Adaptive cooperative control of nonlinear multi-agent systems with uncertain time-varying control directions and dead-zone nonlinearity

Adaptive neural network observer‐based filtered backstepping control for nonlinear systems with fuzzy dead‐zone and uncertainty

“…Throughout the paper, following assumptions are made:Assumption Only output variable of the system is available 14,29,30 Assumption In (), external disturbance $$$ {d}_i $$$ for $$$ i=1,\dots, n $$$ is unknown and bounded, it means that $$$ \left|{d}_i\right|\le {\overline{d}}_i $$$ where $$$ {\overline{d}}_i $$$ is an unknown positive term 2,30,43 Assumption There exist constants $$$ {\overline{f}}_{i,j} $$$ and $$$ {\underset{\_}{f}}_{i,j} $$$ such that: $$$$ {\underset{\_}{f}}_{i,j}\le \frac{\partial {f}_i}{\partial {x}_j}\le {\overline{f}}_{i,j}, $$$$ for $$$ j=1,\dots, i.$…”

“…To solve this problem, Assumption 1 is considered which says that only output variable is measurable and other state variable are estimated by the proposed high gain observer. Assumption 2 is commonly used in References [2, 30, 43], that deals with nonlinear systems with external disturbance and it implies that disturbance is bounded and it does not increase arbitrarily. Assumption 3 usually is used in the design of the nonlinear state observer and it implies that if $$$ {f}_i(x) $$$ is differentiable then it will satisfy $$$ \left|\frac{\partial {f}_i}{\partial {x}_j}\right|=\left|\underset{y_j\to {x}_j}{\lim}\frac{f\left({x}_1,\dots, {y}_j,.\dots, {x}_n\right)-f\left({x}_1,\dots, {x}_j,.\dots, {x}_n\right)}{y_j-{x}_j}\right|\le L $$$ where $$$ L=\max \left\{\;{\underset{\_}{f}}_{i,j},{\overline{f}}_{i,j},2\le i\le n,1\le j\le i\right\}.$…”

“…It destroys the systems performance and even becomes the source of instability. Most of the existing schemes describe dead‐zone by the conventional certain models 1–7 . But, in practice, dead‐zone output is uncertain and it would be better to describe it by a fuzzy value.…”

Event-triggered adaptive neural finite-time tracking control for non-strict feedback stochastic nonlinear systems with unknown time-varying control directions and input nonlinearities

“…Recently, Shahriari-Kahkeshi et al had considered uncertain time-varying control directions for multi-agent systems with dead-zone nonlinearity in. 28 Furthermore, there were some other works to research unknown control directions for stochastic nonlinear systems as follows. The stability of stochastic nonlinear large-scale systems in the strict-feedback form with unknown control directions was studied in, 29 and a Theorem was proposed to directly analyze the stability of the system.…”

“…Recently, Shahriari-Kahkeshi et al had considered uncertain time-varying control directions for multi-agent systems with dead-zone nonlinearity in. 28…”