Adaptive full order sliding mode control for electronic throttle valve system with fixed time convergence using extreme learning machine

Abstract

This paper proposes a novel extreme learning machine (ELM)-based fixed time adaptive trajectory control for electronic throttle valve system with uncertain dynamics and external disturbances. The developed control strategy consists of a recursive full order terminal sliding mode structure based on the bilimit homogeneous property and a lumped uncertainty changing rate upper bound estimator via an adaptive ELM algorithm such that not only the fixed time convergence for both sliding variable and error states can be guaranteed, but also the chattering phenomenon can be suppressed effectively. The stability of the closed-loop system is proved rigorously based on Lyapunov theory. The simulation results are given to verify the superior tracking performance of the proposed control strategy.

Appendix 1

1.1 Proof of Proposition 1

Recalling (34), when $\left| s \right| \ne 0$, it is observed that the estimated values of $\hat{\beta }$ will keep increasing and thus a time instance $t_{1}$ exists ensuring

$$\left( {\mathrm{H}\hat{\beta } + \xi } \right) + \Pi \mathrm{H}\hat{\beta }^{\mu + 1} \left| s \right|^{\mu + 1} + \kappa \left| s \right| > \left| {\dot{T}_{lum} } \right|$$

(A1)

where $\Pi = K^{ - 1} \left( {\eta + K} \right)$. Following (A1), from $t = t_{1}$, the adaptation gain will be significantly large and $\hat{\beta }$ will reach their final values $\hat{\beta }_{{t_{1} + \Delta t}}$ until s decreases to $s = 0$ by the controller (31) in a finite time $\Delta t$. Thus, according to the continuity property, for all t, $\hat{\beta }$ should have an upper bound, which means there exists a positive constant $\beta^{*}$ in (35) to ensure $\hat{\beta } \le \beta^{*}$ is satisfied.

This completes the proof of Proposition 1.