# The Hybrid Adaptive Control of T-S Fuzzy System Based on Niche

Authors: Tong Zhao; Guo-ping Lu; Yun-li Hao; Yi-min Li
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Based on the niche characteristics, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Considering the close degree of Niche as the consequent of adaptive T-S fuzzy control system, the hybrid control law is designed by tracking, continuous supervisory, and adaptive compensation. Adaptive compensator is used in the controller to compensate the approximation error of fuzzy logic system and the effect of the external disturbance. The adaptive law of consequent parameters, which is achieved in this paper, embodies system adaptability as biological individual. It is proved that all signals in the closed-loop system are bounded and tracking error converges to zero by Lyapunov stability theory. The effectiveness of the approach is demonstrated by the simulation results.


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## Body

## 1. Introduction

In recent years, fuzzy technique has gained rapid development in complex nonlinear plants [1]. Fuzzy logic offers human reasoning capabilities to capture nonlinearities and uncertainties, which cannot be described by precise mathematical models. Theoretical proofs of fuzzy models as universal approximators have been presented in the last decade [2, 3]. Fuzzy adaptive control methodologies have emerged in recent years as promising ways to approach nonlinear control problems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge for nonlinear control problems. Recently, fuzzy systems have aroused a great resurgence of interest from part of the control community on the ground that they may be applied to model ill-defined complex systems. Direct and indirect schemes are two staple configurations for adaptive fuzzy controls. It has been established that indirect and direct controls (DAFC and IAFC) are able to incorporate plant knowledge and control knowledge, respectively, to yield stable and robust control systems [4]. The last two decades or so have witnessed a large quantity of results on indirect [5, 6] or direct [7–11] adaptive control (IAC/DAC) using fuzzy systems. However, the fuzzy adaptive controller proposed in [6, 7] ensure that the tracking error asymptotically convergence to zero (or a neighborhood of the origin) if the minimum approximation error is squared integrable along the state trajectory. And those algorithms are just confined to the linear and nonlinear systems whose state variables are assumed to be available for measurement. In many complicated cases, all state variables are not measurable such that output feedback or observer-based adaptive control techniques have to be applied. In [12–15], observer-based IAC and DAC algorithms are proposed for nonlinear systems, respectively. But [6, 12, 13] proposed adaptive control gain is only applicable to nonlinear systems with unknown constant gain. So a hybrid adaptive fuzzy control is needed.In order to exploit the relative advantages of indirect and direct adaptive configurations at the same time, some researchers have developed several hybrid IAC and DAC algorithms [16–18] where a weighting factor, which can be adjusted by the tradeoff between knowledge of the plant and knowledge of the control, is adopted to sum together the control efforts from both the IAC and DAC. However, those schemes have their limitations. Above all, they take full-state feedback, which can be unsuitable for nonlinear systems without state variables available. Moreover, the conventional adaptive controller proposed in [19] combines indirect, direct, and variable structure methods; nonetheless its plants are assumed to be linear systems only. The author of [6] has developed a hybrid adaptive fuzzy control (HAFC) for nonlinear systems. However, input gains are required to be a constant 1 and control gain is an unknown constant. The HAFC algorithm of [20] can just be applied to robot manipulators with bounds estimation whereas that of [21] derives an unsupported HAFC scheme from a faulty Lyapunov derivative. On the other hand, a certain observer-based combined direct and indirect adaptive fuzzy neural controller is developed in [22]. Sensor fault may be in any form, even unbounded, which will make the system failure unavoidably. Paper [23] proposed a reliable observer-based controller, which makes the system work well no matter whether sensor faults occur or not.Among various kinds of fuzzy models, there is a very important class of Takagi-Sugeno (T-S) fuzzy models [24] which have recently gained much popularity because of their special rule consequent structure and success in a functional approximation [6, 25]. In the recent two years, with the stability theory of T-S fuzzy system drawing lots of researchers attention, Yeh et al. [25] proposed stability analysis of interconnected and robustness design of time-delay fuzzy systems using fuzzy Lyapunov method. Moreover, T-S fuzzy model plays an important role in dealing with practical problems, such as oceanic structure [26], Vehicle occupant classification [27], and engineering systems [28], in which T-S fuzzy systems are applied to sensor fault estimation.Upon all the fuzzy control method mentioned above, the consequent rules only contain the mathematical expressions without the practical significances. In this paper, the self-adaptability of the ecosystem is introduced into the fuzzy control method. Ecosystem is a complex large system, the interaction between individuals within the system, the interaction between the system and the outside world, made the internal dynamics of ecological system extremely complex. Biological evolution in complex ecosystems displays a strong nonlinearity [29], and increasing complexity is an indication of biological evolution, which makes it easier for organisms to adapt to environmental changes, and higher adaptability is the performance of a higher degree of evolution, that is, the so-called “survival of the fittest” [30]. Because of the redundancy and stability of ecological system, individuals acquire their own living ability and evolve to the advantage direction. This is why the ecological system could keep balance. The stability of the system is closely related with individuals’ ecological niche. Through such feature, Wang and Yi-min [31] discussed a method of fuzzy control based on niche model. The basic concept and methods of niche are introduced. Using the similarity measures, the capabilities of the living’s self-learning and self-organization are joined in fuzzy system, gaining the fuzzy system which has the meaning of niche. Considering the high-order nonlinearity, complexity, and biological individuals’ adaptability of ecological systems, we combine biological characteristics with adaptive fuzzy control method and raise a new hybrid T-S fuzzy adaptive control way, which contains great biological significance itself. The consequent of fuzzy system is parameter’s ecological niche close degree functions for T-S model. In condition of immeasurable biological individuals’ ecological factors, in [18, 30], the author designs controller and obtains adaptive law with consequent parameter which takes good tracking effect to changing error target function. Fuzzy T-S system of ecological niche reflects the degree of biological exploiting and usage and develops to its favorable way. During this course, the system moves forward by mutual changes, coordination with environment and themselves. This fuzzy adaptive control of niche reflects that the biology of individual organisms in the new environment has a strong adaptability, strong in the biological and practical significance, but also provided a reference value for the artificial cultivation of rare species. Therefore, fuzzy control background gets its biological meaning in this way.


