We detail in this article the importance of a change of strategy for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.

We solve the problem of stabilizing a general class of linear first-order hyperbolic systems using actuation at both boundaries of the spatial domain. We design a novel Fredholm transformation similarly to backstepping approaches to derive a boundary controller and a boundary observer enabling stabilization by output feedback. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence to zero.

For most networked systems found in the literature, the actuated boundary is usually located at one end. In this paper, we first consider the stabilization of a chain of two interconnected subsystems, actuated at the in-between boundary. Each subsystem corresponds to coupled hyperbolic partial differential equations. Such in-domain actuation leads to higher complexity, and represents a significant difference with existing results. Then, starting from a classical controllability condition, we design a state feedback control law for the considered class of systems. The proposed approach is based on the backstepping methodology. However, to deal with the complex structure of the system, we use Fredholm integral transforms instead of classical Volterra transforms. We prove the invertibility of such transforms using an original operator framework. The well-posedness of the backstepping kernel equations defining the transformations is also shown with the same arguments. By using a similar procedure, we are then able to design a Luenberger-type observer. Finally, we use the state estimation in the stabilizing controller to obtain an output-feedback law, and some test cases complete the paper.

In this paper, we present three methods to achieve reliable drillbit angular velocity control for deep drilling operations. We consider a multi-sectional drilling system with the bit off-bottom, which represents the system at the start-up of a drilling operation, e.g., after a connection. The three control procedures are all based on a distributed model for the drilling system. The proposed model has been field validated and considers Coulomb friction between the drillstring and the borehole. The first algorithm we propose combines the industry standard ZTorque controller with a feedforward component. The second procedure is based on a multiplicity-induced-dominancy (MID) design that corresponds to a pole-placement for the downhole state. Finally, the last class of controllers relies on a recursive interconnected dynamics framework. All the controllers are combined with a disturbance rejection procedure whose design is based on a switching-mode approach. These three algorithms are illustrated in simulations with field scenarios on several test-cases. Their complexities, effectiveness and limitations regarding industrial implementation criteria are discussed.

In this paper, we develop a sufficient stability condition for a class of coupled first-order linear hyperbolic PDEs with constant coefficients that appear when considering target systems for backstepping boundary control. Using a backstepping transform, the problem is reformulated as a stability problem for a difference equation with distributed delay. Finding the explicit solution to the backstepping kernels, we derive an explicit sufficient condition depending on the plant coefficients. This stability condition is compared to an existing stability result based on a Lyapunov analysis. Both the proposed and existing sufficient conditions are then contrasted in some examples to a (computationally expensive) numerical approximation of a necessary and sufficient condition for exponential stability to illustrate their conservatism.