An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

TL;DR

This paper introduces a systematic feedback control framework for LTI systems that ensures real-time optimal steady-state performance by integrating state estimation, optimization-based drift computation, and stability analysis via IQCs.

Contribution

It develops a novel IQC-based framework that guarantees optimal steady-state tracking and stability for LTI systems under disturbances, with conditions expressed as LMIs.

Findings

Guarantees optimal steady-state performance in real-time.

Provides conditions for stability and optimality using LMIs.

Demonstrates versatility through multiple examples.

Abstract

Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state. We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.