Optimal Bounding Ellipsoid Research Articles

Asymptotic properties of the OBE parameter estimation algorithm under diffuse initialization

The parameters estimation problem of autoregressive with exogenous inputs models is considered. Our approach is based on the study of the optimal bounding ellipsoid (OBE) recurrent algorithm properties. It is assumed that the shape matrix of the initial ellipsoid in the OBE algorithm is proportional to a large positive parameter. Asymptotic expansions of the relations that describe the OBE algorithm are obtained. The limit recursive algorithm (diffuse) not depending on a large parameter which may lead the OBE algorithm to a divergence is proposed and explored. The robust properties of the proposed algorithm are illustrated by a numerical example. By simulation, it is shown that it may provide better the tracking ability compared to the diffuse least-squares (RLS) algorithm and the diffuse RLS algorithm with the sliding window.

Identification Scheme for Hammerstein Output Error models with Bounded Noise

This technical note presents a method for the identification of Hammerstein Output Error (HOE) models corrupted by bounded noise. The method is based on an iterative scheme using two Optimal Bounding Ellipsoid (OBE) type algorithms. The proposed approach allows the joint estimation of the linear part and of the nonlinear part with a low computational burden. A numerical example is given so as to show the efficiency of the method.

Bounded-error identification for closed-loop systems

This paper presents a scheme for the identification of a system which operates in closed-loop and in the presence of bounded output disturbances. Two algorithms are proposed to solve this identification problem. The first algorithm is an Optimal Bounding Ellipsoid (OBE) type algorithm. This first algorithm is analyzed and sufficient conditions for stability and convergence are established. Relaxation of these conditions leads to a second identification algorithm. The implementation of that second algorithm is realized in an iterative scheme. A numerical example is provided to show the efficiency of the scheme.

Modified quasi-OBE algorithm with improved numerical properties

The quasi-OBE (QOBE) algorithm is a set-membership adaptive filtering algorithm based on the principles of optimal bounding ellipsoid (OBE) processing. This algorithm can provide enhanced convergence and tracking performance as well as reduced average computational complexity in comparison with the more traditional adaptive filtering algorithms such as the recursive least-squares (RLS) algorithm. In this paper, we show that the QOBE algorithm is prone to numerical instability due to the unbounded growth/decay of its internal variables. To tackle this problem, we develop a new set-membership adaptive filtering algorithm by transforming QOBE's internal variables into a new set of internal variables. The new algorithm, called modified quasi-OBE (MQOBE), can be viewed as an exponentially-weighted RLS algorithm with a time-varying forgetting factor, which is optimized at each iteration by imposing a bounded-magnitude constraint on the a posteriori filter output error. The proposed algorithm delivers the same convergence and tracking performance as the QOBE algorithm but with enhanced numerical properties. We demonstrate the improved numerical behavior of the proposed algorithm by simulation examples for a MIMO channel estimation problem.

Steady-state mean squared error and tracking performance analysis of the quasi-OBE algorithm

The quasi-OBE (QOBE) algorithm is a set-membership adaptive filtering algorithm based on the principles of optimal bounding ellipsoid (OBE) processing. This algorithm can provide enhanced convergence and tracking performance as well as reduced average computational complexity in comparison with the more traditional adaptive filtering algorithms such as the recursive least squares (RLS) algorithm. In this paper, we analyze the steady-state mean squared error (MSE) and tracking performance of the QOBE algorithm. For this purpose, we derive energy conservation relation of the QOBE algorithm. The analysis leads to a nonlinear equation whose solution gives the steady-state MSE of the QOBE algorithm in both stationary and nonstationary environments. We prove that there is always a unique solution for this equation. The results predicted by the analysis show good agreement with the simulation experiments.

Low-complexity implementation of quasi-OBE algorithm

The quasi-OBE (QOBE) algorithm is a set-membership adaptive filtering algorithm based on the principles of optimal bounding ellipsoid (OBE) processing. This algorithm can provide enhanced convergence and tracking performance as well as reduced average computational complexity in comparison with the more traditional adaptive filtering algorithms. In this reported work, the QOBE algorithm is implemented using the dichotomous coordinate descent iterations. This significantly reduces QOBE's per-update computational complexity, especially the number of required multiplications, and delivers a trade-off between performance and complexity.

Abstract: P934 DEVELOPING A MATHEMATICAL MODEL TO DETERMINE THE RATES OF LIPOPROTEIN METABOLIC PROCESSES FROM A SINGLE PLASMA SAMPLE

This paper presents set-membership estimation methods for narrowband interference cancellation (NBIC) in differential chaotic shift keying (DCSK) communication. Based on the boundedness of the chaotic carrier, two optimal bounding ellipsoid (OBE) algorithms are formulated, one with known and one with unknown interference parameters. Compared with the Kalman filter and recursive least-squares (RLS) algorithms, the proposed NBIC methods are computationally efficient due to the discerning update features that can estimate interfering signals with only a fraction of the observations. Simulation results illustrate that the proposed NBIC methods can suppress autoregressive (AR) or sinusoidal interference effectively.

