Semi-global fixed/predefined-time RNN models with comprehensive comparisons for time-variant neural computing

Abstract

This paper concerns with the time-variant neural computing in a semi-global sense, taking into account initial conditions located within a region with a definitely finite radius. Both the conventional single and double power-rate RNN models are characterized and the closed-form expressions of the settling time functions are presented for given initial conditions, by which the fixed/predefined-time convergence can be assured in the semi-global sense. Despite asymptotic convergence behavior, the conventional linear RNN model is examined for comparison purposes. Modified RNN models adopt the inverse of the bound, according to the fixed-time convergence results, and the prescribed time can be an adjustable parameter. A novel two-phase RNN model with the pre-specified transition state is proposed, which has not only semi-global fixed/predefined-time stability but also a faster convergence rate than that of the conventional models. The proposed models are applied and compared, through numerical simulation, for time-variant matrix inversion, linear equation solving, and repeatable motion planning of a redundant manipulator in the presence of initial errors.

Appendix A. The proof for Corollary 1

For model (8), it follows that

$$\begin{aligned} {{\dot{\Lambda }}}_{ij} = -\kappa _{1}\Lambda _{ij}^{\alpha }- \kappa _{2} \Lambda _{ij}^{\beta } \end{aligned}$$

From $\Lambda _{ij}(0) > 1$ to $\Lambda _{ij}(t_{1})=1$, we have

$$\begin{aligned} {{\dot{\Lambda }}}_{ij} \ge -\kappa _{1}\Lambda _{ij}- \kappa _{2} \Lambda _{ij}^{\beta } \end{aligned}$$

due to that $\Lambda _{ij}^{\alpha } \le \Lambda _{ij}$. Defining $y =\Lambda _{ij}^{1-\beta }$ yields

$$\begin{aligned} {\dot{y}} + \kappa _{1}(1-\beta )y + \kappa _{2}(1-\beta ) \le 0 \end{aligned}$$

Solving the above linear differential inequality, we obtain

$$\begin{aligned} y(t)\le & {} \left( y(0) + \frac{\kappa _{2}}{\kappa _{1}} \right) e^{- \kappa _{1} (1-\beta )t} - \frac{\kappa _{2}}{\kappa _{1}} \end{aligned}$$

Since $y(t_1)=1$,

$$\begin{aligned} t_1 \ge \frac{1}{\kappa _1 (\beta -1)} \mathrm{ln} \left( \frac{ \left( 1+\frac{\kappa _{2}}{\kappa _{1}}\right) }{ \left( \Lambda _{ij}^{1-\beta }(0) +\frac{\kappa _{2}}{\kappa _{1}}\right) } \right) \end{aligned}$$

(44)

From $\Lambda _{ij}(t_1)=1$ to $\Lambda _{ij}(t_2)=0$, we have

$$\begin{aligned} {{\dot{\Lambda }}}_{ij} \ge - \kappa _{1} \Lambda _{ij}^{\alpha }-\kappa _{2}\Lambda _{ij} \end{aligned}$$

due to that $\Lambda _{ij}^{\beta } \le \Lambda _{ij}$. Defining $y =\Lambda _{ij}^{1-\alpha }$ leads to

$$\begin{aligned} {\dot{y}} + \kappa _{2} (1-\alpha )y + \kappa _{1} (1-\alpha ) \ge 0 \end{aligned}$$

Solving the above differential inequality for $t \ge t_1$, we obtain

$$\begin{aligned} y(t)\ge & {} \left( y(t_1) + \frac{\kappa _{1}}{\kappa _{2}} \right) e^{- \kappa _{2} (1-\alpha ) (t-t_1)} - \frac{\kappa _{1}}{\kappa _{2}} \end{aligned}$$

Since $y(t_1) = 1$ and $y(t_2)=0$,

$$\begin{aligned} t_2 \ge t_1 + \frac{1}{\kappa _{2}(1-\alpha )} \mathrm{ln} \left( 1+ \frac{\kappa _{2}}{\kappa _{1} } \right) \end{aligned}$$

(45)

Hence, the settling time function satisfies

$$\begin{aligned}&T_s(\Lambda _{ij}(0)) \\\ge & {} \frac{1}{\kappa _{1} (\beta -1)}\mathrm{ln} \left( \left( 1+\frac{\kappa _{2}}{\kappa _{1}}\right) / \left( \Lambda _{ij}^{1-\beta }(0) +\frac{\kappa _{2}}{\kappa _{1}}\right) \right) \\&+\frac{1}{\kappa _{2}(1-\alpha )} \mathrm{ln} \left( 1+\frac{\kappa _{2}}{\kappa _{1}} \right) \end{aligned}$$

(46)

This completes the proof.