PID Principles to Obtain Adaptive Variable Gains for a Bi-order Sliding Mode Control
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
PID Principles to Obtain Adaptive Variable Gains for a Bi-order Sliding Mode Control Sergio Alvarez-Rodríguez* and Gerardo Flores Abstract: 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. Keywords: Chattering-effect, Lyapunov stability approach, PID control, sliding mode control.
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INTRODUCTION
Sliding Mode Control (SMC) is used to design robust controllers for a large variety of nonlinear systems (see [1]). SMC strategy uses a discontinuous control law to drive the system state from an arbitrary initial state to the origin along a previously specified trajectory, i.e., the sliding manifold [2]. The main advantages of Sliding Modes (SM) are robustness to parameter uncertainty and invariance to unknown disturbances [3]. However, as it is stated by [4–6] (among others), SMC has a major drawback, which is known as the chattering-effect. This effect is caused when the control signal switches at a high frequency (see [7, 8]), causing several problems in the plant under control such as heat, mechanical wear, noise, and delay in the actuator response. The Twisting Algorithm (TA), is one of the main modalities of the so called High Order Sliding Mode Control (HOSM), which was designed to deal with the chattering-effect (see [9,10]). SMC is capable to modify the amplitude and the shape of the switching control signal by correctly tuning its parameters. To achieve robustness with low levels of chattering,
both the conventional SMC of the first order and the TA of the second order are chosen to set the basis of this work. Further, this study considers the use of proportional integral derivative (PID) control concepts to get adaptability and flexibility properties for the variable gains of the SM controller proposed along this m
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