## 2. Problem Formulation

Consider thenth-order nonlinear system of the following form:
(2.1)x(n)=f(x,x˙,…,x(n-1))+g(x,x˙,…,x(n-1))u+d(t)y=x,
where f and g are unknown but bounded functions, u,y∈R are the input, output of the system, respectively,  and y and d(t) is the external bounded disturbance. X=(x1,…,xn)T=(x,x˙,…,x(n-1))T∈Rn is the state vector where not all xi are assumed to be available for measurement. In order for the system (2.1) to be controllable, it is required that g(X)≠0 for X in a certain controllability region Uc⊂Rn. Since g(X) is continuous, we assume 0<g(X)≤∞ for X∈Uc. The control objective is to design a combined controller, tune the correlation parameters of adaptive law, and force the system output y to follow a bounded reference signal ym(t) under the constraint that all signals involved must be bounded. To begin with, the signal vector y¯m(t), the tracking error vector e, and estimation error vector e¯   are  defined as y¯m=(ym,y˙m,…,y˙m(n-1))T∈Rn,  e=ym-X=(e,e˙,…,e(n-1))T∈Rn,  e¯=y¯m-X¯=(e¯,e¯˙,…,e¯(n-1))T∈Rn, where X¯ and e¯ denote the estimates of X and e, respectively. If the functions f(X) and g(X) are known and the system is free of external disturbance d(t), then we can choose the controller u* to cancel the nonlinearity and design controller. Let k0=(k01,…,k0n)T∈Rn be chosen such that all roots of the polynomial sn+k0nsn-1+⋯+k01 are in the open left half-plane and control law of the certainty equivalent controller is obtained as
(2.2)u*=1g(X)[-f(X)+ym(n)+k0Te].
Substituting (2.2) to (2.1), we get the main objective of the control is limt→∞e(t)=0. However f(X) and g(X) are unknown, the ideal controller (2.2) cannot be implemented, and not all system states X can be measured. So we have to design an observer to estimate the state vector X in the following.


