Tham khảo tài liệu 'crc press mechatronics handbook 2002 by laxxuss episode 3 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1 X X X 0 X 0 0 0 0 0 0 X 0 0 q ọ CỌ ó ó Ó II II II p- p- p X 0 - 0 01 23456789 10 discrete time t FIGURE Step response of the system for different eigenvalues. The output signal y t yh t yp t is shown in Fig. for different values of the eigenvalue h . The transient is given by yh t h and the steady state response by yp t 1. We observed in Eq. that the system eigenvalues define the damping of its transient response but also determine its frequency of oscillation when the eigenvalues have a nonzero imaginary part . The potential problem when resonant modes exist is the same problem we found in the context of continuous-time systems . the system input contains a sine wave or another kind of signal with energy at a frequency close to one of the natural frequencies of the system. The system output still remains bounded although it grows to undesirable amplitudes. Example Consider the discrete-time system described by the state space model 1 0 y t 0 x t The eigenvalues of the system are obtained from Ad x t 1 u t h1 2 e And the associated natural modes present in the transient response are t __ j 4 _ t n n h1 2 e cos 4tl jsin 41 x t 1 0 The natural modes are slightly damped because h1 2 is close to 1 and they show an oscillation of frequency n 4. In the plots shown in Fig. we appreciate a strongly resonant output. The upper plot corresponds to an input u t sin p t . the input frequency coincides with the frequency of the natural modes. In the lower plot the input is a square wave of frequency input signal n 12. In this case the input third harmonic has a frequency equal to the frequency of the natural modes. Effect of Different Sampling Periods We observe in Eq. that Ad and Bd depend on the choice of the sampling period A. This choice determines the position of the eigenvalues of the system too. If we look at the Eq. .
