# Hybrid Adaptive Bionic Fuzzy Control Method

Authors: Yan Li; Yimin Li; Faxiang Zhang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The bionic fuzzy system embeds the biologically active adaptive strategy into the traditional T-S fuzzy system, which increases the active adaptability. On the basis of the researches, an identification model is added to the system, and a hybrid bionic adaptive fuzzy control method is proposed in this paper, which makes the system have biological adaptability and strong anti-interference ability. The adaptive law contains two items: the first one is the general term for adjusting system parameters by using the current state and the second one is a compensation item for the adjustment of system parameters based on the development trend.Lyapunov synthesis method is used to analyze the stability and convergence of system. The design method of fuzzy controller, adaptive laws, and parameter constraints are given. Finally, the effectiveness of the method is verified by simulation of inverted pendulum model.


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## Body

## 1. Introduction

Due to the high complexity of dynamic environment and system structure, the research of nonlinear systems is difficult to achieve accurate description. At the same time, higher requirements on the stability and effectiveness of the controlled system needed to meet certain performance index. By combining the adaptive control strategy and fuzzy control method, the stability and anti-interference of the system can be improved due to the active adaptability of the organism [1, 2]. The niche of individual organism is introduced into the design of fuzzy systems, and adaptive behavior and individual feedback are used to establish the fuzzy control model based on ecological niche [3]. The design method of genetic algorithm and double nested fuzzy control method are given to make optimized controllable niche structure. A tracking feedback control method based on optimum niche is proposed, which is applied in the intelligent greenhouse system to realize the fuzzy control design for optimal ecological system [4]. In [5, 6], the self-organization and self-learning of niche are added to the fuzzy system, and a new fuzzy control method based on niche model is given. At the same time, the universal approximation of system was demonstrated, which achieved good results by using the system function as approximation. In [7], the T-S adaptive fuzzy control method is established by using the niche proximity with normal distribution characteristics as the consequent of fuzzy rules. The research [8] gives H∞ tracking performance designs in both indirect and direct adaptive fuzzy systems.On the basis of the study, in this paper, combining the identification model [9, 10] with bionic fuzzy system and mixing the model error with tracking error, the hybrid bionic fuzzy control method is built. Stability and convergence analyzed byLyapunov synthesis method [11–13], adaptive law, and constraints of the system parameters are given. The two items of bionic adaptive law, respectively, represent the current state and development trend of systems, which makes the system have good active adaptability. Finally, the strong anti-interference and better stability of highly complexity nonlinear system are verified by the simulation [14–18].


## 2. Basic Theory of Fuzzy System

### 2.1. Hybrid Adaptive Fuzzy System

Consider the following nonlinear systems:(1)xn=fx,x˙,…,xn-1+gx,x˙,…,xn-1uy=x.f(x) and g(x) are continuous functions; u,y∈R are the input and output of the system. x=x1,x2,…,xnT=x,x˙,…,xn-1T∈Rn is obtained by measuring the state vector. In order to control the system, when x belongs to a controllable interval Uc⊂Rn,g(x)≠0 might as well set g(x)>0.The ideal controllers in the form(2)u∗=1gx-fx+ymn+kTeset e=ym-y=ym-x,e=e,e˙,…,en-1T,k=kn,kn-1,…,k1T and make the real part of all the roots of polynomial sn+k1sn-1+⋯+kn negative. When t→∞,e(t)→0. That is, the output of the system converges y asymptotically to the ideal output ym.A hybrid adaptive controller is constructed as follows [12]:(3)u=αuc+1-αuD+us+uI.uc is equivalent controller, uD is output controller, us is supervisory controller, uI is adaptive compensation controller, and α∈[0,1] is weighted factor.


### 2.2. Bionic Fuzzy System

#### 2.2.1. Niche Ecostate-Ecorole Theory Function

Considering the ecosystem withn biological unit, the size of the niche of the k biological unit is defined as(4)Nk=sk+Ckpk∑j=1nsj+Cjpj.sj indicates the state of the j biological unit, Cj is the dimensional conversion coefficient, and pj indicates the rate of change of the j biological unit, j=1,2,…,n. If the biological individual has a good state and good development trend, then the niche of the biological individual is relatively large, Nk∈[0,1]. When the niche Nk of the k biological unit is greater, it indicates the greater niche role relatively in the biological unit in the system in study.


