Output feedback boundary control of 2×2 semilinear hyperbolic systems

Abstract

We present a control design for semilinear and quasilinear 2 × 2 hyperbolic partial differential equations with the control input at one boundary and a nonlinear ordinary differential equation coupled to the other. The controller can be designed to asymptotically stabilize the system at an equilibrium or relative to a reference signal. Two related but different controllers for semilinear and general quasilinear systems are presented and the additional challenges in quasilinear systems are discussed. Moreover, we present an observer that estimates the distributed PDE state and the unmeasured ODE state from measurements at the actuated boundary only, which can be used to also solve the output feedback control problem.

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

“…Evaluation of the presented control and estimation laws requires solving PDE systems. For linear systems, explicit formulas for the state-feedback control law could be derived in (Strecker and Aamo, 2016) for the case n = 1. Such explicit formulas are desirable from a computational point of view, and it should be investigated if the same is possible for n > 1.…”

“…Remark 10. Exponential Lyapunov stability of the origin can be proven similarly as in (Strecker and Aamo, 2016). …”

“…However, backstepping is (still) limited to the linear case. Recently, we presented a constructive method for the controller and observer design for semilinear systems (Strecker and Aamo, 2016), and applied it to the so-called heave problem in drilling (Strecker and Aamo, 2017a). In the present paper, we generalize this method to n semilinear systems in series interconnection.…”

“…A detailed proof for the case n = 1 can be found in (Strecker and Aamo, 2016), which needs to be modified by including the coupling conditions (4)-(5). First, the PDE system (1)-(5) (note that we omit the actuation (6)) is transformed into integral equations by integrating them along their characteristic lines, similarly as it is done in the proof of Theorem 9.…”

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

“…Constructive methods for feedback control and state estimation have been developed recently in the form of backstepping for linear systems, see e.g. [11] and subsequent papers, and in [12] for semilinear systems. These papers consider actuation and sensing at one boundary of the domain.…”