An adaptive linear quadratic tracker design for continuous – time systems with completely unknown dynamics

Hari Om Shanker Mishra, Sumit Kumar Jha, Amit Dhawan and Manish Tiwari

International Journal of Systems Science, 2026, vol. 57, issue 2, 311-332

Abstract: The aim of this manuscript is to design an adaptive linear quadratic tracker (LQT) for continuous-time (CT) systems with completely unknown system dynamics. To address these LQT issues, one can utilised a quadratic version of the value function to obtain LQT-Augmented Algebraic Riccati Equation (ARE). We have demonstrated that the value function exhibits quadratic behaviour based on the system state and command generator, and this leads to the LQT Bellman error equation. Gradient-based update rules enable online estimation of the unknown optimum gain parameters. A dynamic state-feed controller can enable continued adaptation and convergence to an optimal controller by utilising input-state data along the trajectory of the system. Utilising an online state derivative estimator helps simplify the process of designing a model-free controller for a completely unknown system. Unlike previous research, we develop the adaptive optimum controller without considering the system dynamics or the initial stabilising strategy. Instead of switching or performing frequent intermittent updates, which may result in stability issues, the controller undergoes continuous changes based on input-state data. A simulation example validates the theoretical contribution of the proposed algorithm, and a Lyapunov-based approach establishes uniform exponential stability of a closed-loop system.

Date: 2026
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DOI: 10.1080/00207721.2025.2503205

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