#### 2.2.2. Bionic Fuzzy System Rules

Considering system (1), f(x) and g(x) are unknown functions, and the fuzzy system is constructed by a series of “if-then” fuzzy rules:Rfi: if x1 is A1i and x2 is A2i … and xn is Ani, thenfi(x)=Nki,i=1,2,…,Mf.Rgi: if x1 is B1i and x2 is B2i … and xn is Bni, thengi(x)=Nki,i=1,2,…,Mg.k=1,2,…,n, Ri represents i fuzzy rule, and Aki and Bki are the fuzzy sets of state vector xk, determined by the Ecostate ski and the Ecorole pki of the k biological unit. Nki=ski+Ckpki/∑m=1nsmi+Cmpmi is the output of the i rule.Using the center-average defuzzifier, product inference engine, singleton fuzzifier, and Gauss membership function, Adaptive fuzzy system based on biological adaptive strategy is as follows:(5)f^xθ1f,θ2f=∑i=1Mski+Ckpki/∑j=1nsji+Cjpji∏j=1nexp⁡-xj-x-ji/δji2∑i=1M∏j=1nexp⁡-xj-x-ji/δji2=θ1fTξx+Ckθ2fTξx,where(6)θ1f=θ1f1,θ1f2,…,θ1fMT,θ1fi=sk∑j=1nsji+Cjpji,θ2f=θ2f1,θ2f2,…,θ2fMfT,θ2fi=pki∑m=1nsmi+Cmpmi,ξx=ξ1x,ξ2x,…,ξMxT,ξix=∐j=1nexp⁡-xj-x¯ji/δji2∑i=1M∐j=1nexp⁡-xj-x¯ji/δji2.Similarly,(7)g^x∣θ1f,θ2f=θ1gTηx+Ckθ2gTηx,(8)uDx∣θ1D,θ2D=θ1DTφx+Ckθ2DTφx,where θ1f,θ1g,θ1D are on behalf of the current state of the biological individuals and θ2f,θ2g,θ2D represent development trend of the biological individuals in the future.


## 2.1. Hybrid Adaptive Fuzzy System

Consider the following nonlinear systems:(1)xn=fx,x˙,…,xn-1+gx,x˙,…,xn-1uy=x.f(x) and g(x) are continuous functions; u,y∈R are the input and output of the system. x=x1,x2,…,xnT=x,x˙,…,xn-1T∈Rn is obtained by measuring the state vector. In order to control the system, when x belongs to a controllable interval Uc⊂Rn,g(x)≠0 might as well set g(x)>0.The ideal controllers in the form(2)u∗=1gx-fx+ymn+kTeset e=ym-y=ym-x,e=e,e˙,…,en-1T,k=kn,kn-1,…,k1T and make the real part of all the roots of polynomial sn+k1sn-1+⋯+kn negative. When t→∞,e(t)→0. That is, the output of the system converges y asymptotically to the ideal output ym.A hybrid adaptive controller is constructed as follows [12]:(3)u=αuc+1-αuD+us+uI.uc is equivalent controller, uD is output controller, us is supervisory controller, uI is adaptive compensation controller, and α∈[0,1] is weighted factor.


## 2.2. Bionic Fuzzy System

### 2.2.1. Niche Ecostate-Ecorole Theory Function

Considering the ecosystem withn biological unit, the size of the niche of the k biological unit is defined as(4)Nk=sk+Ckpk∑j=1nsj+Cjpj.sj indicates the state of the j biological unit, Cj is the dimensional conversion coefficient, and pj indicates the rate of change of the j biological unit, j=1,2,…,n. If the biological individual has a good state and good development trend, then the niche of the biological individual is relatively large, Nk∈[0,1]. When the niche Nk of the k biological unit is greater, it indicates the greater niche role relatively in the biological unit in the system in study.


