Fall 2001 6.3110-5 Dynamic Interpretation ● Since a=TAT-,then T where we have written a which is a column of rows Multiply this expression out and we get that i=1 Assume A diagonalizable, then i=A. a(0)given, has solution A TeAtr-l(o ∑eu{nzO State solution is a linear combination of the system modes v;eai eAit-Determines the nature of the time response Ui- Determines extent to which each state contributes to that mode Bi -Determines extent to which the initial condition excites the modeFall 2001 16.31 10–5 Dynamic Interpretation • Since A = TΛT −1, then eAt = T eΛt T −1 =   | | v1 ··· vn | |     eλ1t ... eλnt     − wT 1 − . . . − wT n −   where we have written T −1 =   − wT 1 − . . . − wT n −   which is a column of rows. • Multiply this expression out and we get that eAt =  n i=1 eλit viwT i • Assume A diagonalizable, then ˙x = Ax, x(0) given, has solution x(t) = eAtx(0) = T eΛt T −1 x(0) =  n i=1 eλit vi{wT i x(0)} =  n i=1 eλit viβi • State solution is a linear combination of the system modes vieλi eλit – Determines the nature of the time response vi – Determines extent to which each state contributes to that mode βi – Determines extent to which the initial condition excites the mode