Davide Mannini

Davide Mannini, James B. Rawlings

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).

Davide Mannini, James B. Rawlings

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).