### 2.2.2. Bionic Fuzzy System Rules

Considering system (1), f(x) and g(x) are unknown functions, and the fuzzy system is constructed by a series of “if-then” fuzzy rules:Rfi: if x1 is A1i and x2 is A2i … and xn is Ani, thenfi(x)=Nki,i=1,2,…,Mf.Rgi: if x1 is B1i and x2 is B2i … and xn is Bni, thengi(x)=Nki,i=1,2,…,Mg.k=1,2,…,n, Ri represents i fuzzy rule, and Aki and Bki are the fuzzy sets of state vector xk, determined by the Ecostate ski and the Ecorole pki of the k biological unit. Nki=ski+Ckpki/∑m=1nsmi+Cmpmi is the output of the i rule.Using the center-average defuzzifier, product inference engine, singleton fuzzifier, and Gauss membership function, Adaptive fuzzy system based on biological adaptive strategy is as follows:(5)f^xθ1f,θ2f=∑i=1Mski+Ckpki/∑j=1nsji+Cjpji∏j=1nexp⁡-xj-x-ji/δji2∑i=1M∏j=1nexp⁡-xj-x-ji/δji2=θ1fTξx+Ckθ2fTξx,where(6)θ1f=θ1f1,θ1f2,…,θ1fMT,θ1fi=sk∑j=1nsji+Cjpji,θ2f=θ2f1,θ2f2,…,θ2fMfT,θ2fi=pki∑m=1nsmi+Cmpmi,ξx=ξ1x,ξ2x,…,ξMxT,ξix=∐j=1nexp⁡-xj-x¯ji/δji2∑i=1M∐j=1nexp⁡-xj-x¯ji/δji2.Similarly,(7)g^x∣θ1f,θ2f=θ1gTηx+Ckθ2gTηx,(8)uDx∣θ1D,θ2D=θ1DTφx+Ckθ2DTφx,where θ1f,θ1g,θ1D are on behalf of the current state of the biological individuals and θ2f,θ2g,θ2D represent development trend of the biological individuals in the future.


## 2.2.1. Niche Ecostate-Ecorole Theory Function

Considering the ecosystem withn biological unit, the size of the niche of the k biological unit is defined as(4)Nk=sk+Ckpk∑j=1nsj+Cjpj.sj indicates the state of the j biological unit, Cj is the dimensional conversion coefficient, and pj indicates the rate of change of the j biological unit, j=1,2,…,n. If the biological individual has a good state and good development trend, then the niche of the biological individual is relatively large, Nk∈[0,1]. When the niche Nk of the k biological unit is greater, it indicates the greater niche role relatively in the biological unit in the system in study.


## 2.2.2. Bionic Fuzzy System Rules

Considering system (1), f(x) and g(x) are unknown functions, and the fuzzy system is constructed by a series of “if-then” fuzzy rules:Rfi: if x1 is A1i and x2 is A2i … and xn is Ani, thenfi(x)=Nki,i=1,2,…,Mf.Rgi: if x1 is B1i and x2 is B2i … and xn is Bni, thengi(x)=Nki,i=1,2,…,Mg.k=1,2,…,n, Ri represents i fuzzy rule, and Aki and Bki are the fuzzy sets of state vector xk, determined by the Ecostate ski and the Ecorole pki of the k biological unit. Nki=ski+Ckpki/∑m=1nsmi+Cmpmi is the output of the i rule.Using the center-average defuzzifier, product inference engine, singleton fuzzifier, and Gauss membership function, Adaptive fuzzy system based on biological adaptive strategy is as follows:(5)f^xθ1f,θ2f=∑i=1Mski+Ckpki/∑j=1nsji+Cjpji∏j=1nexp⁡-xj-x-ji/δji2∑i=1M∏j=1nexp⁡-xj-x-ji/δji2=θ1fTξx+Ckθ2fTξx,where(6)θ1f=θ1f1,θ1f2,…,θ1fMT,θ1fi=sk∑j=1nsji+Cjpji,θ2f=θ2f1,θ2f2,…,θ2fMfT,θ2fi=pki∑m=1nsmi+Cmpmi,ξx=ξ1x,ξ2x,…,ξMxT,ξix=∐j=1nexp⁡-xj-x¯ji/δji2∑i=1M∐j=1nexp⁡-xj-x¯ji/δji2.Similarly,(7)g^x∣θ1f,θ2f=θ1gTηx+Ckθ2gTηx,(8)uDx∣θ1D,θ2D=θ1DTφx+Ckθ2DTφx,where θ1f,θ1g,θ1D are on behalf of the current state of the biological individuals and θ2f,θ2g,θ2D represent development trend of the biological individuals in the future.


