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In this chapter, we have confined our attention to the response of simple control systems that were either first-order or second-order. This means that the transient response can be found by referring to Chaps. 5 and 8. However, if integral action were added to the controller in the system of Fig. 13.6, the overall transfer function would have a third-order polynomial in the denominator. Inversion would require factoring a cubic, which is generally a difficult task. Actually, systems with denominator polynomials of order greater than two are the rule rather than the exception. Hence, we shall develop in forthcoming chapters convenient techniques for studying the response of higher-order control systems. These techniques will he of direct use in control system design. In Chap. 1, PI control of a heated, stirred tank with measurement lag was discussed. It was indicated that incorrect selection of controller parameters could lead to a response with increasing amplitude. These unstable responses can occur in all systems with third- or higher-order polynomials in the denominator of the overall transfer function. In the next chapter, we shall present a concrete definition of stability and begin the development of methods for determining stability in control systems.
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13.1. The set point of the control system shown in Fig. P13.1 is given a step change of 0.1 unit. Determine:
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(a) The maximum value of C and the time at which it occurs. (b) The offset. (c) The period of oscillation. Draw a sketch of C(t) as a function of time.
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13.2. The control system shown in Fig. P13.2 contains a three-mode controller. (a) For the closed loop, develop formulas for the natural period of oscillation r and the damping factor 5 in terms .of the parameters K, 70. q , and 71. For the following parts, TD = 71 = 1 and rt = 2, (b) Calculate l when K is 0.5 and when K is 2. (c) Do 6 and r approach limiting values as K increases, and if so, what am these values (6) Determine the offset for a unit-step change in load if K is 2. (e) Sketch the response curve (C versus r) for a unit-step change in load when K is 0.5 and when K is 2. v) In both cases of part (e) determine the maximum value of C and the time at which it occurs. 13.3. The location of a load change in a control loop may affect the system response. In the block diagram shown in Fig. P13.3, a unit-step change in load enters at either location 1 or location 2. (a) What is the frequency of the transient response when the load enters at location 1 and when the load enters at location 2 (b) What is the offset when the load enters at location 1 and when it enters at location 2 (c) Sketch the transient response to a step change in Ut and to a step change in
FIGURE P13-3
13.4. Consider the liquid-level control system shown in Fig. P13.4. The tanks are noninteracting. The following information is known:
1. The resistances on the tanks are linear. These resistances were tested separately,
and it was found that, if the steady-state flow rate q cfm is plotted against steady-state tank level h ft, the slope of the line dqldh is 2 ft*/min. 2. The cross-sectional area of each tank is 2 ft2. 3. The control valve was tested separately, and it was found that a change of 1 psi in pressure to the valve produced a change in flow of 0.1 cfm. 4. There is no dynamic lag in the valve or the measuring element.
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