Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

Abstract

We detail in this article the development of a robust stabilizing output feedback control law for an underactuated cascade network of n systems of two heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the subsystems is actuated. The proposed approach combines successive backstepping transformations that present a specific cascade structure. With these transformations it is possible to rewrite the original network system as a simple system for which all the in-domain coupling terms have been removed. One can then design a strictly proper stabilizing control law.The proposed control law is proved to be robust to small delays in the actuation. Finally, a boundary observer is designed, enabling stabilization by output feedback.

Stabilization of a hyperbolic PDEs-ODE network using a recursive dynamics interconnection framework

“…One of the main advantages of such an approach is that, provided we can solve each subsystem's tracking problem, and correctly predict each subsystem's state (which is possible for ODEs and hyperbolic PDEs), it becomes straightforward to add new subsystems. This results in a simpler and more generic control design procedure than the one developed in [4].…”

“…The main advantage of the proposed methodology is its potential extension to arbitrary chains of ODEs and PDEs. Moreover, the conditions derived in this paper are less restrictive than those used in [4].…”

“…This problem has already been solved in [4] for the case of a chain of PDEs (no ODE at the end of the chain). The method presented in [4] uses a triangular integral transformation to move most of the in-domain couplings at the actuated boundary. We believe that such an approach could be adjusted to the presence of the ODE.…”

“…However, this would require deriving and solving a new set of complex kernel equations. Generally speaking, the approach proposed in [4] cannot be straightforwardly extended when adding new blocks inside the chain that have a different structure (i.e. adding ODEs in the chain or PDEs that have a different number of states).…”

“…In recent contributions [2], the authors have stabilized systems with a PDE-ODE-PDE structure. Finally, the general problem of the stabilization of an arbitrary number of interconnected scalar PDEs has been solved in [4], introducing a complex backstepping transformation.…”