$\mathscr{L}_{1}$ Adaptive Controller Design for Single-Link Flexible Joint Manipulator with Fuzzy-PID Filter

Smartization filter of L1$\mathcal {L}_{1}$ adaptive controller using ANFIS system optimized with genetic algorithm

“…A system is Considered with the following dynamics [27, 28]. $$\begin{equation} \begin{aligned} \dot{x}(t)&=Ax(t)+B_{m}\omega u(t)+f(t,x(t)).\\ y&=Cx(t).$$…”

“…In other words: $$\begin{eqnarray} \begin{aligned} \dot{x}(t)&= Ax(t)+ B_{m}\omega {\left(-K_{m}^{T}x(t)+ u_{ad}(t)\right)}+f(t,x(t)),\\ \rightarrow \dot{x}(t)&= {\left(A-B_{m}\omega K_{m}^{T}\right)}x(t)+B_{m}\omega u_{ad}(t)+f(t,x(t)), \end{aligned} \nonumber\\ \end{eqnarray}$$where $$B= B_{m}\omega$$ and the matrix of $$A-B K_{m}$$ represents the optimal matrix of the system and is equal to $$A_{m}$$. As a result, the system equations will be equal to [27, 28]: …”

“…In other words: $$\begin{equation} \Vert x_{0} \Vert \le \rho _{0}\le \infty . \end{equation}$$In order to design this controller, the system specified (5) is reformulated as follow [27, 28]: $$\begin{equation} \begin{aligned} \dot{x}(t)&=A_{m}x(t)+B_{m}{\left(\omega u_{ad}(t)+f_{1}(t,x(t))\right)},\\ y&=Cx(t). \end{aligned} \end{equation}$$The following assumptions are considered based on [27, 28]: Assumption Bordering $$f(t,0)$$ For every amount $$t\ge 0$$, There is a value of $$d_{f}$$ so that $$ \Vert f(t,0)\Vert _{\infty }\le d_{f}$$…”

“…The general form of state estimator is as follows [27, 28]: $$\begin{equation} \hat{x}(t)=A_{m}\hat{x}(t)+B_{m}{\left(\hat{\omega }(t)u(t)+\hat{\theta }(t)\Vert x_{t}\Vert _{\infty }+\hat{\sigma }(t)\right)}, \end{equation}$$where $$\hat{\sigma }\in \mathbb {R}^{m},\hat{\theta }\in \mathbb {R}^{m}$$ and $$\hat{w}\in \mathbb {R}^{m}$$ are comparative estimates.…”

A fast L1 adaptive constrained back-stepping controller using barrier Lyapunov function for anuncertain submersible vehicle

“…The main incentive of using L 1 adaptive control is its ability to cope with unmatched and matched uncertainties as well as the possibility of estimating uncertainties and determining the control effort to minimize the influence of these uncertainties on the system closed-loop response. The low-pass filters, introduced in this method, even in the presence of fast adaptation and large amplitude reference inputs, guarantees that the closed-loop response of L 1 adaptive control output remains in the lowfrequency range 21,22 . The primary results on L 1 adaptive control system design are reported in the seminal work in 23 .…”