Time Domain Calculation of System Performance Criteria

In the design of dynamical systems, described by the linear differential equation !_ 0 ai(dix/ dti)=0, integrals of the form Ik= Jo°° tKx2dt, k=0, 1, 2, . . . are often used as criteria of performance. Itis then desirable to be able to calculate these integrals without having to solve explicitly the differential equation of the system. Anke [4] showed that this can be done for integrals of the form Io= Jo'x0dt. This paper uses the results obtained by Anke and a change of variable to calculate the more general performance criteria Ik=Jotx2dt, k=0, 1, 2, . . . .