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

Davide Mannini, James B. Rawlings

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).

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

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 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).
Paper Structure (15 sections, 10 theorems, 143 equations, 4 figures, 1 table)

This paper contains 15 sections, 10 theorems, 143 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup and Finite Horizon Preliminaries
  3. Problem Set Up
  4. Finite Horizon Solution
  5. Solution Regions
  6. Steady-state SiDAR
  7. Steady-state problem
  8. System Classes
  9. Infinite Horizon SiDAR Solution
  10. Numerical Examples
  11. Degenerate vs Nondegenerate Systems: Scalar Cases
  12. Degenerate vs Nondegenerate Systems: Multivariable Cases
  13. Computational Complexity
  14. Summary
  15. Appendix

Key Result

Proposition 1

Let Assumptions 1-3 hold. Consider the following scalar convex optimization subject to the Riccati recursion where for $k \in [0,1,\dots,N-1]$ and terminal condition $\Pi_N = P_f \succeq0$. Given the solution to the scalar convex optimization lsi, $\lambda^*(x_0)$, then

Figures (4)

  • Figure 1: $\mathcal{X}_L(N)$ regions versus horizon length $N$ for systems 1–3: nondegenerate system 1 (top), degenerate system 2 (middle), system 3 violating Assumption 2 (bottom).
  • Figure 2: Recursion $\Pi_{k}(N)$ versus stage $k$ for systems 1–3 with varying horizon lengths. Left: $x_0=0$. Right: $x_0=2\in\mathcal{X}_{NL}(N)$. Top: nondegenerate system 1; Middle: degenerate system 2; Bottom: system 3 violating Assumption \ref{['asst2']}. Dashed lines indicate steady-state value $\overline{\Pi}$.
  • Figure 3: Optimal solution $\lambda^*(N)$ versus horizon length $N$ for systems 1–3 with $P_f=0$. Left column: $x_0 \in \mathcal{X}_L$. Right column: $x_0 \in \mathcal{X}_{NL}$. Rows: nondegenerate system 1 (top), degenerate system 2 (middle), system 3 violating Assumption \ref{['asst2']} (bottom). Dashed lines indicate steady-state value $\overline{\lambda}$.
  • Figure 4: $\mathcal{X}_L(N)$ regions for $2 \times 2$ systems with varying horizon lengths: nondegenerate system 4 (top), degenerate system 5 (bottom).

Theorems & Definitions (20)

  • Proposition 1: Finite horizon SiDAR mannini:rawlings:2026a
  • Definition 1: Solution regions for SiDAR
  • Definition 2: Solution regions for \ref{['eq:ss-problem']}
  • Proposition 2: LMI for steady-state SiDAR
  • Proposition 3: Existence of LMI solution
  • Definition 3: steady-state system classes
  • Remark 1
  • Proposition 4: Convergence: nondegenerate systems
  • Proposition 5: Well-defined limit regions
  • Proposition 6: Convergence: degenerate systems
  • ...and 10 more