Exponential Stability of a Class of Boundary Control Systems

Abstract

We study a class of partial differential equations (with variable coefficients) on a one dimensional spatial domain with control and observation at the boundary. For this class of systems we provide simple tools to check exponential stability. This class is general enough to include models of flexible structures, traveling waves, heat exchangers, and bioreactors among others. The result is based on the use of a generating function (the energy for physical systems) and an inequality condition at the boundary. Furthermore, based on the port Hamiltonian approach, we give a constructive method to reduce this inequality to a simple matrix inequality.Index Terms-Boundary control systems (BCS), partial differential equations (PDEs).

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

“…Different variations around these first results can be found in (Villegas, 2007) and in (Jacob & Zwart, 2012). Well-posedness and stability have been investigated in open-loop and for static boundary feedback control in (Zwart et al, 2010) and (Villegas et al, 2005;Villegas et al, 2009) respectively, and linear dynamic boundary control has been studied in (Ramirez et al, 2014;Augner & Jacob, 2014;Villegas, 2007).…”

“…Following (Villegas et al, 2009;Ramirez et al, 2014), the objective is to interconnect (7) at the boundaries with (1), as shown in Figure 1, such that the closed-loop system is exponentially stable. In Ramirez et al (2014) it is shown that if the finite-dimensional control system is linear, strictly input-passive and exponentially stable, then the closed-loop system is exponentially stable.…”

A Piezoelectric Euler-Bernoulli Beam with Dynamic Boundary Control: Stability and Dissipative FEM

“…In spite of this progress, and due to its widespread technological applications, considerable research on EBB-control problems is still underway: In the more recent papers [22,20] exponential stability of related control systems was established by verifying the Riesz basis property. For the exponential stability of a more general class of boundary control systems (including the Timoshenko beam) in the port-Hamiltonian approach we refer to [49].…”

“…In spite of this progress, and due to its widespread technological applications, considerable research on EBB-control problems is still underway: In the more recent papers [22,20] exponential stability of related control systems was established by verifying the Riesz basis property. For the exponential stability of a more general class of boundary control systems (including the Timoshenko beam) in the port-Hamiltonian approach we refer to [49].We shall analyze an inhomogeneous multi-layered piezoelectric EBB with applied tip mass and moment of inertia, coupled to a dynamic controller that uses only low order boundary measurements. This system was introduced by Kugi and Thull in [31] to independently control the tip position and the tip angle of a piezoelectric cantilever along prescribed trajectories.…”

Energy shaping of boundary controlled linear port Hamiltonian systems

“…Such approach allows to link the variation of the energy within the system to the power flow at its boundary. In (Villegas et al, 2005) and (Villegas et al, 2009) this approach has been used to derive some simple matrix conditions to This work was supported by French ANR sponsored project HAMECMOPSYS under Reference Code ANR-11-BS03-0002 and the LABEX ANR-11-LABX-01-01 insure the exponential or asymptotic stability for a class of linear 1D boundary controlled systems. Port Hamiltonian formulation has also been used to design stabilizing control laws by energy shaping (Macchelli and Melchiorri, 2004).…”