# New Results on the Control for a Kind of Uncertain Chaotic Systems Based on Fuzzy Logic

Authors: Bo Wang; L. L. Chen
Journal: Complexity
 (2019)

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## Abstract

In this paper, the problem of the control for an uncertain nonlinear chaotic system has been studied; based on fuzzy logic, a kind of single-dimensional controller is constructed for the control of the chaotic systems in the situation that uncertainties and unknowns exist; at last some typical numerical simulations are carried out, and corresponding results illuminate the effectiveness of the controller.


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## Body

## 1. Introduction

Nonlinear systems exist in real engineering widely. Since the pioneering work from Lurie in 1944, the research on nonlinear system control has become the challenging issue, and many techniques, such as differential geometry technique [1, 2], sliding mode technique [3–6] and so on, have been proposed to deal with this problem. It can be noted that these approaches are based on multidimensional control. However, in some cases, the single-dimensional controller is more cherished for its simpler structure and more convenient application in practice.As an important branch of nonlinear systems, chaotic system and its control received many attentions, and a lot of related results have been reported so far [7–14]. For instance, in [7], based on output feedback control strategy, a method was presented to realize the control for unified chaotic systems; in [8], the synchronization control for Lü systems with unknown parameters was investigated; in [9], the adaptive control for the synchronization of hyperchaotic systems was studied; in [10], the fuzzy control for Arneodo chaotic system is discussed. However most of these researches focused on just one typical chaotic system. In addition, it is well known that there exist many kinds of uncertainties in practical control system, and the following chaotic system model is studied. (1)x˙i=bi+Δbixi+1+fix-i,1≤i≤n-1x˙n=hnx+Δhnx+fnx+bnuy=x1where x=x1,x2,···,xnT∈Rn, x-i=x1,x2,···,xiT∈Ri, bi are the known system parameters and satisfy 0<bim≤bi≤biM, where bim and biM are the positive scalars, fn(x) and fi(x-i) are the unknown terms, Δbi and Δhn are the uncertainties, hn is the known term, bn is the control parameter, y is the system output, and u is the single-dimensional control input. A lot of chaotic systems can be transformed into the system with the form (1) through topological mapping.As an important technique, fuzzy techniques are very suitable for the research of nonlinear and complex systems (see [15–23] and references therein), and they will be introduced to design the single-dimensional controller for system (1) in this paper. Some simulations will be included to illuminate the effectiveness of the constructed controller.


## 2. Model Description and Preliminaries

It is well known that fuzzy logic system can approximate the nonlinear function. Letfx denote the smooth function and φ(x) denote the fuzzy logic system. There exists the optimal parameter θ∗=argminθ∈Ω0⁡[supx∈Ω⁡fx-φ(x)] for the least approximation error, where Ω0 and Ω are bounded sets of θ andx.Define fuzzy rules as(2)IFx1isF1jand…andxnisFnj,thenφxisBjj=1,2,⋯,NDefine the following fuzzy logic system [16](3)φx=∑j=1Nθj∏i=1nμFijxi∑j=1N∏i=1nμFijxiwhere x=x1,x2,⋯,xnT∈Rn, μFijxi is the fuzzy membership function, θj=maxφ(x)∈R⁡Bjφ(x).Letξx=ξ1x,ξ2x,⋯,ξNxT and θ=θ1,θ2,⋯,θNT; one can get φx=ξTxθ.Hence, iff-k is the continuous function from a compact set, φkx-k can approximate f-k, which means that there exist θk=θ1k,θ2k,···,θNkT and εk>0, such that (4)f-kx-k-ξkTx-kθk≤εk,k=1,2,⋯,n.wherex-k=x1,x2,⋯,xkT,k=1,2,⋯,n.In the paper, the following lemmas are concerned.Lemma 1 (see [24]).
Iff(t)∈L∞∩L2 and f˙(t)∈L∞, one has(5)limt→+∞ft=0


