Introduction to the Modeling and Analysis of Complex Systems

7.4. ASYMPTOTIC BEHAVIOR OF CONTINUOUS-TIME LINEAR DYNAMICAL... 121with X 0 = I. This is a Taylor series-based definition of a usual exponential, but now it isgeneralized to accept a square matrix instead of a scalar number (which is a 1 x 1 squarematrix, by the way). It is known that this infinite series always converges to a well-definedsquare matrix for any X. Note that e X is the same size as X.Exercise 7.10 Confirm that the solution Eq. (7.43) satisfies Eq. (7.40).The matrix exponential e X has some interesting properties. First, its eigenvalues arethe exponentials of X’s eigenvalues. Second, its eigenvectors are the same as X’s eigenvectors.That is:Xv = λv ⇒ e X v = e λ v (7.45)Exercise 7.11 Confirm Eq. (7.45) using Eq. (7.44).We can use these properties to study the asymptotic behavior of Eq. (7.43). As inChapter 5, we assume that A is diagonalizable and thus has as many linearly independenteigenvectors as the dimensions of the state space. Then the initial state of the systemcan be represented asx(0) = b 1 v 1 + b 2 v 2 + . . . + b n v n , (7.46)where n is the dimension of the state space and v i are the eigenvectors of A (and of e A ).Applying this to Eq. (7.43) results inx(t) = e At (b 1 v 1 + b 2 v 2 + . . . + b n v n ) (7.47)= b 1 e At v 1 + b 2 e At v 2 + . . . + b n e At v n (7.48)= b 1 e λ 1t v 1 + b 2 e λ 2t v 2 + . . . + b n e λnt v n . (7.49)This result shows that the asymptotic behavior of x(t) is given by a summation of multipleexponential terms of e λ i(note the difference—this was λ i for discrete-time models).Therefore, which term eventually dominates others is determined by the absolute value ofe λ i. Because |e λ i| = e Re(λi) , this means that the eigenvalue that has the largest real partis the dominant eigenvalue for continuous-time models. For example, if λ 1 has the largestreal part (Re(λ 1 ) > Re(λ 2 ), Re(λ 3 ), . . . Re(λ n )), thenx(t) = e λ 1t ( b 1 v 1 + b 2 e (λ 2−λ 1 )t v 2 + . . . + b n e (λn−λ 1)t v n), (7.50)lim x(t) ≈t→∞ eλ 1t b 1 v 1 . (7.51)