Narrowband interference cancellation based on set-membership estimation in DCSK communication system

This paper presents set-membership estimation methods for narrowband interference cancellation (NBIC) in differential chaotic shift keying (DCSK) communication. Based on the boundedness of the chaotic carrier, two optimal bounding ellipsoid (OBE) algorithms are formulated, one with known and one with unknown interference parameters. Compared with the Kalman filter and recursive least-squares (RLS) algorithms, the proposed NBIC methods are computationally efficient due to the discerning update features that can estimate interfering signals with only a fraction of the observations. Simulation results illustrate that the proposed NBIC methods can suppress autoregressive (AR) or sinusoidal interference effectively.

Multiweight optimization in optimal bounding ellipsoid algorithms

Optimal Bounding Ellipsoid (OBE) algorithms offer an attractive alternative to traditional least-squares methods for identification and filtering problems involving affine-in-parameters signal and system models. The benefits-including low computational efficiency, superior tracking ability, and selective updating that permits processor multi-tasking-are enhanced by multiweight (MW) optimization in which the data history is considered in determining update times and optimal weights on the observations. MW optimization for OBE algorithms is introduced, and an example MW-OBE algorithm implementation is developed around the recent quasi-OBE algorithm. Optimality of the solution is discussed, and simulation studies are used to illustrate performance benefits.

A new NLMS algorithm for slow noise magnitude variation

A set-membership (SM) normalized least-mean-square (NLMS) (SMNLMS) algorithm is developed using SM theory in the class of optimal bounding ellipsoid (OBE) algorithms. This signed version of NLMS algorithm requires a priori knowledge of a bound for the error magnitude, which is unknown in most applications. A very simple algorithm is proposed for the case in which the unknown magnitude of the measurement noise is slowly time-varying. The proposed algorithm is able to extract the noise magnitude information and exploit this magnitude to enhance or accelerate the learning process without risk of overbounding or performance loss due to underbounding. The performance of the proposed algorithm is compared with that of SMNLMS using some simulation examples.

Set-membership identification and filtering for signal processing applications

Optimal bounding ellipsoid (OBE) algorithms comprise a class of novel recursive identification methods for affine-in-parameters system and signal models. OBE algorithms exploit a priori knowledge of bounds on model disturbances to derive a monotonically nonincreasing set of solutions that are feasible in light of the observations and the model structure. In turn, these sets admit criteria for ascertaining the information content of incoming observations, obviating the expense of updating when data are redundant. Relative to classical recursive methods for this task, OBE algorithms are efficient, robust, and exhibit remarkable tracking ability, rendering them particularly attractive for realtime signal processing and control applications. After placing the OBE algorithms in the hierarchy of the broader set-membership identification methods, this article introduces the underlying set-theoretic concepts, compares and contrasts the various published OBE algorithms including the motivation for each development, then concludes with some illustrations of OBE algorithm performance. More recent work on the use of OBE processing infiltering tasks is also included in the discussion. The paper is a survey of a broad and evolving topic, and extensive references to further information are included.

Some Signal Processing Applications of Set Solutions

Optimal bounding ellipsoid (OBE) algorithms provide feasible parameter sets at each time increment of an identification process. These sets contain information that can enhance tracking and classification, and provide a framework for incorporating difficult-to-quantify criteria such as perceptual quality. Some example signal processing applications of OBE parameter sets are presented in this paper.

Performance studies on a ‘quasi-obe’ algorithm for real-time signal processing

A novel algorithm is studied for the identification of linear-in-parameters models. This technique is dubbed Quasi-OBE (QOBE) because it is based on the principles of optimal bounding ellipsoid (OBE) identification, but has other geometric and classic least-squares interpretations that enhance its interpretability and application potential. Convergence behavior of both the central point estimate and measures of the hyperellipsoidal membership-set will be discussed in general terms, particularly in comparison with several conventional OBE algorithms. The QOBE algorithm uses highly-selective updating and exhibits excellent tracking ability in time-varying environments.

BEACON: an adaptive set-membership filtering technique with sparse updates

This paper deals with adaptive solutions to the so-called set-membership filtering (SMF) problem. The SMF methodology involves designing filters by imposing a deterministic constraint on the output error sequence. A set-membership decision feedback equalizer (SM-DFE) for equalization of a communications channel is derived, and connections with the minimum mean square error (MMSE) DFE are established. Further, an adaptive solution to the general SMF problem via a novel optimal bounding ellipsoid (OBE) algorithm called BEACON is presented. This algorithm features sparse updating, wherein it uses about 5-10% of the data to update the parameter estimates without any loss in mean-squared error performance, in comparison with the conventional recursive least-squares (RLS) algorithm. It is shown that the BEACON algorithm can also be derived as a solution to a certain constrained least-squares problem. Simulation results are presented for various adaptive signal processing examples, including estimation of a real communication channel. Further, it is shown that the algorithm can accurately track fast time variations in a nonstationary environment. This improvement is a result of incorporating an explicit test to check if an update is needed at every time instant as well as an optimal data-dependent assignment to the updating weights whenever an update is required.

