PID Principles to Obtain Adaptive Variable Gains for a Bi-order Sliding Mode Control

A new model to obtain adaptive variable gains for a bi-order Sliding Mode Control is proposed in this work. The variable gains for the controller are designed to dynamically adapt their values, using principles of the well-known Proportional-Integral-Derivative control technique, where the magnitude of the tracking error is the signal feedback. According the way to tune parameters, it can become a first-order or a second-order controller. This design takes into account the actuators constraints (operational limits of the plant to control). As a result of the adaptive properties of the proposed scheme, the new controller significantly reduces the energy consumption in control processes, and it rejects the so called chattering-effect, simultaneously maintaining the main robust properties of the Sliding Mode strategy. In order to show the feasibility and effectiveness of the proposed design, simulation results are presented, where the performance of the proposed controller is compared with the conventional Sliding Modes of order one and two, and also with the classical PID controller. A strong stability analysis in the sense of Lyapunov is presented, showing global exponential stability for the equilibrium point of the closed-loop control system when the proposed control design is used.