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Linear-Quadratic optimal control for boundary controlled networks of waves

Anthony Hastir, Birgit Jacob, Hans Zwart

Abstract

Linear-Quadratic optimal controls are computed for a class of boundary controlled, boundary observed hyperbolic infinite-dimensional systems, which may be viewed as networks of waves. The main results of this manuscript consist in converting the infinite-dimensional continuous-time systems into infinite-dimensional discrete-time systems for which the operators dynamics are matrices, in solving the LQ-optimal control problem in discrete-time and then in interpreting the solution in the continuous-time variables, giving rise to the optimal boundary control input. The results are applied to two examples, a small network of three vibrating strings and a co-current heat-exchanger, for which boundary sensors and actuators are considered.

Linear-Quadratic optimal control for boundary controlled networks of waves

Abstract

Linear-Quadratic optimal controls are computed for a class of boundary controlled, boundary observed hyperbolic infinite-dimensional systems, which may be viewed as networks of waves. The main results of this manuscript consist in converting the infinite-dimensional continuous-time systems into infinite-dimensional discrete-time systems for which the operators dynamics are matrices, in solving the LQ-optimal control problem in discrete-time and then in interpreting the solution in the continuous-time variables, giving rise to the optimal boundary control input. The results are applied to two examples, a small network of three vibrating strings and a co-current heat-exchanger, for which boundary sensors and actuators are considered.
Paper Structure (9 sections, 8 theorems, 85 equations, 2 figures)

This paper contains 9 sections, 8 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Well-posedness analysis
  3. LQ-optimal control approach
  4. Control objective
  5. Discrete-time systems approach for LQ-optimal control
  6. Examples
  7. A network of interconnected vibrating strings
  8. Co-current heat exchanger
  9. Perspectives and future works

Key Result

Lemma 1

Under Assumption Assum_M, the two following differential equations with $\zeta\in (0,1]$ and $Q(0) = I = P(0)$, possess unique continuous and continuously differentiable solutions. Moreover, $P$ and $Q$ are invertible and satisfy for all $\zeta\in [0,1]$.

Figures (2)

  • Figure 1: Schematic view of three interconnected vibrating strings. \newlabelfig:ConnectedStrings0
  • Figure 2: Schematic profile view of a co-current heat-exchanger. \newlabelfig:HE0

Theorems & Definitions (24)

  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Remark 2.1
  • Proposition 2.1
  • Proof 3
  • Remark 2.2
  • Lemma 1
  • Proof 4
  • ...and 14 more