A Period-Specific Realization of Linear Continuous-Time Systems

Factorizing the Monodromy Matrix of Linear Periodic Systems

“…The proof of Theorem 9 shares the common realization technique in the period-specific realization by the authors [16]. In the period-specific realization problem, the weighting pattern pattern matrix W (t, p) = C(t)Φ(t, p)B(p) is given and is supposed to be factored as W (t, p) = L 0 (t)R 0 (p) in the globally reduced form.…”

“…There is a T -periodic controllable and observable subsystem if and only if the index q, which correspond to the monodromy matrix detΦ 22 (T, 0) of the unknown subsystem, satisfies q > 0. If this condition q > 0 is satisfied, the computation procedure in the period-specific realization [16] recovers the unknown (C 2 ,Ã 22 ,B 2 )-triple from L 0 (t) and R 0 (p). In other words, the dimension of the system is reduced in advance, and then, the minimal realization (C 2 ,Ã 22 ,B 2 ) is computed based on the computation of the matrix logarithms G 22 and F 22 in the proof of Theorem 9.…”

“…In other words, the dimension of the system is reduced in advance, and then, the minimal realization (C 2 ,Ã 22 ,B 2 ) is computed based on the computation of the matrix logarithms G 22 and F 22 in the proof of Theorem 9. If the condition q < 0 is satisfied, the computation procedure in the period-specific realization [16] generates the non-minimal (C, A, B)-triple by augmenting the system dimension.…”

“…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

“…In all the above generalization to Floquet factorization the resulting homogenous system is still periodic, but it has a form which is more amenable to analysis and controller synthesis. In contrast, Hodaka and Jikuya (2008) show that even when the monodromy matrix has real negative eigenvalues, Floquet-type representations with time-invariant homogenous dynamics may be constructed without the need to introduce period doubling. This is achieved by increasing the state dimension in such a way that the monodromy matrix of the augmented system has a real logarithm.…”