## 3. T-S Fuzzy System Based on Niche

### 3.1. Description of T-S Fuzzy System Based on Niche

Becausef(X) and g(X) are unknown, T-S fuzzy systems are used to approximate them. The basic configuration of T-S [24] system is expressed as
(3.1)Rfl:ifx1isA1l,…,xnisAnl,thenf^l(X)=32-ϕ[λfl-λσfl+σ],Rgl:ifx1isG1l,…,xnisGnl,theng^l(X)=32-ϕ[λgl-λσgl+σ],Rul:ifx1isU1l,…,xnisUnl,thenu^l(X)=32-ϕ[λul-λσul+σ](l=1,…,m).
Here, Ail,Gil,Uil are fuzzy sets, λl=(x1,x2,…,xn) represents the real ecological factor of ecologic niche, and λ=(x1*,x2*,…,xn*) is the ideal ecological factor. Consequent indicates the difference between real state of ecologic niche and ideal one. we choose Gaussian membership function, which satisfies xil-xjl≫δ, where xil,xjl are the centers and σ is the variance of the functions. Then difference could be showed by approach functions as H(A,B)=3/2-ϕ[(λfl-λ)/(σfl+σ)], where ϕ(x)=(1/2π)∫-∞xe-t2/2dt. From approach functions above, we could know consequent is a zero-order T-S mode. Here, we use a0l as 3/2-ϕ[(λfl-λ)/(σfl+σ)] for convenience. Let m be the number of systems with central average defuzzifier, and product inference and singleton fuzzifier can be expressed as(3.2)f^(X∣θf)=∑l=1ma0l(∏i=1nexp⁡(-((xi-x¯il)/σil)2))X~∑l=1m∏i=1nexp⁡(-((xi-x¯il)/σil)2)=∑l=1mξl(X)θfl=ξ(X)Tθf,
where ξl(X)=∏i=1nexp(-((xi-x¯il)/σil)2)X~/∑l=1m∏i=1nexp(-((xi-x¯il)/σil)2), θfl=(a0l,0,…,0)∈R1×(n+1), X~=[1,x1,…,xn]T;ξ(X)=(ξ1(X),…,ξm(X))∈R1×m, θf=(θf1⋮θfm)∈Rm×(n+1); g^(X∣θg)=ξ(X)Tθg and  uD(X∣θD)=η(X)TθD can be expressed similarly.


### 3.2. Niche T-S Model Parameter Optimization Method

Niche fuzzy T-S model based on parameter optimization of backpropagation algorithm known input and output data(x0p,y0p),x0p∈U∈Rn,y0p∈V∈R, the task is to determine the form (3.2) Niche T-S model parameters, To fitting error ep=(1/2)[f^(x0p)-y0p]2 the minimum. Assume M is known, by adjusting λfl,σfl,x¯il,σil, and let ep be the minimum. To facilitate the discussion, use e^,f^,y^ to represent ep,f^(x0p), and y0p. b=∑l=1mZl,Zl=∏i=1nexp(-((xi-x¯il)/σil)2),a=∑l=1m(a0lZl).Using gradient descent to determines the parameters [6]
(3.3)λfl(q+1)=λfl(q)-γ∂e^∂λfl|q,l=1,2,…,m,q=0,1,2,…,γ
determines  the steps.Then(3.4)∂e^∂λfl=∂e^∂f^∂f^∂a∂a∂a0l∂a0l∂λfl=(f^-y^)⋅12πe-((λfl-λ)/(σfl+σ))2/2⋅1σfl+σ⋅1b⋅ZlX. Then,(3.5)λfl(q+1)=λfl(q)-γ(f^-y^)⋅12πe-((λfl(q)-λ)/(σfl(q)+σ))2/2⋅1σfl(q)+σ⋅1b⋅ZlX.
Equally,
(3.6)σfl(q+1)=σfl(q)-γ(y^-f^)⋅12πe-((λfl(q)-λ)/(σfl(q)+σ))2/2⋅λfl(q)-λ(σfl(q)+σ)2⋅1b⋅ZlXadjustable parameters x¯il,σil with the same way:(3.7)x¯il(q+1)=x¯il(q)-γ∂e^∂x¯il|q=x¯il(q)-γ(f^-y^)(f^-Nb)⋅2(xi-x¯il(q))(σil(q))2Xσil(q+1)=σil(q)-γ∂e^∂σil|q=σil(q)-γ(f^-y^)(f^-Nb)⋅2(xi-x¯il(q))2(σil(q))3X,
where  N=∑l=1m(3/2-ϕ((λfl-λ)/(σfl+σ))),λgl,σgl,λul and σul can be optimized directly.


