No abstract

U S I N G a few simple rules, it is possible to design a compensator which will provide a feedback control system with a pair of complex conjugate roots of a preselected value. T h i s can be done while simultaneously m a i n t a i n i n g any specified value of gain (K p , K v , K a , etc). I n general, the characteristic equation of the closedloop system is specified bywhere GH(s) is the loop transfer function. T h e roots of this equation are readily evaluated using root-locus techniques. Assume that the designer has evaluated GH(s) for a given system, has determined the m i n i m u m acceptable gain, and has determined the s-plane coordinates at which he desires to locate a pair of complex conjugate roots without altering the steady-state accuracy. If the desired location of a root is r 9 then r S-PLANE V /S\ 1 p ^ΊΓ c -VJ W , \ \ h V X /$)$\fs\ 1 ijw * < J ' Fig. 1. S-plane diagram for the derivation of equations.is not on the root locus of GH(s), the phase angle of GH(s) is not 18(W° at r, b u t is some other n u m b e r which may be computed by using the spirule, or by substituting the ^-co-ordinates of r in the equation for GH(s). T h e angle, Φ, which a compensator must provide to place point r on the compensated root locus is obtained by subtraction. T h e "root-locus gain" at s z= r may also be computed from the original GH(s) G = r n (r + Pi) (r + P 2 ) (r + zi) (r + z 2 )(2)Note that the spirule may also be used to evaluate G.T h e vector ratio contributed by the compensator is(3)