$H^\infty$-control for a class of boundary controlled hyperbolic PDEs

TL;DR

This work addresses suboptimal $H^ olinebreak[4]\infty$ control for a class of boundary-controlled hyperbolic PDEs by establishing an equivalent infinite-dimensional discrete-time representation. It then leverages finite-dimensional discrete-time $H^ olinebreak[4]\infty$ theory, via a Kalman-Szeg\H{o}-Popov-Yakubovich Riccati framework, to synthesize a stabilizing controller expressed as a linear fractional transformation. The main contributions include a rigorous well-posedness analysis, a constructive controller design in the discrete-time setting, and a continuous-time realization that preserves $H^ olinebreak[4]\infty$ performance, demonstrated on a boundary controlled vibrating string. The approach converts boundary unbounded operators into multiplication operators, enabling practical boundary controllers for infinite-dimensional plants with significant implications for networked and distributed systems engineering.

Abstract

A solution to the suboptimal $H^\infty$-control problem is given for a class of hyperbolic partial differential equations (PDEs). The first result of this manuscript shows that the considered class of PDEs admits an equivalent representation as an infinite-dimensional discrete-time system. Taking advantage of this, this manuscript shows that it is equivalent to solve the suboptimal $H^\infty$-control problem for a finite-dimensional discrete-time system whose matrices are derived from the PDEs. After computing the solution to this much simpler problem, the solution to the original problem can be deduced easily. In particular, the optimal compensator solution to the suboptimal $H^\infty$-control problem is governed by a set of hyperbolic PDEs, actuated and observed at the boundary. We illustrate our results with a boundary controlled and boundary observed vibrating string.

Figures (4)

• Figure 1: Open-loop system $\Sigma$ with inputs $d$ and $u$ and outputs $z$ and $y$.
• Figure 2: Closed-loop system resulting from the interconnection of $\Sigma$ and $\Sigma_c$.
• Figure 3: Closed-loop system resulting from the interconnection of $\Sigma$ and $\Sigma_c$ with $\Sigma_c$ being the left fractional transformation of $\Sigma_g$ and $\Sigma_Q$.
• Figure 4: Representation of $\Vert\mathbf{G}_{cl}^{opt}(e^{i\theta})\Vert$ for $\theta\in [0,2\pi]$ and $\sigma\in[6.1,13.1]$.

Theorems & Definitions (20)

• Proposition 1
• proof : Proof.
• Lemma 1
• proof : Proof.
• Remark 1
• Proposition 2
• proof : Proof.
• Definition 1
• Remark 2
• Definition 2