1. Introduction
Along with the development of sensor technology, electro mechanical system and digital communication network, both the basis structure and run mode of engineering control system changed enormously [
1,
2]. These components of the engineering control systems are more intelligent, they will never act as the single role of controlled object or controller [
3,
4], instead an agent integrates sense, compute, execution and communication together. However, it also brings more uncertainty to the system, such as time variation and nonlinear characteristics. Therefore, the investigation of agent based fuzzy T-S multi-model system by the way of data information of the process is studied.
Many works have mentioned the idea of agent based modelling [
5,
6,
7,
8]. Different from the traditional model, the agent based modelling is formed by agents that lie in different layers [
9]. There exists the interaction between the agents, and the input signals for an agent may be affected by the feedback information of others. Affecting other agents and affected by the others at the same time, the essence of the cooperation and interaction between different agents is the communication process of data information [
10]. The communication channels between agents form the real time dynamic network. Within the network, each agent can execute the task alone or collaborate with other agents to obtain a fixed goal [
11,
12]. However, the control-oriented models of engineering systems call for further investigation.
Few attempts have been initiated in the study of the agent based fuzzy T-S multi-model system. In [
13], a multi-agent consensus problem with an active leader and variable interconnection topology is studied. A neighbor-based local controller together with a neighbor-based state-estimation rule is given for each autonomous agent to track the leader. Then, a cooperative output regulation problem for linear uncertain multi-agent problem is studied in [
14], and it shows a simple way of cooperation of the system if all subsystems of the agents have the same nominal dynamics. In [
15], the fuzzy modelling and consensus problem of multi-agent nonlinear systems is analyzed. The T-S fuzzy models are proposed to express the system with variable structures. However, the agent based fuzzy multi-model strategy remains preliminary because how to construct the agents and lead all the agents in a multi-model system to reach a consensus is still a problem. More endeavors should be conducted in order to use the modelling strategy widely.
Starting from the basic concepts of agent and fuzzy T-S model, the agent based fuzzy T-S multi-model (ABFT-SMM) system is proposed in this paper. In this system, the agent that is described with a dynamic equation will be the local part of the multi-model, and the consensus of the agents accomplished by the center average defuzzifier. Moreover, we proved the ABFT-SMM could approximate any linear or nonlinear systems at arbitrary accuracy.
The structure of the rest of this paper is as following. In
Section 2, several concepts of agents and fuzzy T-S model are introduced. In
Section 3, the method for building agent based fuzzy multi-model is described in detail. In section 4, the approximation capability of the agent based fuzzy multi-model system is analyzed. Three different application examples are given in
Section 5. Lastly, the conclusion is drawn in
Section 6.
3. Agent Based Fuzzy T-S Multi-Model System
It has been proved that the fuzzy T-S model can approximate any nonlinear systems at arbitrary accuracy in [
17] and [
20], At the same time, it has been proved that the second kind of fuzzy T-S model has the similar approximation capability with the first kind of fuzzy T-S model [
21]. Here we use the linearized consequent of the fuzzy T-S model to express the dynamical equation of the agent. Combine formula (2) with (6), the agent based fuzzy model can be written as:
where
is the rule of agent
i , and
is the fuzzy subset,
is the consequent parameter of linear sub model.
The center average defuzzifier is adopted to the agent based fuzzy model, and then the agent based fuzzy T-S multi-model will be described as:
where
y is the system action which affected by the agents that instead the different work conditions,
N is the number of the agents,
is the dynamical equation of agent
i and its linearized expression is like the output of (8), the expression of
is corresponding to the membership value of agent
i and it can be calculated from Gaussian bells function as (10).
where
is
dimensional row vector,
n is the number of the agent states,
m is the number of the past input states that relevant to current agent output,
is the central variable of the Gaussian function, and
is used to determine the width of the Gaussian function. Moreover,
can be gotten in the following way
where
is the
neighbor center of the current center
, and constant
is used to determine the impact between deferent agents. Usually, the variation range of
is not significant, and the value of
can be chosen offline with the prior knowledge.
As analysed, in the modeling process, agents are h-consensus to the system output. A single agent is equivalent to a local network model. The cooperation among agents is accomplished by their membership value.
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:
where
,
is the
variable that effect agent
i, here
. 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
, there exists
such that
; (iii)
Z vanishes at no point of
U, that is, for each
there exists
such that
; then for any real continuous function
on
U and arbitrary
ϵ, there exists
such that
.
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
, and then
can be written as:
Then,
can be expressed as the form of (15),
where,
can be seen as the input Gaussian membership function,
is the dynamic equation of the agent. Obversely, (15) own the same structure with (12), it means that
. Similarly,
It is easy to know that
. At the same time, when
which is also in the same form with (12), so
. Then, we said that
Y is an algebra.
Secondly, we prove that
Y separates points on
U. Construct a required fuzzy system
, let
be two arbitrary points and
. The parameters of
in the form of (12) are chosen as
. The established fuzzy system is as follows:
and we can get (19) and (20) from (18)
Since , then we can know , then it is easy to get that . Therefore, Y separates points on U.
At last, we prove that
Y vanishes at no point of
U. It can be known from (12) that there always be
when
, hence
Combining the proof and Stone-Weierstrass theorem, the conclusion of theorem 1 can be obtained. While it is just the continuous time condition, the approximate ability will be discussed for the discrete system following:
Corollary 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 discrete function on a compact set at arbitrary accuracy.
Proof of Corollary 1: Assume
is a square integrable function in compact set
, and
, combine with theorem 1, the following equation can be gotten
Then, it is easy to know that .
More details about the approximate characteristic of the fuzzy model can be found in [
20], while it mentioned the traditional fuzzy system. Here, we draw the collusion form theorem 1 and corollary 1 that ABFT-SMM can approximate any linear or nonlinear system at arbitrary accuracy, and it means that the ABFT-SMM system is a universal approximator. Thus, the proposed method could be widely used in a number of fields, such as behavior modeling, adaptive nonlinear control, fault detection and diagnostics. This paper could be a theoretical foundation for the implementation of ABFT-SMM systems in practical applications.