PID Control of α-Order Systems in Fractal Time

This paper presents a novel proportional–integral–derivative (PID) control framework for first α-order systems evolving in fractal time. The main contribution is the extension of classical control theory to systems exhibiting anomalous temporal scaling by employing local fractal derivatives. In contrast to fractional-order PID (FOPID) approaches, which primarily model memory effects, the proposed fractal PID framework captures time-scaling behavior arising in non-smooth environments, such as viscoelastic friction and irregular contact surfaces. The closed-loop dynamics are formulated as a second α-order fractal differential equation, from which a characteristic equation is derived to establish conditions for asymptotic stability. It is shown that, for a constant reference input and positive controller gains, the tracking error converges to zero as t→∞. In addition, a quantitative performance analysis demonstrates that the fractal-order α governs temporal stretching: smaller values of α lead to increased rise and settling times and reduced oscillation frequency. The effectiveness of the proposed approach is illustrated through applications to a thermal system with fractal heat input and robotic actuators operating in irregular environments. These results highlight the potential of fractal-time control as a systematic framework for modeling and controlling dynamical systems with non-integer temporal structure.