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A definition of stability for a control system has been presented and discussed. This definition was translated into a simple mathematical criterion relating stability to the location of roots of the characteristic equation. Briefly, it was found that a control system is stable if all the roots of its characteristic equation lie in tbe left half of the complex plane. The Routh criterion, a simple algebraic test for detecting roots of a polynomial lying in the right half of the complex plane, was presented and applied to control system stability analysis. This criterion suffers from two limitations: (1) It is applicable only to systems with polynomial characteristic equations, and (2) it gives no information about the actual location of the roots and, in particular, their proximity to the imaginary axis. This latter point is quite important, as can be seen from Fig. 14.2 and the results of Example 14.3. The Routb criterion tells us only that for K, < 10 the system is stable. However, from Fig. 14.2 it is clear that the value K, = 9
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produces a response that is undesirable because it has a response time that is too long. In other words, the controlled variable oscillates too long before returning to steady state. It will be shown later that this happens because for K, = 9 there is a pair of roots close to the imaginary axis. In the next chapter tools will be developed for obtaining more information about the actual location of the roots of the characteristic equation. This will enable us to predict the form of the curves of Fig. 14.2 for various values of K,. The advantage of these tools is that they are graphical and are easy to apply compared with standard algebraic solution of the characteristic equation. There are two distinct approaches to this problem: root-locus methods and frequency-response methods. The former ate discussed in Chap. 15 and the latter in Chaps. 16 and 17. These groups of chapters am written in parallel, and the reader may study one or both groups in either order. As a guide to making this decision, here am some general comments concerning the two approaches. Root-locus methods allow rapid determination of the location of the roots of the characteristic equation as functions of parameters such as K, of Fig. 14.1. However, they ate difficult to apply to systems containing transportation lags. Also, they require a reasonably accurate knowledge of the. theoretical process transfer function. Frequency-response methods are an indirect solution to the location of the roots. They utilize the sinusoidal response of the open-loop transfer function to determine values of parameters such as K, that keep these toots a safe distance from the right half plane. The actual transient response for a given value of K, can be only crudely approximated. However, frequency-response methods are easily applied to systems containing transportation lags and may be used with only experimental knowledge of the unsteady-state process behavior. A mastery of control theory requires knowledge of both methods because they ate complementary. However, the reader may choose to study only frequency response and still be adequately prepared for most of the material in the remainder of this book. The choice of studying only root locus will be more restrictive in terms of preparation for subsequent chapters. In addition, much of the literature on process dynamics relies heavily on frequency-response methods.
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14.1. Write the characteristic equation and construct the Routh array for the control
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system shown in Fig. P14.1. Is the system stable for (a) Kc = 9.5, (b) Kc = 11, (c) K, = 12
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FIGURE P14-1
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FIGURE Pl4-2
4.2 By means of the Routh test, determine the stability of the system shown in Fig. P14.2 when K, = 2. 4. In the control system of Prob. 13.6, determine the value of gain (psi/ F) that just 9 causes the system to be unstable if (a) 70 = 0.25 min, (b) 70 = 0.5 min. 14.4. Prove that, if one or mom of the coefficients (ao, a 1, . . . , an) of the characteristic equation [Eq. (14.9)] is negative or zero, then there is necessarily an unstable root. Hint: First show that allao is minus the sum of all the rqots, u2/uo is plus the sum of all possible products of two roots, u/uo is (- 1) times the sum of all possible products of j roots, etc. 14.5. Prove that the converse statement of Prob. 14.4, i.e., that an unstable root implies that one or mom of the coefficients will be negative or zero, is untrue for all n > 2. Hint: To prove that a statement is untrue, it is only necessary to demonstrate a single counterexample. 14.6. Deduce an extension of the Routh criterion that will detect the presence of roots with real parts greater than --(+ for any specified cr > 0. 14.7. Show that any complex number s satisfying 1s 1 1 yields a value of < 1+s z== that satisfies Re(z) > 0
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