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Robust and Gain-Scheduling ${\cal H}_2$ Control Techniques for LFT Uncertain and Parameter-Dependent Systems

Fen Wu

TL;DR

This work develops a unified robust and gain-scheduled ${\mathcal H}_2$ control framework for uncertain and parameter-dependent systems modeled via linear fractional transformations. By introducing an intermediate variable and leveraging LMIs, the approach yields convex synthesis conditions for both robust state-feedback and LPV gain-scheduled output-feedback while preserving the white-noise interpretation of ${\mathcal H}_2$ performance. The resulting methods reduce conservatism relative to ${\mathcal H}_\infty$ designs and provide certified robustness across structured uncertainties and scheduling parameters. Numerical examples on a two-disk system and an active magnetic bearing demonstrate significantly improved disturbance rejection and robustness across varying conditions, validating the practical impact of the proposed framework.

Abstract

This paper addresses the robust ${\cal H}_2$ synthesis problem for linear fractional transformation (LFT) systems subject to structured uncertainty (parameter) and white-noise disturbances. By introducing an intermediate matrix variable, we derive convex synthesis conditions in terms of linear matrix inequalities (LMIs) that enable both robust and gain-scheduled controller design for parameter-dependent systems. The proposed framework preserves the classical white-noise and impulse-response interpretation of the ${\cal H}_2$ criterion while providing certified robustness guarantees, thereby extending optimal ${\cal H}_2$ control beyond the linear time-invariant setting. Numerical and application examples demonstrate that the resulting robust ${\cal H}_2$ controllers achieve significantly reduced conservatism and improved disturbance rejection compared with conventional robust ${\cal H}_\infty$-based designs.

Robust and Gain-Scheduling ${\cal H}_2$ Control Techniques for LFT Uncertain and Parameter-Dependent Systems

TL;DR

This work develops a unified robust and gain-scheduled control framework for uncertain and parameter-dependent systems modeled via linear fractional transformations. By introducing an intermediate variable and leveraging LMIs, the approach yields convex synthesis conditions for both robust state-feedback and LPV gain-scheduled output-feedback while preserving the white-noise interpretation of performance. The resulting methods reduce conservatism relative to designs and provide certified robustness across structured uncertainties and scheduling parameters. Numerical examples on a two-disk system and an active magnetic bearing demonstrate significantly improved disturbance rejection and robustness across varying conditions, validating the practical impact of the proposed framework.

Abstract

This paper addresses the robust synthesis problem for linear fractional transformation (LFT) systems subject to structured uncertainty (parameter) and white-noise disturbances. By introducing an intermediate matrix variable, we derive convex synthesis conditions in terms of linear matrix inequalities (LMIs) that enable both robust and gain-scheduled controller design for parameter-dependent systems. The proposed framework preserves the classical white-noise and impulse-response interpretation of the criterion while providing certified robustness guarantees, thereby extending optimal control beyond the linear time-invariant setting. Numerical and application examples demonstrate that the resulting robust controllers achieve significantly reduced conservatism and improved disturbance rejection compared with conventional robust -based designs.
Paper Structure (8 sections, 2 theorems, 41 equations, 2 figures, 3 tables)

This paper contains 8 sections, 2 theorems, 41 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Robust ${\cal H}_2$ Performance Analysis
  3. Robust ${\cal H}_2$ State-Feedback Control
  4. Gain-Scheduling ${\cal H}_2$ Output-Feedback Control
  5. Illustrative Examples
  6. Two-Disk System
  7. Active Magnetic Bearing System
  8. Conclusion

Key Result

Theorem 1

The closed-loop system is robustly stabilizable by a state-feedback controller and achieves a robust ${\cal H}_2$ norm less than $\gamma$ if there exist positive-definite matrices $P_-, P_+ \in {\cal S}_+^{n \times n}$, $Q \in {\cal S}_+^{n_d \times n_d}$, a scaling matrix $X \in {\cal D}$, and rect The resulting robust state-feedback controller is $u(k) = M V^{-1} x(k)$.

Figures (2)

  • Figure 1: Weighted open-loop interconnection for the magnetic bearing system.
  • Figure 2: Time-domain simulation of gain-scheduling ${\cal H}_2$ control

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2