Robust Control Design of Underactuated 2 × 2 PDE-ODE-PDE Systems

Abstract

In this paper, we design a robust stabilizing controller for a system composed of two sets of linear heterodirectional hyperbolic PDEs, with actuation at one boundary of one of the PDEs, and couplings at the middle boundary with ODEs in a PDE-ODE-PDE configuration. The system is underactuated since only one of the PDE systems is actuated. The design approach employs a backstepping transformation to move the undesired system couplings to the proximal boundary (where the actuation is located). We can then express this target system as a time-delay neutral system for which we can design an appropriate control law to obtain an exponentially stable target system.

Robustification of stabilizing controllers for ODE-PDE-ODE systems: A filtering approach

“…Thus, w-stability implies delay-robust stability in the sense of [33]. It also includes robustness w.r.t some uncertainties (but not all, since it has been shown in [5] that uncertainties in the transport velocities cannot be modeled by approximate identities).…”

“…We now consider the case where the control input acts at the connection point between the PDE and ODE (B X ≡ 0). This case has not been well studied in the literature and most of the contributions do not consider the ODE X 0 [8,5,21]. However, in all these works, canceling the PDE reflection term Qu(t, 0) simplified the analysis, and the resulting control laws verify the conditions of Theorem 10.…”

Output-feedback control of an underactuated network of interconnected hyperbolic PDE–ODE systems

“…However, it required solving a complex set of kernel equations and could not be easily extended to 45 more complex systems (since new target systems would be needed). The backstepping approach has also been used to stabilize PDEs interconnected with ODEs in chain structures and, in particular, to design control laws stabilizing an ODE-PDE-ODE structure [17,18,19,20,21], and 50 PDE-ODE-PDE structures [22,23]. More recently, this approach has been implemented in [24] to delay-robustly stabilize a system of hyperbolic PDEs coupled with an ODE in an n+m PDE-ODE structure.…”