Chapter3 Modeling of Digital Control Systems G s ( s ) = 3 1 1 = - s ( s + 3 ) s s + 3 This is the transform of the continuous-time function shown in Figure 3.13, which must be sampled, shifted, and z -transformed to obtain the desired transfer function. Using the modified z -transforms obtained in Section 2.7, the desired transfer function is 1 z - 1 e - 0 . 3 × 0 . 9 - G ZAS ( z ) = 4 z z - 1 z - e - 0 . 3 0 . 237 z + 0 . 022 z - 0 . 741 - 0 . 763 ( z - 1 ) = z - 4 = 4 z - 0 . 741 z ( z - 0 . 741 ) { { } } g s (t) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 FIGuRE 3.13 Continuous time function g s (t) . 0.5 1 1.5 2 Time s 3.7 THE CLOSED-LOOP TRANSFER FuNCTION Using the results of Section 3.5, the digital control system of Figure 3.1 yields the closed-loop block diagram of Figure 3.14. The block diagram includes a compara- tor, a digital controller with transfer function C(z), and the ADC-analog subsystem- DAC transfer function G ZAS (z). The controller and comparator are actually computer programs and replace the computer block in Figure 3.1. The block diagram is identical to those commonly encountered in s-domain analysis of analog systems