Disturbance Attenuation Regulator I-B: Signal Bound Convergence and Steady-State

TL;DR

This paper analyzes the convergence and steady-state behavior of the signal bound disturbance attenuation regulator (SiDAR), introducing a steady-state version with a tractable LMI and classifying system types based on solution properties.

Contribution

It develops a steady-state SiDAR with an efficient LMI formulation and characterizes convergence properties for different system classes.

Findings

Finite horizon solution converges to steady-state for nondegenerate systems.

Steady-state properties depend on system classification as degenerate or nondegenerate.

Numerical examples demonstrate convergence and turnpike phenomena.

Abstract

This paper establishes convergence and steady-state properties for the signal bound disturbance attenuation regulator (SiDAR). Building on the finite horizon recursive solution developed in a companion paper, we introduce the steady-state SiDAR and derive its tractable linear matrix inequality (LMI) with $O(n^3)$ complexity. Systems are classified as degenerate or nondegenerate based on steady-state solution properties. For nondegenerate systems, the finite horizon solution converges to the steady-state solution for all states as the horizon approaches infinity. For degenerate systems, convergence holds in one region of the state space, while a turnpike arises in the complementary region. When convergence holds, the optimal multiplier and control gain are obtained directly from the LMI solution. Numerical examples illustrate convergence behavior and turnpike phenomena. Companion papers address the finite horizon SiDAR solution and the stage bound disturbance attenuation regulator (StDAR).