Convergence analysis of the quasi‐OBE algorithm and related performance issues

Abstract

Quasi-OBE' (QOBE) is an adaptive set identification and filtering algorithm which is based on the principles of optimal bounding ellipsoid processing, but which has other geometric and classic leastsquares interpretations which greatly enhance its application potential. In particular, because of its unusual optimization criterion, the ellipsoidal membership set associated with QOBE is more likely to retain (i.e. to move in the parameter space with) the system model's 'true parameters,' say h * , when those parameters are time varying. Moreover, in the unlikely event that h * moves outside the set, the integrity of the point-set estimation remains intact, and the estimator provably converges under known conditions. The consistency of the set estimation can be restored at any time using typical 'rescue procedures' if desired. Understanding convergence performance is very critical to successful QOBE application. Convergence analysis of both the central point estimate and measures of the hyperellipsoidal membership set is presented. The main results give conditions for point estimate convergence, and show that set convergence to a point is not possible. Implications of these convergence results for practical application are discussed.

Fault diagnosis based on set membership identification using output‐error models

“…In [56], the optimization problem stands for the computation of the minimum volume ellipsoid that outer bounds the intersection of the prior ellipsoid and a data-hyperstrip, generated by considering an equation-error model. The fact that the OVE algorithm was specifically designed following a geometrical approach [57] (other ellipsoidbased SMI algorithms rely on a recursive least-square type of computation [58,59]) facilitated (i) the proper customization of the OVE algorithm in order to compute the minimum volume ellipsoid that outer bounds the intersection of a prior ellipsoid and a data-hypersector following the approach in [55] and (ii) checking the intersection of an ellipsoid and a data-hypersector, which requires more consistency tests than checking the intersection of an ellipsoid and a data-hyperstrip.…”

Robust Bounding Ellipsoidal Adaptive Constrained least-squares algorithm and its performance analysis

“…The optimal bounding ellipsoids (OBE) algorithms approximate the membership function by tightly outer-bounding it with ellipsoids in the associated parameter space [7][8][9] that are well-established SM adaptive filter algorithms. A thorough review of the OBE family can be found in [10].…”

“…It reduces the power consumption and computational complexity. The objective of a SM filter is to estimate set membership function, which defines the SM filter's performance specification [5,6,7].The optimal bounding ellipsoids (OBE) algorithms approximate the membership function by tightly outer-bounding it with ellipsoids in the associated parameter space [7][8][9] that are well-established SM adaptive filter algorithms. A thorough review of the OBE family can be found in [10].…”

Fault Detection and Diagnosis enhanced by a reconfiguration control scheme for a micro-electrostatic actuator

“…The proposed FDD method relying on Set Membership Identification [9], aims at capturing the microactuator's failure modes that occur abruptly. Under the knowledge of the bounds of the noise that corrupts the measurement data, the goal of SMI is the computation of the feasible parameter set containing the system's nominal parameter vector using geometric-based SMI-techniques [10,11]. The hyper-ellipsoid bounding the feasible parameter set and its support-orthotope are computed [12].…”