A Floquet-like factorization for linear periodic systems

Factorizing the Monodromy Matrix of Linear Periodic Systems

“…for certain matrix-valued functionsP 11 ∈ C 1 (R, R n1×n1 ), (9). Then, the determinant of its monodromy matrix is positive.…”

“…The periodic Kalman canonical decomposition and the Floquet factorization (or the Floquet-like factorization by the authors) [9,10,11] have been independently investigated; specifically, the periodic Kalman canonical decomposition in [8] is based on the factorization of matrixvalued functions due to Sibuya and the Floquet factorization (or the Floquet-like factorization) is based on the computation of matrix logarithms. The proof of Theorem 9 is based on the computation of matrix logarithms and is analogous to the computation procedure of the Floquetlike factorization.…”

“…The condition is reduced to the positiveness of determinants of the block diagonal elements of the upper triangular matrix; therefore, the computational difficulty has been significantly reduced. The relations to the Floquet-like factorization [9,10,11] and the period-specific realization [16] are also discussed; the proposed computation algorithm simultaneously achieves the Floquet-like factorization as well as the periodic Kalman canonical decomposition and shares the common realization technique. Similar arguments are also discussed for the other types of decompositions.…”

Floquet-Type Representations of Linear Periodic Systems – General Case

“…For continuous-time systems, Floquet factorization consists of finding the matrix logarithm of the monodromy matrix, i.e., the system state transition matrix Φ(T, 0) over one period. Although a Floquet factorization always exists (Jikuya and Hodaka 2009), it need not be real. In particular, when the monodromy matrix has real-valued negative eigenvalues, there are cases when no real-valued Floquet factorization exists.…”

“…Yakubovich and Starzhinskii (1975) and Montagnier, Paige and Spiteri (2003) consider a Floquettype system characterization which, although 2T -periodic, has the property that the solutions can be found from computations on a single period of length T . Jikuya and Hodaka (2009) introduce a factorization, where periodicity of the system matrix is preserved, but it can be expressed as a Fourier series containing only one frequency component.…”