A Fast Finite-Time Neural Network Control of Stochastic Nonlinear Systems

Command‐filter‐based fast finite‐time composite adaptive neural control for nonlinear systems with input dead‐zone

“…Under the virtual control law (46), the composite adaptation law (47) and the compensate signal (48), the derivative of V 2 can be given by…”

“…Consider the nonlinear system (1). By employing the virtual control laws (29), (46) and ( 63), the actual control law (80), the composite adaptation laws (30), (47), (64), and (81) and with the help of the command-filter ( 16), Assumptions 1-3 and suitable parameters, we can gain that the tracking error z 1 = x 1 − y d converges to a small neighborhood around zero in fast finite-time and all signals in the closed-loop system are bounded in fast finite-time. The tracking error z 1 satisfies…”

“…Lemma (Reference 46 ) Consider the system $$$ \dot{\chi}=f\left(\chi, u\right) $$$ , assuming that $$$ V\left(\chi \right) $$$ is a positive definite smooth function. If there exist constants $$$ {K}_1>0 $$$, $$$ {K}_2>0 $$$, $$$ 0<\frac{\beta }{2}<1 $$$ , and $$$ 0<\rho <\infty $$$ such that $$$$ \dot{V}\left(\chi \right)\le -{K}_1V\left(\chi \right)-{K}_2V{\left(\chi \right)}^{\frac{\beta }{2}}+\rho, $$$$ then system $$$ \dot{\chi}=f\left(\chi, u\right) $$$ is practical almost fast finite‐time stable.…”

Adaptive fuzzy fixed‐time control for non‐triangular structural stochastic switching nonlinear systems

“…Different from asymptotic stability, finite‐time stability can realize state variables rapidly converge to equilibrium within finite‐time interval. Therefore, finite‐time stability problem has received increasingly attention 24‐31 . Furthermore, a large number of meaningful works have proposed the idea of fixed‐time control (FTC) which achieved that the settling‐time is independent of the initial conditions, which makes the FTC algorithm more advanced and more practical.…”

“…Theorem 1. : For the stochastic system (1) under Assumptions 1 and 2, the virtual control signals (26) (42), the controller (54), and the adaptive law (55) are designed. Combining with the bounded initial conditions, the controlled systems are SGPFS, and z 1 converges into a small region near zero.…”