Recursive algorithms for Bayes smoothing with uncertain observations

Recursive algorithms for the Bayes solution of fixed-interval, fixed-point, and fixed-lag smoothing under uncertain observations are presented. The Bayes smoothing algorithms are obtained for a Markovian system model with Markov uncertainty, a model more general than the one used in linear smoothing algorithms. The Bayes fixed-interval smoothing algorithm is applied to a Gauss-Markov example. The simulation results for this example indicate that the MSE performance of the Bayes smoother is significantly better than that of the linear smoother.     

Discrete-time fixed-lag smoothing algorithms

Kalman filtering results are applied to yield alternative computationally stable fixed-lag smoothing algorithms including reduced order and minimal order fixed-lag smoothers. The reduced order smoothing algorithms are new and clearly have advantages over the more familiar algorithms. The properties of such fixed-lag smoothers are also studied.     

A distributed algorithm for H-infinity fLxed-lag smoothing

1IntroductionThe problemof estimating the values of a random(orstochastic) process given observations of a related randomprocess is encountered in many areas of science andengineering, i .e . communication, control , signalprocessing,economics and statistics [1] .There are twowell_known approaches for such problems .One is based onanL2or an H2criterion,where the estimators are designedto minimize thelinear meansquared error ,for example ,theWiener filtering and Kalmanformulation [2] .The oth…     

Recursive smoothing algorithms for the estimation of signals from uncertain observations via mixture approximations

Considering discrete-time systems with uncertain observations when the signal model is unknown, but only covariance information is available, and the signal and the observation additive noise are correlated and jointly Gaussian, we present recursive algorithms for suboptimal fixed-point and fixed-interval smoothing estimators. To derive the algorithms, we employ a technique consisting in approximating the conditional distributions of the signal given the observations by Gaussian distributions, taking successive approximations of the mixtures of normal distributions. The expectation of these approximations provides us with the suboptimal estimators. In a numerical simulation example, the performance of the proposed estimators is compared with that of linear ones, via the sample mean square values of the corresponding estimation errors.     

An efficient fixed-lag smoothing algorithm for discrete linear systems

The modified Bryson-Frazier fixed interval smoothing algorithm [6], is an addendem to the Kalman filter. This algorithm when applied to the problem of fixed-lag smoothing is computationally more efficient than the algorithms recently reported in refs. [1–3]. Features of the algorithm are ease of implentation, computational efficiency, reduced storage requirements, and stability.     

Bayes smoothing algorithms for segmentation of binary images modeled by markov random fields

A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel, given the total noisy image, in a recursive way. The a posteriori distribution together with a criterion of optimality then determine a Bayes estimate of the scene. The algorithm presented is an extension of a 1-D Bayes smoothing algorithm to 2-D and it gives the optimum Bayes estimate for the scene value at each pixel. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field, a special class of Markov random fields, and the Bayes smoothing algorithm is applied on overlapping strips (horizontal/vertical) of the image consisting of several rows (columns). It is assumed that the signal (scene values) vector sequence along the strip is a vector Markov chain. Since signal correlation in one of the dimensions is not fully used along the edges of the strip, estimates are generated only along the middle sections of the strips. The overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image. The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables.     

A unified setting for decoupling with preview and fixed-lag smoothing in the geometric context

Exact decoupling with preview, perfect tracking of previewed references, unknown-input state observation with fixed lag, and left inversion with fixed lag are considered from a unifying perspective where exact decoupling with preview is the basic problem. Necessary and sufficient constructive conditions for decoupling with finite preview are proved in the geometric framework. Structural and stabilizability conditions are considered separately and the use of self-bounded controlled invariant subspaces allows the dynamic compensator with the minimal unassignable dynamics to be straightforwardly derived. A steering along zeros technique is devised to guarantee decoupling with stability also in the presence of unstable unassignable dynamics of the minimal self-bounded controlled invariant.     

Duality of linear fixed-point smoothing and fixed-time control

This paper considers the duality of linear fixed-point smoothing and fixed-time quadratic regulator problems in which the quadratic cost functional is positive definite with respect to control. It is further shown that the dual of the fixed-point smoothing algorithms in coloured measurement noise is the fixed-time regulator in which the quadratic cost functional is restricted to be non-negative definite.     

Monotonic convergence of fixed-point algorithms for ICA

We re-examine a fixed-point algorithm proposed by Hyvarinen for independent component analysis, wherein local convergence is proved subject to an ideal signal model using a square invertible mixing matrix. Here, we derive step-size bounds which ensure monotonic convergence to a local extremum for any initial condition. Our analysis does not assume an ideal signal model but appeals rather to properties of the contrast function itself, and so applies even with noisy data and/or more sources than sensors. The results help alleviate the guesswork that often surrounds step-size selection when the observed signal does not fit an idealized model.     

Recursive method of smoothing curvature of path in path planning problems for wheeled robots

Path planning problem is considered as follows. Wheeled robot is manually guided over the required path; coordinates of the path are measured by GNSS receiver. Following the initial path in automatic mode makes it necessary to construct a path which satisfies certain smoothness requirements and curvature constraints. Small measurement errors for the coordinates of the corresponding points may substantially modify the curvature of the constructed path (thus, making it inapplicable for control). A recursive method of smoothing curvature of the curve (composed of uniform cubic B-splines) is proposed; the method can be used in real-time under constraints imposed on available random-access memory.     