## 3. Main Results

For convenience, letΔgix-i+1=Δbixi+1,1≤i≤n-1, and Δgn=Δhn(x).Step 1.
Define the tracking errore1=y-yd; yd is the desired trajectory.
For the first subsystem of system (1), the virtual variable α1 is introduced, such that(6)e˙1=x˙1-y˙d=b1x2-b1α1+b1α1+f1x-1+Δg1x-2-y˙d=b1e2+b1α1+f1x-1+Δg1x-2-y˙dwhere e2=x2-α1.Step 2.
For the second subsystem of system (1), the virtual variable α2 is introduced, such that(7)e˙2=x˙2-α˙1=b2x3-b2α2+b2α2-α˙1+f2x-2+Δg2x-3=b2e3+b2α2-α˙1+f2x-2+Δg2x-3where e3=x3-α2.Step k (k<n). Fork-th subsystem of system (1), the virtual variable αk is introduced, such that(8)e˙k=x˙k-α˙k-1=bkxk+1-bkαk+bkαk-α˙k-1+fkx-k+Δgkx-k+1=bkek+1+bkαk-α˙k-1+fkx-k+Δgkx-k+1where ek+1=xk+1-αkStep n.For then-th subsystem of system (1), one can get(9)e˙n=x˙n-α˙n-1=bnu+hnx+fnx+Δgnx-α˙n-1where en=xn-αn-1Then, the following tracking error dynamic system can be derived(10)e˙i=biei+1+biαi-α˙i-1+fix-i+Δgix-i+1t,1≤i≤n-1e˙n=-α˙n-1+hnx+fnx+Δgnx+bnuwhere α0=ydThe object of this paper is to design a controller, such that(11)limt→+∞et=0Choose the first Lyapunov function as(12)V1=12b1e12then(13)V˙1=1b1e1e˙1=1b1e1b1e2+b1α1+f1+Δg1-y˙d=e1e2+α1+1b1e1f1+Δg1-y˙d=e1e2+α1+f-1where(14)f-1=f1+Δg1-y˙db1.Letα1=-λ1e1-φ1, λ1>0, where φ1=ξ1Tx-2θ1 is used to approximate the nonlinear function f-1, then(15)V˙1=-λ1e12+e1e2+e1f-1-φ1Choose the second Lyapunov function as(16)V2=V1+12b2e22Letα2=-λ2e2-e1-φ2, λ2>0, where φ2=ξ2Tx-3θ2 is used to approximate the nonlinear function f-2, then(17)V˙2=V˙1+1b2e2e˙2=V˙1+e2e3+α2+f2+Δg2-α˙1b2=-λ1e12+e2e1+e3+α2+f-2+e1f-1-φ1=-λ1e12+e2e3-λ2e22+e1f-1-φ1+e2f-2-φ2=-∑i=12λiei2+∑i=12eif-i-φi+e2e3where(18)f-2=f2+Δg2-α˙1b2.Letαk-1=-λk-1ek-1-ek-2-φk-1, λk-1>0, where φk-1=ξk-1Tx-kθk-1 is used to approximate the nonlinear function f-k-1, then(19)V˙k-1=-∑i=1k-1λiei2+∑i=1k-1eif-i-φi+ek-1ekChoose thek-th Lyapunov function (k<n) as(20)Vk=Vk-1+12bkek2Hence(21)V˙k=V˙k-1+1bkeke˙k=V˙k-1+ekek+1+αk+1bkfk+Δgk-α˙k-1=-∑i=1k-1λiei2+∑i=1k-1eif-i-φi+ekek+1+αk+ek-1+1bkfk+Δgk-α˙k-1=-∑i=1k-1λiei2+∑i=1k-1eif-i-φi+ekek+1+αk+ek-1+f-kwhere(22)f-k=1bkfk+Δgk-α˙k-1.Letαk=-λkek-ek-1-φk, λk>0, where φk=ξkTx-k+1θk is used to approximate the nonlinear function f-k, then(23)V˙k=-∑i=1kλiei2+∑i=1keif-i-φi+ekek+1It is consistent with our notation thatαn-1=-λn-1en-1-en-2-φn-1, λn-1>0, where φn-1=ξn-1Tx-nθn-1 is used to approximate the nonlinear function f-n-1, then(24)V˙n-1=-∑i=1n-1λiei2+∑i=1n-1eif-i-φi+en-1enChoose then-th Lyapunov function as(25)Vn=Vn-1+12bnen2Hence(26)V˙n=V˙n-1+enu+1bnf-n-α˙n-1=-∑i=1n-1λiei2+∑i=1n-1eif-i-φi+en-1en+enu+1bnfn+Δgn-α˙n-1=-∑i=1n-1λiei2+∑i=1n-1eif-i-φi+enu+en-1+1bnfn+Δgn-α˙n-1where(27)f-n=1bnfn+Δgn-α˙n-1Suppose thatφn=ξnTxθn approximate the nonlinear function f-n and that is based on Lyapunov theory, then the following theoretical result can be obtained.Theorem 2.
