PI Controllers for 1-D Nonlinear Transport Equation

Abstract

In this paper, we introduce a method to get necessary and sufficient stability conditions for systems governed by 1-D nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified.

Forwarding-Lyapunov design for the stabilization of coupled ODEs and exponentially stable PDEs

“…Assumption 4. The pair (S, HQ) is detectable 1 , with Q given by Assumption 2 and H defined as in (6).…”

“…where, in the last line, we have used the definition of H given in (6). Then, using the Young's inequality, one obtains, for every ν > 0…”

“…A really well known example of the internal model approach is the integral action controller for tracking or rejection of constant signals, see, e.g. [3], [6], [24], [27]. For more sophisticated exosystems represented, for instance, by the combination of a finite number of linear oscillators, there exist some results devoted to abstract systems, i.e., systems described by operators, see, for instance, [17], [18].…”

Proportional Integral Regulation Control of a One-Dimensional Semilinear Wave Equation

“…More recently, the problem of PI boundary control of linear hyperbolic systems has been reported in a number of works [12,19,27,48]. This research direction has then been extended to the case of nonlinear transport equations [11,14,20,23,37,45]. The case of the boundary regulation control of the Neumann trace for a linear reaction-diffusion in the presence of an input delay was considered in [29].…”

Regulation of a reaction-diffusion equation with bounded observation

“…Regulation control of finite-dimensional systems are very classical problems that have been widely investigated [1,6]. This work considers the problem of regulation of distributed parameter systems [2,3,14,16,17]. This class of systems succeeds to model many dynamical systems, such as heat dynamics, chemical reactors, fluid mechanical systems, among many other potential applications (see [4] for a general reference).…”