A novel extended precise integration method based on Fourier series expansion for periodic Riccati differential equations. (9th January 2017)

• Record Type: Journal Article
• Authors: Tan, Shu‐Jun; Zhou, Wen‐Ya; Peng, Hai‐Jun; Wu, Zhi‐Gang
• Is Part Of: Optimal control applications and methods. Volume 38:Number 6(2017)
• Page Start: 896
• Page End: 907
• Publication Date: 2017-01-09
• Subjects: doubling algorithm -- Fourier series expansion -- periodic Riccati differential equation -- periodic system -- precise integration method; Control theory -- Periodicals; Mathematical optimization -- Periodicals; 629.8312
• DOI: 10.1002/oca.2298
• Languages: English
• ISSNs: 0143-2087

Abstract

Summary: A new, reliable algorithm for nonnegative, stabilizing solutions for the periodic Riccati differential equation (PRDE) is proposed based on Fourier series expansion and the precise integration method (PIM). Taking full advantages of periodicity, we expand coefficient matrices of the underlying linear time‐varying periodic Hamiltonian system associated with the PRDE in Fourier series, and a novel extended PIM for the transition matrix of linear time‐varying periodic systems is developed by combining the doubling algorithm with the increment‐storage technique. This method needs to compute the matrix exponential and its related integrals only once for all evenly divided subintervals, which greatly improves the computational efficiency. Further, by introducing the Riccati transformation, a fast recursive formula for the PRDE is derived based on the block form of the transition matrix computed by the extended PIM. Finally, two numerical examples are presented to verify the numerical accuracy and efficiency of the proposed algorithm with compared results.