Low‐complexity ISS state estimation approach with bounded disturbances

Abstract

This paper presents a low-complexity input-to-state stable ellipsoidal outer-bounding state estimation approach with unknown but bounded disturbances. The bounds on the noise are specified by ellipsoids. The feasible set is updated through computing the Minkowski sum and intersection of two ellipsoids. At the observation stage, the observation noise bounding ellipsoid is replaced by a parallelotope containing it. Then, each observation update is transformed into multiple consecutive iterations to intersect ellipsoid with strips, which significantly reduces its per-update computational complexity.Furthermore, an adaptive selection scheme of the parameters is derived to ensure the stability of the estimation error. As a result, the proposed approach entails stability and delivers a trade-off between performance and complexity.

Stability of Linear Set-Membership Filters

“…Alice and Bob employ the classical filtering framework in Algorithm 1, while Carol use the OIT-inspired filtering framework in Algorithm 2. 10 From Fig. 2, we can see that the difference of the initial conditions chosen by these three designers are corrected by the measurements y 0:6 , and the estimates converge to the true conditional range x 6 |y 0:6 .…”

“…The most related work is on the uniform boundedness of the estimate, which guarantees that the size of estimate does not increase unboundedly with time. For ellipsoidal SMFs, [8] proposed an input-to-state stable algorithm that ensures the uniform boundedness of the estimate; since in [8] the input-to-state stability required to solve a timeinefficient polynomial equation at each time step, [9] developed an SMF with increased efficiency based on minimizing an important upper bound; then, [10] provided a parallelotopebounding technique to reduce the complexity, which was further improved by [11]. For polytopic SMFs, [12] proposed a zonotopic Kalman filter, where a sufficient condition (called robust detectability) for the uniform boundedness of the estimate was given; in [13], a zonotopic SMF was designed for linear parameter-varying systems, and an upper bound on the radius of the estimate was derived by solving linear matrix inequalities (LMIs); based on [13], the article [14] proposed a zonotopic SMF for switched linear systems, where an LMIbased upper bound was obtained for the radius of the estimate.…”

Bounded-error constrained state estimation of LTV systems in presence of sporadic measurements

“…(87) where λ k > 1 and W k SPSD, are defined in (76a) and (76b) and both bounded. Basing on the same reasoning as done in Lemma 3 in[22], it can be shown that for any vectors a, b ∈ IR n and any matrices A, B ∈ IR n×n , (a + b) T (A + B) † (a + b) ≤ a T A † a + b T B † b (88)…”

“…Remark 3.1 It is worth noting that the volume minimization problem arg min µi det † (Q ki ) has an explicit solution here. If the unknown input vector was bounded by an ellipsoid, as was the case in [11,13,15,22], rather than by an interval-like set, such as a zonotope, µ vol ki would be the unique positive root of an n−order polynomial. Nevertheless, considering that the pseudo-inverse of a n × n matrix is needed at each time step k, in line with (11i), the trace minimization is more appealing, at least from the computational point of view.…”

“…Consequently, in contrast with all the algorithms in the literature, [11,12,13,14], that minimize the size of the ellipsoid Ȇ(β) and where the stability issue was not addressed, the optimal value of β chosen here is the one for which the input-to-state stability of the estimation algorithm to be derived, could be fulfilled, by minimizing some quadratic measure of the estimation error vector 4 in the worst noise case, embodied by ς, in the manner of [15,22], inspired by [27,35,36]. (24b) and its minimum is achieved at…”