## 3.1. Description of T-S Fuzzy System Based on Niche

Becausef(X) and g(X) are unknown, T-S fuzzy systems are used to approximate them. The basic configuration of T-S [24] system is expressed as
(3.1)Rfl:ifx1isA1l,…,xnisAnl,thenf^l(X)=32-ϕ[λfl-λσfl+σ],Rgl:ifx1isG1l,…,xnisGnl,theng^l(X)=32-ϕ[λgl-λσgl+σ],Rul:ifx1isU1l,…,xnisUnl,thenu^l(X)=32-ϕ[λul-λσul+σ](l=1,…,m).
Here, Ail,Gil,Uil are fuzzy sets, λl=(x1,x2,…,xn) represents the real ecological factor of ecologic niche, and λ=(x1*,x2*,…,xn*) is the ideal ecological factor. Consequent indicates the difference between real state of ecologic niche and ideal one. we choose Gaussian membership function, which satisfies xil-xjl≫δ, where xil,xjl are the centers and σ is the variance of the functions. Then difference could be showed by approach functions as H(A,B)=3/2-ϕ[(λfl-λ)/(σfl+σ)], where ϕ(x)=(1/2π)∫-∞xe-t2/2dt. From approach functions above, we could know consequent is a zero-order T-S mode. Here, we use a0l as 3/2-ϕ[(λfl-λ)/(σfl+σ)] for convenience. Let m be the number of systems with central average defuzzifier, and product inference and singleton fuzzifier can be expressed as(3.2)f^(X∣θf)=∑l=1ma0l(∏i=1nexp⁡(-((xi-x¯il)/σil)2))X~∑l=1m∏i=1nexp⁡(-((xi-x¯il)/σil)2)=∑l=1mξl(X)θfl=ξ(X)Tθf,
where ξl(X)=∏i=1nexp(-((xi-x¯il)/σil)2)X~/∑l=1m∏i=1nexp(-((xi-x¯il)/σil)2), θfl=(a0l,0,…,0)∈R1×(n+1), X~=[1,x1,…,xn]T;ξ(X)=(ξ1(X),…,ξm(X))∈R1×m, θf=(θf1⋮θfm)∈Rm×(n+1); g^(X∣θg)=ξ(X)Tθg and  uD(X∣θD)=η(X)TθD can be expressed similarly.


