Variable-Structure Systems and Sliding-Mode Control by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9783030366216
Publisher: Springer International Publishing

(b)if additionally, (8.19) is such that

(8.21)

then in the sliding motion x(t) converges exponentially to .

The proof of this theorem is given in Sects. 8.5.2 and 8.5.3.

8.5.1 On the Closed-Loop Solutions

The time derivative of the sliding variable is given by

(8.22)

Note that the right-hand side of (8.22) depends on x, and the right-hand side of (8.4) in closed-loop with (8.15) depends on . Moreover, the right-hand side of (8.22) is discontinuous in . Thus, (8.22), (8.4) constitute a system of discontinuous functional differential equations. Nevertheless, the existence and uniqueness of solutions of such a system can still be guaranteed by means of standard definitions as explained below.

First, let us rewrite (8.4) and (8.22) as Follows:

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