Suboptimal control with quadratic performance index for linear systems with heavy-tailed disturbances

Linear Quadratic Gaussian (LQG) theory has become one of the most successful control design methods in modern control theory, prized for its practical engineering utility and mathematical tractability. Based on the assumption of Gaussian white noise, it derives closed-form solutions for optimal controllers and estimators. However, with the growing complexity of control environments and detection equipment, inevitably introduces significant outliers, challenging the optimality and practical applicability of conventional LQG control. To address this limitation, this paper proposes a novel control strategy: the Linear Quadratic Student’s t (LQST) controller. This approach substitutes the traditional Gaussian noise model with the heavy-tailed Student’s t distribution, enabling more effective handling of prevalent outliers in engineering applications and significantly enhancing control system robustness. The controller design leverages the Bellman dynamic programing principle combined with Student’s t-distribution-based filtering techniques, ensuring effectiveness and stability in complex settings. The proposed LQST controller retains the desirable feedback structure and separation principle of classical LQG controllers while offering superior inherent robustness. Simulation experiments validate that the LQST controller significantly outperforms traditional LQG controllers in environments characterized by heavy-tailed process and measurement noise.