Example2.1Calculateerby using the Laplacetransformmethod.01A-3.2SolutionThe characteristicmatrix andits inverse matrix are calculated ass(sI - A) =22s+3s+3s+31(s +1)(s +2)(s +1)(s +2)(sI - A)-1-2-2s(s +3)+2SS(s +1)(s+2)(s+1)(s +2)
Example2.1CalculateeA by using the Laplace transform01method..A=-3-2Solution1$+3s+31(s +1)(s + 2)(s+1)(s +2)(sI - A)-1-2-2s(s +3)+2SS(s +1)(s +2)(s +1)(s +2)Taking the inverse Laplace transform, the matrix exponentialfunctioncanbe obtainedas.e4t = L-[(sI - A)-]21-1-1s +2-21s+2-21s+1s+12e=L-12-222-12e-+2e-2t+2e-2t2$+2S+2_s+1s+1
2.2.3 Similarity Transformation MethodCase1A can be diagonalized.Consider a n dimension system, governed by state spaceX = AX + Budescriptiony = CX + DuIfAcanbediagonalized,thereisanonsingulartransformation X(t) = PX(t)which transform the general state description into the diagonalcanonical form, such as.X= AX+Buy=cX +Du
2.2.3 Similarity Transformation Method y CX Du X AX Bu y CX Du X AX Bu
2.2.3 Similarity Transformation MethodCase1A can be diagonalized.X= AX+Buy=CX+Du0[AWhere A=P-1AP=4=is a diagonal matrix30and.0e40
2.2.3 Similarity Transformation Method y CX Du X AX Bu