By Wan E.A.

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8 Types of Control - PID Proportional Integral D(s) = k u(k) = ke(t): (47) Zt D(s) = T1 ks u(t) = Tk e(t)dt (48) D(s) = k TD s (49) i i 0 ) Zero steady state error ( may slow down dynamics) Derivative ! increases damping and improves stability. Example: M damping G(s) = s2 +1 as = s (s 1+ a) x x 31 With derivative Pole-Zero Cancellation x x Closed loop pole trapped here But what about initial conditions ? 0 1 + s+k a ! 1y+(0)k 6= 0 a Thus, the e ect due to initial conditions is not negligible. 9 Steady State Error and Tracking The reference input to a control system is often of the form: k r(t) = kt !

1. { For 1k G(s) = 0, as k ! 1, G(s) must approach 0. Typically, this should happen at the zeros of G(s), but, it could also take place if there are more poles than zeros. , let: b(s) = sm + b1sm;1 + ::::: (69) a(s) sn + a1 sn;1 + ::::: So, if n > m then as s ! 1, G(s) ! 0. That is, as s becomes large, the poles and zeros approx. cancel each other. Thus 1 + kG(s) 1 + k (s ; 1)n;m = pni ;; mzi where, pi = poles, zi = zeros, and is the centroid. (70) (71) { There are n ; m asymptotes, where n is the number of zeros and m is the number of poles, and since 6 G(s) = 180o, we have: (n ; m) l = 180o + (l)360o For instance, n ; m = 3 ) l = 60o 180o 300o.

42 { Figure 3(b) shows another common situation when two complex-conjugate root loci ap{ { { { proach the real axis, meet at the breakaway point and then depart in opposite directions along the real axis. In general, a breakaway point may involve more than two root loci. Figure 3(c) illustrates such a situation when the breakaway point represents a fourthorder root. A root locus diagram can have more than one saddle point. They need not always be on the real axis and due to conjugate symmetry of root loci, the saddle points not on the real axis must be in complex-conjugate pairs.

### Control systems.Classical,neural,and fuzzy by Wan E.A.

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