Uniqueness in time limited input reconstruction. (15th March 2023)

• Record Type: Journal Article
• Authors: Bernal, Dionisio; Ulriksen, Martin D.
• Is Part Of: Mechanical systems and signal processing. Volume 187(2023)
• Publication Date: 2023-03-15
• Subjects: Input reconstruction -- Impact loads -- Observability -- Controllability -- Uniqueness; Structural dynamics -- Periodicals; Vibration -- Periodicals; Constructions -- Dynamique -- Périodiques; Vibration -- Périodiques; Structural dynamics; Vibration; Periodicals; 621
• DOI: 10.1016/j.ymssp.2022.109900
• Languages: English
• ISSNs: 0888-3270

Abstract

Highlights: A necessary condition for the numbers of inputs that can be uniquely reconstructed from a set of outputs is presented. A sufficient condition for uniqueness based on the rank of a matrix that accounts for the input–output arrangement is given. It is shown that the number of inputs that can be reconstructed can be larger than the number of outputs when inputs are of short duration. Abstract: It is shown in this paper that an upper bound for the number of inputs that can be uniquely reconstructed from m outputs is r max = m n a - 2 + γ n a - 1 - 1, where n a is an upper bound on the number of time stations that support the inputs and γ = N + R a n k D c, with N being the system order and D c the input feedthrough matrix (which may be zero). An alternative phrasing being that a lower bound for the number of sensors needed to reconstruct r inputs of limited duration is m min = r n a - 1 - γ n a - 2 - 1 . These results hold on the premise that the system is linear and observable, that the inputs are accurately reconstructed by a first-order hold and that the initial condition is known. At the limits, r max or m min, the problem is typically poorly conditioned, but the situation improves away from the bounds and the anticipated robustness can be readily checked. It is shown that for some conditions the number of inputs that can be robustly reconstructed can exceed the number of outputs, sometimes by a significant margin.