# Fuzzy Adaptive Control of Uncertain MIMO Chaotic Systems with Unknown Control Direction

Authors: Hua Zhang
Journal: Complexity
 (2021)

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## Abstract

This paper presents an adaptive controller for MIMO chaotic systems with system uncertainties and unknown control direction. In the controller design, the matrix decomposition theory is used, and we decompose the control gain matrix into a positive matrix, a diagonal matrix whose diagonal entries are +1 or −1, and a unity upper triangular matrix. To handle the unknown control direction (i.e., the unknown sign of the control gain matrix), we use the Nussbaum-type function. In addition, we propose an adaptation law named proportional integral (PI) law to update the parameters of the fuzzy system. The stability of the controlled system is proven strictly. Finally, simulation results are presented.


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## Body

## 1. Introduction

As we all know, most systems are nonlinear and multivariable in the practical control engineering, especially chaotic systems. Chaos is an intrinsic characteristic of nonlinear dynamical systems and a universal phenomenon of nonlinear systems, and there are some vital features such as the sensitive dependency on the initial condition, intrinsic randomness, and irregular order. Chaotic systems can be encountered in many fields, such as mathematics, finance, and physics. According to the characters of dynamic systems, there are usually four classes of chaos systems: temporal chaotic systems, spatiotemporal chaotic systems, spatial chaotic systems, and functional chaotic systems. So, the problem of control chaotic with multi-input and multioutput (MIMO) has received a great interest. Chaotic system control problem was proposed firstly in [1]. After that, an increasing number of methods have been researched and successfully exploited in the control of chaotic system [2], including sliding mode control [3–5], neural adaptive control [6], and fuzzy adaptive control [7, 8]. Due to the universal approximation theorem [9], the problem of fuzzy adaptive controller designing has aroused an increasing interest. In the last two decades, fuzzy control has a great influence on the control engineering field due to its systematic and efficient frame.There are two different ways about designing a fuzzy adaptive controller: direct method and indirect method. In the direct method [10], the ideal controller is approximated by the fuzzy system, and the parameters of the fuzzy system are adjusted for achieving control requirements. And in another method [11], the unknown part of the actual system is estimated by the fuzzy system, and then the controller of the system is designed based on these estimated values. By using an indirect method, a problem that calculating the inverse of the control gain matrix is impossible when the matrix is a singular matrix. In order to avoid this problem, an algorithm was proposed in [12], and it is suggested to make the estimated parameters of the fuzzy system inside a compact set where the singularity problem does not exist. However, in [13], it is said that this approach needs a priori knowledge about the set of the parameters and no general approach for obtaining such a set. Another approach about solving this problem is proposed in [14], which uses the regularized inverse matrix of the estimated matrix of the fuzzy system.Note that a basic assumption is required in the above papers: the sign of the control gain matrix should be known in advance. However, in fact, for most of nonlinear systems, it is very difficult to measure their control direction in practical applications. Therefore, it is very important to solve this problem. In order to control the chaotic systems to meet our objective, a function is proposed in [15] named Nussbaum-type function which can be used to solve this problem. Moreover, two assumptions are produced for facilitating the stability analysis of the closed-loop system and the controller designing: a nonlinear system with a lower triangular control structure and the control gain is bounded.Motivated by the above discussion, in this paper, an adaptive fuzzy controller is implemented for uncertain chaotic systems with unknown control direction. The stability analysis of the closed-loop system is given strictly. Finally, a numerical simulation example is presented for testing the performance. The main contributions of this paper are as follows. (1) Motivated by the theorem of matrix decomposition proposed in [16], the control gain is decomposed into three matrices: a symmetric positive definite matrix, a diagonal matrix with diagonal entries being +1 or −1, and a unity upper triangular matrix. (2) The Nussbaum-type function [17] is used to estimate the real sign of the gain matrix. (3) A PI adaptation law is proposed to update the fuzzy parameters.The structure of this work is listed as follows. InSection 2, the problem description is given. In Section 3, the fuzzy logic system is introduced, and some basic lemmas are also given. In Section 4, the concrete controller design procedure is given, and the stability of the controlled system is proven strictly. Simulation results are included in Section 5. Finally, Section 6 summarizes this work, and future research directions are also discussed.


