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This paper proposes new stability analysis and convergence results applied to the Iterative Feedback Tuning (IFT) of a class of Takagi–Sugeno–Kang proportional-integral-fuzzy controllers (PI-FCs). The stability analysis is based on a convenient original formulation of Lyapunov’s direct method for discrete-time systems dedicated to discrete-time input affine Single Input-Single Output (SISO) systems. An IFT algorithm which sets the step size to guarantee the convergence is suggested. An inequality-type convergence condition is derived from Popov’s hyperstability theory considering the parameter update law as a nonlinear dynamical feedback system in the parameter space and iteration domain. The IFT-based design of a low-cost PI-FC is applied to a case study which deals with the angular position control of a direct current servo system laboratory equipment viewed as a particular case of input affine SISO …