CHAPTER2 TIME RESPONSE OF THE LTISYSTEM·CONTENT>2.1 TimeResponseof theLTI HomogeneousSystem> 2.2 Calculation of the Matrix ExponentialFunction> 2.3 State Transition Matrix> 2.4 Time Response of the LTI System
CHAPTER2 TIME RESPONSE OF THE LTI SYSTEM • CONTENT 2.1 Time Response of the LTI Homogeneous System 2.2 Calculation of the Matrix Exponential Function 2.3 State Transition Matrix 2.4 Time Response of the LTI System
2.2 Calculation of the Matrix ExponentialFunctionFrom the previous discussion, we have known that the matrixexponential function plays an important role in the study of theresponsefortheLTI system.+Considerthe response of the LTIhomogeneous systemX(t) = AX(t)X(t) = Φ(t)X(O)X(t) =Φ(t- t.)X(to)Φ(t -t) = eA(t-fo)Φ(t) = eArorIn this section, we will introduce four methods to calculate thematrixexponentialfunction
2.2 Calculation of the Matrix Exponential Function
2.2.1 Direct MethodBased on the definition of the matrix exponential functionA't?At3At=I+At+2!3!we can obtain the matrix exponential function theoretically. Itshould be noted that this power series will not, generally, give aclosed-form solution and it is mainly used on computersimulation
2.2.1 Direct Method
2.2.2 Laplace Transform MethodConsiderthe homogeneous state equation shown asX(t) = AX(t)with the initial condition X(t.) = X(O)where X(O) denotesthe initial state vector evaluatedat t = 0Taking the Laplace transform, we havesX(s) - X(O) = AX(s)(sI - A)X(s) = X(O)X(s) = (sI - A)-1 X(0)The free response of the LTI homogeneous system can beobtained by taking the inverse Laplace transform on both side +
2.2.2 Laplace Transform Method
2.2.2 Laplace Transform MethodConsider the homogeneous state equation shown asX(t) = AX(t)with the initial condition X(t.) = X(O)X(s) = (sI - A)-1 X(0)L-'[(sI - A)-1X(t) =x(0)X(t) = (t)X(O)Φ(t) = eAre4t =L-"[(sI - A)-]
2.2.2 Laplace Transform Method
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
2.2.3 Similarity Transformation MethodCase1A can be diagonalized.00 beIn this case, the matrix exponential functioncancalculatedas.t02P-Af-P-1AtD-1=P.et.P-1=PCe20
2.2.3 Similarity Transformation Method