A consistently convergent OBE algorithm with automatic estimation of error bounds

Conventional optimal bounding ellipsoid (OBE) algorithms require a priori knowledge of error bounds which is unknown in most applications. Conservative (overestimated) error bounds used in practice may lead to inconsistent parameter estimation. This paper presents an enhanced OBE algorithm that is proven to be consistently convergent without a priori knowledge of error bounds. Only a lower bound of the 'tail probability' of the disturbance process is required. Simulations comparing conventional and the enhanced OBE algorithm is provided.

Set-membership adaptive equalization and an updator-shared implementation for multiple channel communications systems

This paper considers the problems of channel estimation and adaptive equalization in the novel framework of set-membership parameter estimation. Channel estimation using a class of set-membership identification algorithms known as optimal bounding ellipsoid (OBE) algorithms and their extension to tracking time-varying channels are described. Simulation results show that the OBE channel estimators outperform the least-mean-square (LMS) algorithm and perform comparably with the RLS and the Kalman filter. The concept of set-membership equalization is introduced along with the notion of a feasible equalizer. Necessary and sufficient conditions are derived for the existence of feasible equalizers in the case of linear equalization for a linear FIR additive noise channel. An adaptive OBE algorithm is shown to provide a set of estimated feasible equalizers. The selective update feature of the OBE algorithms is exploited to devise an updator-shared scheme in a multiple channel environment, referred to as updator-shared parallel adaptive equalization (USHAPE). U-SHAPE is shown to reduce the hardware complexity significantly. Procedures to compute the minimum number of updating processors required for a specified quality of service are presented.

Stochastic Convergence of Optimal Bounding Ellipsoid Algorithms

Given appropriate persistency of excitation, the ellipsoidal seta associated with optimal bounding ellipsoid (OBE) algorithms with an interpretable optimization (volume) criterion, converge to a point under the condition that the model disturbance process visit the error bounds infinitely often. The spectral properties of the disturbance are not primary in the convergence behavior of OBE algorithms, rendering these identifiers distinctly different from the structurally similar RLS. Rigorous proof of the sufficiency of the "infinite visitation" property is given for both independent and correlated noise. In the colored noise case, it is shown that a mixing condition on the observations will assure convergence if the OBE algorithm is slightly modified. Examples of simulations are used to illustrate the theoretical developments.

Do Interpretable Optimal Bounding Ellipsoid Algorithms Converge? Part I - The Long- Awaited Set-Convergence Proof

Optimal bounding ellipsoid (OBE) algorithms with interpretable optimization (volume or trace) criteria converge under the condition that the disturbance process visit the error bounds infinitely often. This result is proven in two logical steps. First, the set-membership-stochastic approximation (SM-SA) is presented. SM-SA is a new volume minimization OBE algorithm for which convergence of both the point-estimator and the membership set is proven. Next, it is argued that the salient SM-SA convergence properties apply to all volume-based OBE algorithms. This paper is concerned with proof of convergence for the case of i.i.d. disturbances. In the companion paper (Part II), less constrained white and colored noises are considered, yielding very interesting relationships between OBE algorithms and RLS.

Do Interpretable Optimal Bounding Ellipsoid Algorithms Converge? Part II - OBE Vs. RLS: Clearing The Smoke

Unlike its close relative, RLS, the set-membership - stochastic approximation (SM-SA) algorithm, and hence any volume-minimizing optimal bounding ellipsoid (OBE) algorithm, does not depend upon second moment properties of the model disturbances for convergence. Rather the frequency with which the noise approaches arbitrary neighborhoods of the noise bounds determines convergence behavior. SM-SA will converge without bias even in the presence of colored noise if the noise approaches its bounds infinitely often.

Least-square identification with error bounds for real-time signal processing and control

Set-membership (SM) identification, which refers to a class of algorithms using certain a priori knowledge about a parametric model to constrain the solutions to certain sets, is considered. The focus is on a class of SM-based techniques that are of particular interest in applications requiring real-time processing. The optimal bounding ellipsoid (OBE) algorithms are interpreted as a blending of the classical least-square error minimization approach with knowledge of bounds on model errors arising from SM considerations. Using this interpretation, a general framework embracing all currently used OBE algorithms is developed, and strategies for adaptation and for implementation on parallel machines are discussed. Computational complexity benefits are considered for the various algorithms. The treatment is tutorial, leaving many of the formal details to an appendix that presents an archival theoretical treatment of the key results. A second appendix gives an overview of current research in the general SM identification field.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>