Prescribed-time convergent and noise-tolerant Z-type neural dynamics for calculating time-dependent quadratic programming

Abstract

Neural-dynamics methods for solving quadratic programming (QP) have been studied for decades. The main feature of a neural-dynamics solver is that it can generate a continuous path from the initial point, and the final path will converge to the solution. In particular, the Z-type neural dynamics (ZND) that has emerged in recent years shows that it can completely converge to the ideal solution for the real-time-dependent QP in the ideal situation, i.e., without noise. It is worth noting that noise substantially influences the accuracy of neural-dynamics models in the process of solving the problems. Nevertheless, the existing neural-dynamics methods show limited capacity of noise tolerance, which may seriously affect their application in practical problems. By exploiting the Z-type design formula and two nonlinear activation functions, this work proposes a prescribed-time convergent and noise-tolerant ZND (PTCNTZND) model for calculating real-time-dependent QPs under noisy environments. Theoretical analyses of the PTCNTZND model show that it can be accelerated to prescribed-time convergence to the time-dependent optimal solution, and has natural anti-noise ability. The upper bound of the convergence time is also derived theoretically. Finally, the performance of the PTCNTZND model was verified by experiments, and the results substantiate the excellent robustness and convergence characteristics of the proposed PTCNTZND model for calculating real-time-dependent QPs, as compared with the existing ZND models.

Appendix

For PTCNTZND model (4) with AF (6), by defining a Lyapunov function candidate $w(t)=|u_{j}(t)|$, we have

$$\begin{aligned} {\dot{w}}(t)\le -\alpha \bigg (r_{1}w^{i}(t)+ r_{2}w^{1/i}(t)\bigg ). \end{aligned}$$

(16)

For inequality (16), because $0<i<1$ and $1/i>1$, we obtain

$$\begin{aligned} {\dot{w}}(t)\le -\alpha r_{2}w^{1/i}(t) \end{aligned}$$

(17)

if $w(t)>1$, and

$$\begin{aligned} {\dot{w}}(t)\le -\alpha r_{1}w^{i}(t) \end{aligned}$$

(18)

for $w(t)\le 1$. If $w(0)>1$, inequality (17) guarantees $w(t)\le 1$ for

$$\begin{aligned} t\ge \frac{1}{\alpha r_{2}(1/i-1)}. \end{aligned}$$

If $w(t_0)\le 1$, it can be derived from (18) that $w(t)=0$ for

$$\begin{aligned} t\ge t_0+\frac{1}{\alpha r_{1}(i-1)}. \end{aligned}$$

Therefore, for any initial states $w(0)=|u_{j}(0)|$, when

$$\begin{aligned} t \ge \frac{1}{\alpha r_{1}(1-i)}+\frac{1}{\alpha r_{2}(1/i-1)}, \end{aligned}$$

we have $w(t)=|u_{j}(t)|=0$. It means that the convergence time of the jth subsystem of the PTCNTZND model is

$$\begin{aligned} t_{j}\le \frac{1}{\alpha r_{1}(1-i)}+\frac{1}{\alpha r_{2}(1/i-1)}. \end{aligned}$$

The proof is thus completed.