Fuzzy Finite-Time Dynamic Surface Control for Nonlinear Large-Scale Systems

Abstract

In this paper, a new finite-time dynamic surface control (DSC) method is proposed to enhance the convergence rate and robustness of a DSC system on the basis of the traditional infinite-time stability viewpoint for application to nonlinear large-scale systems. By defining a novel recursive finite-time virtual error variable, a virtual controlled stabilizing functions are derived, and the finite-time control performance is ensured by means of the finite-time-based Lyapunov stability theorem. An adaptive fuzzy logic system is adopted to approximate the unknown functions. The applicability of the proposed control strategy is verified by simulation and an experiment for a mobile manipulator and an articulated manipulator.

Appendix

Consider the following Lyapunov function:

$$ V_{s} = \sum\limits_{{i = 1}}^{m} {\sum\limits_{{j = 1}}^{{n_{i} }} {V_{{si,j}} } } $$

(61)

$$ {V_{s_{i,j}} = \frac{1}{2}s_{i,j}^{2}}. $$

(62)

By differentiating (52) with respect to time, and substituting the previous results, we obtain the following:

$$ \begin{aligned} {\dot{V}}_{\rm s} & < -\sum\limits_{i = 1}^{\rm m} \left(\sum\limits_{j = 1}^{n_{i}} {(k_{i,j} - 2)s_{i,j}^{2}} + \sum\limits_{j = 1}^{n_{i}} \left|{G_{i,j} ({\bar{x}}_{i,j})} \right|c_{i,j} s_{i,j} s_{i,j}^{\gamma_{i,j}} \right) + \frac{1}{4}\sum\limits_{i = 1}^{\rm m} \sum\limits_{j = 1}^{n_{i}} \left\| {\tilde{W}}_{Mi,n_{i} }^{\text{T}} \chi_{i,n_{i}} \right\|_{\rm max}^{2}\\ &\quad + \frac{1}{4}\sum\limits_{i = 1}^{\rm m} \sum\limits_{j = 1}^{n_{i}} \left\| {\tilde{\rho}}_{i,j} \right\|_{\rm max}^{2} \le - \sum\limits_{i = 1}^{\rm m} \left({S_{i}^{\text{T}}} K_{i} S_{i} + S_{i}^{\text{T}} \zeta_{i} S_{i}^{\gamma_{i} } \right) + \sum\limits_{i = 1}^{\rm m} {\mu_{s_i} } \\ \end{aligned} , $$

(63)

where $S_{i} = [s_{i,1}, s_{i,2}, \ldots, s_{i,n_{i}}]^{\text{T}}, \quad K_{i} = {\text{diag}}(k_{i,1} - 2, k_{i,2} - 2, \ldots, k_{i,n_{i}} - 2), \quad k_{i,j} >2$, for $i = 1, \ldots, m, \quad j = 1, \ldots, n_{i}, \zeta_{i} = {\text{diag}} \left(c_{i,1} \left|{G_{i,1}} \right|c_{i,2} \left|{G_{i,2} } \right|, \ldots ,c_{i,n_{i}} \left|{G_{i,n_{i}}} \right| \right)$, ${\tilde{W}}_{Mi} = \left[{\tilde{W}}_{Mi,1}, {\tilde{W}}_{Mi,2}, \ldots, {\tilde{W}}_{Mi,n_{i}} \right]^{\text{T}},$ $\mu_{s_{i}} = \sum\limits_{j = 1}^{n_{i}} \left\|{\tilde{W}}_{Mi,n_{i}}^{\text{T}} \chi_{i,n_{i}}\right\|_{\max}^{2}/4 + \sum\limits_{j = 1}^{n_{i}}\left\|{\tilde{W}}_{Mi,n_{i}}^{\text{T}} \chi_{i,n_{i}}\right\|_{\max}^{2}/4,\,\chi_{i} = [\chi_{i,1},\chi_{i,2},\ldots\chi_{i,n_{i}}]^{\text{T}},$ and ${\tilde{\rho}}_{i} = \left[{\tilde{\rho}}_{1} ,{\tilde{\rho}}_{2} , \ldots ,{\tilde{\rho}}_{n_{i}} \right]^{\text{T}}$. Using the notation [24]

$$ S_{i}^{\gamma_{i}} = \left[s_{i,1}^{\gamma_{i,1}}, s_{i,2}^{\gamma_{i,2}}, \ldots, s_{i,n_{i}}^{\gamma_{i,n_{i}}} \right]^{\text{T}} , $$

(64)

the following inequality is obtained as

$$ \begin{aligned} - S_{i}^{\text{T}} \zeta_{i} S_{i}^{\gamma_{i}} &= - \left[{s_{i,1} ,s_{i,2} , \ldots ,s_{i,n_{i}}} \right]\zeta_{i} \left[{s_{i,1}^{{\gamma_{i,1} }} ,s_{i,2}^{{\gamma_{i,2} }} , \ldots ,s_{{i,n_{i} }}^{{\gamma_{i,n_{i}}}}} \right]^{\text{T}} \\ & < - \lambda_{\rm min} (\zeta_{i} )\sum\limits_{j = 1}^{{n_{i} }} {s_{i,j}^{{\gamma_{i,j} + 1}} } \\ & = - \lambda_{\rm min} (\zeta_{i} )\sum\limits_{j = 1}^{{n_{i} }} {\left( {s_{i,j}^{2} } \right)^{{(r_{i,j} + 1)/2}} } \\ & = - \beta_{2i} V_{s_i}^{{(\gamma_{i} + 1)/2}} \end{aligned} , $$

(65)

where ${\beta_{1i}} = {\lambda_{\rm min}} (K_{i})$ and ${\beta_{1i}} = {\lambda_{\rm min}} (K_{i})$. Therefore, (52) can be written as

$$ {\dot{V}}_{s_i} < - \beta_{1i} V_{s_i} - \beta_{2_i} V_{s_i}^{(\gamma_{i} + 1)/2} + \mu_{s_i} . $$

(66)

Equation (66) can then be rewritten in the following two forms:

$${\dot{V}}_{s_i} + \left({\beta_{1i}} - \frac{\mu_{s_i}}{V_{s_i}} \right)V_{s_i} + {\beta_{2_i}} V_{s_i}^{(\gamma_{i} + 1)/2} < 0, $$

(67)

$$ {\dot{V}}_{s_i} + \beta_{1i} V_{s_i} + \left({\beta_{2i} - \frac{{\mu_{s_i} }}{{V_{s_i}^{{(\gamma_{i} + 1)/2}}}}} \right)V_{s_i}^{{(\gamma_{i} + 1)/2}} < 0 . $$

(68)

From (67) and (68), if $\beta_{1i}$ and $\beta_{2i}$ are selected such that $\beta_{1i} > \mu_{s_{i} } /V_{s_{i} }$ and $\beta_{2i} > \mu_{s_{i}} /Vs_{i}^{(\gamma_{i} + 1)/2}$, respectively, the finite-time stability is then guaranteed by Lemma 2, and the tracking errors will reach the regions $\left\| {S_{i} } \right\| < \Delta = \min (\Delta_{1} ,\Delta_{2} )$ of the neighborhood in S _i  = 0 and

$$ \Delta_{1} < \sqrt{\frac{2\mu_{s_i}}{\beta_{1i}}} , $$

(69)

$$ {\Delta_{2}} < \left({\frac{2 \mu_{si}} {\beta_{2i}}} \right)^{1/({\gamma_{i}} + 1)} $$

(70)

in finite time.