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The purpose of this section is to translate the stability definition into a more simple criterion, one that can be used to ascertain the stability of control systems of the form shown in Fig. 14.3.
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FIGURE 14-3 Basic single-loop control system.
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CHARACTERISTIC EQUATION. From the block diagram of the control system
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(Fig. 14.3), we obtain by the methods of Chap. 12 C =
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G2 GG R+ u 1 + GlG2H 1 + G1G2H
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(14.4)
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In order to simplify the nomenclature, let G = GiG2H. We call G the open-loop transferfunction because it relates the measured variable B to the set point R if the feedback loop of Fig. 14.3 is disconnected from the comparator (i.e., if the loop is opened). In terms of the open-loop transfer function G, Eq. (14.4) becomes
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SR+-$J
(14.5)
In principle, for given forcing functions R(s) and U( S), Eq. (14.5) may be inverted to give the control system response. To determine under what conditions the system represented by Eq. (14.5) is stable, it is necessary to test the response to a bounded input. Suppose a unit-step change in set point is applied. Then GW (s) GIG:! C(s) = --1 = l+Gs s(s - rl)(s - 12) . . . (s - r,) where rl,r2, . . . . rn are the n roots of the equation 1 + G(s) = 0 (14.7) (14.6)
and F(s) is a function that arises in the rearrangement to the right-hand form of Eq. (14.6). Equation (14.7) is called the characteristic equation for the control system of Fig. 14.3. For example, for the control system of Fig. 14.1 the step response is C(s) = GG2 s(1 + G)
= (71s + l)(QS + 1) which may be rearranged to
C(s) =
s /I 1+ ,
(71s + l)(QS + l)(T3S + 1)
Kc(T3s + 1) s[T1T273s3 -I- (7172 + 7173 + ~2~3)s~ -I- (71 -I- 72 -I- 73)s + (1 -i- Kc)]
This is equivalent to
C(s) = Kc(73s 4s - rd(s + l)/T,T273 - r2M - r3)
where r 1, r2, and r3 are the roots of the characteristic equation
717273s~
(7172
T2T3)S2
+ 72 + 73)s + (1 + K,) = 0
(14.8)
LINEAR CLOSED-LOOP SYSTEMS
Evidently, for this case the function F(s) in Eq. (14.6) is F(s) =
(71s + l)(QS +
715-2 73
1)(73s
+ 1)
In Chap. 3, the qualitative nature of the inverse transforms of equations such as Eq. (14.6) was discussed. It was shown that (see Fig. 3.1 and Table 3.1), if there are any of the roots ~-1, r2, . . . , rn in the right half of the complex plane, the response C(t) will contain a term that grows exponentially in time and the system is unstable. If there are one or more roots of the characteristic equation at the origin, there is an s in the denominator of Eq. (14.6) (where m 2 2) and the response is again unbounded, growing as a polynomial in time. This condition specifies m as greater than or equal to 2, not 1, because one of the s terms in the denominator is accounted for by the fact that the input is a unit-step (l/s) in Eq. (14.6). If there is a pair of conjugate roots on the imaginary axis, the contribution to the overall step response is a pure sinusoid, which is bounded. However, if the bounded input is taken as sin w t, where o is the imaginary part of the conjugate roots, the contribution to the overall response is a sinusoid with an amplitude that increases as a polynomial in time. It is evident from Eq. (14.5) that precisely the same considerations apply to a change in U. Therefore, the definition of stability for linear systems may be translated to the following criterion: a linear control system is unstable if any roots of its characteristic equation are on, or to the right of, the imaginary axis. Otherwise the system is stable. It is important to note that the characteristic equation of a control system, which determines its stability, is the same for set-point or load changes. It depends only on G(s), the open-loop transfer function. Furthermore, although the rules derived above were based on a step input, they are applicable to any input. This is true, first, by the definition of stability and, second, because if there is a root of the characteristic equation in the right half plane, it contributes an unbounded term in the response to any input. This follows from Eq. (14.5) after it is rearranged to the form of Eq. (14.6) for the particular input. Therefore, the stability of a control system of the type shown in Fig. 14.3 is determined solely by its open-loop transfer function through the roots of the characteristic equation.
Example 14.1. In terms of Fig. 14.3, a control system has the transfer functions 0.5s + 1 G1 3 los (PI (stirred controller) tank)
G2 = 1
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