## 3.2. Niche T-S Model Parameter Optimization Method

Niche fuzzy T-S model based on parameter optimization of backpropagation algorithm known input and output data(x0p,y0p),x0p∈U∈Rn,y0p∈V∈R, the task is to determine the form (3.2) Niche T-S model parameters, To fitting error ep=(1/2)[f^(x0p)-y0p]2 the minimum. Assume M is known, by adjusting λfl,σfl,x¯il,σil, and let ep be the minimum. To facilitate the discussion, use e^,f^,y^ to represent ep,f^(x0p), and y0p. b=∑l=1mZl,Zl=∏i=1nexp(-((xi-x¯il)/σil)2),a=∑l=1m(a0lZl).Using gradient descent to determines the parameters [6]
(3.3)λfl(q+1)=λfl(q)-γ∂e^∂λfl|q,l=1,2,…,m,q=0,1,2,…,γ
determines  the steps.Then(3.4)∂e^∂λfl=∂e^∂f^∂f^∂a∂a∂a0l∂a0l∂λfl=(f^-y^)⋅12πe-((λfl-λ)/(σfl+σ))2/2⋅1σfl+σ⋅1b⋅ZlX. Then,(3.5)λfl(q+1)=λfl(q)-γ(f^-y^)⋅12πe-((λfl(q)-λ)/(σfl(q)+σ))2/2⋅1σfl(q)+σ⋅1b⋅ZlX.
Equally,
(3.6)σfl(q+1)=σfl(q)-γ(y^-f^)⋅12πe-((λfl(q)-λ)/(σfl(q)+σ))2/2⋅λfl(q)-λ(σfl(q)+σ)2⋅1b⋅ZlXadjustable parameters x¯il,σil with the same way:(3.7)x¯il(q+1)=x¯il(q)-γ∂e^∂x¯il|q=x¯il(q)-γ(f^-y^)(f^-Nb)⋅2(xi-x¯il(q))(σil(q))2Xσil(q+1)=σil(q)-γ∂e^∂σil|q=σil(q)-γ(f^-y^)(f^-Nb)⋅2(xi-x¯il(q))2(σil(q))3X,
where  N=∑l=1m(3/2-ϕ((λfl-λ)/(σfl+σ))),λgl,σgl,λul and σul can be optimized directly.


## 4. Hybrid Controller with Supervisory and Compensation  Control Scheme

The overall control law is constructed as(4.1)u=αuI(X¯)+(1-α)uD(X¯∣θ)+us+uc,
where uI is an indirect controller, uD is the output of the T-S controller, us is the supervisory control to force the state within the constraint set, α∈[0,1] is a weighting factor, and uc is the compensate controller of adaptive control. Since X cannot be available and f(X) and g(X) are unknown, we replace the functions f(X),g(X), and error vector  e  by estimation functions f^(X¯),g^(X¯), and e¯. The certainty equivalent controller can be rewritten as
(4.2)u*=1g(X)[-f(X)+ym(n)+k0Te¯].
The indirect control law is written as  uI=(1/g^(X¯))[-f^(X¯)+ym(n)+k0Te¯]. Applying (4.2), (4.1) to (2.1), the error dynamic equation is
(4.3)e˙=Ae-Bk0Te¯+B{α[f^(X¯)-f(X)+(g^(X¯)-g(X))uI]}+(1-α)g(X)(u*-uD)+Bg(X)(us+uc)-Bd,defining   kc=(kcn,…,kc1) as the observer vector, the observation errors are defined as e~=e-e¯  from (4.2) and [24] and we get
(4.4)e~˙=Λe~+B{α[f^(X¯)-f(X)+g^(X¯)-g(X)uI]+(1-α)g(X)(u*-uD)}-Bg(X)(us+uc)-Bd,where
(4.5)Λ=A-kcCT=[-kcn100⋯00-kc(n-1)010⋯00⋮⋮⋮⋮⋱⋮⋮-kc2000⋯01-kc1000⋯00],A=[0100⋯000010⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯010000⋯00],B=[00⋮01],C=[10⋮00].
Let A¯=A-Bk0T be strictly Hurwitz matrix; so there exists a positive definite symmetric n×n  matrix P, P¯ which satisfies the Lyapunov equation ΛTP+PΛ=-Q, A¯TP¯+P¯A¯=-Q¯,   where Q¯ and Q are arbitrary  n×n definite symmetric matrix. Let Ve¯=(1/2)e¯TPe¯. Since Q¯ and kc are designer by the designer, we can choose Q¯ and kc, such that V˙e¯≤0. Hence, Ve¯ is a bounded function and there exists a constant value V¯e¯, such that Ve¯≤V¯e¯.