## 3. Bionic Hybrid Controller

A nonlinear system is introduced in parallel with system (1):(9)x^˙1=x2,x^˙2=x3⋮x^˙n=f^x∣θ1f,θ2f+g^x∣θ1g,θ2gu.Define model error as(10)ε=x^˙nt-x˙nt.From system (9) and system (1), it can be obtained that(11)ε=f^x∣θ1f,θ2f-fx+g^x∣θ1g,θ2g-gxu.The model error of the original system and the identification model combined as(12)ε=ϕ1fTξx+Cϕ2fTξx+ϕ1gTηxu+ϕ2gTCηxu+ω,ϕ1f=θ1f-θ1f∗,ϕ2f=θ2f-θ2f∗,ϕ1g=θ1g-θ1g∗,ϕ2g=θ2g-θ2g∗using f^(x∣θ1f,θ2f),g^(x∣θ1f,θ2f),uD(x∣θ1D,θ2D) replace f^(x),g^(x), and uD(x); the minimum approximation error is(13)ω=αf^x∣θ1f∗,θ2f∗-fx+g^x∣θ1g∗,θ2g∗-gxuc+1-αgxu∗-uDx∣θ1D,θ2D.Tracking error can be written as(14)e˙=Λce+bcαf^x∣θ1f,θ2f-fx+g^x∣θ1g,θ2g-gxuc+bc1-αgxu∗-uDx∣θ1D,θ2D-bcgxus+uI.Following designus satisfies V˙≤0, and design supervision controller as(15)us=VV-psgn⁡eTPbcαg1xf^x∣θ1f,θ2f+f1xg^xθ1g,θ2g+g2xuc+f1x+ymn+kTe+VV-psgn⁡eTPbc1-αuDx∣θ1D,θ2D+g1xf1x+ymn+kTe+g2xuI.Choose the compensation controller as(16)uI=c0sgn⁡eTPbc.c0 is positive number; then the fuzzy controller of the closed loop system is(17)u=α1g^x∣θ1g,θ2g-f^x∣θ1f,θ2f+ymn+k0Te+1-αuDx∣θ1D,θ2D+us+uI.If V≤V-, the supervisory controller is 0; if the system tends to separate, V≥V-, using us to ensure V≤V-.


## 4. The Design of Adaptive Laws

Adjusting optimal parameters of hybrid fuzzy system based on Bionicθ1f∗,θ2f∗,θ1g∗,θ2g∗,θ1D∗,θ2D∗(18)θ1f∗,θ2f∗=arg⁡minθ1f∈Ω1f,θ2f∈Ω2f⁡supx∈uI⁡fx∣θ1f,θ2f-fx,θ1g∗,θ2g∗=arg⁡minθ1g∈Ω1g,θ2g∈Ω2g⁡supx∈uI⁡gx∣θ1g,θ2g-gx,θ1D∗,θ2D∗=arg⁡minθ1D∈Ω1D,θ2D∈Ω2D⁡supx∈uI⁡ux∣θ1D,θ2D-ux(19)Ω1f=θ1fl:θ1fl≤M1f,Ω2f=θ2fl:θ2fl≤M2f,Ω1g=θ1gl:δ≤θ1gl≤M1g,Ω2g=θ2gl:δ≤θ2gl≤M2g,Ω1D=θ1Dl:θ1Dl≤M1D,Ω2D=θ2Dl:θ2Dl≤M2D.Ω1f,Ω2f,Ω1g,Ω2g,Ω1D,Ω2D are, respectively, the constraint set of θ1f,θ2f,θ1g,θ2g,θ1D,θ2D and M1f,M2f,M1g,M2g,M1D,M2D,δ are given constant.Consider the followedLyapunov function:(20)V=12eTPe+α2γ1fθ1f-θ1f∗Tθ1f-θ1f∗+α2γ2fθ2f-θ2f∗Tθ2f-θ2f∗+α2γ1gθ1g-θ1g∗Tθ1g-θ1g∗+α2γ2gθ2g-θ2g∗Tθ2g-θ2g∗+1-α2γ1Dθ1D-θ1D∗Tθ1D-θ1D∗+α2γ2Dθ2D-θ2D∗Tθ2D-θ2D∗.P is a positive definite matrix and satisfies theLyapunov equation, and Q is an arbitrary positive definite matrix.The bionic adaptive laws of the identification system are as follows:(21)θ_˙1f=-γ1fγε+eTPbcξx,θ_˙2f=-γ2fγε+eTPbcCξx,θ_˙1g=-γ1gγε+eTPbcηxu,θ_˙2g=-γ2gγε+eTPbcCηxu,θ_˙1D=-γ1Dγε+gxeTPbcφx,θ_˙2D=-γ2Dγε+gxeTPbcCφx,where θ1f,θ1g,θ1D represent the current state and θ2f,θ2g,θ2D represent the development trend of systems, which makes the systems have good active adaptability.


