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Disturbance Attenuation Regulator I-A: Signal Bound Finite Horizon Solution

Davide Mannini, James B. Rawlings

TL;DR

This work addresses finite-horizon disturbance attenuation for linear systems under a signal-bounded disturbance budget, extending SiDAR to arbitrary initial states. It combines dynamic programming with a Lagrange multiplier to decouple the aggregate budget from the state dynamics, yielding Riccati recursions $\Pi_k(\lambda)$ parameterized by a scalar $\lambda$ and a per-stage scalar convex optimization over $\lambda$. The resulting optimal policy is nonlinear in the state, with a clear partition of the state space into a linear region $\mathcal{X}_L(\alpha)$ and a nonlinear region $\mathcal{X}_{NL}(\alpha)$, where the policy reduces to standard $H_{\infty}$ control in the former. The methodology provides a constructive, recursive solution for arbitrary initial conditions, with monotonicity and ellipsoidal geometry of the solution regions and a tractable mechanism for online implementation via a single scalar optimization at each stage. The work lays groundwork for extensions to steady-state, stage-bound disturbances, and infinite-horizon problems, as explored in companion papers.

Abstract

This paper develops a generalized finite horizon recursive solution to the discrete time signal bound disturbance attenuation regulator (SiDAR) for state feedback control. This problem addresses linear dynamical systems subject to signal bound disturbances, i.e., disturbance sequences whose squared signal two-norm is bounded by a fixed budget. The term generalized indicates that the results accommodate arbitrary initial states. By combining game theory and dynamic programming, we derive a recursive solution for the optimal state feedback policy valid for arbitrary initial states. The optimal policy is nonlinear in the state and requires solving a tractable convex scalar optimization for the Lagrange multiplier at each stage; the control is then explicit. For fixed disturbance budget $α$, the state space partitions into two distinct regions: $\mathcal{X}_L(α)$, where the optimal control policy is linear and coincides with the standard linear $H_{\infty}$ state feedback control, and $\mathcal{X}_{NL}(α)$, where the optimal control policy is nonlinear. We establish monotonicity and boundedness of the associated Riccati recursions and characterize the geometry of the solution regions. A numerical example illustrates the theoretical properties. This work provides a complete feedback solution to the finite horizon SiDAR for arbitrary initial states. Companion papers address the steady-state problem and convergence properties for the signal bound case, and the stage bound disturbance attenuation regulator (StDAR).

Disturbance Attenuation Regulator I-A: Signal Bound Finite Horizon Solution

TL;DR

This work addresses finite-horizon disturbance attenuation for linear systems under a signal-bounded disturbance budget, extending SiDAR to arbitrary initial states. It combines dynamic programming with a Lagrange multiplier to decouple the aggregate budget from the state dynamics, yielding Riccati recursions parameterized by a scalar and a per-stage scalar convex optimization over . The resulting optimal policy is nonlinear in the state, with a clear partition of the state space into a linear region and a nonlinear region , where the policy reduces to standard control in the former. The methodology provides a constructive, recursive solution for arbitrary initial conditions, with monotonicity and ellipsoidal geometry of the solution regions and a tractable mechanism for online implementation via a single scalar optimization at each stage. The work lays groundwork for extensions to steady-state, stage-bound disturbances, and infinite-horizon problems, as explored in companion papers.

Abstract

This paper develops a generalized finite horizon recursive solution to the discrete time signal bound disturbance attenuation regulator (SiDAR) for state feedback control. This problem addresses linear dynamical systems subject to signal bound disturbances, i.e., disturbance sequences whose squared signal two-norm is bounded by a fixed budget. The term generalized indicates that the results accommodate arbitrary initial states. By combining game theory and dynamic programming, we derive a recursive solution for the optimal state feedback policy valid for arbitrary initial states. The optimal policy is nonlinear in the state and requires solving a tractable convex scalar optimization for the Lagrange multiplier at each stage; the control is then explicit. For fixed disturbance budget , the state space partitions into two distinct regions: , where the optimal control policy is linear and coincides with the standard linear state feedback control, and , where the optimal control policy is nonlinear. We establish monotonicity and boundedness of the associated Riccati recursions and characterize the geometry of the solution regions. A numerical example illustrates the theoretical properties. This work provides a complete feedback solution to the finite horizon SiDAR for arbitrary initial states. Companion papers address the steady-state problem and convergence properties for the signal bound case, and the stage bound disturbance attenuation regulator (StDAR).
Paper Structure (11 sections, 9 theorems, 118 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 9 theorems, 118 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. SiDAR Set Up
  3. Dynamic Programming
  4. SiDAR Solution
  5. Two-stage Solution
  6. Finite Horizon Solution
  7. Optimal Policy and Implementation
  8. Solution Regions and Properties
  9. Numerical Example
  10. Summary
  11. Appendix

Key Result

Proposition 1

Let Assumptions 1-3 hold. Consider the following scalar convex optimization where Given the solution to the scalar convex optimization eq:scalar-opt, $\lambda^*(x_0)$, and terminal condition $P_f \succeq0$, then

Figures (2)

  • Figure 1: Graphical construction of $\lambda_1$ via the fixed point of $\left| G'\Pi_1(\lambda)G \right|$ (curves) against the identity $y=\lambda$ (solid black). Dashed vertical lines mark $\lambda_2\coloneqq\left| G'P_fG \right|=0.5$ and the resulting $\lambda_1$. Blue:$\lambda_2\ge \left| G'\Pi_1(\lambda_2)G \right|$, so the admissibility condition $\lambda\ge \left| G'\Pi_1(\lambda)G \right|$ already holds for all $\lambda\ge\lambda_2$ and we set $\lambda_1=\lambda_2$. Orange:$\lambda_2<\left| G'\Pi_1(\lambda_2)G \right|$, so there is a unique $\lambda_1>\lambda_2$ with $\lambda_1=\left| G'\Pi_1(\lambda_1)G \right|$; for all $\lambda\ge\lambda_1$ the condition $\lambda\ge \left| G'\Pi_1(\lambda)G \right|$ holds.
  • Figure 2: Optimal control $u^*_0(x_0, b_0)$ for nondegenerate scalar system with $N=10$. Top: contour lines with shaded region $\mathcal{X}_L(b_0)$ where the policy is linear in $x_0$ for a fixed $b_0$. Middle: cross-section $u^*_0(x_0, 1)$ versus $x_0$ with shaded region $\mathcal{X}_L(1)$. Bottom: cross-section $u^*_0(0.5, b_0)$ versus $b_0$ with shaded region indicating $\mathcal{X}_L$ for fixed $x_0=0.5$. Note that $b_0 = \alpha$.

Theorems & Definitions (21)

  • Remark 1
  • Proposition 1: Two-stage SiDAR \ref{['1dsignaldp']}
  • Proposition 2: Finite horizon SiDAR \ref{['signaldp-w']}
  • Proposition 3: Derivative of the SiDAR value function
  • Remark 2
  • Remark 3: Comparison with LQR
  • Remark 4: Implementation and time consistency
  • Remark 5: Computational implementation
  • Remark 6: Budget update
  • Definition 1: Solution regions for SiDAR
  • ...and 11 more