Dimensional analysis approach to dominant three-pole placement in delayed PID control loops

PID control loops with time delay are characterized by infinite number of poles but the pole assignment technique for adjusting the controller parameters can be applied to placing three poles only. The dominance of these poles is therefore an essential condition for this application. A novel approach to this problem involves applying dimensional analysis theory to obtain a generalized model of the control loop and then to perform a parameter tuning for its dimensionless representation. A one-row dimensional matrix results from the assumption of the usual dimensionless interpretation of both control error and actuating signals of the controller. Dimensionless similarity numbers of the so-called swingability and laggardness are introduced to specify the plant dynamics in the controller synthesis. A trio of numbers is assigned to become the dominant zeros of the characteristic quasi-polynomial of the control loop and the corresponding PID parameter adjustment is derived in the form of uniform formulae. The tuning of the proper damping and the real pole position ratios is provided by means of an IAE optimization technique. A dominance degree notion is introduced and an argument increment criterion is proposed to check the dominance of any of the pole placement cases. The quality of the disturbance rejection response is taken as the general criterion in the design of the time delay plant control.