By Wan E.A.
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6 Summary The Root Locus technique presents a graphical method of investigating the roots of the characteristic equation of a linear time-invariant system when one or more parameters vary. The steps of construction listed above should be adequate for making a reasonably adequate plot of the root-locus diagram. The characteristic-equation roots give exact indication on the absolute stability of the system. The zeros of the closed-loop transfer function also govern the dynamic performance of the system.
Also, by adding the zero, we can move the locus to a position having closed-loop roots and damping ratio 0:5 We have "compensated" then dynamics by using D(s) = s + 2. The trouble with choosing D(s) based on only a zero is that the physical realization would contain a di erentiator that would greatly amplify the high-frequency noise present from the sensor signal. Furthermore, it is impossible to build a pure di erentiator. Try adding a pole at a high frequency, say at s = ;20, to give: 2 D(s) = ss++20 (85) The following gure shows the resulting root loci when p = 10 and p = 20.
U(t) = ;1 h( ) u(t ; )dt Y (s) = H (s) U (s) 20 (10) (11) Derivatives y_ $ sY (s) ; y (0) (12) y(n) + an;1 y (n;1) + : : : + a1y_ + a0y = bmum + bm;1um;1 + : : : + b0u (13) m m;1 Y (s) = bmssn ++a bm;s1ns;1 + +: : :: :+: +a b0 U (s) (14) Y (s) = H (s) = b(s) U (s) a(s) (15) n;1 for y (0) = 0. 2 Poles and Zeros Consider the system, H (s) = s2 2+s 3+s 1+ 2 b(s) = 2 (s + 21 ) ! zero : ;21 a(s) = (s + 1) (s + 2) ! poles : ; 1 ;2 21 2 x x o H (s) = s;+11 + s +3 2 h(t) = ;e;t + e;2t The response can be empirically estimated by an inspection of the pole-zero locations.