Disturbance Attenuation Regulator II: Stage Bound Finite Horizon Solution

TL;DR

This paper presents a recursive finite horizon solution for the stage bound disturbance attenuation regulator in discrete-time systems, accommodating arbitrary initial states and providing explicit nonlinear state feedback policies.

Contribution

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

Findings

Explicit nonlinear feedback control law derived.

Steady-state solution formulated as a linear matrix inequality.

Numerical examples demonstrate effectiveness and properties of the solution.

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.