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Non-singular Terminal Sliding Mode Funnel Control of Servo Systems With Input Saturation  221


                               Substituting (14.21)into(14.17), then the closed-loop error dynamics
                            can be derived as
                                                                     1
                                                                         p/q
                                               T
                                    s ˙ 2 = F φ 	 F [ ˜ W φ(X) + ε − μsgn(s 2 )]− |s 2 | sgn(s 2 )  (14.23)
                                                                     β
                            14.3.3 Stability Analysis
                            In this section, the boundedness of all signals and the stability of the system
                            (14.23) in both the reaching phase and the sliding phase will be provided.
                            To prove finite-time convergence, we first present the following lemma:
                            Lemma 14.1. [19] Assume that there exists a continuous positive definite func-
                            tion V(t) satisfying the following inequality:

                                                       γ
                                               ˙ V(t) + nV (t) ≤ 0,                   (14.24)
                                                                 ∀t > t 0
                            where n > 0, 0 <γ < 1 are all positive constants. Then, for any given t 0,V(t)
                            satisfies the following inequality:

                                     V  1−γ  (t) ≤ V 1−γ  (t 0 ) − n(1 − γ)(t − t 0 ), t 0 ≤ t ≤ t s

                            and


                                                   V(t) ≡ 0, ∀t ≥ t s
                            where t s is given by

                                                           V  1−γ  (t 0 )
                                                    t s ≤ t 0 +
                                                            n(1−γ)
                               Please refer to [19] for a detailed proof of the above lemma.
                               Now, the main results of this chapter can be summarized as follows:

                            Theorem 14.1. Consider the motor servo system (14.6) with unknown non-
                            linear saturation (14.2), the non-singular terminal sliding manifold (14.20), feed-
                            back control (14.21), and the adaptive law (14.22) are used, then:
                            1) All signals in the closed-loop system are bounded.
                            2) The non-singular terminal sliding manifold s 2 can converge to zero in finite
                                                                            T
                                time provided that the parameter is set as μ>P N +
 ˜ W φ(X)
.
                            3) The tracking error e can be retained within a prescribed boundary (14.10).
                            Proof. 1) Choose the following Lyapunov function as

                                                     1  2  1
                                                       s + W   W
                                                V =  2k m 2  2  ˜ T  −1 ˜             (14.25)
                            where k m =|F φ 	 F | > 0.
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