ferential equation (3.12) 3.2.1 Floquet theory Here, all coefficients bk k(t) E R -> R are T-periodic in t. In a modern setting [Lam97], this is rephrased as studying the solutions of a system of n first-order differential equa- tions (3.13) y(t) = c(t)Tx(t)+d(t)u(t) (3.14) where A(t) ER-Rnn,b(t),c(t)ER- - > R", and d(t) E R -> R are T-periodic matrix-valued functions. The formulation (3.13)-(3.14) is called a state-space repre- sentation of (3.12). Floquet theory proves that it is possible to determine a T-periodic transformation of variables z(t) = P t x (t) (3.15) with P(t) R Rnxn, that transforms (3.13)-(3.14) into a state-space representation with constant matrix A', or (3.16) y(t) = (3.17) The dynamic behavior, e.g. settling times and stability, of the LPTV system is therefore completely determined by the eigenvalues of A'. These eigenvalues are called the Floquet exponents of the LPTV system. The Floquet exponents and the transformation matrix P(t) can be obtained from the fundamental matrix P(t,T) E RxR -> Rnxn satisfying =I. . (3.18) Here, I E Rnxn is the unity matrix. Given P(t,T), we find A' = In(d(T,0)) (3.19) P(t)-1=D(t,0)exp(-At) (3.20) The matrix P(T,0) is called the monodromy matrix. Note that P(t,0) can be found by integrating (3.18) over the interval [0,T]. A' and P(t) can also be computed us- ing frequency-domain techniques [Dem03]. These techniques solve the time-varying eigenvalue problem (3.21) Numerically, the latter method is better conditioned than the direct computation of D(t,0). PRIOR ART 3.2.2 Lifting A second way to deal with LPTV behavior involves converting (3.13)-(3.14) to a time- invariant discrete-time system [Bam92]. This is called lifting. If we introduce the processes uk(t)=u(t+kT) for E [0,T], then we find (3.22) Here, xk==(xT) = and = P(T,0) with P(t,T) satisfying (3.18). The properties of continuous-time LPTV systems can now be analyzed using time-invariant discrete-time techniques. For example, stability of the LPTV system (3.13)-(3.14) is equivalent to the stability of (