## 5. Hybrid Adaptive Control of Niche

We will develop the hybrid adaptive control such that the closed-loop outputy(t) follows ym(t). Let us replace f^(X¯),g^(X¯), and uD(X¯) by f^(X¯∣θf),g^(X¯∣θg), and uD(X¯∣θD), respectively. Therefore, the error dynamics (4.3) can be rewritten as
(5.1)e~˙=Λe~+B{α[f^(X¯∣θf)-f(X)+g^(X¯∣θg)-g(X)uI]Λe~+B+(1-α)g(X)(u*-uD(X¯∣θD))}+Bg(X)(us+uc)-Bd.
Let Ve~=(1/2)e~TPe~ then using (5.1), we have
(5.2)V˙e~≤-12e~TQe~+|e~TPB|{α[|f^(X¯∣θf)|+|f(X)|+|g^(X¯∣θg)u|I+|g(X)uI|]-12e~TQe~+|e~TPB|+|(1-α)g(X)u*|+|(g(X)uD(X¯∣θD))|}+|e~TPB|[(1-α)|g(X)α+uc|]-e~TPBg(X)us.In order to designus so that V˙e~≤0, we need the following assumption.Assumption 5.1.
We could find three functions asfu(X),gu(X),gl(X) and get |f(X)|≤fu(X)≈fu(X¯) and 0<gl(X¯)≈gl(X)≤g(X)≤gu(X)≈gu(X¯), in which X∈Uc and fu(X)≈fu(X¯)<∞,gu(X)≈gu(X¯)<∞. This is due to the fact that we can choose kc to let X≈X¯. Also external disturbance is bounded. We design |d(t)|≤dm.From Assumption5.1, we choose the supervisory control us as
(5.3)us=(Ve~V¯)psgn⁡(e~TPB)[αgl(X)|f^(X¯∣θf)|+fu(|g^(X¯∣θg)|+gu|uI|)+(fu(X)+|ym(n)|+|kTe|)]+(Ve~V¯)psgn⁡(e~TPB)(1-α)([|ud|]+1gl(X)(fu(X)+|ym(n)|+|k0Te¯|)+gu|uc|+dm).
And we choose uc=cosgn(e~TPB), and co is a nonnegative constant. Considering the case Ve~>V¯ and substitute (5.3) into (5.2), we obtain V˙e~≤-(1/2)e~TQe~≤0. Therefore, if the closed-loop system with the fuzzy controller  u  as
(5.4)u=αg^-1(X¯∣θg)[-f^(X¯∣θf)+|ym(n)|+|k0Te¯|]+(1-α)uD(X¯∣θD)+us+ucworks well in the sense that the error is not too large, if Ve~≤V¯, then the supervisory control us is zero. If the system tends to diverge, that is, Ve~>V¯, then be us gins to force Ve~≤V¯.


