Stabilization of linear discrete-time control systems with the Caputo convolution type difference fractional variable-order operator

Abstract

The paper focuses on the stabilization of fractional variable-order linear discrete-time systems with the Caputo difference operator of the convolution type. The Z-transform is a powerful and widely used tool for analyzing the stability of linear control systems. It provides insight into the behavior of the system solutions, particularly in the discrete-time domain. Given the fundamental role of stability in control theory and its applications in automation, a central objective is the stabilization of control systems. Stabilization refers to the process of determining a suitable state-feedback law that ensures the asymptotic stability of the system. In situations where the open-loop system is not asymptotically stable, stability can often be achieved by introducing a properly designed state-feedback controller. This controller is typically constructed based on the eigenvalue spectrum of the closed-loop system matrix. The conditions for asymptotic stability, derived via the Z-transform approach, provide practical criteria for eigenvalue placement and serve as a guide for ensuring the desired system behavior.