invariant stochastic systems is considered in this paper, leading to a class of simple detection algorithms based on the explicit computation of the generalized likelihood ratio. While most of the already known methods perform the detection by observing the residuals of the Kalman-Bucy filter, here the output of the system is directly used for the detection, obtaining a larger class of algorithms which seems particularly convenient when state estimation is not required. Though the present approach is suitable only for time-invariant systems, its great flexibility allows the algorithm to be tightened to a particular application by changing such fundamental parameters as time-delay and observation interval. The practical application of the method to power systems perturbation detection is developed in the second part of the paper, where simulation results are presented as well. The discussion of the application shows as a precise qualification of the performance in terms of false alarm rate and probability of detection yields simple design criteria for the algorithm.

Design of linear functional niters for linear stochastic systems

This paper considers the problem of designing a linear functional filter (LFF) to estimate a linear function of the state of a linear stochastic system. The LFF is a linear unbiased filter and is generally of lower order than the Kalman-Bucy filter. When the LFF is unbiased in the so-called linear-quadratic-Gaussian (LQG) problem, it is shown that the separation properties are established for the quadratic cost function and the stability. In the steady-state case, the minimal order, a canonical form and the optimal design are obtained for a single LFF which estimates a scalar-valued linear function of the state. Finally, an illustrative example is given to show the effectiveness of the LFF.

Basic optimal estimation and control problems in Hilbert space

The recently developed mathematical framework of Hilbert resolution space valued random processes is used to formulate and solve an abstract quadratic optimization problem. By particularizing the description of the operators appearing in the statement and solution formula of the problem, one rediscovers and generalizes most of the classical estimation and control theory problem statements and results. These include, among others, the Wiener smoothing prediction filter, the Kalman regulator, the Kalman-Bucy filter, the stochastic control separation principle and the more recent Youla-Jabr-Bongiorno optimal servo problem solution.

Bounding filter: A simple solution to lack of exact a priori statistics

Weiner and Kalman—Bucy estimation problems assume that models describing the signal and noise stochastic processes are exactly known. When this modeling information, i.e., the signal and noise spectral densities for the Weiner filter and the signal and noise dynamic system and disturbing noise representations for Kalman—Bucy filtering, is inexactly known, then the filter's performance is suboptimal and may even exhibit apparent divergence. In this paper a system is designed whereby the actual estimation error covariance is bounded by the covariance calculated by the estimator. Therefore, the estimator obtains a bound on the actual error covariance which is not available, and also prevents, its apparent divergence. The bounding filter can be of lower order than the original stochastic models; hence, a technique is devised of reducing the order of the filtering system and concurrently obtaining a figure of merit for its performance. For many cases, the design conditions devised for the steady-state Weiner filter apply to transient Kalman—Bucy filter performance.