Tham khảo tài liệu 'ogata - modern control engineering part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Table 5-1 Operational-Amplifier Circuits That May Be Used as Compensators 270 Chapter 5 Basic Control Actions and Response of Control Systems x 0 v í G .v -----Figure 5-50 A -v z .y Stable linear time-invariant system. G s p s __ 7Ơ Ơ 52 - 5 5 The Laplace-transformed output y s is then m G S x . x S 5_31 where A 5 is the Laplace transform of the input x t . It will be shown that after waiting until steady-state conditions are reached the fre-quency response can be calculated by replacing 5 in the transfer function by JJ. It will also be shown that the steady-state response can be given by G j w Me Af 0 where M is the amplitude ratio of the output and input sinusoids and Ị is the phase shift between the input sinusoid and the output sinusoid. In the frequency-response test the input frequency Ư is varied until the entire frequency range of interest is covered. The steady-state response of a stable linear time-invariant system to a sinusoidal input does not depend on the initial conditions. Thus we can assume the zero initial condition. If Y S has only distinct poles then the partial fraction expansion of Equa-tion 5-31 yields y s G s X s G s 1-7 I a a b . . 5 JOJ 5 joj 5 5ị 2 5 s2 V 5 5 32 ỏ -r where a and the bi where i 1 2 . are constants and a is the complex conjugate of a. The inverse Laplace transform of Equation 5-32 gives ----------y t ae ut -I- ãel ưl 4- b e Sỵl -I- b2eS2 bne Sĩit t 0 5-33 For a stable system -51 -52 . . -Sr have negative real parts. Therefore as t approaches infinity the terms e sir e r . and approach zero. Thus all the terms on the right-hand side of Equation 5-33 except the first two drop out at steady state. _If Y s involves multiple poles Sj of multiplicity m7 then y t will involve terms such as thie sir hj 0 1 2 . . m 1 . For a stable system the terms thje sir approach zero as t approaches infinity. Thus regardless of whether the system is of the distinct-pole type the steady-state response becomes yss t .
