Analytic Matrix Descriptions for Time-Delay Systems Enable Generalization of Polynomial Matrix Description Results
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The analysis of complex systems frequently relies on effective mathematical descriptions, and researchers continually seek methods to represent these systems with greater accuracy and flexibility. Rafikul Alam and Jibrail Ali, both from the Department of Mathematics at IIT Guwahati, have developed an analytic matrix description, a powerful new framework for understanding linear time-invariant time-delay systems. This approach extends the well-established polynomial matrix description, allowing scientists to analyse systems where past states influence present behaviour, a common feature in many real-world applications. By generalizing key results from polynomial matrix descriptions, this work provides a robust tool for characterizing and controlling complex time-delay systems, potentially leading to advances in fields ranging from robotics to network control. Meromorphic Systems, Stability and Controllability Analysis This paper presents a significant generalization of linear systems theory, extending it to systems described by meromorphic matrices. This advancement moves beyond traditional rational matrices, allowing for the analysis of more complex system dynamics, including those found in systems with time delays, transcendental functions, or singularities. The research establishes a robust framework for analyzing the stability, controllability, and observability of these generalized systems, offering new tools for understanding complex dynamics. The foundation of this work lies in meromorphic matrices, which are complex differentiable everywhere except at isolated points called poles. This broader class of system descriptions allows for the modelling of a wider range of phenomena than traditional rational matrices. The study introduces Meromorphic Function Descriptions (MFDs) and Analytic Matrix Descriptions (AMDs), which generalize existing methods for representing system dynamics in a state-space form. These descriptions are crucial for simplifying analysis and design. Scientists established canonical forms for transfer functions, AMDs, and state matrices, providing a standardized way to represent and analyze these systems. The research also defines criteria for determining when two systems are equivalent, enabling comparisons and simplifications. Characterizing AMDs of least order is also achieved, minimizing the complexity of system representations. The study investigates the zeros and poles of the system, including their structural indices, and their relationship to stability and controllability. The concept of decoupling zeros, which identifies specific types of input and output interactions, is extended to these meromorphic systems. The framework demonstrates its applicability to time-delay systems, which are often described by meromorphic matrices. The research thoroughly examines the conditions for controllability and observability of these generalized systems, providing a comprehensive analysis of their dynamic behaviour. This work provides a general framework encompassing a wide range of linear systems, including those with time delays, transcendental functions, and singularities. Time-Delay System Analysis via Holomorphic Functions Scientists developed a comprehensive framework for analyzing time-delay systems (TDS), building upon established methods for linear time-invariant (LTI) systems. The study begins by formulating the dynamics of a TDS, describing how the rate of change of state variables depends on current and past values of both state variables and control inputs. Researchers constructed a system matrix and a transfer function, which are holomorphic matrix-valued functions, essential for understanding the system’s behaviour.
The team established a connection between the zero output of the TDS and the eigenvalues and eigenvectors of both the system matrix and the transfer function, demonstrating that analyzing these properties reveals crucial information about the system’s stability and response. A key distinction was made between the transfer function of a standard LTI system and the transfer function of a TDS, which can exhibit either a finite number of poles and an isolated essential singularity, or an infinite number of poles and a non-isolated essential singularity. This difference necessitates new approaches for defining concepts like least order and McMillan degree, which are vital for analyzing LTI systems. To address this, scientists investigated specific examples of TDS, including systems with single control delays and single state delays, deriving expressions for their respective system matrices and transfer functions. These analyses revealed the complexity of the transfer functions, particularly the presence of essential singularities, and motivated the development of a generalized framework for analyzing these systems. Researchers extended concepts from rational matrix theory, such as coprime matrix-fraction descriptions and canonical forms, to the realm of meromorphic matrices, enabling a more comprehensive analysis of TDS. Time-Delay System Analysis via Analytic Matrices Scientists have developed a comprehensive framework for analyzing time-delay systems (TDS) by introducing the concept of an analytic matrix description (AMD), which extends the established methodology of polynomial matrix descriptions (PMDs) used for standard linear time-invariant systems. The research establishes a system matrix and a corresponding transfer function for TDS, enabling a detailed analysis of system behaviour through matrix descriptions. Experiments reveal that the transfer function can be either a transcendental meromorphic matrix with a finite number of poles and an isolated essential singularity, or an infinite number of poles and a non-isolated essential singularity, depending on the specific TDS configuration.
The team demonstrated that the zero output of a TDS can be analyzed by examining the eigenvalues and eigenvectors of both the system matrix and the transfer function, mirroring the approach used for conventional linear systems. Results show that the system matrix and transfer function are crucial for understanding the dynamics of TDS, providing a means to characterize system behaviour through matrix representations. Specifically, the research addresses TDS with both single control delays and single state delays, revealing how the properties of the transfer function differ in each case. Measurements confirm that for a TDS with a single control delay, the transfer function exhibits a finite number of poles and an isolated essential singularity, while a TDS with a single state delay results in an infinite number of poles and a non-isolated essential singularity. The study extends the established concepts of least order and McMillan degree, traditionally used for rational transfer functions, to the more complex case of meromorphic transfer functions in TDS. This breakthrough delivers a powerful analytical tool for understanding and controlling time-delay systems.
Meromorphic Matrices Preserve Rational Spectral Properties This research presents a theoretical framework for analysing meromorphic matrix descriptions (AMDs) and matrix fraction descriptions (MFDs), extending established results from rational matrices to the more general case of meromorphic matrices. Scientists successfully demonstrated that key properties, including coprime MFDs, least order AMDs, equivalence criteria, canonical forms, and characterization via transfer functions, hold true for these extended descriptions. Furthermore, the work establishes a clear relationship between the zeros and poles of AMDs and their rational counterparts, confirming that the fundamental spectral properties are preserved.
The team also investigated structural indices of zeros and poles, providing a deeper understanding of the system’s dynamic behaviour. Importantly, the findings demonstrate that controllability and observability of linear time-invariant time-delay systems can be determined by the properties of the associated pair of matrices, simplifying analysis of these complex systems. The authors acknowledge that the current work focuses on theoretical development and future research could explore numerical methods for implementing these analytical results, particularly in the context of time-delay systems. This advancement provides a powerful tool for analysing a broader class of linear dynamical systems, including those encountered in time-delay applications. 👉 More information 🗞 Analytic matrix descriptions with application to time-delay systems 🧠 ArXiv: https://arxiv.org/abs/2512.06957 Tags: