Tham khảo tài liệu 'ogata - modern control engineering part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | where M 1 1-4 1 . a - -tan 1 2 2 - As given by Equation 8-6 for 0 c the maximum value of M occurs at the frequency O r where 8-11 0Jr cưnVl - 2 2 cư Vcos 20 8-10 The angle 0 is defined in Figure 8-80. The frequency tír is the resonant frequency. At the resonant frequency the value of M is maximum and is given by Equation 8-7 rewritten M _ 1__1__ r 2 V1 - g sin 20 where Mr is defined as the resonant peak magnitude. The resonant peak magnitude is related to the damping of the system. The magnitude of the resonant peak gives an indication of the relative stability of the system. A large resonant peak magnitude indicates the presence of a pair of dominant closed-loop poles with small damping ratio which will yield an undesirable tran-sient response. A smaller resonant peak magnitude on the other hand indicates the absence of a pair of dominant closed-loop poles with small damping ratio meaning that the system is well damped. Remember that U r is real only if . Thus there is no closed-loop resonance if . The value of Mr is unity only if . See Equation 8-8 . Since the values of Mr and Cứr can be easily measured in a physical system they are quite useful for checking agreement between theoretical and experimental analyses. It is noted however that in practical design problems the phase margin and gain margin are more frequently specified than the resonant peak magnitude to indicate the degree of damping in a system. Correlation between step transient response and frequency response in the standard second-order system. The maximum overshoot in the unit-step response of the standard second-order system as shown in Figure 8-79 can be exactly correlated 0 Figure 8-80 Definition of the angle Ớ. 550 Chapter 8 Frequency-Response Analysis to the resonant peak magnitude in the frequency response. Hence essentially the same information of the system dynamics is contained in the frequency response as is in the transient response. For a unit-step input .