## 5. Analysis of System Stability and Convergence

It will affect the stability when the system is disturbed. In this paper, a hybrid adaptive fuzzy controller is designed, and the adaptive law is given. The following analysis is conducted for the performance of fuzzy controller. To ensureθ1f∈Ω1f,θ2f∈Ω2f,θ1g∈Ω1g,θ2g∈Ω2g,θ1D∈Ω1D,θ2D∈Ω2D, parametric projection method is designed to satisfy the parameter vector.In terms off^(x∣θ1f,θ2f),(22)θ˙1f=-γ1fγε+eTPbcξxwhenθ1f<M1forθ1f=M1f,eTPbcξxθ1fT≥0P-γ1fγε+eTPbcξxwhenθ1f=M1f,eTPbcξxθ1fT<0θ˙2f=-γ2fγε+eTPbcξxwhenθ2f<M2forθ2f=M2f,eTPbcξxθ2fT≥0P-γ2fγε+eTPbcξxwhenθ2f=M2f,eTPbcCξxθ2fT<0which satisfies the projection operator(23)P-γ1fγε+eTPbcξx=-γ1fγε+eTPbcξx+γ1fγε+eTPbcθ1fTθ1fξxθ1f2.In terms of g^(x∣θ1g,θ2g), if one element of θ1g,  θ2g satisfies θ1gi=δ,θ2gi=δ, then(24)θ˙1gl=-γ1gγε+eTPbcηlxuγε+eTPbcηlxu<00γε+eTPbcηlxu≥0θ˙2gl=-γ2gγε+eTPbcCηlxuγε+eTPbcCηlxu≤00γε+eTPbcCηlxu>0.Otherwise,(25)θ˙1g=-γ1gγε+eTPbcηxwhenθ1g<M1gorθ1g=M1g,eTPbcηxθ1gT≥0P-γ1gγε+eTPbcηxwhenθ1g=M1g,eTPbcηxθ1gT<0θ˙2g=-γ2gγε+eTPbcCηxwhenθ2g<M2gorθ2g=M2g,eTPbcCηxθ2gT≥0P-γ2gγε+eTPbcCηxwhenθ2g=M2g,eTPbcCηxθ2gT<0which satisfies the projection operator(26)P-γ1gγε+eTPbcηx=-γ1gγε+eTPbcηx+γ1gγε+eTPbcθ1gTθ1gηxθ1g2.In terms of uDx∣θ1D,θ2D,(27)θ˙1D=-γ1Dγε+eTPbcgxφxwhenθ1D<M1Dorθ1D=M1D,eTPbcgxφxθ1DT≥0P-γ1Dγε+eTPbcgxφxwhenθ1D=M1DoreTPbcgxφxθ1DT<0θ˙2D=-γ2Dγε+eTPbcgxφxwhenθ2D<M2Dorθ2D=M2D,eTPbcgxφxθ2DT≥0P-γ2Dγε+eTPbcgxφxwhenθ2D=M2D,eTPbcgxφxθ2DT<0which satisfies the projection operator(28)P-γ1Dγε+eTPbcφx=-γ1Dγε+eTPbcφx+γ1Dγε+eTPbcgxθ1DTθ1Dφxθ1D2.Theorem 1.
Set the constraints set ofΩ1f,  Ω2f,  Ω1g,  Ω2g,  Ω1D,  Ω2D which are given by (19); if the initial value of the parameter satisfies θ1f0∈Ω1f,θ2f0∈Ω2f,θ1g(0)∈Ω1g,  θ2g(0)∈Ω2g,θ1D(0)∈Ω1D,θ2D(0)∈Ω2D, to any t≥0, adaptive laws can ensure θ1f(t)∈Ω1f,  θ2f(t)∈Ω2f,  θ1g(t)∈Ω1g,θ2g(t)∈Ω2g,  θ1D(t)∈Ω1D,θ2D(t)∈Ω2D.Proof.
SetV1f=1/2θ1fTθ1f, if formula (24) satisfies the first condition, when θ1f=M1f,  θ1f≤M1f. Otherwise, V˙1f=-γ1f(γε+eTPbcθ1fT)ξ(x)≤0, and θ1f≤M1f; if formula (24) satisfies the second condition, then θ1f=M1f, and V˙1f=-γ1f[γε+eTPbc]θ1fTξ(x)+γ1f[γε+eTPbc]θ1fTθ1fξ(x)/θ1f2θ1fTξ(x)=0; therefore, in this case, θ1f≤M1f, due to the initial conditions θ1f(0)≤M1f, to any t≥0,  θ1f(t)≤M1f. Similarly, it can be proved that θ2f(t)≤M2f,  θ1g(t)≤M1g,  θ2g(t)≤M2g,  θ1D(t)≤M1D, and  θ2D(t)≤M2D.
It can be obtained by formula (24) that if θ1gl=δ,  θ˙1gl≥0, then there is θ1gl≥δ,  θ1g≥δ. Similarly, θ2g≥δ.Theorem 2.
