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A Convex Parameterization of Controllers Constrained to use only Relative Measurements

Walden Marshall, Bassam Bamieh, Emily Jensen

TL;DR

The paper addresses optimal controller design when sensors provide only relative measurements, a setting that challenges standard convex approaches, especially under additional network-structure constraints. It introduces a relative-feedback parameterization, showing that relative measurements permit a convex reformulation over a linear subspace S_rel(C2) via an auxiliary transfer R (and its Q-parameterization), with a recoverability link to the original controller K. By leveraging quadratic invariance, the authors show how network constraints can be integrated into the same convex framework, enabling tractable design in distributed settings. A numerical ring-graph example demonstrates practical gains and computational simplifications (e.g., circulant Q), indicating the approach's potential for scalable, distributed control with relative sensing.

Abstract

The optimal controller design problem for systems equipped with sensors that measure only relative, rather than absolute, quantities is considered. This relative measurement structure is formulated as a design constraint; it is demonstrated that the resulting constrained controller design problem can be written as a convex program. Certain additional network structural constraints can be incorporated into this formulation, making it especially useful in distributed or networked settings. An illustrative example highlights the advantage of the proposed methodology over the standard formulation of the output feedback controller design problem. A numerical example is provided.

A Convex Parameterization of Controllers Constrained to use only Relative Measurements

TL;DR

The paper addresses optimal controller design when sensors provide only relative measurements, a setting that challenges standard convex approaches, especially under additional network-structure constraints. It introduces a relative-feedback parameterization, showing that relative measurements permit a convex reformulation over a linear subspace S_rel(C2) via an auxiliary transfer R (and its Q-parameterization), with a recoverability link to the original controller K. By leveraging quadratic invariance, the authors show how network constraints can be integrated into the same convex framework, enabling tractable design in distributed settings. A numerical ring-graph example demonstrates practical gains and computational simplifications (e.g., circulant Q), indicating the approach's potential for scalable, distributed control with relative sensing.

Abstract

The optimal controller design problem for systems equipped with sensors that measure only relative, rather than absolute, quantities is considered. This relative measurement structure is formulated as a design constraint; it is demonstrated that the resulting constrained controller design problem can be written as a convex program. Certain additional network structural constraints can be incorporated into this formulation, making it especially useful in distributed or networked settings. An illustrative example highlights the advantage of the proposed methodology over the standard formulation of the output feedback controller design problem. A numerical example is provided.
Paper Structure (12 sections, 8 theorems, 46 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Set Up
  3. Notation & Preliminaries
  4. Relative Measurement Structure
  5. Network Control Structure
  6. Motivating Example
  7. A Relative Feedback Parameterization
  8. A Convex Formulation of Controller Design for Relative Measurement Systems
  9. Incorporating Network Structure
  10. Numerical Example
  11. Controller Design
  12. Conclusion

Key Result

Theorem 1

Let ${\boldsymbol{R}}$ be a relative LTI system. If $\mathcal{A}_{\rm meas.}(C_2)$ corresponds to a connected graph, then there is a controller ${\boldsymbol{u}} = {\boldsymbol{K}}{\boldsymbol{y}}$ for which ${\boldsymbol{K}}C_2 = {\boldsymbol{R}}.$ More generally, if $\mathcal{A}_{\rm meas.}(C_2)$ where the vector ${\boldsymbol{x}}^{(i)}$ contains the subset of states contained in the $i^{\rm th

Figures (2)

  • Figure 1: Graph corresponding to relative measurement structure of \ref{['eq:exmp_rel_structure']}
  • Figure 2: Cost (with $\gamma=0.5$, $\alpha=0.6$) per node ($\frac{1}{n}||\mathcal{F}({\boldsymbol{P}}, {\boldsymbol{K}}||$) for the $\mathcal{H}_2$ optimal controller solving \ref{['eq:modelmatching']}.

Theorems & Definitions (13)

  • Example 1
  • Definition 1
  • Theorem 1
  • Corollary 1
  • Example 2
  • Proposition 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • Example 3
  • ...and 3 more