# Mamdani-Type Fuzzy-Based Adaptive Nonhomogeneous Synchronization

Authors: J. R. Pulido–Luna; J. A. López–Rentería; N. R. Cazarez–Castro
Journal: Complexity
 (2021)

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## Abstract

The aim of this work is the design of an adaptive controller based on Mamdani-type fuzzy inference systems. The input control is constructed with saturation functions’ fuzzy-equivalents, which works as the adaptive scheme of the controller. This control law is designed to stabilize the error system to synchronize a pair of chaotic nonhomogeneous piecewise systems. Finally, an illustrative example as numerical evidence is developed.


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## Body

## 1. Introduction

The synchronization phenomena among dynamical systems are a widely studied topic in the last decades due to the vast amount of applications in science and engineering [1–3]. In the related literature, dynamical systems and synchronization applications in many fields can be found, from biology [4, 5], mechanical systems [6–9], chemistry [10], physics [11, 12], fuzzy modeling [13–16] to secure communications [17–19], among many others. In general, it is said that a set of dynamical systems achieve synchronization if trajectories in each system approach a common trajectory.Among the systems studied in synchronization, the ones that stand out are the chaotic systems; chaotic systems exhibit more complex dynamics, and they must satisfy the next conditions according to Devaney’s definition of chaos [20]: i sensitive dependence to initial conditions, ii dense periodic orbits, and iii must be transitive. Many works consider the problem of chaos synchronization; in [21], the authors synchronize chaotic systems by linking them with common signals. Chua et al. [22] explore the synchronization phenomena in Chua’s circuit, proven to be the simplest electronic circuit to exhibit chaotic behaviour; on the contrary, Femat and Solís-Perales [23] discuss several phenomena involved with chaos synchronization, and a feedback controller is implemented to illustrate such synchronization. In [24], chaos synchronization between two coupled chaotic dynamical systems is presented. Conditions for global asymptotic synchronization are presented and a new method for the analysis of the stability of the synchronization is reported, but different techniques and applications are still developed for this type of system in the last years. In [25], the authors present the design of a rule-based controller for a class of master-slave chaos synchronization, and unlike traditional methods, the control law obtained from this method has less maximum magnitude of the control signal and reduces the actuator saturation phenomenon in mechanical systems. AL-Azzawi and Aziz [26] present the synchronization between two nonhomogeneous hyperchaotic systems. A nonlinear control is used to achieve synchronization and also report a stability analysis of the error dynamics system using Lyapunov’s second method and Cardano’s method. In [27], higher-order adaptive PID controllers as a new generation of PID controllers for chaos synchronization is designed, and in [28], the authors study the collected dynamics of a n-coupled piecewise linear systems with different numbers of scrolls.Along the synchronization schemes and controllers designed, the adaptive type has been proven useful depending on the application [29–32], and this kind of controller allows the optimization of the energy necessary to accomplish the synchronization between systems.Wu et al. [29] study the synchronization of two chaotic systems which are not identical and use adaptive controllers to adjust the parameters of the systems such that the two systems will synchronize. In [25], the design of a rule-based fuzzy controller for a class of master-slave chaos synchronization is presented; however, the whole control action is substituted by the fuzzy controller, while Xi et al. propose an adaptive robust finite-time control method based on a global sliding surface for the synchronization of a class of chaotic systems in [32].The main contribution of this work is the design of an adaptive synchronization scheme based on a Mamdani-type inference system, an equivalence with saturation functions is used, and the stability of the error system is proven using Lyapunov stability theory. Both, master and slave systems can be described in as many pieces as necessary. In order to provide evidence of energy optimization, a comparison between fixed gain and adaptive gain in the same synchronization scheme is reported in a satisfactory fashion.The rest of this work is organized as follows. Section2 presents the problem statement and the systems description. Section 3 introduces the basic concepts of synchronization and Mamdani fuzzy inference system (FIS). Section 4 presents the main results with an example as numeric evidence; finally, in Section 5, the conclusions are presented.


## 2. Problem Statement

Consider a classic master-slave synchronization scheme as(1)Master:x˙=fx,(2)Slave:y˙=gy+u,where both the master and slave system are piecewise switching systems [33–36] of the form(3)z˙=Az+B,with(4)B=B1ifz∈D1B2ifz∈D2⋮⋮⋮Bjifz∈Dj,that undergo chaotic behaviour, where x,y∈ℝ3 are the state variables, Bj=bj1,bj2,bj3T, which are the switching laws of the piecewise chaotic systems, and u∈ℝ3 is the actuator in charge of achieve the synchronization of the slave over the master. Moreover, the domains Dj satisfy ℝ3=∪j=1mDj and ∩j=1mDj=∅ for m>1. The objective of this work is to design the controller u as an adaptive controller of the form(5)u=u1ify∈E1u2ify∈E2⋮⋮⋮ukify∈Ek,for k=1,2,…,n, using a Mamdani-based fuzzy inference system for each uk to synchronize the master and slave systems, optimizing the energy usage for the control action. Consequently, there are three major objectives to develop in this work:(1)
The equivalence relation between the fuzzy-based controller and the saturation functions(2)
Achieve adaptive synchronization between systems(3)
Ensure asymptotic stability of the error system via Lyapunov’s theory


