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

TL;DR

The paper tackles the discrete-time adaptive state tracking problem for MIMO plants with unknown dynamics by introducing a least-squares adaptive law that minimizes an accumulative estimation error. It develops a LS-based indirect adaptive controller, including a tailored cost function, parameterization, and projection-enabled updates, and proves stability and tracking under the proposed scheme. The method is applied to a 3-robot system with a collision-avoidance mechanism based on repulsive fields, and simulations show improved tracking performance and safe operation, surpassing gradient-based approaches. This work provides a practical and theoretically grounded alternative to Lyapunov-based discrete-time designs, with direct relevance to autonomous multi-robot coordination and real-time adaptive control in discrete-time implementations.

Abstract

This paper develops an adaptive state tracking control scheme for discrete-time systems, using the least-squares algorithm, 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.

Figures (5)

• Figure 1: Schematic diagram of robot positions.
• Figure 2: System response for the control input containing $F_{ri}(t)$. (a) Robot 1 tracking error components vs. steps of iteration. (b) Robot 2 tracking error components vs. steps of iteration. (c) Robot 3 tracking error components vs. steps of iteration. (d) Estimation error norm $\|\epsilon(t)\|_2$ vs. steps of iteration
• Figure 3: Robot trajectories controlled by modified control signal $U_{i}(t)$ (solid lines) or tracking control signal $U_{oi}(t)$ (dashed lines), with asterisk markers ($*$) indicating the central points of the robots' initial positions and circles enclosing colored product markers ($\times$) representing their final positions.
• Figure 4: Shortest distance between robots comparison between the control input with or without collision avoidance mechanism.
• Figure 5: Tracking error norm comparison between the control scheme using least-squares or gradient algorithm (gain parameter of the gradient adaptive law is set as 1.9).

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 3.1
• Proposition 3.1
• Proposition 3.2
• Theorem 3.1
• Remark 4.1
• Proposition 4.1