Forλn>0, ri>0 and ki>0, based on the controller(28)u=-λnen-en-1-hnxbn-ξnTxθnand the adaptive law(29)θ˙i=rieiξi-2kiθi,i=1,2,⋯,nthen the output of chaotic system (10) can track the desired trajectory.Proof.
Based on (25), construct Lyapunov function as(30)V=Vn+∑i=1n12riθ-iTθ-iwhere θ-i=θi∗-θi.
Combined with (28), it can be concluded that (31)V˙=-∑i=1nλiei2+∑i=1neif-i-φi-∑i=1n1riθ-iTθ˙i=-∑i=1nλiei2+∑i=1neif-i-θi∗Tξi+∑i=1neiθi∗Tξi-θiTξi-∑i=1n1riθ-iTθ˙i=-∑i=1nλiei2+∑i=1neif-i-θi∗Tξi+∑i=1neiθ-iTξi-∑i=1n1riθ-iTθ˙i≤-∑i=1nλiei2+∑i=1nθ-iTeiξi-1riθ˙i+∑i=1neiεi
Define(32)S=-∑i=1nλiei2+∑i=1nθ-iTeiξi-1riθ˙i+∑i=1neiεi
Supposing thatai=λi-1/2, it can be derived(33)λi=ai+12
Hence(34)S=-∑i=1naiei2-12∑i=1nei2+∑i=1nθ-iTeiξi-1riθ˙i+∑i=1neiεi
Consider(35)-12∑i=1nei2+∑i=1neiεi≤-12∑i=1nεi2
and with adaptive law (29), one can get(36)S≤-∑i=1naiei2+∑i=1nθ-iTeiξi-1ririeiξi-2kiθi+12∑i=1nεi2=-∑i=1naiei2+∑i=1n2kiriθi∗-θiTθi+12∑i=1nεi2=-∑i=1naiei2+∑i=1nkiri2θi∗Tθi-2θiTθi+12∑i=1nεi2
Consider(37)θi∗Tθi∗+θiTθi≥2θi∗Tθi
Then(38)2θi∗Tθi-2θiTθi≤θi∗Tθi∗-θiTθi
One can derive(39)V˙≤-∑i=1naiei2+∑i=1nkiri-θiTθi+θi∗Tθi∗+12∑i=1nεi2=-∑i=1naiei2+∑i=1nkiri-θiTθi-θi∗Tθi∗+∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2
Consider(40)θ-iTθ-i=θi∗-θiTθi∗-θi=θi∗Tθi∗-2θi∗Tθi+θiTθi≤2θi∗Tθi∗+2θiTθi
Then(41)-12θ-iTθ-i≥-θiTθi-θi∗Tθi∗
One can derive(42)V˙≤-∑i=1naiei2-∑i=1nki2riθ-iTθ-i+∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2≤-∑i=1nai2bim2biei2-∑i=1nki2riθ-iTθ-i+∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2
Choosingλi>1/2, one can obtain ai>0.
Let(43)a0=min⁡2bimai,ki:i=1,2,···,nb0=∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2
Then(44)V˙≤-a0∑i=1n12biei2+∑i=1n12riθ-iTθ-i+b0=-a0V+b0
The solution of differential equationV˙=-a0V+b0 is(45)Vt=V0exp⁡-a0t+b0exp⁡-a0texp⁡a0t-1a0
Considering (44), it can be derived that(46)Vt≤V0-b0a0exp⁡-a0t+b0a0≤V0exp⁡-a0t+b0a0≤V0+b0a0
Define(47)d0=min⁡ai:i=1,2,···,n.
From (42), one can obtain(48)V˙≤-∑i=1naiei2-∑i=1nki2riθ-iTθ-i+∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2≤-minai∑i=1nei2+∑i=1n2kiriθi∗Tθi∗+12∑i=1nεi2=-d0∑i=1nei2+b0
Hence, whene>b0/d01/2, one can get V˙<0, which means e(t)∈L∞.
Integrating both sides of inequality (48) from 0 toT, one can get(49)∫0TV˙tdt≤-∫0Td0∑i=1nei2sds+Tb0
Consider(50)∫0TV˙tdt=VT-V0
One can get(51)VT-V0≤-d0∑i=1n∫0Tei2sds+Tb0
Hence(52)∑i=1n∫0Tei2sds≤1d0V0-VT+Tb0which means e(t)∈L2. From error dynamic system (10), it can be concluded that e˙(t)∈L∞. Accordingly based on Lemma 1, one can get limt→+∞⁡e(t)=0, which means the achievement of the track control. The proof of Theorem 2 is thus completed.


