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To develop the closed-loop transfer functions for a cascade control system, consider the general block diagram shown in Fig. 18.3. In this diagram, the load disturbance U enters between two blocks of the plant and the inner loop encloses this load disturbance. To determine the transfer function C/R, the inner loop is reduced to one block by the method shown in 12. The result is shown in Fig. 18.3b, and the block diagram of Fig. 18.3b can be used to give the result
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FIGURE 18-3
Block diagram for cascade control for set-point change.
To obtain the transfer function relating output to load, C/U, the block diagram of Fig. 18.3~ is nxrranged by placing the transfer function G ,G t in the feedback paths of the primary and secondary loops; the new arrangement is shown in Fig. 18.4a. Since R = 0 for the case under consideration, the block diagram can be redrawn as shown in Fig. 18.4b. This diagram, which has the same form as the one in Fig. 18.3a, can now be reduced to the form shown in Fig. 18.4~. Application of the rules of 12 to Fig. 18.4~ finally gives
C G3 GO -=u GGc2 1 + GaGc,HlG3
(18.2)
where G, is the same as given in Eq. (18.1).
Example 18.1. To compare conventional control with cascade control, consider the
conventional control system of Fig. 18.5~ in which a third-order process is under PI
control. A cascade version of this single-loop control system is shown in Fig. 18% in which an inner-loop having proportional control encfoses the load disturbance U. To obtain a response of the conventional control system for use in comparison with the response of the cascade system, the block diagram of Fig. 18.5~ was simulated on a computer. The values of K, and q were chosen by trial and error to give the response to a step, change in set point shown as Curve I of Fig. 18.6; this response, which has a decay ratio of about 4, was obtained with K, = 2.84 and ~1 = 5. The Ziegler-Nichols settings (Kc = 3.65 and q = 3.0 ) gave a set-point response that was too oscillatory. Having obtained satisfactory controller settings
ADVANCED CONTROL STRATujlES
FIGURE 18-4
Block diagram for cascade control for load change.
(K, = 2.84 and q = 5.0) the response of the system to a step change in U of 4 units is shown as Curve II of Fig. 18.7. The load response for no control (i.e., K, = 0) is also shown as Curve I for comparison. The cascade control system of Fig. 18Sb was also simulated to obtain a load response. The controller gain K,, of the inner loop was chosen arbitrarily to be 10.0. This value was chosen to be high in order to obtain a fast-responding inner loop, a desirable situation for cascade control. Because of the introduction of the inner loop, the dynamics of the control system have changed and it is necessary to tune the primary controller parameters for a good response to a step change in set point. By trial and error, primary controller settings of K,, = 1 .O and q = 0.63 were found that produced the response to a unit step in set point, shown as Curve II in Fig. 18.6. The use of Ziegler-Nichols settings produced a less desirable response. Using the controller parameters found from the step change in set point (Kc, = 1.0 , q = 0.63), the response of the cascade system to a step change in load of 4 units was obtained and is shown as Curve III of Fig. 18.7. As shown
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FIGURE 18-5 Block diagrams for Example 18.l:(a) Single-loop conventional control (b) cascade control.
in Fig. 18.7, the load response for the cascade control system is far superior to the
load response of the conventional control system. The maximum deviation of the
cascade response has been reduced by a factor of about four and the frequency of
oscillation has nearly doubled.
Cascade control is especially useful in reducing the effect of a load disturbance that moves through the control system slowly. The inner loop has the effect of reducing the lag in the outer loop, with the result that the cascade system responds more quickly with a higher frequency of oscillation. Example 18.2 will illustrate this effect of cascade control.
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