## 2. Problem Description

A class of nonlinear MIMO chaotic systems is considered as follows:(1)X˙=AX+fX+GXu,where X=x1,x2,…,xnT∈Rn is the state vector, u=u1,…,unT∈Rn is the input vector, A is an unknown n×n-dimension matrix, fX=f1x,…,fnx∈Rn, and GX=g11x⋯g1nx⋮⋱⋮gn1x…gnnx are unknown continuous nonlinear functions.Let us denoteXd=xd1,…,xdnT∈Ωxd⊂Rn as the ideal trajectory, where Ωxd is a known bounded compact set, and the purpose of our work is to design a controller such that the output trajectory can approximate the ideal trajectory Xd with a certain precision. Furthermore, all signals involved can remain bounded.Remark 1.
In this paper, system (1) is investigated. However, it should be emphasized that system (1) can represent a large scale of nonlinear systems (chaotic system or not). In addition, most chaotic systems can be expressed by (1). That is, our control method is valid for most existed chaotic systems. One can see that, in the simulation part, a special chaotic system is given for an example.Then, the tracking error can be defined as(2)e1=xd1−x1,⋮en=xdn−xn.Let us denote thatE=e1,…,enT; then, (2) can be written as(3)E=Xd−X,and E˙ can be written as(4)E˙=Xd.−X˙.Substituting system (1) into (4), we can get(5)E˙=Xd.−AX+fX+GXu.Therefore, we can exploit (5) to process the problem of the fuzzy controller design, and then the stability analysis can also be carried out.


## 3. Basic Knowledge of the Fuzzy Logic System (FLS)

A complete FLS includes four parts: fuzzy inference engine, fuzzifier, fuzzy rules, and defuzzifier [18], whose i-th fuzzy rule isRi: if x1 is B1i and ,…, and xn is Bni, then f^isfi,with B1i,B2i,…,Bni being some fuzzy sets and fi representing the singleton. The FLS is expressed by(6)f^x¯=∑i=1lfi∏j=1nνBjixj∑i=1l∏j=1nνBjixj=θTψx¯,with νBjixj being the membership function, and θT=f1,…,fl is an adjustable variable, with(7)ψix¯=∏j=1nνBjixj∑i=1l∏j=1nνBjixj.


