Solvability, stability and its application in the P–M synchronization problem of discrete-time fractional order singular systems

Abstract

Fractional order singular systems are an important class of systems characterized by algebraic constraints combined with fractional-order dynamic behaviors. This paper focuses on discrete-time fractional order singular systems (DFOSSs), introducing the concept of the index to analyze their structural properties. Using the Drazin inverse, we establish a lemma that decomposes DFOSSs into simpler subsystems, forming the basis for deriving solvability and stability conditions. These results are achieved through techniques from fractional calculus and singular systems. Additionally, we provide an explicit solution formula for DFOSSs, enabling practical computation. A control strategy is then proposed to achieve $P\text{–}M$ synchronization, a method that synchronizes different dimensions within the same master–slave system, surpassing traditional synchronization approaches. To demonstrate the utility of our findings, practical applications in electrical circuits are presented, showcasing the effectiveness of our methods. This study offers a comprehensive framework for analyzing and controlling DFOSSs, bridging theoretical insights with real-world applications.