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A System Level Approach to LQR Control of the Diffusion Equation

Addie McCurdy, Andrew Gusty, Emily Jensen

Abstract

The continuous-time, infinite horizon LQR problem for the diffusion equation over the unit circle with fully distributed actuation is considered. It is well-known that the solution to this problem can be obtained from the solution to an operator-valued algebraic Riccati equation. Here, it is demonstrated that this solution can be equivalently obtained by solving an $H_2$ control problem through a closed-loop design procedure that is analogous to the "System Level Synthesis" methodology previously developed for systems over a discrete spatial domain and/or over a finite time horizon. The presented extension to the continuous spatial domain and continuous and infinite-horizon time setting admits analytical solutions that may complement computational approaches for discrete or finite-horizon settings. It is further illustrated that spatio-temporal constraints on the closed-loop responses can be incorporated into this new formulation in a convex manner.

A System Level Approach to LQR Control of the Diffusion Equation

Abstract

The continuous-time, infinite horizon LQR problem for the diffusion equation over the unit circle with fully distributed actuation is considered. It is well-known that the solution to this problem can be obtained from the solution to an operator-valued algebraic Riccati equation. Here, it is demonstrated that this solution can be equivalently obtained by solving an control problem through a closed-loop design procedure that is analogous to the "System Level Synthesis" methodology previously developed for systems over a discrete spatial domain and/or over a finite time horizon. The presented extension to the continuous spatial domain and continuous and infinite-horizon time setting admits analytical solutions that may complement computational approaches for discrete or finite-horizon settings. It is further illustrated that spatio-temporal constraints on the closed-loop responses can be incorporated into this new formulation in a convex manner.

Paper Structure

This paper contains 16 sections, 8 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation & Mathematical Preliminaries
  3. Problem Set Up
  4. Controller Design for the Diffusion Equation
  5. Closed-Loop Formulation of $H_2$ Problem
  6. Analytical Solution to Closed-Loop Formulation of $H_2$ Problem
  7. Implementation of Controller via Closed-Loop Solution
  8. Equivalence to LQR Control
  9. Incorporation of Spatio-Temporal Decay Rate Constraints
  10. Conclusion
  11. Proof of Proposition \ref{['prop:projection']}
  12. Proof of cost equivalence in Proposition \ref{['prop:LQR_equivalence']}
  13. Proof of Lemma \ref{['lem:KStabilizing_PhiStable']}
  14. Proof of Lemma \ref{['lem:op_constraint']}
  15. Proof of Lemma \ref{['lem:Controller_Implementation']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $K$ be a spatially-invariant and internally stabilizing controller for eq:olODE. Then the operator-valued transfer functions eq:Phi are spatially-invariant, strictly proper, well-defined on $\{s \in \mathbb{C};~ {\rm Re}(s) \ge 0 \},$ and satisfy Condition eq:Hinf_condition can be equivalently verified pointwise in the spatial frequency domain as:

Figures (2)

  • Figure 1: Block diagram of the dynamic controller implementation \ref{['eq:control_implementation']} with noisy measurements and exogenous disturbance: $w$ is additive noise to the state $\psi$, and $w$ is additive noise to the control signal $u$.
  • Figure 2: Spatio-temporal convolution kernels corresponding to the transfer functions in Fig. \ref{['fig:controller_implementation_diagram']}. Plot (a) shows the yellow block's static, $h_{\Phi^u}(\xi)$, and dynamic $f_{\Phi^u}(t, \xi)$ kernels. Plot (b) shows the blue block's dynamic kernel $f_{\Phi^\psi}(t, \xi)$.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Theorem 1
  • ...and 12 more