Minimum time observer designs for n+m linear hyperbolic systems with unilateral, bilateral or pointwise in-domain sensing

Abstract

In this paper we derive minimum time convergent observers for systems of linear coupled first-order 1-D hyperbolic PDEs, that use either unilateral (single boundary measured), bilateral (both boundaries measured) or pointwise in-domain sensing. First, a Volterra integral transformation is combined with a Fredholm integral transformation to derive a minimum time unilateral observer for systems. Then, it is shown that an system with bilateral sensing can be transformed to an system with unilateral sensing via an invertible coordinate transformation. The bilateral observer is subsequently obtained from the minimum time unilateral observer, and it is shown that it converges in a theoretical minimum time for bilateral sensing. In a similar fashion, the observer using pointwise in-domain measurement is derived using the same techniques. The performances of the bilateral and unilateral observers are demonstrated in simulations.