## 6. Design of Adaptive Law

In order to adjust the parameters in the fuzzy logic system, we have to derive adaptive laws. Hence, the optimal parameters estimatesθf*,θg*, and θD* are defined as
(6.1)θf*=argmin⁡θf∈Ωf[sup⁡X∈ΩX,X¯∈ΩX¯|f^(X¯∣θf)-f(X)|],θg*=argmin⁡θg∈Ωg[sup⁡X∈ΩX,X¯∈ΩX¯|g^(X¯∣θg)-g(X)|],θD*=argmin⁡θD∈ΩD[sup⁡X∈ΩX,X¯∈ΩX¯|u*(X)-u(X¯∣θD)|],
where Ωf,Ωg,ΩD,ΩX, and ΩX¯, are compact sets of suitable bounds on θf,θg,θD,X¯, and X, respectively, and they are defined as
(6.2)Ωf={θf:‖θf‖≤Mf},Ωg={θg:‖θg‖≤Mg},ΩD={θD:‖θD‖≤MD},ΩX¯={X¯:‖X¯‖≤MX¯},ΩX={X:‖X‖≤MX},
where Mf,Mg,MD,MX¯ and MX, are positive constants. Define the minimum approximation errors as
(6.3)ω=α[f^(X¯∣θf*)-f(X)+g^(X¯∣θg*)-g(X)uI]+(1-α)g(X)(u*(X)-u(X¯∣θD*))-d(t).Now consider the Lyapunov function(6.4)V=12e~TPe~+12Γf(θf-θf*)T(θf-θf*)+α2Γg(θg-θg*)T(θg-θg*)+(1-α)2ΓD(θD-θD*)T(θD-θD*),
where  Γf=[τf0⋱τfn](n+1)(n+1)  and Γg  and ΓD is expressed similarly. We get the adaptive law as
(6.5)θ˙f=-Γfξ(X¯)BTPe~,θ˙g=-Γgξ(X¯)BTPe~uI,θ˙D=-ΓDη(X¯)g(X)BTPe~.So
(6.6)(θ˙f)T=[a˙0la˙1l⋮a˙nl]=ξ(X¯)BTPe~[τf00⋮0].
Furthermore, the adaptive law of niche factors is derived as follows. We derived the adaptive law of real ecologic factors, see Formula (6.7), (6.8), which represented the real niche always develop towards the ideal one. It reflected the compensation of the control system to the external disturbance:
(6.7)λ˙f(q+1)=-12πe-((λf(q)+λ)/(σf(q)+σ))2/21σf(q)+στf0ξ(X¯)BTPe~,σ˙f(q+1)=12πe-((λf(q)+λ)/(σf(q)+σ))2/2λf(q)+λ(σf(q)+σ)2τf0ξ(X¯)BTPe~.
The same way we have
(6.8)λ˙g(q+1)=-12πe-((λg(q)+λ)/(σg(q)+σ))2/21σg(q)+στg0BTPe~uI,σ˙g(q+1)=12πe-((λg(q)+λ)/(σg(q)+σ))2/2λg(q)+λ(σg(q)+σ)2τg0BTPe~uI,λ˙D(q+1)=-12πe-((λD(q)+λ)/(σD(q)+σ))2/21σD(q)+στD0η(X¯)g(X)BTPe~,σ˙D(q+1)=12πe-((λD(q)+λ)/(σD(q)+σ))2/2λD(q)+λ(σD(q)+σ)2τD0η(X¯)g(X)BTPe~.Applyinguswe have V˙≤-(1/2)e~TQe~+e~TPBω.Design parameters of, vector adaptive law ofθf,θg,θD, stability, and performance analysis are similar to those of [18]; here we omit it.


