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A Scalable Procedure for $\mathcal{H}_{\infty}-$Control Design

Amit Kumar, Prasad Vilas Chanekar

TL;DR

This work introduces a scalable gradient-based procedure for $\mathcal{H}_{\infty}$-control design by leveraging an algebraic Riccati equation constraint and a novel Armijo-inspired step-size rule. The method replaces the nonconvex BMI with an ARE-based formulation and uses a tight upper-bound objective $f_{\eta}$ to enable tractable gradient computations, achieving $\mathcal{O}(n^3)$ per-iteration complexity. Empirical results on benchmark engineering systems show that the proposed ARE-based method matches or outperforms SDP-based gradient descent in final performance $\gamma^*$ while dramatically improving scalability, with SDP failing on large-scale problems where the ARE-based approach remains feasible. Overall, the paper advances scalable $\mathcal{H}_{\infty}$ synthesis by combining ARE constraints, gradient-based optimization, and adaptive step-size strategies, offering practical benefits for large systems.

Abstract

This paper proposes a novel gradient based scalable procedure for $\mathcal{H}_{\infty}-$control design. We compute the gradient using algebraic Riccati equation and then couple it with a novel Armijo rule inspired step-size selection procedure. We perform numerical experiments of the proposed solution procedure on an exhaustive list of benchmark engineering systems to show its convergence properties. Finally we compare our proposed solution procedure with available semi-definite programming based gradient-descent algorithm to demonstrate its scalability.

A Scalable Procedure for $\mathcal{H}_{\infty}-$Control Design

TL;DR

This work introduces a scalable gradient-based procedure for -control design by leveraging an algebraic Riccati equation constraint and a novel Armijo-inspired step-size rule. The method replaces the nonconvex BMI with an ARE-based formulation and uses a tight upper-bound objective to enable tractable gradient computations, achieving per-iteration complexity. Empirical results on benchmark engineering systems show that the proposed ARE-based method matches or outperforms SDP-based gradient descent in final performance while dramatically improving scalability, with SDP failing on large-scale problems where the ARE-based approach remains feasible. Overall, the paper advances scalable synthesis by combining ARE constraints, gradient-based optimization, and adaptive step-size strategies, offering practical benefits for large systems.

Abstract

This paper proposes a novel gradient based scalable procedure for control design. We compute the gradient using algebraic Riccati equation and then couple it with a novel Armijo rule inspired step-size selection procedure. We perform numerical experiments of the proposed solution procedure on an exhaustive list of benchmark engineering systems to show its convergence properties. Finally we compare our proposed solution procedure with available semi-definite programming based gradient-descent algorithm to demonstrate its scalability.
Paper Structure (8 sections, 1 theorem, 23 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 8 sections, 1 theorem, 23 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Solution Procedure
  5. Examples
  6. Example $1$
  7. Example $2$
  8. Conclusion

Key Result

Lemma 4.1

(Gradient of $f_{\eta}$ with respect to $K$). Consider optimization problem prob3. Then for $K^0\in \mathcal{K}_{\beta}$, the gradient of $\mathcal{L}$ with respect to $K$ is with where $A_1 = A_c+\frac{1}{\beta}B_1B_1^{\top}P_{\infty}$.

Figures (1)

  • Figure 1: $\nabla_K f$ and $\nabla_K f_{\eta}$ at $K^0\in\mathcal{K}_{\beta}$

Theorems & Definitions (3)

  • Lemma 4.1
  • proof
  • Remark 4.2