A Least-Squares Control Strategy for Asymptotic Tracking and Disturbance Rejection Using Tikhonov Regularization and Cascade Iteration

This paper presents a comprehensive strategy for addressing tracking and disturbance rejection for both lumped and distributed parameter systems, focusing on infinite-dimensional input and output spaces. Building on the geometric theory of regulation, the proposed methodology employs a cascade algorithm coupled with Tikhonov regularization to derive control laws that improve tracking accuracy iteratively. Unlike traditional optimal control approaches, the framework minimizes the limsup in time of the tracking error norm, rather than with respect to a quadratic cost function. It is important to note that this work also includes applicability to over- and under-determined systems. We provide theoretical insights, detailed algorithmic formulations, and numerical simulations to demonstrate the effectiveness and generality of the method. Results indicate that the cascade controls asymptotically approximate the classical optimal control solutions, with limitations addressed through rigorous error analysis. Applications include diverse scenarios with both finite and infinite-dimensional input and output spaces, showcasing the versatility of the approach.