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Disturbance Attenuation Regulator II: Stage Bound Finite Horizon Solution

Davide Mannini, James B. Rawlings

TL;DR

The paper tackles finite-horizon robust control for linear systems under per-stage disturbances bounded by $|w_k|^2 \le \alpha_k$ by formulating StDAR as a deterministic minmax problem. It develops a recursive dynamic-programming solution that yields nonlinear, state-dependent feedback through online convex optimization of Lagrange multipliers, producing Riccati-like recursions $\Pi_k(\boldsymbol{\lambda}_k)$ and explicit stationarity conditions. For constant stage bounds, it derives a steady-state formulation realized as a tractable LMI with complexity $O(n^3)$, and provides numerical examples showing notable qualitative differences from SiDAR, including nonmonotonic turnpike behavior. Overall, the work generalizes disturbance-attenuation control to arbitrary initial states, offering practical, robust controllers that handle time-varying stage bounds without assuming coordinated disturbances across time.

Abstract

This paper develops a generalized finite horizon recursive solution to the discrete time stage bound disturbance attenuation regulator (StDAR) for state feedback control. This problem addresses linear dynamical systems subject to stage bound disturbances, i.e., disturbance sequences constrained independently at each time step through stagewise squared two-norm bounds. The term generalized indicates that the results accommodate arbitrary initial states. By combining game theory and dynamic programming, this work derives a recursive solution for the optimal state feedback policy. The optimal policy is nonlinear in the state and requires solving a tractable convex optimization for the Lagrange multiplier vector at each stage; the control is then explicit. For systems with constant stage bound, the problem admits a steady-state optimization expressed as a tractable linear matrix inequality (LMI) with $O(n^3)$ complexity. Numerical examples illustrate the properties of the solution. This work provides a complete feedback solution to the StDAR for arbitrary initial states. Companion papers address the signal bound disturbance attenuation regulator (SiDAR): the finite horizon solution in Part~I-A and convergence properties in Part~I-B.

Disturbance Attenuation Regulator II: Stage Bound Finite Horizon Solution

TL;DR

The paper tackles finite-horizon robust control for linear systems under per-stage disturbances bounded by by formulating StDAR as a deterministic minmax problem. It develops a recursive dynamic-programming solution that yields nonlinear, state-dependent feedback through online convex optimization of Lagrange multipliers, producing Riccati-like recursions and explicit stationarity conditions. For constant stage bounds, it derives a steady-state formulation realized as a tractable LMI with complexity , and provides numerical examples showing notable qualitative differences from SiDAR, including nonmonotonic turnpike behavior. Overall, the work generalizes disturbance-attenuation control to arbitrary initial states, offering practical, robust controllers that handle time-varying stage bounds without assuming coordinated disturbances across time.

Abstract

This paper develops a generalized finite horizon recursive solution to the discrete time stage bound disturbance attenuation regulator (StDAR) for state feedback control. This problem addresses linear dynamical systems subject to stage bound disturbances, i.e., disturbance sequences constrained independently at each time step through stagewise squared two-norm bounds. The term generalized indicates that the results accommodate arbitrary initial states. By combining game theory and dynamic programming, this work derives a recursive solution for the optimal state feedback policy. The optimal policy is nonlinear in the state and requires solving a tractable convex optimization for the Lagrange multiplier vector at each stage; the control is then explicit. For systems with constant stage bound, the problem admits a steady-state optimization expressed as a tractable linear matrix inequality (LMI) with complexity. Numerical examples illustrate the properties of the solution. This work provides a complete feedback solution to the StDAR for arbitrary initial states. Companion papers address the signal bound disturbance attenuation regulator (SiDAR): the finite horizon solution in Part~I-A and convergence properties in Part~I-B.
Paper Structure (15 sections, 13 theorems, 120 equations, 2 figures, 1 algorithm)

This paper contains 15 sections, 13 theorems, 120 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. StDAR Set Up
  3. Dynamic Programming
  4. StDAR Solution
  5. Two-stage Solution
  6. Finite Horizon Solution
  7. Optimal Policy and Implementation
  8. Steady-state StDAR
  9. Steady-state problem
  10. Numerical Examples
  11. StDAR vs SiDAR Comparison
  12. Stage bound vs Signal bound Policy Comparison
  13. Computational Complexity
  14. Summary
  15. Appendix

Key Result

Proposition 1

Let Assumptions 1--3 hold and define Consider the convex optimization eq:2stage-opt where, for $\lambda_1\in\mathbb{R}$ Given $\lambda_1$, define Given the solution to the convex optimization eq:2stage-opt, $(\lambda_0^*,\lambda_1^*)$, and terminal condition $P_f \succeq0$, then

Figures (2)

  • Figure 1: $\Pi_k$ as a function of stage $k$ for the finite horizon StDAR \ref{['stagedp-w']} (left) and SiDAR \ref{['signaldp']} (right), for $N=20$ (top) and $N=100$ (bottom). The dashed black line represents the $H_{\infty}$ steady-state value $P_{\infty}=1.2$; the solid black line represents the LQR value $P_{LQR}=0.55$.
  • Figure 2: Optimal control policies for scalar system with $N=20$. Top: stage bound $u^*_0(x_0)$ with $\alpha_k = 0.05$ for all $k$. Bottom: signal bound $u^*_0(x_0, b_0)$ with budget $b_0=\alpha=1$. Shaded region in bottom panel indicates $\mathcal{X}_L$ where SiDAR policy is linear in $x_0$.

Theorems & Definitions (19)

  • Proposition 1: Two-stage StDAR
  • Proposition 2: Finite horizon StDAR \ref{['stagedp-w']}
  • Remark 1: Parametric dependence on stage bound
  • Remark 2: Comparison with LQR
  • Remark 3: Implementation and time consistency
  • Remark 4: Computational implementation
  • Remark 5
  • Proposition 3: LMI for steady-state StDAR
  • Proposition 4: Existence of LMI
  • Remark 6: Non-monotonic turnpike
  • ...and 9 more