Abstract
This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.
Highlights
THERE has been a growing interest for utilization of Fractional Calculus (FC) in engineering and science problems because of its promises of better describing real world objects and phenomenon [1,2,3,4,5]
It was suggested that real world objects can be more accurately modeled and analyzed by using FC because real world objects do not have to precisely comply with integer order system models: Majority of them may exhibit fractionalty even at a low degree [1,2,5,6,7,8,9,10]
An important remark to notice when the second order Linear Time Invariant (LTI) system is stable, the real coefficient commensurate order system with / 1 is stable. (See Property 3 in Appendix) This allows checking the stability of the commensurate order systems in the form of equation (11) by using root equivalence of second order systems as follows
Summary
THERE has been a growing interest for utilization of Fractional Calculus (FC) in engineering and science problems because of its promises of better describing real world objects and phenomenon [1,2,3,4,5]. Extensive researches on LTI systems offered well established mathematical background, simplified solutions, experimentally proven methods for the characterization and analysis of real systems In these analyses, LTI system models can be expressed in many forms such as differential equations, state space models and transfer function forms [18]. This study investigates properties of a special class of commeasure order systems This class of system models has the same characteristic root set for different commeasure order ( R ). It is observed that the root equivalency formed by order shifts the roots of the characteristic equation in complex plane and draws a root trajectory with respect to the commensurate order This trajectory represents root locus of root equivalency class of commensurate order systems in complex domain and properties of these trajectories can provide useful information for system analysis and design. In order to transform the complex coefficient root equivalent systems into real coefficient commensurate order systems, we use complex conjugate root addition method
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