Fast and robust fixed-point algorithms for independent componentanalysis

Independent component analysis (ICA) is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible. We use a combination of two different approaches for linear ICA: Comon's information theoretic approach and the projection pursuit approach. Using maximum entropy approximations of differential entropy, we introduce a family of new contrast functions for ICA. These contrast functions enable both the estimation of the whole decomposition by minimizing mutual information, and estimation of individual independent components as projection pursuit directions. The statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model, and it is shown how to choose contrast functions that are robust and/or of minimum variance. Finally, we introduce simple fixed-point algorithms for practical optimization of the contrast functions     

Parallel algorithms for the iterative solution of sparse least-squares problems

In this paper, we are concerned with a generalized Gauss-Seidel approach to sparse linear least-squares problems. Two algorithms, related to those given by Schechter (1959), for the solution of linear systems are presented and their parallel implementation is discussed. In these procedures, which can be viewed as an alternative ordering of the variables in the SOR methods, the variables are divided into nondisjoint groups. Numerical results, obtained on CRAY X-MP/48, are presented and discussed.     

When does the H/sup /spl infin// fixed-lag smoothing performance saturate for a finite smoothing lag?

A notable difference between the H/sup 2/ and H/sup /spl infin// smoothing is that the achievable performance in the latter problem might "saturate" as the function of the smoothing lag in the sense that there might exist a finite smoothing lag for which the achievable performance level is the same as for the infinite smoothing lag. In this note, necessary and sufficient conditions under which such a saturation occurs are derived. In particular, it is shown that the H/sup /spl infin// performance saturates only if the H/sup /spl infin// norm of the optimal error system is achieved at the infinite frequency, i.e., if the worst case disturbance for the infinite smoothing lag case can be arbitrarily fast and thus in a sense unpredictable.     

On the H fixed-lag smoothing: how to exploit the information preview

The problem of the continuous-time H^∞ fixed-lag smoothing over the infinite horizon is studied. The first solution to the problem is derived in terms of one algebraic Riccati equation of the same dimension as in the filtering case and the mechanism by which the performance improvements with respect to the H^∞ filtering occur is clarified. It is shown that the H^∞ smoother exploits the information preview in an “H² manner”.     

Parallel filtering and smoothing algorithms

This paper develops algorithms for filtering and smoothing for parallel computers. Numerical results are presented and implementation details are discussed. In the example it is illustrated that parallel methods have better convergence properties than nonparallel methods for nonlinear problems.     

Recursive algorithms for the representation of parametric curves and surfaces

Recursive algorithms for the representation of parametric curves and surfaces are presented which are based upon a geometric property of the de Casteljau algorithm. The algorithms work with triangular and pyramidal arrays that provide an easy handling of the curve and the surface ‘in a large’ design.     

Implementation of two-level algorithms using fixed-point arithmetic

The effect of using fixed-point arithmetic in the digital implementation of two-level control algorithms is examined in this paper. Analytical expressions are developed to predict the change in the expected minimum cost and associated matrices. It is shown that there is a favourable match between the analytical predictions and averaged simulation experiments. The use of finite-precision machines increases the expected theoretical minimum cost and makes the two-level algorithm become slow and thus require excessive iterations to converge.     

Computational complexity and solution algorithms for flowshop scheduling problems with the learning effect

In this paper, we analyze the two-machine flowshop problem with the makespan minimization and the learning effect, which computational complexity was not determined yet. First, we show that an optimal solution of this problem does not have to be the ‘permutation’ schedule if the learning effect is taken into consideration. Furthermore, it is proved that the permutation and non-permutation versions of this problem are NP-hard even if the learning effect, in a form of a step learning curve, characterizes only one machine. However, if both machines have learning ability and the learning curves are stepwise then the permutation version of this problem is strongly NP-hard. Furthermore, we prove the makespan minimization problem in m-machine permutation proportional flowshop environment remains polynomially solvable with identical job processing times on each machine even if they are described by arbitrary functions (learning curves) dependent on a job position in a sequence. Finally, approximation algorithms for the general problem are proposed and analyzed.     

Polynomial fixed-point smoothing of uncertainly observed signals based on covariances

In this article, the least-squares νth-order polynomial fixed-point smoothing problem of uncertainly observed signals is considered, when only some information about the moments of the processes involved is available. For this purpose, a suitable augmented observation equation is defined such that the optimal polynomial estimator of the original signal is obtained from the optimal linear estimator of the augmented signal based on the augmented observations and, hence, a recursive algorithm for this linear estimator is deduced. The proposed estimator does not require the knowledge of the state-space model of the signal, but only the moments (up to the 2νth one) of the signal and observation noise, as well as the probability that the signal exists in the observations.     

Fuzzy smoothing algorithms for variable structure systems

A variable structure system (VSS) is a control system implementing different control laws in different regions of the state space divided by a set of boundary manifolds. The control input switches from one control law to another when the state crosses the boundary manifolds. In general, the control input may not be smooth when switching at these boundary manifolds and may excite high frequency dynamics. This paper proposes two fuzzy rule based algorithms for smoothing the control input. The merits of these fuzzy smoothing control algorithms are illustrated by two examples: a semiactive suspension system based on optimal control and a direct drive robot arm under discrete time sliding mode control. The controller design for these two examples is a blend of traditional control theoretic approaches and fuzzy rule based approaches