Minimum Time Bilateral Observer Design for 2x2 Systems of Linear Hyperbolic PDEs - With Application to Oil Well Drilling State Estimation for Improved Kick Handling

We consider state estimation of 1D systems of 2x2 linear first-order coupled hyperbolic PDEs, consisting of two distributed states propagating information in opposite directions but interacting across the spatial domain. These types of systems can be used to model various technical and physical phenomena. In most practical situations only measurements from boundary points are available, and the interior states must from these be deduced. We derive a bilateral boundary observer, utilizing sensing at both boundaries of the spatial domain, to provide state estimates of the 2x2 system which are correct within a finite and theoretically minimal amount of time. This optimal convergence time is smaller than the corresponding lowest possible convergence time for unilateral boundary observers, being observers only using measurements from a single boundary point. As a first step in the derivation of the 2x2 bilateral observer, we derive for a 1D system of 2+2 linear first-order coupled hyperbolic PDEs, consisting of four distributed states coupled point-wise across the domain, of which two transmit information leftwards and the other two information rightwards, a boundary observer which relies on measurements from a single boundary that is collocated with actuation. The design is achieved by using infinite dimensional backstepping through applying a Volterra integral transformation composed with a Fredholm integral transformation. We show that this observer converges within the theoretical lower bound for convergence time of unilateral boundary observers for 2+2 systems. Next, we split the spatial domain of the 2x2 system at an interior point, and transform it to a 2+2 system by re-scaling of the sub-domains and redefining boundary conditions. A bilateral observer for the 2x2 system is subsequently derived from the previously derived minimum time unilateral observer for the 2+2 system. How the splitting point should be chosen to achieve minimum time convergence for the bilateral 2x2 observer is shown. To demonstrate the efficiency of the 2x2 bilateral observer, we implement 2x2 systems and their corresponding minimum time bilateral observers in simulations. These are bench-marked against 2x2 unilateral boundary observers only sensing a single boundary point, and it is found that the bilateral observers have superior performance to the unilateral observers. As an application of the theory derived, it is next shown how the 2x2 system can be used to model the pressure and flow dynamics of drilling fluid during oil well drilling. A scenario where the observers are used for state estimation during kick handling is considered. The scenario is simulated and it is shown that an observer using both topside and downhole measurements during drilling has a more efficient response with respect to correctly estimating the states of the drilling fluid than an observer only utilizing topside measurements, something that can contribute to safer kick handling