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Interior-Conic Polytopic Systems Analysis and Control

Alex Walsh, James Richard Forbes

TL;DR

The polytopic conic controller is demonstrated on a heat exchanger in simulation, and compared to existing LPV control design techniques.

Abstract

Linear parameter varying (LPV) analysis and controller synthesis theory rooted in the small gain and passivity framework currently exist. The study of conic systems encompasses both small gain and passivity properties, and herein, analysis and controller synthesis for polytopic conic systems is considered. Linear matrix inequality constraints are given to provide conic bounds for polytopic systems. In addition, given controller conic bounds, a control synthesis method is introduced. The polytopic conic controller is demonstrated on a heat exchanger in simulation, and compared to existing LPV control design techniques.

Interior-Conic Polytopic Systems Analysis and Control

TL;DR

The polytopic conic controller is demonstrated on a heat exchanger in simulation, and compared to existing LPV control design techniques.

Abstract

Linear parameter varying (LPV) analysis and controller synthesis theory rooted in the small gain and passivity framework currently exist. The study of conic systems encompasses both small gain and passivity properties, and herein, analysis and controller synthesis for polytopic conic systems is considered. Linear matrix inequality constraints are given to provide conic bounds for polytopic systems. In addition, given controller conic bounds, a control synthesis method is introduced. The polytopic conic controller is demonstrated on a heat exchanger in simulation, and compared to existing LPV control design techniques.

Paper Structure

This paper contains 14 sections, 5 theorems, 40 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Conic Systems
  5. Conic Polytopic Systems
  6. System Architecture
  7. Conic Bounds for Polytopic Systems
  8. Determining Conic Bounds
  9. Design of Polytopic Conic Controllers
  10. Numerical Example
  11. System Description
  12. Robust Control Design
  13. Numerical Results
  14. Concluding Remarks

Key Result

Theorem 2.1

Consider the negative feedback interconnection of two square systems, $\boldsymbol{\mathcal{G}} _1: \mathcal{L}_{2e} \to \mathcal{L}_{2e}$ and $\boldsymbol{\mathcal{G}} _2: \mathcal{L}_{2e} \to \mathcal{L}_{2e}$, shown in Fig. fig:IOsysBD, where $\mbf{y}_i = \boldsymbol{\mathcal{G}} _i \mbf{u}_i$ f

Figures (8)

  • Figure 1: Input-output system block diagram.
  • Figure 2: Standard problem block diagram with an uncertainty block.
  • Figure 3: Upper and lower LFTs for controller synthesis.
  • Figure 4: Plant conic boundary and plant Nyquist plot. Embedded plot is a zoomed-in version to emphasize the plant's Nyquist plot at the vertices.
  • Figure 5: Plant conic boundary and plant Nyquist plot with minimum conic radius.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 2.1: Conic Sector Theorem Zames1966Joshi2002
  • Corollary 2.1: Small Gain Theorem Brogliato:2007ys
  • Remark 1
  • Lemma 3.1
  • Theorem 3.1
  • Corollary 3.1