Abstract
Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because previous work has focused on the definition and computationally efficient implementation of arithmetic operations and extensions of nonlinear standard functions, where all arguments are replaced by thick ellipsoids, this paper introduces novel operators for specifically evaluating quasi-linear system models with bounded parameters as well as for the union and intersection of thick ellipsoids. These techniques are combined in such a way that a discrete-time state observer can be designed in a predictor-corrector framework. Estimation results are presented for a combined observer-based estimation of state variables as well as disturbance forces and torques in the sense of an unknown input estimator for a hovercraft.
Highlights
Predictor-corrector approaches for the model-based estimation of state variables and disturbances were presented in several previous research activities [1,2]
An extension of the technique for computing Dikin ellipsoids to a scenario in which two ellipsoids with different centers are intersected, is based on the following threestage procedure: Step 1 Determine the common center point for the desired inner and outer bounds of the intersection that must be included in all ellipsoids to be intersected; Step 2 Determine initial approximations of the shape matrices for the inner and outer bounds according to Section 3.2.1; Step 3 For non-empty inner bounds, correct the outer enclosure so that the inner and outer ellipsoid surfaces become parallel to each other and, form a thick ellipsoid ((E))
The correction step of the thick ellipsoid state estimator at the point k + 1 is given by the application of Theorem 5 with the following step-by-step procedure: 1. Determine the inner shape matrix on the basis of Equation (47) according to
Summary
Predictor-corrector approaches for the model-based estimation of state variables and disturbances were presented in several previous research activities [1,2]. The lack of information concerning domains that belong to the exact sets of reachable states makes it impossible to distinguish between the phenomena of wide state enclosures as a consequence of uncertainty or the result of wide bounds due to pessimism in numerical evaluations The latter aspect is well-known in the frame of interval analysis and denoted as overestimation that can be traced back to the so-called dependency effect (numerous variables are treated as independent despite underlying physical or mathematical correlations) or to the wrapping effect (complex-shaped solution sets are replaced conservatively by more simple outer bounds which are subsequently propagated further) [11,12,13,14].
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