Hybrid adaptive fuzzy systems form (1), where f(x) and g(x) are all unknown functions, and the equivalent controller uc, the output controller uD, the supervisory controller us, and the adaptive compensation controller uI are, respectively, designed according to formulas (3), (8), (15), and (16). The adaptive law is given by (21), and adaptive fuzzy control system must have the following characteristics:(1)
All parameters and state variables are bounded:(29)θ1fl≤M1f,θ2fl≤M2f,θ1gl≤M1g,θ2gl≤M2gθ1Dl≤M1D,θ2Dl≤M2D,xt≤MxandM1f,  M2f,  M1g,  M2g,  M1D,  M2D,  Mx are given positive constant.(2)
The boundary of tracking error and the model error is given by the minimum approximation error setϕ=ϕf+ϕguTξ(x); then(30)∫0teτ2dτ+∫0tϕτ2dτ≤a+b∫0tωτ2dτ,∀t≥0,∫0teτ2dτ+∫0tετ2dτ≤a+b∫0tωτ2dτ,∀t≥0.(3)
If square ofω is integrable, it means that ∫0∞ω(τ)2dt<∞; then limt→∞⁡e(t)=0,  limt→∞⁡ε(t)=0.Proof.
θ
1
f
l
≤
M
1
f,  θ2fl≤M2f,  θ1gl≤M1g,  θ2gl≤M2g,  θ1Dl≤M1D, and  θ2Dl≤M2D have been proved in Theorem 1, and the following is to prove that x≤Mx.
While designing the controllerus, set V<V-,  V- is a given constant, set λPmin is the minimum eigenvalue of P, and due to V=1/2eTPe, it can be obtained that(31)V1/2≥λPmin21/2e≥λPmin21/2x-ym.Therefore, V≤V- equals x≤ym+2V-/λPmin1/2, and to satisfy x≤Mx, it can choose (32)V-=λPmin2Mx-supt≥0⁡ym2.Proof.
Combine the adaptive laws (21), (22), (24), (25), and (26) into the followingLyapunov function:(33)V˙=-12eTQe+eTPbcω+αγ1fϕ1fTθ˙1f+γ1feTPbcξx+αγ2fϕ2fTθ˙2f+γ2feTPbcCξx+αγ1gϕ1gTθ˙1g+γ1geTPbcηxu+αγ2gϕ2gTθ˙2g+γ2geTPbcCηxu+1-αγ1Dϕ1DTθ˙1D+γ1DeTPbcgxφx+1-αγ2Dϕ2DTθ˙2D+γ2DeTPbcgxCφx.Then(34)V˙=-12eTQe+eTPbcω+ϕ1fT-γεξx+ϕ2fT-γεCξx+∑pϕ1gpeTPbcηpxu+∑pϕ1gp-γεηpxu+∑qϕ2gqeTPbcCηqxu+∑qϕ2gq-γεCηqxu+ϕ1DT-γεφx+ϕ2DT-γεCφxdue to that,(35)∑pϕ1gp-γεηpxu+∑qϕ1gq-γεηqxu=∑l=1Mϕ1gl-γεηlxu,∑pϕ2gp-γεCηpxu+∑qϕ2gq-γεCηqxu=∑l=1Mϕ2gl-γεCηlxu,ϕ1fT-γεξx+ϕ1gT-γεηxu+ϕ2fT-γεCξx+ϕ2gT-γεCηxu=-γϕ2-λϕω,and there is(36)ϕ=ϕ1fTξx+Cϕ2fTξx+ϕ1gTηxu+ϕ2gTCηxu=ε-ω.Set(37)e~=eϕ,Q~=Q002γ,P~n=Pbc-γso it can be obtained that(38)V˙≤-12eTQe+eTPbc-γϕω-γϕ2=-12e~TQ~e~+e~TP~nω≤-12λQ~mine~2+12e~2+12P~nω2=-λQ~min-12e~2+12P~nω2to any t≥0,(39)∫0te~τ2dτ≤2λQ~min-1V0-Vt+P~n2λQ~min-1∫0tωτ2dτ≤2λQ~min-1V0+P~n2λQ~min-1∫0tωτ2dτand because of e~(t)2=e(t)2+[ε(t)]2,  P~n2=Pbc2+γ2, it can be proved by(40)a=2V0min⁡λQmin,2γ-1,b=Pbc2+γ2min⁡λQmin,2γ-1noting that {λQmin,2γ}-1>0.Proof.
Ifω∈L2, we can obtain e∈L2 by Theory 2, because the variables are bounded; then e˙∈L∞, byBarbalat lemma (if e∈L2∩L∞ and e˙∈L∞, then limt→∞⁡e(t)=0); we can get limt→∞⁡e(t)=0.