## 7. Simulation

Form combined fuzzy adaptive control of two-dimensional predation system [32] as
(7.1)dx1dt=x1(a-cx1-bx12-h(x))-x2αx121+βx12dx2dt=x2(-q+kαx121+βx12),
where  x1(t) means total number of food at the time of t  and x2(t) denotes the total number of predators; a,b,c,q,k,α,β are the regular numbering ecology, and k is transforming factor,  q represents death ratio of predators. h(x) is the function of Holling’s functional responses, and x2(αx12/(1+βx12)) is the third kind of Holling’s functional responses. The demonstration of the two-dimensional predator system without the controller is shown in Figure 1.Figure 1
Two-dimension predator system without the controller.In order to reach an ideal ecologic balance in this two-dimension predators system, we get a way to control it, whereh(x)=0.0012x13,a=1.0,b=0.0014,c=0.06,q=0.2,k=0.08,α=5.0,β=1.2.Then(7.2)dx1dt=x1[(1+1.2x12)(1-0.06x1-0.0014x12-0.0012x13)-x1x2]dx2dt=x2(0.16x12-0.2)+uy=x1.
In order to establish the direct relation between output  y  and controller  u, we need the derivation of  y. After derivating twice, we get
(7.3)y¨=x¨1=x1[(1+1.2x12)(0.8-0.06x1-0.0014x12-0.0012x13)-x1x2]2-x12x2(0.8-0.06x1+1.3586x12-0.0732x13-0.00168x14-0.00144x15-x1x2)+2.3972x12-0.2196x13-0.00672x14-0.0072x15-0.06x1-x12u.
Here, we  command
(7.4)f(X)=x1[(1+1.2x12)(0.8-0.06x1-0.0014x12-0.0012x13)-x1x2]2-x12x2(0.8-0.06x1+1.3586x12-0.0732x13-0.00168x14-0.00144x15-x1x2)+2.3972x12-0.2196x13-0.00672x14-0.0072x15-0.06x1g(X)=-x12.
Then (7.2) and (7.3) could be shown as
(7.5)y¨=f(X)+g(X)u.
If hoping to apply adaptive fuzzy controlling system here, we have to firstly confirm the boundary of fu,gu,  and  gl. From [29], we know that   0.2<|x1|<5.5,1.5<|x2|<5, then
(7.6)|f(x1,x2)|≤2.7205e+0.03≈2.7205e+0.03=fu(x¯1,x¯2)0<gl(x¯1,x¯2)≈gl(x1,x2)=0.04≤|g(x1,x2)|≤30.25=gu(x1,x2)≈gu(x¯1,x¯2).
According to the above, we find out that the scope of f(x1,x2)  is larger than  g(x1,x2). Therefore, we choose τf0=50,τg0=1,τD0=15. Then we select Mf=7.7,Mg=31, MD=31,  ε=0.5,  |u|≤5. The proposal method of this paper does not need to predefine the reference signal, but to reach the balance of each individual self-adaptively. Conveniently, we suppose kc=[89184] (in this way, s2+k1s+k2 could be stable) and k0=[44],   we select Q=[10131328], from ΛcTP+PΛc=-Q, then we solve P=[29-14-147]. Q^ as [40252530], and A^=[01-4-4] so we have P^=[15555]; when letting λmin=2.93, we could get V¯=2λmin=5.86. In this way, we get the ideal =0,σ=1. The adaptive law comes out as follows:
(7.7)λ˙f(q+1)=-502πe-(λf(q)/(σf(q)+1))2/21σf(q)+1ξ(X¯)BTPe~σ˙f(q+1)=502πe-(λf(q)/(σf(q)+1))2/2λf(q)(σf(q)+1)2ξ(X¯)BTPe~λ˙g(q+1)=-12πe-(λg(q)/(σg(q)+1))2/21σg(q)+1BTPe~uIσ˙g(q+1)=12πe-((λg(q)+0)/(σg(q)+1))2/2λg(q)(σg(q)+1)2BTPe~uIλ˙D(q+1)=-152πe-(λD(q)/(σD(q)+1))2/21σD(q)+1η(X¯)g(X)BTPe~σ˙D(q+1)=152πe-(λD(q)/(σD(q)+1))2/2λD(q)(σD(q)+1)2η(X¯)g(X)BTPe~.
We choose x1(0)=0.3,x2(0)=1.0, then the trend of the prey and predator numbers with T-S controller show as in Figures 2 and 3. The trend of the T-S controller is shown in Figure 4.Figure 2
The trend of the prey numbers with T-S controller.Figure 3
The trend of the predator numbers with T-S controller.Figure 4
The trend of the T-S controller.From the simulation figures, we can see that the prey and predator numbers reach a stable status in a short period of time under the control of the proposed method in this paper. The creature individuals show a characteristic of self-adaptation according to outside changes.From the figure of the trend of the T-S controller, it also reaches an ideal status to maintain the overall balance of the two-dimension predator system. The ecological system is optimized with the use of this control method.


## 8. Conclusion

For the ecological niche, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Let the close degree of Niche which contains parameters as the consequent of adaptive T-S fuzzy control system, then designs the hybrid control law by tracking, continuous supervisory and adaptive compensation. Using gradient descent to optimize the parameters, we get the adaptive law of consequent parameters, embodying biological individual’s ability of adaptability. Based on Lyapunov stability theory, it is proved that all signals in the closed-loop system are bounded and tracking error converges to zero. This paper shows that the fuzzy methods provide good results in practical engineering problems. The performance of the developed approach is illustrated by simulation, on two-dimension predation system model.



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*Source: 158720-2012-07-02.xml*