## 3. Preliminaries

In this section, the basic concepts of synchronization and fuzzy inference system design will be presented; this concept will be convenient to understand how it is possible to achieve the principal objective of this work.


### 3.1. Synchronization Scheme

Consider the master-slave synchronization scheme defined in [37] as(6)Master:x˙=fx,Slave:y˙=gy+u,with x,y∈ℝn as the system states and(7)u=−Pe+fx−gy,where P=diagp12,p22,p32. The synchronization is achieved when the master and slave system synchronize, i.e., when the synchronization objective,(8)limt⟶∞yt−xt=0,is reached.Definition 1.
(see [3]). A system of the form x˙=fx is called master system if its flow xt is independent. A system of the form y˙=gy+u is called slave system of the master system if its flow yt is constrained by the flow of the master system.
From the previous definition, a slave system is constrained by a master system via a specific condition, which means that a slave system will be controlled towards a master system under a specific control law. This phenomenon is called the synchronization of the slave and master systems under such a specific condition [3, 38].


### 3.2. Fuzzy Inference System Design

The design of the fuzzy inference systems considers an IF-THEN fuzzy rules based on the form [39](9)Ri:IFeisMi,THENpisHi,where e is the error between two states, and the input of the membership functions is(10)M−1e=1,ife<Φ−1,1Φ−1e,ifΦ−1≤e≤Φ0,0,ife>Φ0,M0e=−1Φ−1e+1,ifΦ−1≤e<Φ0,−1Φ1e+1,ifΦ0≤e≤Φ1,0,elsewhere,M1e=0,ife<Φ0,1Φ1e,ifΦ0≤e≤Φ1,1,ife>Φ1,with Φ−1=−Φ1, while the output membership functions are singletons, defined as(11)H−1p=1,ifp=Ω−1,0,elsewhere,H0p=1,ifp=Ω0,0,elsewhere,H1p=1,ifp=Ω1,0,elsewhere.The input and output membership functions are depicted in Figure1. Now, consider the Mamdani-type fuzzy inference system, with product in the antecedent and center of average as the defuzzifier method. Then, the fuzzy inference system is written as(12)pe=∑i=−11Mie∑j=−11MjeΩi=∑i=−11XieΩi,where Ωi refers the crisp value of the output regarding of e. The function Xie must satisfy the following conditions [40, 41]:(i)
Xie is a locally Lipschitz continuous and bounded function(ii)
Xi0=0(iii)
Xie=Xi−eFigure 1
Membership functions: (a) input; (b) output.
(a)(b)The designed rule base matrix is given in Table1.Table 1
Fuzzy rule base.
InputeOutputpeM−1H−1M0H0M1H1Only two rules are fired at the same time [41, 42], for any value of e. Moreover, it satisfies(13)∑i=−11Xie=1,which is a linear convex combination.


## 3.1. Synchronization Scheme

Consider the master-slave synchronization scheme defined in [37] as(6)Master:x˙=fx,Slave:y˙=gy+u,with x,y∈ℝn as the system states and(7)u=−Pe+fx−gy,where P=diagp12,p22,p32. The synchronization is achieved when the master and slave system synchronize, i.e., when the synchronization objective,(8)limt⟶∞yt−xt=0,is reached.Definition 1.
(see [3]). A system of the form x˙=fx is called master system if its flow xt is independent. A system of the form y˙=gy+u is called slave system of the master system if its flow yt is constrained by the flow of the master system.
From the previous definition, a slave system is constrained by a master system via a specific condition, which means that a slave system will be controlled towards a master system under a specific control law. This phenomenon is called the synchronization of the slave and master systems under such a specific condition [3, 38].


