Ali J. Koshkouei, Mohammad R. Zarrabi, Mohammad H. Farahi, and Sohrab Eﬀati

Inﬁnite-horizon optimal control problem, measure theory, optimal control, sliding mode control, sliding surface design

In this paper, a sliding mode control (SMC) design approach using embedding process, measure theory and optimal control techniques is proposed. In this method, a state trajectory starting from a given initial point reaches a deﬁnite point on a surface, the so-called sliding surface, in a ﬁnite time, and then tends to the origin along the sliding surface (SS). A measure theory approach with embedding process is used to solve such a problem in two phases. In the ﬁrst phase, using an appropriate measure, an embedding is constructed to solve an appropriate optimal control problem such that the system trajectory reaches the SS in a ﬁnite time, and in the second phase, using measure theory, a control is usually designed such that the system trajectory tends to the origin along the SS. It is well-known that the main disadvantage of the use of the traditional SMCs is the chattering phenomenon. The proposed SMC is piecewise continuous and chattering free. Also, the sliding mode reaching point is selected prior to the design of the SMC and it is not necessary to verify the sliding mode reaching condition as it is guaranteed during the design process. In addition, a numerical example is presented to illustrate the effectiveness of the proposed method.

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