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27.1. Derive D(z) for the control system shown in Fig. P27.1 for a unit-step change in
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R and for a response in which C is returned to the set point in one sampling period and remains there at sampling instants. Notice that Cd is l/(z - 1) for this problem. Express the manipulated variable m in terms of present and past values of e and m. Plot m(t) and c(t) during the first few periods.
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27.2. (a) Determine the pulse transfer function, D(z), for the system shown in Fig. P27.2 if the input disturbance is a unit-step function and if the output is to reach the set point one sampling period after the disturbance occurs. Plot the manipulated variable. Notice that Cd is l/(z - 1) for this problem. (b) If the input is r(t) = tu(t), plot the output if the D(z) of part (a) is used.
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27.3. The sampled-data system shown in Fig. P27.3 uses the following algorithm 1 z(z - b) D(z) = (1 - b) (z + l)(z - 1)
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For the process r = 1, a = 1, T = a7 = 1. If a unit-step enters as a load change (i.e., U(S) = l/s), determine C(z). Plot the continuous response c(t). Determine values of c(t) at t = 1, 1.5, 2, 3, and 4. Determine me(t) for t = 0, 1, 2, 3, and 4.
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27.4. Determine the minimal prototype response for a unit-step change in load for the control system shown in Fig. P27.4 for the following plant transfer functions: (a) Gp = eeTSl(s + 1) (b) Gp = 1/(2s + 1) (c) G, = e-2Ts/s (6) Gp = l/(s* + 0.4s + 1) Express your result as Cd = 70 + 7112 -l + 1722 -2 + . *. Give numerical values of 70, 71, 712, . . .
DESIGN
SAMPLED-DATA
CONTROLLERS
27.5. (a) For the sampled-data process shown in Fig. P27.5~ show that G(z) =
where b = ebTf7 a = 2 - l/b K is proportional to K,
K(z - a) z(z - l)(z - b)
For T = 2, determine the value of K, for which the system becomes unstable. Use the root locus method. For Kc = 1, determine c(O). (b) For the process in Fig. P27.5b, use for D(z) the following PI equivalent: *
D(z) = Kcz-a! -
z - l
71 where (r = q + T
For Kc = 1, ~1 = 1, and r = 2, determine c(nT) and compare with c(nT) ofpart( . (c) For the continuous control process shown in Fig. P27.5, determine the ultimate value of K,. Compare this value of K, with that of part (a) to see the effect of
R=u (4 ywl+ 4
1 -e -Ts s - K,(
+r$) -
-TS e -c rs+
FIGURE P27.5
STATE-SPACE METHODS
PART
CHAPTER
STATE-SPACE REPRESENTATION OFPHYSICAL SYSTEMS
Up.to this point, we have described dynamic physical systems by means of differential equations and transfer functions. Another method of description, which is widely used in all branches of control theory, is the state-space method. In fact, other disciplines of engineering (e.g., electrical engineering) introduce the statespace description before the transfer function description. The reader who plans to go beyond an introductory course in control or read from other engineering disciplines should be familiar with state-space methods. In the chapters of this part of the book, the state-space method will be developed and compared with the transfer function method. It is much easier to start with the transfer function method and then develop the state-space method. The mathematical background needed for the transfer function approach involves differential equations and Laplace transforms. The additional mathematical background needed for the state-space method involves matrix algebra. Nearly all students today receive information on matrices in their mathematics courses. For those who are rusty in this topic, it is recommended that they review some of the fundamental matrix operations. A brief review of matrix algebra is given in Appendix 28A. The transfer function approach is sufficient to calculate the response of linear control systems. The state-space approach is especially valuable in the field of optimal control of linear or nonlinear systems. The concepts developed in this part of the book will be used in the next part on nonlinear control. 431
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