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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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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