## 4. Designing a Fuzzy Adaptive Controller and Stability Analysis

Before starting the main work of this paper, we need to introduce some assumptions and lemmas in advance.Proof. The proof of this lemma can be seen in the literature [19]. □Lemma 1.
According to Costa et al. [19], it can be concluded that any real matrix GX∈Rn×n whose leading principal minors are nonzero can be decomposed into the form as follows:(8)GX=GsXDTX,where GsX∈Rn×n is a symmetric positive definite matrix, D∈Rn×n is a diagonal matrix whose diagonal element is −1 or +1, and TX∈Rn×n is a unity upper triangular matrix.Assumption 1.
(1)
GX is a positive definite or negative definite matrix, and the sign of GX is unknown(2)
GsX and d/dtGsX are continuous(3)
∂gijx/∂yiri−1=0, ∀i=1,2,…n and ∀j=1,2,…,nRemark 2.
(1)
What we should mention is that the sign ofGX is unknown in this paper; this means that there are two cases about GX: the diagonal element is −1 or +1.(2)
In order to obtain our control objective, a Nussbaum function is used in the design of the controller, and some important properties of this function are listed as follows:(a)
lims⟶∞sup1/s∫0sNζdζ=+∞(b)
lims⟶∞inf1/s∫0sNζdζ=−∞(3)
There are some examples about the Nussbaum function from the literature:(9)N1ζ=ζ2cosζ,N2ζ=ζcosζ,N3ζ=cosπ2ζeζ2,and in this paper, we will use the functionN3ζ, and it is easy to know that ζ˙=Σi=1nθiTψiz.Now, we will introduce another lemma about the Nussbaum function that will be used in the following stability analysis.Proof. The proof of this lemma is given in Ge and Jing Wang [20]. □Lemma 2.
Vt and ζt are of C1 whose definition domain is 0,tf, with Vt≥0∀∈0,tf. Let Nt be a Nussbaum function; then,(10)Vt≤c0±∫0tgNζ±1ζ˙dτ,with g>0, c0 being a suitable constant, and ∫0tgNζ±1ζ˙dτ being bounded on 0,tf.Because matrixA is unknown, the function fx and the sign of GX are not known. We need to use FLSs to design our controllers.Remark 3.
As we all know, there are two methods about designing a fuzzy adaptive control system: the direct method and the indirect method. The most important difference between two methods is the control object: the direct method is using the fuzzy system to estimate the ideal controller, and the indirect method is using the fuzzy system to estimate the unknown part of the system such as matrixA, the function fx, and the sign of GX. In this paper, we will exploit the indirect method to design the fuzzy controller that we need.By using Lemma1, matrix GX can be decomposed into GsXDTX. Substituting it into system (5), we can get(11)G1XE˙=G1XXd.−AX+fX−DTX−D−Du=FX,u−Du,where G1X=Gs−1X and FX,u=G1XXd.−AX+fX−DTX−D. Let us denote αz=1/2G1.XE+FX,u; then, (11) can be transformed into the following form:(12)G1XE˙=−12G1.XE+αz−Du.Remark 4.
By checking the expressing form ofαz and FX,u, we can find that the structure of vector z is an upper triangular control structure. The element of vector z can be selected by(13)z1=X,E,u1,…,un,z2=X,E,u2,…,un,⋮zn−1=X,E,un−1,un,zn=X,E,un.Define the compact sets as follows:(14)Dzi=X,E,ui,…,un,i=1,…,n.Remark 5.
Noting thatE and ui are functions of vector X and Xd is bounded, we can know that zi=X.Now, we will use the fuzzy system to approximateαz. We can use the FLS as follows:(15)αi^zi,θi=θiTψizi.Let us denote the ideal parameters ofθi as(16)θi∗=argminθi=supzi∈Dziαizi−αi^zi,θi.We should notice that the ideal parameters ofθi are just introduced for analysis, and we do not need to know their true value.Let us define(17)θi˜=θi−θi∗,as the error of the parameter estimation; then, we have(18)εizi=αizi−αi^zi,θi∗.According to Chang [21], we can assume that our fuzzy system is not contrary to the universal approximate principle on the compact set Dz, and we assume this set is big enough. Then, our input vector of the fuzzy system can remain within Dz under the closed-loop system. In summary, we can assume that the error of the parameter estimation is bounded on the compact set Dz, i.e., for any z∈Dz,(19)εizi≤εi,with εi being an arbitrarily small positive constant.Then, let us denote(20)α^z,θ=θTψz=α1^z1,θ1,…,αn^zn,θn,εz=ε1z1,…,εnzn,ε=ε1,…,εn.Then, we have(21)α^z,θ−αz=α^z,θ−α^z,θ∗+α^z,θ∗−αz=α^z,θ−α^z,θ∗−εz=θ˜ψz−εz,with θ˜ψz=θ1˜ψz1,…,θn˜ψzn. Now, in order to satisfy our control purpose, the controller can be designed in the following form:(22)u=NζθTψz+K1E+K0SignE,with(23)K0=diagk01,k02,…,k0n,K1=diagk11,k12,…,k1n,and k0i>0,i=1,…,n, and k1i>0,i=1,…,n.To update parameterθi, we can design a PI adaptation law:(24)θi=∫0t−σiγ0ieiθi+γ0ieiψizidτ−γ1iδi,with δi=σieiθi−eiψizi and γ0i,γ1i,σi>0 being constants.Proof. Let us denote(25)V1=12ETG1XE,V2=∑i=1n12γ0iθi˜+γ1iδiTθi˜+γ1iδi.Remark 6.
Note that there are some related fuzzy control methods which have been proposed recently, for example, in [22–24]. However, the proposed method is different from the above literature because a PI law is designed in this paper. We should mention the difference between the two terms in PI law (24), σiγ0ieiθi and δi. The purpose of the term σiγ0ieiθi is to ensure that parameter θi is bounded, and another one is a proportional term, which can ensure that parameter θi has a fast convergence.Theorem 1.
Consider system (5). Using controller (22) with PL law (24), the following properties can be ensured:(1)
All signals involved are bounded(2)
The tracking errors are asymptotically stableThen, we consider the following Lyapunov function:(26)V=V1+V2,First, through the above analysis, we can get(27)G1XE˙=−12G1.XE+αz−Du=−12G1.XE−K1E−K0SignE−θ˜ψz+εz+θTψz+K1E+K0SignE−Du.Substituting controller (22) into (27), we have(28)G1XE˙=−12G1.XE−K1E−K0SignE−θ˜ψz+εz+θTψz+K1E+K0SignE1+gNζ,with g=dii,∀i=1,…,n.MultiplyingET to both sides of the above equation, we have(29)ETG1XE˙=−12ETG1.XE−∑i=1nk1iei2−∑i=1nk0iei−∑i=1neiθi˜Tψizi+∑i=1neiεizi+∑i=1neiθiTψizi+k1iei+k0isignei1+gNζ.Then, we have(30)V1.=12ETG1.XE+ETG1XE˙=−∑i=1nk1iei2−∑i=1nk0iei−∑i=1neiθi˜Tψizi+∑i=1neiεizi+∑i=1neiθiTψizi+k1iei+k0isignei1+gNζ=−∑i=1nk1iei2−∑i=1nk0iei−∑i=1neiθi˜Tψizi+∑i=1neiεizi+1+gNζζ˙.Next, let us analyzeV2 as(31)V2.=∑i=1n1γ0iθi˜.+γ1iδi.T+θi˜+γ1iδi=∑i=1n1γ0i−σiγ0ieiθi+γ0ieiψiziTθi˜+γ1iδi=∑i=1n−δiTθi˜+γ1iδi=∑i=1nθi˜T−σi|ei|θi+eiψizi−γ1iδi2=∑i=1n−σieiθi˜Tθi−γ1iδi2+eiθi˜Tψizi≤∑i=1nσiei−12θi˜2+12θi∗2+eiθi˜Tψizi−γ1iδi2.From (30) and (31), we have(32)V˙=V1.+V2.≤−∑i=1nk1iei2−∑i=1nk0iei+∑i=1neiεizi−12∑i=1nσieiθi˜2+12∑i=1nσieiθi∗2+1+gNζζ˙≤−∑i=1nk1iei2−∑i=1nk0iei+∑i=1neiεi+12∑i=1nσieiθi∗2+1+gNζζ˙.If we choosek0i≥εi+1/2σiθi∗2, then we can get(33)V˙≤−∑i=1nk1iei2+1+gNζζ˙.By using Lemma2, integrating the above integral over 0,t, we have(34)Vt≤Vt+∫0t∑i=1nk1iei2dτ≤V0+∫0tgNζ±1ζ˙dτ.According to Lemma2, we can know that Vt, ζt, and ∫0tgNζ±1ζ˙dτ are bounded on 0,tf. When tf⟶∞, the conclusion is also true (the proof can be seen in Xu et al. [25]). So, ei is bounded. From (34), we can conclude that ∫0t∑i=1nk1iei2dτ is bounded, i.e., ei∈L2. In (27), all signals involved on the right side are bounded, and note that the functions G1X and G˙x are continuous functions, so it can be easy to show that ei.∈L∞. Finally, we can conclude that ei⟶0 as t⟶∞ by using the Barbalat lemma (see Slotine et al. [26]).Remark 7.
There is still another choice ofζ˙ and controller u. In Yao and Pan [27], Table 1 summarizes this choice, and the proof is similar to the proof of Theorem 1. This means all choices of Table 1 in [27] can guarantee that all signals involved in this closed-loop system are bounded, and the errors converge to zero.Remark 8.
(1) For eliminating the chattering effect caused by robust termSignei of the controller, we can use some smooth functions to replace it, such as tanh or arctan. (2) Because the value of θi∗ is unknown, the choice of parameters k0i is difficult. However, we can use an adaptation law k˙0i=γ2i|ei| to replace it, where γ2i>0.Overall, the framework of this paper is shown in Figure1.Figure 1
Block diagram of this work.


