In this paper a new sliding mode control algorithm for the third-order non-linear, time-varying, uncertain system subject to unknown disturbance is proposed. Since the conventional input constraints expressed by inequalities are often impractical, in this paper the situation is considered where the input signal is subject to elastic constraint. It is assumed that the threshold value of the system input signal is known and exceeding this value is undesirable but possible if justified by essential improvement of the system performance. The proposed algorithm employs a time-varying switching plane. At the initial time the plane passes through the point determined by the system initial conditions in the error state space and afterwards moves with a constant velocity to the origin of the space. The plane is designed in such a way that fast error convergence is achieved using limited control input. By this means, the reaching phase is eliminated, insensitivity of the system to external disturbance is ensured from the very beginning of the control action, and fast, monotonic error convergence to zero is achieved. Moreover, it is demonstrated that the conventional input constraint expressed by an inequality is a limit case of the elastic constraint considered in the paper.Keywords: sliding mode control, variable structure systems, time-varying switching planes, switching surface design INTRODUCTIONcontrol, namely the desired dynamic behaviour of the system, is not obtained for some time from the In recent years much of the research in the area of beginning of its motion. Furthermore, usually for the control systems theory focused on the design of a given initial conditions, there is a trade-off between discontinuous feedback which switches the structure the short reaching phase and the fast system response of the system according to the evolution of its state in the sliding phase. In order to overcome these vector. This technique, usually called sliding mode problems the idea of the time-varying switching control, provides an effective and robust means lines applied for the sliding mode control of the of controlling non-linear plants [1][2][3][4]. The main second-order systems was introduced in references advantage of this technique is that once the system[5] to [7] and further discussed in references [8] state reaches a sliding surface, the system dynamics and [9]. The control algorithms proposed in the remain insensitive to a class of parameter variations papers [8] and [9] eliminate the reaching phase and disturbances. and guarantee a fast error convergence rate for the However, robust tracking is assured only after the second-order uncertain systems with arbitrary initial system state hits the sliding surface; i.e. the robustconditions. Further results on the application of the ness is not guaranteed during the reaching phase.time-varying switching lines for the sliding mode Provided a conventional time-invariant sliding plane control of the second-order systems have recently is considered, the advantage of the slid...
In this paper the design of a time-varying switching plane for the sliding-mode control of the third-order system subject to velocity, acceleration and input signal constraints is considered. Initially, the plane passes through the system representative point in the error state space and then it moves with a constant velocity to the origin of the space. Having reached the origin the plane stops moving and remains fixed. The plane parameters (determining angles of inclination and the velocity of its motion) are selected to ensure the minimum integral absolute error (IAE) without violating velocity, acceleration and input signal constraints. The following cases of the switching plane parameters selection are considered. Firstly, the IAE is minimized subject to each of the three constraints separately. Then, every pair of the constraints is considered, and finally all three constraints are taken into account simultaneously. In all the cases the switching plane is chosen in such a way that the reaching phase is eliminated, insensitivity of the system with reference to the external disturbances and the model uncertainty is guaranteed from the very beginning of the proposed control action and fast, monotonic error convergence to zero is achieved.
In this paper, a new sliding mode control algorithm for a third-order uncertain, non-linear and time-varying dynamic system subject to unknown disturbance and input constraint is proposed. The algorithm employs a time-varying switching plane. At the initial time tt0, the plane passes through the point determined by the system initial conditions in the error state space. Afterwards, the plane moves with a constant velocity to the origin of the space. Two methods of switching plane parameter selection are presented. The first one ensures the minimization of the integral of the absolute error (IAE) over the whole period of the control action and the second one minimizes the integral of the time multiplied by the absolute error (ITAE). In both cases the input constraint is satisfied. The switching plane is chosen in such a way that the reaching phase is eliminated, insensitivity of the system to the external disturbance and the model uncertainty is guaranteed from the very beginning of the proposed control action and fast, monotonic error convergence to zero is achieved.
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