Fa2004 16.3339-2 Dynamic Interpretation Since a= Tat-. then where we have written which is a column of rows Multiply this expression out and we get that At Assume A diagonalizable, then A a(0) given, has solution (t)=er 0)=Tet( ∑e:{nxrO)h ∑ State solution is a linear combination of the system modes vier 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 mode� � � Fall 2004 16.333 9–2 Dynamic Interpretation • Since A = TΛT −1, then ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ T e At | | λ1t ... − w . . 1 − e = TeΛt T −1 = ⎣ v ⎦ 1 · · · vn ⎦ ⎣ ⎦ ⎣ . | | λnt T e − wn − where we have written ⎡ ⎤ − T w . 1 − T −1 = ⎣ . . ⎦ − T wn − which is a column of rows. • Multiply this expression out and we get that n At λit T e = e viwi i=1 • Assume A diagonalizable, then x˙ = Ax, x(0) given, has solution x(t) = eAtx(0) = TeΛt T −1 x(0) n = eλit vi{wi T x(0)} i=1 n λit = e viβi i=1 • 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