## 4. Numerical Simulation

First the following uncertain Arneodo system is considered.(53)x˙1=b1x2+f1x˙2=b2+Δb2x3+f2x˙3=h3+Δh3+f3+b3uwhere(54)b1=1,f1=0.3sin⁡x1,b2=1,Δb2=0.02,f2=0.1cos⁡x1x2,h3=c3x13-c0x1-c1x2-c2x3,Δh3=0.1x1,f3=0.2cos⁡x1sin⁡x3,b3=5,c0=-5.4,c1=3.5,c2=1,c3=-1,k1=1,k2=1.5,k3=1.5,r1=1.5,r2=2,r3=2,λ1=2.5,λ2=5,λ3=5,α1=-λ1x1-yd+y˙d,α2=-λ2x2-α1-x1-yd-ξ2Tx-2θ,u=-λ3x3-α2-x2-α1-ξ3Tx-3θ,ξ1jx¯1=μF1jx1∑j=19μF1jx1,ξ2jx¯2=μF1jx1μF2jx2∑j=19μF1jx1μF2jx2,ξ3jx=μF1jx1μF2jx2μF3jx3∑j=19μF1jx1μF2jx2μF3jx3,μFi1xi=exp⁡-0.5xi+22,μFi2xi=exp⁡-0.5xi+1.52,μFi3xi=exp⁡-0.5xi+12,μFi4xi=exp⁡-0.5xi+0.52,μFi5xi=exp⁡-0.5xi2,μFi6xi=exp⁡-0.5xi-0.52,μFi7xi=exp⁡-0.5xi-12,μFi8xi=exp⁡-0.5xi-1.52,μFi9xi=exp⁡-0.5xi-22,ξ1x¯1=ξ11x¯1,ξ12x¯1,···,ξ19x¯1T,ξ2x¯2=ξ21x¯2,ξ22x¯2,···,ξ29x¯2T,ξ3x¯3=ξ31x¯3,ξ32x¯3,···,ξ39x¯3T.Let desired trajectoryyd=sin⁡2πt, initial value x0=2,0,0T, and the simulation results are displayed in Figures 1–7.Figure 1
Chaotic attractor of Arneodo system.Figure 2
State response of x1 of Arneodo system.Figure 3
State response of x1 of uncertain Arneodo system.Figure 4
Fuzzy membership function.Figure 5
State response of control input.Figure 6
State response ofyd andy.Figure 7
State response of position tracking error.Remark 3.
Figure1 displays the chaotic attractor of Arneodo system. Figure 2 displays the state response of x1 of Arneodo system. From Figures 1 and 2, it can be seen that Arneodo system has the complicated dynamical behavior. Figure 3 displays the state response of variablex1 of uncertain Arneodo system. It can be seen that the existence of unknowns and uncertainties makes Arneodo system unstable.Remark 4.
Figure4 displays the fuzzy membership function. Figure 5 displays the state response of control input. Figure 6 displays the state response ofyd andy. Figure 7 displays the state response of position tracking error. From Figures 4–7, it can be seen that for uncertain Arneodo system, the position tracking can be achieved during 0.5 second based on the designed controller.


## 5. Conclusion

In this paper, based on fuzzy logic, a single-dimensional controller has been constructed for the control of a kind of uncertain chaotic systems. Some typical examples have been employed and corresponding simulation results have illuminated the effectiveness of proposed controller.



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