This paper develops an adaptive state tracking control scheme for discrete-time systems, using least-squares algorithms, as the new solution to the long-standing discrete-time adaptive state tracking control problem to which the Lyapunov method (well-developed for the continuous-time adaptive state tracking problem) is not applicable. The new adaptive state tracking scheme is based on a recently-developed new discrete-time error model which has been used for gradient algorithm based state tracking control schemes, and uses the least-squares algorithm for parameter adaptation. The new least-squares algorithm is derived to minimize an accumulative estimation error, to ensure certain optimality for parameter estimation. The system stability and output tracking properties are studied. Technical results are presented in terms of plant-model matching, error model, adaptive law, optimality formulation, and stability and tracking analysis. The developed adaptive control scheme is applied to a discrete-time multiple mobile robot system to meet an adaptive state tracking objective. In addition, a collision avoidance mechanism is proposed to prevent collisions in the whole tracking process. Simulation results are presented, which verify the desired system state tracking properties under the developed least-squares algorithm based adaptive control scheme.

Qianhong Zhao and Gang Tao

Preprint: A Discrete-Time Least-Squares Adaptive State Tracking Control Scheme with A Mobile-Robot System Study

The solid circles in different colors denote the robots' positions at different steps of iteration. 

The dashed circles in different colors denote the desired trajectories of robots. (For Robot 3, its desired trajectory is designed to stop at the point (0,0).) 

The solid lines in different colors denote the trajectories of the robots.

G. Tao, ``Discrete-Time adaptive state tracking control schemes using gradient algorithms,'' arXiv: 2308.02484 [eess.SY], 2023 (https://arxiv.org/abs/2308.02484).

G. Tao and Q. H. Zhao, ``Koopman system approximation based optimal control of multiple robots---Part I: Concepts and formulations," arXiv: 2305.03777 [eess.SY], 2023 (https://arxiv.org/abs/2305.03777).

Q. H. Zhao and G. Tao, ``Koopman system approximation based optimal control of multiple robots---Part II: Simulations and evaluations," arXiv: 2305.03778 [eess.SY], 2023 (https://arxiv.org/abs/2305.03778).