# Prescribed Performance Synchronization Control of Chaotic Systems with Unknown Control Gain Signs

Authors: Xin Ma; Fang Zhu
Journal: Journal of Control Science and Engineering
 (2021)

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## Abstract

For a class of uncertain nonlinear chaotic systems with unknown control gain signs and saturated input, by means of Nussbaum function, a scheme of finite-time prescribed performance synchronization control is proposed. Here, Nussbaum function is used to eliminate the influence of unknown control gain signs, and fuzzy logic systems are used to estimate unknown functions. Lyapunov theory is used to prove that all synchronization errors converge to a predefined small performance range under the designed control method. Finally, simulation results are provided to illustrate the feasibility of the proposed method.


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## Body

## 1. Introduction

Since chaotic synchronization was first discovered in 1990, chaotic synchronization has always been a concern to scholars. Chaotic synchronization control is widely used in engineering, communication, and other fields. For different chaotic systems, many scholars have proposed some synchronization control schemes [1–10], such as fuzzy control, sliding mode control, backstepping control, adaptive control, and robust control. For example, based on the Takagi–Sugeno fuzzy model, Kumar and Khan [4] designed the adaptive synchronization of chaotic systems by utilizing linear matrix inequality. For the synchronization problems of fractional-order chaotic systems, Tabasi and Balochian [7] designed a fractional-order adaptive sliding mode controller. For chaotic systems, Dai et al. [10] designed a noise-suppression zeroing neural network, which has better synchronization control performance. As we all know, nonlinear systems will inevitably have uncertainties and external disturbances, if the systems also have unknown control gain signs and input saturation, then the controller design will become more challenging.To eliminate the influence of unknown control gain signs, many scholars have proved the effectiveness of the Nussbaum-type function method [11–18]. For example, Boulkroune [12] constructed a direct adaptive controller by using the Nussbaum function and the adaptive fuzzy system. Khettab et al. [13] introduced fractional Nussbaum gain to deal with unknown control directions and used the fuzzy logic set to identify the fractional-order chaotic system. Chen [16] introduced a heightened version of the Nussbaum function and discussed the necessity of using it. Using backstepping technique and Nussbaum gain method, Zhang et al. [18] proposed an adaptive control method. However, the Nussbaum gain function can only ensure the boundedness of the signal in the closed-loop system.For the past few years, many literatures have studied prescribed performance control (PPC) schemes for nonlinear systems with unknown control directions [19–30]. A new finite time performance function is given for uncertain nonlinear systems in [20]. Wang and Yang [22] proposed a prescribed performance feedback tracking control scheme. Shi et al. [23] designed adaptive fuzzy predetermined performance controller by using symmetric matrix decomposition technique. Zhang and Yang constructed a state feedback fault-tolerant control method by introducing a new error transformation function in [25]. In [29], Xiang and Liu transformed the controlled system into an equivalent system and constructed a fuzzy adaptive-prescribed performance control technique to make tracking error draw near the predefined neighborhood.Inspired by the above works, for uncertain nonlinear chaotic systems with unknown control gain signs and input saturation, we complete the following works. (1) The synchronization of two uncertain chaotic systems is realized. (2) The synchronous controller is constructed by Nussbaum function, fuzzy logic systems, and finite time-prescribed performance function.This brief is organized as follows. In Section2, the description of problem and preparatory works are given, which is mainly to deduce the described system and give some assumptions and lemmas. In Section 3, a prescribed performance synchronization control scheme of chaotic systems is addressed. In Section 4, the simulation results are given. A short conclusion is presented in Section 5.


