Fall 2001 16.3112-8 Modal ests Earlier examples showed the relative simplicity of testing observ ability/controllability for system with a decoupled A matrix There is, of course, a very special decoupled form for the state-space model: the Modal Form(8-5) Assuming that we are given the model Ar+B Ca+ du and the A is diagonalizable(A- TAT-)using the transformation 1 based on the eigenvalues of a. note that we wrote which is a column of rows Then define a new state so that =tz. then i=t-i=T(Ac+ Bu)=(T- AT)z+r- bu Az+T- Bu y= C+ Du= CTz+ du˙ Fall 2001 16.31 12—8 Modal Tests • Earlier examples showed the relative simplicity of testing observ￾ability/controllability for system with a decoupled A matrix. • There is, of course, a very special decoupled form for the state-space model: the Modal Form (8—5) • Assuming that we are given the model x˙ = Ax + Bu y = Cx + Du and the A is diagonalizable (A = TΛT −1) using the transformation T =  | |  v1 · · · vn | |   based on the eigenvalues of A. Note that we wrote: T −1 =   − w1 T − . . . − wn T −   which is a column of rows. • Then define a new state so that x = T z, then z = T −1 x˙ = T −1 (Ax + Bu)=(T −1 AT)z + T −1 Bu = Λz + T −1 Bu y = Cx + Du = CTz + Du