A Game-Theoretic Framework for PI/PID Controller Selection in a Coupled Two-Tank Hydraulic System

This paper addresses loop-wise PI/PID controller structure selection in a coupled two-tank hydraulic system. Existing studies generally prescribe controller architectures or compare predefined configurations after synthesis, leaving this problem insufficiently explored when each choice depends energetically on the neighboring loop. Although exact feedback linearization decouples the auxiliary tracking dynamics, the reconstructed physical pump inputs retain nonlinear hydraulic coupling. Consequently, reducing the control energy of one loop may increase the effort required by the other, creating conflicting local objectives that motivate a non-cooperative game-theoretic formulation. The plant is feedback-linearized, PI and PID controllers are parameterized by pole placement, and their assignment is posed as a finite static game in which each player minimizes pump energy cost. For the nominal symmetric configuration, (PID, PID) is the unique Nash equilibrium, although (PI, PI) yields a lower aggregate control energy cost (291.4134 vs. 291.8184), showing that unilateral optimality does not coincide with collective energy minimization. PID reduces settling time from 5.253 s to 3.422 s (34.9%) but increases maximum overshoot from 13.55% to 20.82% (53.7%). Sensitivity analyses show that tuning, leakage, coupling, and initial conditions can reverse best responses, while a ±10% parametric mismatch analysis preserves convergent tracking with small steady-state errors, although the equilibrium may change.