all $t \ge 0$, the truncated $\mathcal{L}_\infty$ norm and the $\mathcal{L}_\infty$ norm \[, Section 5\] are defined as $\norm{x_\tau(t)}_{\mathcal{L}_\infty} \triangleq \max_{1\le i \le n}(\sup_{0\le t \le \tau}|x_i(t)|)$ and $\norm{x(t)}_{\mathcal{L}_\infty} \triangleq \max_{1\le i \le n}(\sup_{t \ge 0}|x_i(t)|)$, respectively.

 

Preliminaries
=============

Consider the nonlinear uncertain dynamical system given by \_(t)&=&A\_x\_(t)+B\_u(t) + B\_ \_(x\_(t)), x\_(0)=x\_, \[unc:sys\] where $x_\rom{p}(t)\in\IR^{n_\rom{p}}$ is the accessible state vector, $u(t)\in\IR^m$ is the control input, $\delta_\rom{p}:\IR^{n_\rom{p}}\rightarrow\IR^m$ is an *uncertainty*, $A_\rom{p}\in\IR^{n_\rom{p} \times n_\rom{p}}$ is a known system matrix, $B_\rom{p}\in\IR^{n_\rom{p} \times m}$ is a known control input matrix, and $\Lambda \in \IR^{m \times m}_+\cap \mathbb{D}^{m \times m}$ is an *unknown* control effectiveness matrix. Furthermore, we assume that the pair $(A_\rom{p},  B_\rom{p})$ is controllable and the uncertainty is parameterized as \_(x\_)&=&W\_\_(x\_), x\_\^[n\_]{}, \[unc:prm\] where $W_\rom{p}\in \IR^{s\times m}$ is an *unknown* weight matrix and $\sigma_\rom{p}:\IR^{n_\rom{p}} \rightarrow \IR^s$ is a known basis function[^1] of the form $\sigma_\rom{p}\bigl(x_\rom{p}\bigl)=\bigl[\sigma_\rom{p_1}\bigl(x_\rom{p}\bigl), \sigma_\rom{p_2}\bigl(x_\rom{p}\bigl),\ldots,  \sigma_\rom{p_s}\bigl(x_\rom{p}\bigl)\bigl]\mT$.

To address command following, let $c(t)\in\IR^{n_\rom{c}}$ be a given bounded piecewise continuous command and $x_\rom{c}(t)\in\IR^{n_\rom{c}}$ be the integrator state satisfying \_(t)&=&E\_x\_(t)-c(t), x\_(0)=x\_, \[int:state\] where $E_\rom{p}\in\IR^{n_\rom{c} \times n_\rom{p}}$ allows to choose a subset of $x_\rom{p}(t)$ to be followed by $c(t)$[^2]. Now, (\[unc:sys\]) can be augmented with (\[int:state\]) as (t)&=&Ax(t)+Bu(t)+BW\_\_(x\_(t))+B\_c(t), x(0)=x\_0, \[unc:sys2\] where $x(t)\triangleq\bigl[x_\rom{p}\mT(t),  x_\rom{c}\mT(t)\bigl]\mT\in\IR^n$, $n=n_\rom{p}+n_\rom{c}$, is the 