## 4 Universal Approximation Characteristics of Agent Based Fuzzy T-S Mutil-Model Systems  
In this section, the approximation characteristic of ABFT-SMM system will be discussed. Without loss of generality, the continuous system of ABFT-SMM will be discussed first. With (1), (9), the expression of the continuous system of ABFT-SMM can be expressed as follows:  
\[\dot{x}(t)=\frac{\sum\limits_{i=1}^{N}\lambda_{i}(x_{i}(t))f_{i}(x_{i}(t),u_{ i}(t))}{\sum\limits_{i=1}^{N}\lambda_{i}(x_{i}(t))} \tag{12}\]  
where \(\lambda_{i}(x_{i}(t))=\prod\limits_{l=1}^{in+m+1}\exp(-(\frac{n-l}{s_{i}^{2}} )^{2})\), \(z_{l}\) is the \(l^{th}\) variable that effect agent \(i\), here \(l=1,...,\)\((in+m+1)\). Now we can get the following theorem:  
Theorem 1: An ABFT-SMM system with product inference engine, singleton fuzzifier, center average defuzzifier, and Gaussian membership functions can uniformly approximate any linear or nonlinear continuous function on a compact set at arbitrary accuracy. To prove this theorem, the famous Stone-Weierstrass theorem will be used. Stone-Weierstrass Theorem [22]: Let \(Z\) be a set of real continuous functions on a compact set \(U\). If, (i) \(Z\) is an algebra, that is, the set \(Z\) is closed under addition, multiplication, and scalar multiplication; (ii) \(Z\) separates points on \(U\), that is, for every \(x,y\in U,x\
eq y\), there exists \(f\in Z\) such that \(f(x)\
eq f(y)\); (iii) \(Z\) vanishes at no point of \(U\), that is, for each \(x\in U\) there exists \(f\in Z\) such that \(f(x)\
eq 0\); then for any real continuous function \(g(x)\) on \(U\) and arbitrary \(\epsilon\), there exists \(f\in Z\) such that \(\sup\limits_{x\in U}|f(x)-g(x)|<\epsilon\). Proof of theorem 1: Let \(Y\) be the set of all fuzzy systems in the form of (12). We now show that \(Y\) is an algebra, \(Y\) separates points on the compact set \(U\), and \(Y\) vanishes at no point of \(U\). Firstly, we will prove \(Y\) is an algebra. Let \(f_{1},f_{2}\in Y\), and then \(f_{1},f_{2}\in Y\) can be written as:  
\[f_{1}=\frac{\sum\limits_{i=1}^{N1}\lambda_{1i}(\varphi(k))f_{1i}(x_{1i}(t),u_{ 1i}(t))}{\sum\limits_{i=1}^{N1}\lambda_{1i}(\varphi(k))} \tag{13}\]  
\[f_{2}=\frac{\sum\limits_{i=1}^{N2}\lambda_{2i}(\varphi(k))f_{2i}(x_{2i}(t),u_{ 2i}(t))}{\sum\limits_{i=1}^{N2}\lambda_{2i}(\varphi(k))} \tag{14}\]  
Then, \(f_{1}(x)+f_{2}(x)\) can be expressed as the form of (15),  
\[f_{1}+f_{2}=\frac{\sum\limits_{l_{1}=1}^{N1}\sum\limits_{l_{2}=1}^{N2}2 \lambda_{1i}\lambda_{2i}(f_{1i}+f_{2i})}{\sum\limits_{l_{1}=1}^{N1}\sum \limits_{l_{2}=1}^{N2}2\lambda_{1i}\lambda_{2i}} \tag{15}\]  
where, \(2\lambda_{1i}\lambda_{2i}\) can be seen as the input Gaussian membership function, \(f_{1i}+f_{2i}\) is the dynamic equation of the agent. Obversely, (15) own the same structure with (12), it means that \(f_{1}(x)+f_{2}(x)\in Y\). Similarly,  
\[f_{1}f_{2}=\frac{\sum\limits_{l_{1}=1}^{N1}\sum\limits_{l_{2}=1}^{N2}\lambda_ {1i}\lambda_{2i}(f_{1i}f_{2i})}{\sum\limits_{l_{1}=1}^{N1}\sum\limits_{l_{2}=1 }^{N2}\lambda_{1i}\lambda_{2i}} \tag{16}\]  
It is easy to know that \(f_{1}f_{2}\in Y\). At the same time, when \(c\in R\)  
\[cf_{1}=\frac{\sum\limits_{l=1}^{N1}c\lambda_{1i}f_{1i}}{\sum\limits_{l=1}^{N1 }c\lambda_{1i}} \tag{17}\]  
which is also in the same form with (12), so \(cf_{1}(x)\in Y\). Then, we said that \(Y\) is an algebra. Secondly, we prove that \(Y\) separates points on \(U\). Construct a required fuzzy system \(f(x)\), let \(z^{0},\)\(k^{0}\in U\) be two arbitrary points and \(z^{0}\
eq k^{0}\). The parameters of \(f(x)=\dot{x}(t)\) in the form of (12) are chosen as \(N=2,f_{1}=0,f_{2}=1,s_{i}^{l}=1,z_{i}^{*1}=z_{i}^{0},z_{i}^{*2}=k_{i}^{0}(i=1,2,l=1,2,...,n)\).