## 5. Numerical Simulation

In this section, the effectiveness of the controller that we designed in this paper is shown. A chaotic financial system is used for simulation.The MIMO chaotic financial system is given by(35)x1.=x3+x2−ax1+1+x12u1t,x2.=1−bx2−x12+1+x22u2t,x3.=−x1−cx3+1+x32u3t,where a,b, and c are system parameters. Let us denote X=x1,x2,x3T and u=u1,u2,u3T; then, system (35) can be expressed as follows:(36)X˙=AX+fX+GXu,where(37)A=−a010−b0−10−c,fX=f1Xf2Xf3X=x1x2−x12+10,GX=1+x120001+x220001+x32.The ideal trajectories arexd1=sint, xd2=sint, and xd3=sint, and the initial conditions are X0=1,−1,2, θ1=0, θ2=0, θ3=0, and ζ0=0.In the simulation example, we use three fuzzy systems. The input of each system is the state variable of the system. For each input (that is, each system state variable), we define four fuzzy membership functions in the interval [−7, 7]. The design parameters in this simulation are selected as follows:a=0.78, b=0.2, c=1.5, k1i=0.5i=1,2,3, k1i=0.5i=1,2,3, γ0i=200i=1,2,3, and σi=0.02i=1,2,3.The simulation results are shown in Figures2–6. Figure 2 shows the trajectory of xi,i=1,2,3, and the tracking error ei,i=1,2,3. One can see that the performance is very good. Figure 3 shows the controller uit,i=1,2,3. In Figures 4–6, the results of parameters θ1, θ2, and θ3, respectively, are presented.Figure 2
Simulation results forxit and eit in (a) tracking of xd1, (b) tracking of xd2, (c) tracking of xd3, (d) time response of e1, (e) time response of e2, and (f) time response of e3.
(a)(b)(c)(d)(e)(f)Figure 3
Simulation results foru1t, u2t, and u3t.Figure 4
Simulation results forθ1.Figure 5
Simulation results forθ2.Figure 6
Simulation results forθ3.


## 6. Conclusions

In this study, a novel fuzzy adaptive controller for the MIMO chaotic system with unknown control direction has been proposed. A type of function called Nussbaum function has been used in the designing of the controller. We also proposed a PI law to update the fuzzy parameters. Finally, in the simulation results, we can see that all signals involved in this closed system are bounded, and the tracking errors all converge to zero ast⟶∞.



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