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The regulation of the contact force between the pantograph and the overhead supply line (the catenary) has been recognized as one of the main problems to face with in order to speed up the velocity in high-speed railways systems [13, 16, 17, 19, 23]. Taking into account usual pantograph and catenary parameters, resonant modes are excited corresponding to train speed larger than 200 km/h, and this can limit the railway system performance. [ABSTRACT FROM AUTHOR]

Chattering is the most problematic issue in sliding mode control applications [30], [31], [14], [33], [36]. Among the well known approaches based on smooth approximations of the discontinuities [11],[29] and asymptotic state observers [10, 33], the use of the high order sliding mode control approach can attenuate the chattering phenomenon significantly [15],[21], [2],[4], [5],[3],[22],[27]. [ABSTRACT FROM AUTHOR]

In the last two decades the theory of variable structure control (VSC) with sliding modes (SM) [1-4] has been applied in the field of electrical drives [5, 6], principally due to the robustness and high dynamic performance of the resulting schemes. [ABSTRACT FROM AUTHOR]

In this paper a collection of control algorithms which are able to give rise to 2-sliding modes have been presented, and for each of them the sufficient convergence conditions are given. Furthermore, the real sliding behaviour is briefly considered, and, in some cases, the upper bound of the convergence time is given. 2-sliding mode control seems to be an effective tool for the control of uncertain nonlinear systems since it overcomes the main drawbacks of the classical sliding mode control, i.e., the chattering phenomenon and the large control effort. Its real implementation implies very simple control laws and assures an improvement of the sliding accuracy with respect to real 1-sliding mode control. The main difficulty in using 2-sliding mode controllers is the tuning of the parameters which characterize the various algorithms. Their values depend on the bounds of the uncertain dynamics and on the chosen sliding manifold, and only sufficient conditions for the convergence to the sliding behaviour are known. These conditions are very conservative, and, in practice, the parameters are heuristically tuned. It depends on the engineer's experience to define which of the presented algorithms is more suitable for the specific control problem, even if, in the authors' opinion, the super-twisting and the sub-optimal ones seem to be able to cover a large class of control problems with a remarkable implementation easiness. [ABSTRACT FROM AUTHOR]