NT1330 Unit 3 Assignment 1

where $x_i,i=1,2, cdots ,n$ are the states, $underline{x}_i=[x_1,cdots,x_i]^{T} in{R}^i$, $i=1,2, cdots ,n $, $uin {R}$ is the input, and $f_i(cdot)$,$i=1,2, cdots ,n $ are the unknown smooth nonlinear functions which satisfy the global Lipschitz condition. It is assumed that the output $y(cdot)$ is sampled at instants $t_k,k=1,2, cdots ,n$, which represent the sampling instants. $T=t_{k+1}-t_k$ is the sampling interval which is a positive constant. The output signal is available for the observer at instants $t_k+ au_k$, where $ au_k$ are the transmission delays and satisfy $0 leqslant au_k leqslant T$.

egin{remark} label{rem:1} It is assumed that a transmission delay takes place at instant $t_k$, then the sampled data is used for observer at instants $t_k+

The knowledge base consists of a collection of fuzzy if-then rules of the following form: $R^{l}$: if $x_1$ is $F_1^{l}$ and $x_2$ is $F_2^{l}$ and $ldots$ and $x_n$ is $F_n^{l}$, then is $G^{l},~l=1,2, cdots ,n$, where $x=[x_1,cdots,x_i]^{T}$ and $y$ are the FLS input and output, respectively. Fuzzy sets $F_i^{l}$ and $G^{l}$, associated with the fuzzy functions $mu_{{F_i}^{l}}(x_i)$ and $mu_{{G}^{l}}(y)$, respectively. $N$ is the rules inference number.

\Through singleton function, center average defuzzification and product inference cite{shaocheng2000fuzzy}, the FLS can be expressed as:

For any continuous function $f(x)$ defined on a compact set $Omegain R^n$, there exists a fuzzy system $y(x) = heta ^T xi (x)$ and any positive constant $delta$, such that: where there exists an ideal parameter vector $ heta ^*$ to estimate the value of the vector $ heta$. The optimal parameter vector $ heta ^*$ can be defined as:

According to Lemma 1, in order to estimate the