Switching Plane Design for the Sliding Mode Control of Systems with Elastic Input Constraints

Abstract

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...

Shifted Switching Plane Design for Third-Order Systems with Elastic and Conventional Input Constraints

“…In order to find the optimal switching plane parameters when the controlled plant is subject to the elastic input constraint, we minimise a new criterion representing the sum of ITAE and a penalty function for using excessive values of the control signal. The results presented in this paper are novel compared with our earlier work [3] because two different control quality criteria are minimised in the papers. In this paper we minimise ITAE, whereas in [3] IAE was taken into account.…”

“…The results presented in this paper are novel compared with our earlier work [3] because two different control quality criteria are minimised in the papers. In this paper we minimise ITAE, whereas in [3] IAE was taken into account.…”

“…A similar approach has been adopted in [11] where a nonlinear, time-varying switching line, i.e., a changing-shape parabola, was introduced. Furthermore, the idea of using time-varying switching planes for sliding mode control of third-order systems subject to input constraints has recently been proposed in a paper [3]. In that paper the switching plane was designed in such a way that the integral of the system absolute error (IAE) was minimised.…”

“…In that paper the switching plane was designed in such a way that the integral of the system absolute error (IAE) was minimised. The approach introduced in the paper [3] helps obtain good dynamic performance and insensitivity of the controlled systems with respect to external disturbance and parameter uncertainties from the very beginning of the control action.…”

“…The problem considered in this work is formulated in Section 2. Section 3 presents the control law proposed in [3] and then-for clarity-gives a brief description of the system performance. The new time-varying switching plane design procedure is described in Section 4.…”