## 6. Simulation

Example 1.
The inverted pendulum problem is studied by the hybrid adaptive fuzzy control method:(41)x˙1=x2,x˙2=gsin⁡x1-mlx22sin⁡x1/mc+ml4/3-mcos2⁡x1/mc+m+cos⁡x1/mc+ml4/3-mcos2⁡x1/mc+muand g=9.8 m/s2, mc=1 kg, m=0.1 kg, l=0.5 m. Choosing KT=k1k2=12,Ck=1,   Q=diag⁡1010, by solving equation ΛcTP=PΛc=-Q, then P=15555.
The membership function is(42)μM11x1=exp⁡-x1+π/6π/102,μM12x1=exp⁡-x1π/102,μM13x1=exp⁡-x1-π/6π/102,μM21x2=exp⁡-x2+π/6π/102,μM22x2=exp⁡-x2π/102,μM23x2=exp-x2-π/6π/102.
Set the initial conditionsx(0)=-π/9,π/12T, the original system, and control system with internal disturbance (system state input x is substituted by (1+δ1)x,  δ1=0.5) and external disturbance (δ2=0.01).From Figures1–3, when the system is disturbed, the hybrid bionic adaptive controller proposed in this paper makes the system have more anti-interference and good convergence. In addition, this paper makes a comparison when the output trajectory has both internal and external disturbances; it can be seen that the fuzzy system has achieved good results. From Figure 4(a), when Ck=1, the system has good convergence and biological adaptability than system controlled by traditional method. From Figure 4(b), by comparing the output trajectories, it can be obtained that the bionic hybrid fuzzy control has better result.Figure 1
x
1 and x2 with internal disturbance.Figure 2
x
1 and x2 with external disturbance.Figure 3
x
1 and x2 with both internal and external disturbances.Figure 4
(a)
x
1 with both internal and external disturbances (when Ck chooses different values)
(b)
Sliding mode control output trajectoriesx1,x2Example 2.
Consider the following uncertain system:(43)ω˙=-ω+x12+x22,x˙1=x2,x˙2=-x-2x2+Δfx,v,t+Δhω+u,where ω∈R is the unmodelled dynamics of the system. Δh(ω) is the related uncertain item of the unmodelled dynamics ω.
Set(44)Δhω=2ω,Δfx,v,t=b1+b2cos⁡2tx12x22a1+a2sin⁡t+a3x12+a4x22.a,b are unknown constants, satisfying the fact that a1≥a2+1>0,  a3≥1,  a4≥1.
In stimulation, seta1=2,a2=0.5,  a3=1.5,  a4=1.5,  b1=1,b2=2.From Figure5, to the uncertain system with disturbance, the hybrid bionic adaptive controller proposed makes the system have more anti-interference and good convergence. From Figure 6, when Ck=1, the system has good convergence and biological adaptability than system controlled by traditional method.Figure 5
x
1 and x2 with internal disturbance.Figure 6
x
1 with both internal and external disturbances (when Ck chooses different values).


## 7. Conclusion

In this paper, the hybrid fuzzy control method combined model error with tracking error to design the adaptive laws. Compared with traditional method, it has faster convergence rate. In addition, the biological characteristics are embedded in the bionic fuzzy system, which improved the condition that traditional adaptive law just considers the current state of system. The method proposed in this paper makes system have advantages of anti-interference and strong self-adaptability.



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*Source: 9341073-2017-10-18.xml*