Finite-time adaptive fuzzy command filtered control for nonlinear systems with indifferentiable non-affine functions

Abstract

This paper mainly addresses the finite-time tracking problem of pure-feedback systems with indifferentiable non-affine functions. A novel adaptive fuzzy finite-time command filtered scheme is proposed by remodeling the non-affine functions. The control scheme consists of the fuzzy logic systems with modified minimal learning parameter technique, the command filtered scheme and the finite-time error compensation system. The finite-time convergences of tracking error and all closed-loop signals are proved by using the finite-time stability theory. The contributions of this paper are that the new controller can achieve finite-time convergence, relax the restriction conditions that the non-affine functions must be derivative and strictly positive or negative and reduce the computational load significantly. Finally, a simulation example is given to validate the effectiveness of the developed control technique.

Appendix A

The derivative of ${{\alpha }_{1,1}}$ is described as

$$\begin{aligned} {{{\dot{\alpha }}}_{1,1}}=\frac{\partial {{\alpha }_{1,1}}}{\partial {{x}_{1}}}{{g}_{1}}\left( {{{{\bar{x}}}}_{1}},{{x}_{2}} \right) +\frac{\partial {{\alpha }_{1,1}}}{\partial {{y}_\mathrm{r}}}{{{\dot{y}}}_\mathrm{r}}+\frac{\partial {{\alpha }_{1,1}}}{\partial {\hat{\vartheta }}}\dot{{\hat{\vartheta }}}. \end{aligned}$$

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On the one hand, it can be inferred from Assumption 2 that ${{g}_{1}}\left( {{{{\bar{x}}}}_{1}},{{x}_{2}} \right)$, ${{y}_\mathrm{r}}$ and ${{{\dot{y}}}_\mathrm{r}}$ are bounded functions. On the other hand, $\dot{{\hat{\vartheta }}}$ is a bounded function because ${\hat{\vartheta }}$ is used to estimate weight constant $\vartheta$. Therefore, ${{{\dot{\alpha }}}_{1,1}}$ is a bounded function.

Taking the derivative of ${{\alpha }_{1,2}}$ yields

$$\begin{aligned} \begin{aligned} {{{{\dot{\alpha }}}}_{1,2}}&=-{{\mu }_{1}}y_\mathrm{r}^{\left( 2 \right) }\tanh \left( \frac{{{v}_{1}}{{{{\dot{y}}}}_\mathrm{r}}}{{{r}_{1}}} \right) -{\hat{\gamma }}{{\left( \tanh \left( \frac{{{v}_{1}}}{{{r}_{1}}} \right) \right) }^{\left( 1 \right) }} \\&\quad -\dot{{\hat{\gamma }}}\tanh \left( \frac{{{v}_{1}}}{{{r}_{1}}} \right) -{{\mu }_{1}}{{\dot{y}}_\mathrm{r}}{{\left( \tanh \left( \frac{{{v}_{1}}{{{{\dot{y}}}}_\mathrm{r}}}{{{r}_{1}}} \right) \right) }^{\left( 1 \right) }}. \\ \end{aligned} \end{aligned}$$

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Because the hyperbolic tangent function and its first derivative are bounded functions, and Assumption 2 implies that $y_\mathrm{r}^{\left( 2 \right) }$ are bounded function. Therefore, ${{{\dot{\alpha }}}_{1,2}}$ is a bounded function. In summary, ${{{\dot{\alpha }}}_{1}}$ is a bounded function; therefore, for ${{\alpha }_{1}}$, there exists Lipschitz constant ${{L}_{1}}$. Repeating the same analysis process, we can prove that for ${{\alpha }_{K}}$, $k=2,3,\ldots ,n-1$, there exists Lipschitz constant ${{L}_{K}}$.