Output feedback boundary control of series interconnections of 2 × 2 semilinear hyperbolic systems

Abstract

In this paper, we present an output feedback controller for systems consisting of n 2 × 2 semilinear hyperbolic systems in series interconnection where actuation and sensing are restricted to one boundary. The output-feedback control law consists of a state-feedback controller combined with an observer. The control and estimation laws are based on the dynamics on the characteristic lines of the hyperbolic system, and achieve stabilization of the origin or tracking at one location, and full state estimation, respectively, globally and in minimum time. We demonstrate the controller performance in a numerical example, and apply the controller to a relevant disturbance rejection problem in oil well drilling.

Two-sided boundary control and state estimation of 2 × 2 semilinear hyperbolic systems

“…Likewise, the state is estimated via reconstructing the past state on the characteristic line along which the measurements evolve, and is to the best of our knowledge the first constructive observer design for such systems using sensing at both boundaries. The present paper is a continuation of recent results using this method [12], [17], and it would be interesting to see which other cases this approach can be applied to, such as general heterodirectional systems with variable numbers of actuators and sensors at each boundary.…”

“…Replacing v t (x,t + φ u (x)) in the latter equation by (27) yields (17). Repeating the same steps for (ū 2 ,v 2 ) gives (18)-(19).…”

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

“…Combining the operators F and F 1 , a direct consequence of this theorem is that system (37)-(40) and (5)-(8) have equivalent stability properties. Thus, we only have to find a control law U (t) that ensures the exponential stability of (37)- (40). This is straightforward due to Assumption 2.…”

“…Regarding, the boundary conditions in x = 0 (equations (A.13)-(A.16) and (A.25)-(A.28)), they are required to guarantee that the boundary conditions (33)- (34) and (39)- (40) hold. More precisely, there are integral terms in (33)-(34) that do not appear in (39)- (40). Thus, the kernel boundary conditions in (0, ξ) are used to remove these integral terms.…”

“…Moreover, even if the proposed method is straightforward to implement, it may require solving a set of PDEs online, which is computationally expensive. More recently in [40], the authors have considered the output feedback control of a semilinear hyperbolic system with a structure analogous to the one considered in this paper, using the dynamics on the characteristic lines. Again however, the proposed state-feedback control law requires solving a set of PDEs online.…”