DESIGN COMPENSATION FOR LINEAR CONTROL SYSTEM
A control system might need some adjustments in order that several conflicting requirements can be adequately met. These adjustments are called compensations.
A compensator can be inserted either in the forward or feedback paths, or in both the paths. In hardware control systems, the compensators are some physical devices that are electrical, mechanical, or hydraulic/pneumatic. In digital control systems, wherein the control laws are computer programs, the compensators are also computer programs and work equivalently to the HW devices. A lead compensator can be used to provide a phase lead between the output and the input. Similarly lag and lead-lag compensators are used as the case may be. The main idea is to modify the frequency response of the open loop system/TF such that the performance of the compensated closed loop system is satisfactory. Table C2 gives an overview of various approaches used for control system design.
The method is based on the fact that it is possible to adjust the location of the poles of the closed loop TF by varying the loop gain. The root locus sketches the movement in the s-plane of the poles of the closed loop TF as open loop gain or some parameter is varied from zero to infinity. Let the characteristic polynomial of the closed loop system be represented as KP(s) + Q(s) = 0, with K as some parameter to be varied from zero to infinity to obtain the plots of the roots of this equation leading to the root locus. It can be represented in the form: 1 + KP(s)/Q(s) = 0 or Kq(| = — 1 = 1ff180°. This will yield the basic conditions to be satisfied by all the points on the root locus: (1) the angle condition—at a point on the root locus the algebraic sum of the angles of vectors drawn to it from the open loop poles and zeros is an odd multiple of 180° and (2) the magnitude condition—at a point on the root locus the value of K is given as
product of lengths of the vectors from poles
product of lengths of the vectors from zeros
The angle condition tells us if any point in the s-plane lies on the root locus, and the magnitude condition gives the value of K for which this point will be a root of the characteristic equation. Important properties of the root locus are as follows: (1) it is symmetrical about the real-axis of the s-plane, (2) a root locus branch starts from each open loop pole and terminates at each open loop zero or infinity, (3) the number of branches that terminate at infinity is equal to the number of open loop poles less the number of open loop poles, (4) the sections of the real axis to the left of an odd number of poles and zeros are part of the locus, for K > 0, and (5) if the number of poles and zeros are odd and to the right of a point, then this point is on the root locus.