## 2. System Descriptions and Preparatory Works

The main system considered in this paper is(1)x˙1t=f1xx˙2t=f2x⋯x˙nt=fnx,where xit represents the state of master system, xt=x1t,x2t,…,xntT is the state vector of the master system, and fix is unknown smooth function, i=1,2,…,n.The corresponding slave system is described by(2)y˙1t=g1y+d1t+λ1satu1ty˙2t=g2y+d2t+λ2satu2t⋯y˙nt=gny+dnt+λnsatunt,where yit represents the state of slave system, dit is unknown external disturbance, giy is unknown smooth function, λi is control gain with unknown sign, and satuit is the control input affected by saturation type nonlinearity, i=1,2,…,n. Here, satuit is described as(3)satuit=Msignuit,uit≥M,uit,uit<M.A fuzzy adaptive PPC scheme is designed as the objective of this paper to meet the following requirements:P1) The synchronization errorei=yi−xi can be limited within predefined rangeP2) The designed controller can ensure the boundedness of all signals of the closed-loop systemWe make the following assumptions to meet our objective.Assumption 1.
giy and fix are unknown but bounded.Assumption 2.
dit is a bounded external disturbance.Assumption 3.
The input coefficientλi is unknown.Lete=e1,e2,…,enT, and one gets the synchronization error dynamic system:(4)e˙1=g1y+d1t+λ1satu1t−f1xe˙2=g2y+d2t+λ2satu2t−f2x⋯e˙n=gny+dnt+λnsatunt−fnx.The Nussbaum functionN⋅ is given, which has the following properties:(5)limt⟶+∞sup1t∫0tNξdξ=+∞,limt⟶+∞inf1t∫0tNξdξ=−∞,where N⋅ is used to handle unknown control gain signs, and the following lemma will be used in conjunction with Nussbaum function.Lemma 1 (see [29]).
LetN⋅ be an even Nussbaum function and Vt and ζt be smooth function with Vt≥0, t∈0,tf. The following inequality holds:(6)0≤Vt≤a0+e−a1t∫0tϖxτNζ+1ζ˙ea1τdτ,and then, Vt,∫0tϖxτNζζ˙dτ, and ζt must be bounded on t∈0,tf. a1>0, and a0 and a1 are suitable constants, and ϖxt is defined on D=h−,h+ with 0∉D.


## 3. Control Design and Stability Analysis

The finite-time performance constraints are set as follows to maintain the transient and steady-state performance ofei:(7)−ϒit<ei<ϒit,i=1,2,…,n,where ϒit is defined as(8)ϒit=γi,0−ttfe1−tf/tf−t+γi,1,0≤t<tf,γi,1,tf≤t<+∞,where γi,0 and γi,1 are positive design parameters.By introducing the error transformation functionΩiρi, we transform the constrained synchronization error system (4) into an unrestraint equivalent transformation system. Select the error transformation function as Ωiρi=eρi−e−ρi/eρi+e−ρi; let(9)ei=ϒitΩiρi,where ρi is transformation error, i=1,2,…,n.Lemma 2.
(see [29]). ei can be limited to (7) if ρi is bounded.According to (9), one has(10)ρ˙i=rie˙i+Ξi,where ri=1/ϒit∂Ωiρi/∂ρi and Ξi=−ϒitΩiρi, i=1,2,…,n.Using fuzzy logic systems (FLSs), let(11)giy=ηgi∗Tϕgiy+ρgiy,fix=ηfi∗Tϕfix+ρfix,where ηgi∗andηfi∗ are the optimal approximation vectors, ϕgiyandϕfix are the basis function vectors, and ρgiyandρfix are fuzzy estimation errors which are bounded.Substituting (4) and (11) into (10), then we have(12)ρ˙i=rigiy−fix+dit+λisatui+Ξi=riηgi∗Tϕgiy+ρgiy−ηfi∗Tϕfix−ρfix+dit+λisatui−λiui+λiui+Ξi=riηgi∗Tϕgiy−ηfi∗Tϕfix+d¯it+λiui+Ξi,where d¯it=dit+ρgiy−ρfix+λisatui−ui and d¯it≤d¯i∗, d¯i∗>0 is an unknown constant.Then, the error transformation dynamic system can be given as(13)ρ˙1t=r1ηg1∗Tϕg1y−ηf1∗Tϕf1x+d¯1t+λ1u1+Ξ1ρ˙2t=r2ηg2∗Tϕg2y−ηf2∗Tϕf2x+d¯2t+λ2u2+Ξ2⋯ρ˙nt=rnηgn∗Tϕgny−ηfn∗Tϕfnx+d¯nt+λnun+Ξn.According to (13), the controller is designed as(14)ui=ui1+ui0,ui1=1λiη^fiTϕfix−η^giTϕgiy−kiρi−Ξi,ui0=Nζiu¯i0,u¯i0=d¯^i,ζ˙i=ρi⋅u¯i0.The parameter adaptation lawsη^˙gi, η^˙fi, and d¯^˙i are given as(15)η^˙gi=kgiρiϕgiy−bgiη^gi,η^˙fi=kfi−ρiϕfix−bfiη^fi,d¯^˙i=kdiρi−bdid¯^i,where kgi,kfi,kdi,bgi,bfi,andbdi are all positive constants.We construct Lyapunov function as(16)V=12riρi2+12η˜giTkgi−1η˜gi+12η˜fiTkfi−1η˜fi+12kdid¯˜i2,where η˜gi=ηgi∗−η^gi,η˜fi=ηfi∗−η^fi,andd¯˜i=d¯i∗−d¯^i.Since(17)η˜giTη^gi=η˜giTηgi∗−η˜gi=η˜giTηgi∗−η˜giTη˜gi≤−12η˜giTη˜gi+12ηgi∗2,similarly, we have(18)η˜fiTη^fi=η˜fiTηfi∗−η˜fi=η˜fiTηfi∗−η˜fiTη˜fi≤−12η˜fiTη˜fi+12ηfi∗2,d¯˜id¯^i=d¯˜id¯i∗−d¯˜i=−d¯˜i2+d¯˜id¯i∗≤−12d¯˜i2+12d¯i∗2.Therefore,V˙ can be obtained as(19)V˙=−r˙i2ri2ρi2+1riρiρ˙i+η˜giTkgi−1η˜˙gi+η˜fiTkfi−1η˜˙fi+1kdid¯˜id¯˜˙i=−r˙i2ri2ρi2+ρiηgi∗Tϕgiy−ηfi∗Tϕfix+d¯it+λiui+Ξi+η˜giTkgi−1η˜˙gi+η˜fiTkfi−1η˜˙fi+1kdid¯˜id¯˜˙i≤−r˙i2ri2ρi2−kiρi2+ρiη˜giTϕgiy−ρiη˜fiTϕfix+λiρiNζiu¯i0+ρid¯i∗+η˜giTkgi−1η˜˙gi+η˜fiTkfi−1η˜˙fi+1kdid¯˜id¯˜˙i≤−r˙i2ri2ρi2−kiρi2+ρiη˜giTϕgiy−ρiη˜fiTϕfix+λiNζiζ˙i+ρid¯^i+ρid¯˜i−η˜giTρiϕgiy−bgiη^gi−η˜fiT−ρiϕfix−bfiη^fi−d¯˜iρi−bdid¯^i=−r˙i2ri2ρi2−kiρi2+λiNζi+1ζ˙i+η˜giTbgiη^gi+η˜fiTbfiη^fi+bdid¯˜id¯^i≤−r˙i2ri2+kiρi2+λiNζi+1ζ˙i−12bgiη˜giTη˜gi−12bfiη˜fiTη˜fi+12bgiηgi∗2+12bfiηfi∗2−12bdid¯˜i2+12bdid¯i∗2,where b=min2rir˙i/2ri2+ki,bgi⋅λminkgi,bfi⋅λminkfi,bdikdi, and R1=1/2bgiηgi∗2+1/2bfiηfi∗2+1/2bdid¯i∗2. Then, we can obtain(20)V˙≤−bV+R1+λiNζi+1ζ˙i.The two sides of (20) are multiplied by ebt, and the result is(21)dVebtdt≤R1ebt+ebtλiNζi+1ζ˙i.By directly integrating the above inequalities, we obtain(22)Vt≤V0e−bt+R1b1−e−bt+e−bt∫0tλiNζi+1ζ˙i⋅ebτdτ≤V0e−bt+R1b+e−bt∫0tλiNζi+1ζ˙i⋅ebτdτ.From Lemma1, we can obtain that Vt, ∫0tλiNζi+1ζ˙idτ, and ζt are bounded on 0,tf. Similar to the discussion in [31], the boundedness of ρi,η^gi,η^fi, and d¯^i on 0,+∞ can be obtained. Using Lemma 2, because ρi is bounded, so −ϒit<ei<ϒit can be guaranteed.Theorem 1.
Consider the master chaotic system (1) and slave system (2) with unknown control gain signs, and under the condition that assumptions 1–3 hold, the synchronization controller (14) with parameter adaptive laws (15) can guarantee the realization of objectives P1 and P2.Remark 1.
If the control gainλi in this paper is replaced by the function χy, similar conclusions can also be obtained. However, to avoid singularity, we need to replace 1/λi in ui1 with χ^y/ϱ+χ^2y, where ϱ is a small positive constant.Remark 2.
In [5], the feedback control method was employed to realize the state synchronization of two different chaotic systems, but this synchronization method can only satisfy the steady-state performance. In the actual synchronous control, it is expected that the steady-state performance and instantaneous performance can be satisfied at the same time. Therefore, the finite-time PPC method used in this paper can achieve this aim. It can be said that the work of this paper is a continuation of [5].Remark 3.
In [19], the initial value of the traditional performance function ϒit needs to be satisfied:(23)−ϒi0<ei0<εiϒi0,ifei0<0,or(24)−εiϒi0<ei0<ϒi0,ifei0>0,where 0≤εi≤1. This means that it is necessary to design the controller repeatedly to adapt to the change of the sign of initial synchronization error.Remark 4.
Theorem1 shows that, for t>tf, the synchronization error eit is limited within a specific range −γi,1,γi,1. Meanwhile, the presetting time tf can be changed according to the actual needs.


## 4. Numerical Simulations

Consider the classical financial system as the master system:(25)x˙1=x3+x2−0.6x1,x˙2=1−0.6x2−x12,x˙3=−x1−0.9x3.The corresponding slave system is as follows:(26)y˙1=y3+y2−6y1+d1t+λ1satu1,y˙2=1−0.1y2−y12+d2t+λ2satu2,y˙3=−y1−y3+d3t+λ3satu3,where λ1=1,λ2=1,λ3=−1,d1t=sint,d2t=2sint,andd3t=3sint.The synchronization error isei=yi−xi,i=1,2,3.Letϒ1t=ϒ2t=ϒ3t=pt, where(27)pt=2.95−ttfe1−tf/tf−t+0.05,0≤t<tf,0.05,tf≤t<+∞.The control inputs are designed as(28)satui=15signui,ui≥15,ui,ui<15,i=1,2,3.Gaussian membership function is taken as(29)ϕκ=exp−12κ+7.5−2.5i1.22,κ=x1,x2,x3,y1,y2,y3;i=1,2,3,4,5,6,7,where the fuzzy set is defined on the interval −5,5.Simulation is executed withx0=2,−1,−2.5T,y0=1,0.5,−1.2T, η^gi0=η^fi0=0,d¯^i0=0.1, kgi=kfi=5,bgi=bfi=0.1, and Nζ=ζ2cosζ,ζ0=0.02.In order to demonstrate the effectiveness of the presented control scheme (14), we first take tf=5, and simulation results are exhibited in Figures 1 and 2 .Figure 1
Trajectories of synchronization errorse1, e2, and e3 at tf=5.Figure 2
Trajectories ofsatu1, satu2, and satu3 at tf=5.Figure1 gives the trajectories of synchronization errors when tf=5, and we can find that synchronization errors converge to −0.05,0.05 within 5 s and remain in this region after 5s. The saturation inputs are illustrated in Figure 2. We can also find that satui tends to stabilize after 3 s.Sequentially, we taketf=2, and the comparison results are illustrated in Figures 3 and 4 .Figure 3
Trajectories of synchronization errorse1, e2, and e3 at tf=2.Figure 4
Trajectories ofsatu1, satu2, and satu3 at tf=2.Figure3 gives the trajectories of synchronization errors when tf=2, and we can find synchronization errors converge to −0.05,0.05 within 2 s and remain in this region after 2s. From Figure 4, obviously, because the limited time is shortened, the fluctuation of saturation input is larger than that of the previous control effect, but it still tends to be stable after 3 s.To sum up, through the simulation results, we can find that our control time can be set according to the actual situation, and the control effect is better.


## 5. Conclusion

A finite-time synchronization control problem for a class of uncertain chaotic systems with unknown control direction is investigated. Based on finite-time performance function, PPC, Nussbaum function, and FLSs algorithm, an adaptive fuzzy synchronization control method is designed, which ensures that the synchronization errors satisfy the given performance and converge to the predefined region in finite time, and all the signals in the closed-loop system remain bounded. Finally, numerical simulations verify the effectiveness of the proposed control method.



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