## 3.2. Fuzzy Inference System Design

The design of the fuzzy inference systems considers an IF-THEN fuzzy rules based on the form [39](9)Ri:IFeisMi,THENpisHi,where e is the error between two states, and the input of the membership functions is(10)M−1e=1,ife<Φ−1,1Φ−1e,ifΦ−1≤e≤Φ0,0,ife>Φ0,M0e=−1Φ−1e+1,ifΦ−1≤e<Φ0,−1Φ1e+1,ifΦ0≤e≤Φ1,0,elsewhere,M1e=0,ife<Φ0,1Φ1e,ifΦ0≤e≤Φ1,1,ife>Φ1,with Φ−1=−Φ1, while the output membership functions are singletons, defined as(11)H−1p=1,ifp=Ω−1,0,elsewhere,H0p=1,ifp=Ω0,0,elsewhere,H1p=1,ifp=Ω1,0,elsewhere.The input and output membership functions are depicted in Figure1. Now, consider the Mamdani-type fuzzy inference system, with product in the antecedent and center of average as the defuzzifier method. Then, the fuzzy inference system is written as(12)pe=∑i=−11Mie∑j=−11MjeΩi=∑i=−11XieΩi,where Ωi refers the crisp value of the output regarding of e. The function Xie must satisfy the following conditions [40, 41]:(i)
Xie is a locally Lipschitz continuous and bounded function(ii)
Xi0=0(iii)
Xie=Xi−eFigure 1
Membership functions: (a) input; (b) output.
(a)(b)The designed rule base matrix is given in Table1.Table 1
Fuzzy rule base.
InputeOutputpeM−1H−1M0H0M1H1Only two rules are fired at the same time [41, 42], for any value of e. Moreover, it satisfies(13)∑i=−11Xie=1,which is a linear convex combination.


## 4. Main Results

Consider the synchronization error ase=y−x. The error dynamics are defined as(14)e˙=gy−fx+u,where the controller u is(15)u=−Pe+fx−gy,with P=diagp12,p22,p32. In the following, given the sectorial functions provided from the FIS (15), it is necessary to consider the matrix of adapting P=diagp12e1,p22e2,p32e3. Then, we can establish the following result.Lemma 1.
The controlleru=−Pe+fx−gy can be expressed as(16)u=−Ω11sate1Φ112e1Ω12sate2Φ122e2Ω13sate3Φ132e3+fx−gy.Proof.
For eachl=1,2,3, it is possible to rewrite (15) as(17)plel=Ω1lΦ1lel,ifel≤Φ1l,Ω1l,ifel>Φ1l.
Thus, each FIS can be written as a piecewise linear function as follows:(18)plel=Ω1lelΦ1l,ifel≤Φ1l,1,ifel>Φ1l,=Ω1lelΦ1l,ifelΦ1l≤1,1,ifelΦ1l>1,=Ω1lsatelΦ1l.
Substituting (18) in each pl of (15), it is possible to rewrite u as(19)u=−Ω11sate1Φ112e1Ω12sate2Φ122e2Ω13sate3Φ132e3+fx−gy,as previously stated.
Once the piecewise controller is described, whose fuzzy component is the candidate to work as the adaptive scheme, in the following result, we formalize this assertion.Theorem 1.
The synchronization scheme formed by (1) and (2) achieves synchronization under the adaptive controller (16). Thus, the error system (14) is asymptotically stable.Proof.
Consider the quadratic Lyapunov candidate function:(20)Ve=12eTPe.
From Lemma1, it is known that(21)Pe=Ω11sate1Φ12e1Ω12sate2Φ12e2Ω13sate3Φ12e3.
The time derivative of (20) is computed as(22)V˙e=12∂V∂e1+∂V∂e2+∂V∂e3,=eTe˙,=eTgy−fx+u.
Substituting the adaptive control law (16) in (22),(23)V˙e=eTgy−fx−Ω11sate1Φ12e1Ω12sate2Φ12e2Ω13sate3Φ12e3+fx−gy,=−eTΩ11sate1Φ12e1Ω12sate2Φ12e2Ω13sate3Φ12e3,=−eTΩ11sate1Φ112000Ω12sate2Φ122000Ω13sate3Φ132e,=−eTPe<0,which ensures the negativeness of (22) and consequently the asymptotic stability of the error system.
The following example illustrates the effectiveness and performance of the controller design, which comes to corroborate what we claim and prove.Example 1.
According to [43], we synthesized the following master system:(24)x˙=010001−a1−a2−a3x+−β1−β2a1β3+a2β1+a3β2,where a1=3/2, a2=1, a3=1, and(25)β1=5ifx2≥0,−5ifx2<0,β2=14ifx3≥7,0ifx3<7,β3=4ifx1≥0,−4ifx1<0.
In Figure2, the 3D phase portrait plot of system (24) is depicted under initial conditions x0=0,4,10T, and we can see that the master system presents a 8-scroll chaotic attractor in three different directions. The projections in the planes x1,x2, x1,x3, and x2,x3 for the system are shown in Figure 3.
Now, following the same methodology used to design the master system, consider the slave system as(26)y˙=010001−a˜1−a˜2−a˜3y+00a˜1β˜3+u,with a˜i=3/5, for i=1,2,3, and(27)β˜3=4ify1≥0,−4ify1<0.
Unlike the master system, the slave system only presents a double scroll chaotic attractor by taking the initial condition pointy0=1,−2,−5T, whose projections in the planes y1,y2, y1,y3, and y2,y3 for the slave system, when u=0,0,0T, are shown in Figure 4.
The control lawu is designed according to Lemma 1, and its explicit expression is given by(28)u=−Pe+fx−gy,=−Ω11sate1Φ112e1Ω12sate2Φ122e2Ω13sate3Φ132e3+010001−a1−a2−a3x,+−β1−β2a1β3+a2β1+a3β2−010001−a˜1−a˜2−a˜3y−00a˜1β˜3,where Ω1l=10 and Φ1l=1/10, for l=1,2,3. The projections in the planes y1,y2, y1,y3, and y2,y3 for the synchronized slave system are shown in Figure 5. It is clear that the slave system now exhibits the trajectories of the master system and went from a double scroll chaotic attractor to a 8-scroll chaotic attractor.
The errors between the master and slave systems are depicted in Figure6, and these errors tend asymptotically to the origin as intended.Figure 2
3D phase portrait of the master chaotic system.Figure 3
Plane projections for the master system: (a)x1,x2; (b) x1,x3; (c) x2,x3.
(a)(b)(c)Figure 4
Slave system plane projections: (a)y1,y2; (b) y1,y3; (c) y2,y3.
(a)(b)(c)Figure 5
Synchronized slave system plane projections: (a)y1,y2; (b) y1,y3; (c) y2,y3.
(a)(b)(c)Figure 6
Error dynamics.


