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c) Fig. 5 34 Root-Iocii for the controlsystem of Example 5.9, continued. Here, we have three unknowns k, zc, and Pc and one relation as above. To begin with, let us assume k ~ 679.086 (uncompensated gain). Then, we can choose one of the two remauung unknowns arbitrarily. Let us choose zc = 0.05 so that we get Pc = 0.0044. The open-loop transfer function of the Iag-compensated system is given by k(s + 0.05) G(s) = ~s +0 0044 (s +2)(s +5)(s +12) Now we can draw the root-locus as shown in Fig. 5:34c and determine the value of the gain and closed-loop poles for operating with < = 0.2. We get k - 670.3530 and p = -16.2019, -1.3796 -1- j6.8304, and -0.0433. Thus, the pole locations are similar to those observed for the PI compensator. With this new value of the gain k = 670.3530, the steady-state error is slightly changed. We have e Kp = p-, * 2z~k5 , 5 *12 0.05 * 670.3530 = 0 0044 *2*5 *12 - 63.4804 which is close to the target value of 0.01502. Hence, we need not repeat the design process. 7 I i { 1 e(oo) - ~+ K p 1 . =~+63480~ - 0.0155 498 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL Fig.535 Unit-step responses ofExample 5.9. Now to test the designs of PI- and lag-compensated systems, we have obtained unit-step responses of the basic, PI-, and Iag-compensated systems as shown in Fig. 5.35. We observe that the design objectives are met. The transient response of the PI- and lag-compensated systems are almost identical to that of the basic system. Furthermore, as t assumes large values, the steady-state error for the PI compensator approaches zero and that for the lag compensator approaches the target value of 0.01502. Example 5.10 For the following system, design 1) a PD compensator and 2) alead-lag compen- sator so that the peak time is reduced by a factor of 3, while the percent overshoot remains unchanged at 25. 8%. k G(s) = s~s + 3~s + 5~ Solution. PD compensator. The first step is to draw the root-locus of the basic system as shown in Fig. 5.36a. Using Eq. (5.75), we find that the damping ratio Lhat corresponds to 25.38qro overshoot is equal to 0.4. For the basic (uncompensated) system, the value of the gain that corresponds to < = 0.4 is equal to 28.3825, and LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW 499 Root~Locus of Basic System Real Axis a) T ------ -~'- - -- - , jco ┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━┓ www.aero.cn 航空翻译 www.aviation.cn 本文链接地址：动力机械和机身手册3
