Control of 2 × 2 Linear Hyperbolic Systems: Backstepping-Based Trajectory Generation and PI-Based Tracking

International audienceWe consider the problems of trajectory generation and tracking for general 2 × 2 systems of first-order linear hyperbolic PDEs with anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of 2 × 2 linear hyperbolic systems. We apply our approach to a wave PDE with indefinite in-domain and boundary damping. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove exponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel " non-diagonal " Lyapunov functional. We demonstrate that the proposed PI control law compensates in the output the effect of in-domain and boundary disturbances. We illustrate our results with numerical examples

Diffusion and robustness of boundary feedback stabilization of hyperbolic systems

We consider the problem of boundary feedback control of single-input-single-output (SISO) one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up of two interconnected transport equations and provided with anti-located boundary sensing and actuation. The main contribution is to show that the robustness of the control against delay uncertainties is recovered as soon as an arbitrary small diffusion is present in the system. Our analysis also reveals that the effect of diffusion on stability is far from being an obvious issue by exhibiting an alternative simple example where the presence of diffusion has a destabilizing effect instead.Comment: 21 page

Stabilization of underactuated linear coupled reaction-diffusion PDEs via distributed or boundary actuation

This work concerns the exponential stabilization of underactuated linear homogeneous systems of $m$ parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either internally or from the right boundary and in which the diffusion coefficients are distinct. For the distributed control case, a proportional-type stabilizing control is given explicitly. After applying modal decomposition, the stabilizing law is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where the calculation of the stabilization law is independent of the arbitrarily large number of these modes. This is achieved by solving generalized Sylvester equations recursively. For the boundary control case, the proposed controller is dynamic under appropriate sufficient conditions on the coupling matrix (reaction term). A dynamic extension technique is first performed via trigonometric change of variables that places the control internally. Then, modal decomposition is applied followed by a state transformation of the ODE system which must be stabilized in order to be written in a form in which a dynamic law can be established. For both distributed and boundary control systems, a constructive and scalable stabilization algorithm is proposed, as the choice of the controller gains is independent of the number of unstable modes and only relies on the stabilization of the reaction term. The present approach solves the problem of stabilization of underactuated systems when in the presence of distinct diffusion coefficients. The problem is not directly solvable, similarly to the scalar PDE case.Comment: arXiv admin note: substantial text overlap with arXiv:2202.0880

Exponential stability of general 1-D quasilinear systems with source terms for the C1C^1 norm under boundary conditions

We address the question of the exponential stability for the $C^{1}$ norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic $C^{1}$ Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the $C^{1}$ norm. We show that the existence of a basic $C^{1}$ Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the $C^{1}$ norm and we show that the interior condition is also necessary to the existence of a basic $C^{1}$ Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the $C^{p}$ norm, for any $p\geq1$.Comment: 33 page

Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation

This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations