On the Relationship of Optimal State Feedback and Disturbance Response Controllers

2023/04/07 by Runyu Zhang, Zhang, Runyu, Yang Zheng +5
Engineering · Mathematics · #Advanced Control Systems Optimization #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2304.03831

openalex publication_date 2023/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the relationship between state feedback policies and disturbance response policies for the standard Linear Quadratic Regulator (LQR). For open-loop stable plants, we establish a simple relationship between the optimal state feedback controller u_t=K_⋆ x_t and the optimal disturbance response controller u_t=L^(H)_⋆;1w_t-1+⋯+L^(H)_⋆;Hw_t-H with H-order. Here x_t, w_t, u_t stands for the state, disturbance, control action of the system, respectively. Our result shows that L_⋆,1^(H) is a good approximation of K_⋆ and the approximation error ‖K_⋆ - L_⋆,1^(H)‖ decays exponentially with H. We further extend this result to LQR for open-loop unstable systems, when a pre-stabilizing controller K₀ is available.

Related

• A Note on Linear Quadratic Regulator and Kalman Filter
• The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems
• The Power of Linear Controllers in LQR Control
• Linear Quadratic Regulators: A New Look
• Guaranteed Stability Margins for Decentralized Linear Quadratic Regulators