Robust Filtering for Linear Systems With Convex-Bounded Uncertain Time-Varying Parameters

Abstract

This note addresses the design of robust filters for linear systems with a state-space model subject to time-varying uncertain parameters with limited variation. The uncertain parameters and their rate of variation are assumed to belong to a given convex-bounded polyhedral domain. A method based on a parameter-dependent Lyapunov function is proposed for designing a linear stationary asymptotically stable filter with a guaranteed average error variance, irrespective of the uncertain parameters. The proposed design is formulated in terms of linear matrix inequalities.

An IQC approach to robust estimation against perturbations of smoothly time-varying parameters

“…With a simplified version of the recent result in [15] we infer that (8) and (5) imply stability of A and, with the KYP-lemma, the validity of (2), which in turn proves robust stability of the feedback interconnection of G qp and ∆ r . Now in order to prove robust H 2 -performance, let us describe the closed-loop trajectories of system (4) for an impulsive input in the direction of w η as    ẋ…”

“…We especially focus on the recent results of [8], [9] where the proposed H 2 -filter design methods are based on PDLFs that depend quadratically on the uncertain parameters. It should be mentioned that the stochastic interpretation of the H 2 -norm, which is used in [7], [8], [9], is equivalent to the signal based interpretation we are using such that we are allowed to compare our results.…”

“…We especially focus on the recent results of [8], [9] where the proposed H 2 -filter design methods are based on PDLFs that depend quadratically on the uncertain parameters. It should be mentioned that the stochastic interpretation of the H 2 -norm, which is used in [7], [8], [9], is equivalent to the signal based interpretation we are using such that we are allowed to compare our results. We hence took the following numerical example from [8], [9] Figure 2 displays the squared H 2 -norm from w → e (W = I) for the plant (21) with |δ(t)| ≤ 1.4 and for increasing values of µ (length of basis) µ and |υ| (rate-bound) for the methods from [8], [9] and those in this paper.…”

“…The lines "IQC, µ = 0" and "PDLF 0" correspond to the standard quadratic stability tests where δ(t) is allowed to vary arbitrarily fast. The lines "PDLF 1" and "PDLF 2" represent the results of [8] and respectively use a Lyapunov function that depends affinely and quadratically on δ. The line "PDLF 3" corresponds to the results of [9] and is not a complete solution in terms of LMIs, due to a bilinear term.…”

“…In [5] and [6], methods were suggested for systems that depend affinely on uncertain time-invariant parameters. Only recently, in [7], [8], [9], these methods were generalized to the case in which the parameters are allowed to be time-varying. Despite the fact that the results were considerably improved, they all still might yield conservative results.…”