### 4.1. Energy Optimization

One of the advantages of using the adaptive synchronization proposed in this work is the optimization of the energy used to induce the dynamics of the master system in the slave system. Comparing the magnitude ofplel, l=1,2,3, against the magnitude of a fixed value shows clearly that the energy consumption is vastly improved.At first, high values ofplel are injected in the fuzzy actuator in order that the error el reaches its following adaptive domain interval Al=−Φ1l,Φ1l, and then, the value of plel decays until the error reaches the next interval of adaptation. Hence, the power stills decaying whilst the error system tends to zero.Figure7(a) shows the behaviour of the adaptive p1e1 (blue line) and the behaviour of a fixed not adaptive value of p1 (dotted red line). This behaviour is also presented among the adaptives p2e2 and p3e3 (blue lines) and their respective fixed no adaptive values p2 and p3 (in red dots), depicted in Figures 7(b) and 7(c), respectively.Figure 7
Evolution ofplel trough time: (a) p1e1; (b) p2e2; (c) p3e3.
(a)(b)(c)The energy optimization can be modified according the application where it is planned to be used by modifying the adapting intervalAl of each plel. This can be better appreciated in Figures 8(a)–8(c).Figure 8
Comparative between the adaptation domains for (a)Φ1l=0.1; (b) Φ1l=0.25; (c) Φ1l=0.5.
(a)(b)(c)


## 4.1. Energy Optimization

One of the advantages of using the adaptive synchronization proposed in this work is the optimization of the energy used to induce the dynamics of the master system in the slave system. Comparing the magnitude ofplel, l=1,2,3, against the magnitude of a fixed value shows clearly that the energy consumption is vastly improved.At first, high values ofplel are injected in the fuzzy actuator in order that the error el reaches its following adaptive domain interval Al=−Φ1l,Φ1l, and then, the value of plel decays until the error reaches the next interval of adaptation. Hence, the power stills decaying whilst the error system tends to zero.Figure7(a) shows the behaviour of the adaptive p1e1 (blue line) and the behaviour of a fixed not adaptive value of p1 (dotted red line). This behaviour is also presented among the adaptives p2e2 and p3e3 (blue lines) and their respective fixed no adaptive values p2 and p3 (in red dots), depicted in Figures 7(b) and 7(c), respectively.Figure 7
Evolution ofplel trough time: (a) p1e1; (b) p2e2; (c) p3e3.
(a)(b)(c)The energy optimization can be modified according the application where it is planned to be used by modifying the adapting intervalAl of each plel. This can be better appreciated in Figures 8(a)–8(c).Figure 8
Comparative between the adaptation domains for (a)Φ1l=0.1; (b) Φ1l=0.25; (c) Φ1l=0.5.
(a)(b)(c)


## 5. Conclusions

An adaptive synchronization scheme was provided using as adaptation law a Mamdani-type fuzzy inference system. An equivalence is given to express the fuzzy inference system as saturation functions and evidence is provided to ensure the stability of the error system between the master and the slave systems.



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