Direct Data Driven Control Using Noisy Measurements

TL;DR

This paper presents a novel direct data-driven control framework for solving the linear quadratic regulator (LQR) under disturbances and noisy state measurements that guarantees mean-square stability (MSS) and optimal performance by leveraging convex optimization techniques that incorporate noise statistics directly into the controller synthesis.

Abstract

This paper presents a novel direct data-driven control framework for solving the linear quadratic regulator (LQR) under disturbances and noisy state measurements. The system dynamics are assumed unknown, and the LQR solution is learned using only a single trajectory of noisy input-output data while bypassing system identification. Our approach guarantees mean-square stability (MSS) and optimal performance by leveraging convex optimization techniques that incorporate noise statistics directly into the controller synthesis. First, we establish a theoretical result showing that the MSS of an uncertain data-driven system implies the MSS of the true closed-loop system. Building on this, we develop a robust stability condition using linear matrix inequalities (LMIs) that yields a stabilizing controller gain from noisy measurements. Finally, we formulate a data-driven LQR problem as a semidefinite program (SDP) that computes an optimal gain, minimizing the steady-state covariance. Extensive simulations on benchmark systems -- including a rotary inverted pendulum and an active suspension system -- demonstrate the superior robustness and accuracy of our method compared to existing data-driven LQR approaches. The proposed framework offers a practical and theoretically grounded solution for controller design in noise-corrupted environments where system identification is infeasible.

Figures (8)

• Figure 1: Average closed-loop cost comparison for all methods on Example 1 (Active Suspension System) with DDLCLQR regularization parameter $\alpha = 0$. DDNSMLQR achieves the lowest cost, while other methods suffer from poor stability or suboptimal gains.
• Figure 2: Average closed-loop cost for all methods on Example 1 with DDLCLQR using $\alpha = 100$. The increased regularization improves stability for DDLCLQR but leads to higher cost compared to DDNSMLQR.
• Figure 3: Average cost comparison on Example 1 with DDDPLQR using $W = 10^{-4}I$ and DDLCLQR using $\alpha = 1$. Tuning these parameters improves the performance of both methods.
• Figure 4: Quanser Qube-Servo 2 platform used for simulating the rotary inverted pendulum dynamics in Example 2.
• Figure 5: Average closed-loop cost comparison for all methods on Example 2 with DDLCLQR regularization parameter $\alpha = 0$. The absence of regularization results in instability for several approaches, while DDNSMLQR maintains robust performance.

Theorems & Definitions (12)

• Definition 1: Mean Square Stability (MSS)
• Lemma 1
• Remark 1
• Theorem 1: MSS Preservation
• proof
• Theorem 2: Data-driven MSS
• proof
• Remark 2
• Theorem 3: Data-driven LQR Using Noisy Measurements
• proof