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

TL;DR

This paper presents a recursive finite horizon solution for a disturbance attenuation regulator in linear systems, accommodating arbitrary initial states and distinguishing control policies based on state space regions.

Contribution

It introduces a generalized recursive solution combining game theory and dynamic programming for the SiDAR problem with arbitrary initial states.

Findings

Optimal control policy is nonlinear and explicit, solved via convex scalar optimization.

State space is partitioned into linear and nonlinear control regions.

Theoretical properties of Riccati recursions are established and illustrated.

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 $\alpha$, the state space partitions into two distinct regions: $\mathcal{X}_L(\alpha)$, where the optimal control policy is linear and coincides with the standard linear $H_{\infty}$ state feedback control, and $\mathcal{X}_{NL}(\alpha)$, 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).