Formal modeling and verification of fractional order linear systems.

This paper presents a formalization of a fractional order linear system in a higher-order logic (HOL) theorem proving system. Based on the formalization of the Grünwald-Letnikov (GL) definition, we formally specify and verify the linear and superposition properties of fractional order systems. The proof provides a rigor and solid underpinnings for verifying concrete fractional order linear control systems. Our implementation in HOL demonstrates the effectiveness of our approach in practical applications.

[1]  Suman Saha,et al.  On the Selection of Tuning Methodology of FOPID Controllers for the Control of Higher Order Processes , 2011, ISA transactions.

[2]  MA John Harrison PhD.  Theorem Proving with the Real Numbers , 1998, Distinguished Dissertations.

[3]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[4]  Kai Diethelm,et al.  A fractional calculus based model for the simulation of an outbreak of dengue fever , 2013 .

[5]  Yangquan Chen,et al.  Robust controllability of interval fractional order linear time invariant systems , 2006, Signal Process..

[6]  I. Podlubny.  Fractional differential equations , 1998 .

[7]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.

[8]  F. Mainardi.  Fractional Calculus , 2018, Fractional Calculus.

[9]  Y. Chen,et al.  Design, implementation and application of distributed order PI control. , 2013, ISA transactions.

[10]  S. Momani,et al.  Numerical comparison of methods for solving linear differential equations of fractional order , 2007 .

[11]  Michael Norrish,et al.  A Brief Overview of HOL4 , 2008, TPHOLs.

[12]  Bitao Zhang,et al.  Enhanced robust fractional order proportional-plus-integral controller based on neural network for velocity control of permanent magnet synchronous motor. , 2013, ISA transactions.

[13]  Umair Siddique,et al.  Formal analysis of fractional order systems in HOL , 2011, 2011 Formal Methods in Computer-Aided Design (FMCAD).

[14]  Yangquan Chen,et al.  Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems , 2012, Autom..

[15]  J. A. Tenreiro Machado,et al.  Tuning of PID Controllers Based on Bode’s Ideal Transfer Function , 2004 .

[16]  Shi Li-ku.  Formalization of Real Binomial Coefficient in HOL4 , 2014 .

[17]  Paweł Woźny,et al.  Generalized Bernstein Polynomials , 2004 .

[18]  Junesang Choi,et al.  Certain summation formulas involving harmonic numbers and generalized harmonic numbers , 2011, Appl. Math. Comput..

[19]  B. Vinagre,et al.  IIR approximations to the fractional differentiator/integrator using Chebyshev polynomials theory. , 2013, ISA transactions.

[20]  J. Trujillo,et al.  A unified approach to fractional derivatives , 2012 .

[21]  Yangquan Chen,et al.  Robust stability check of fractional order linear time invariant systems with interval uncertainties , 2005, IEEE International Conference Mechatronics and Automation, 2005.

[22]  Ivo Petrás,et al.  Modeling and numerical analysis of fractional-order Bloch equations , 2011, Comput. Math. Appl..