# Design of Robust Adaptive Fuzzy Controller for a Class of Single-Input Single-Output (SISO) Uncertain Nonlinear Systems

Authors: Jieqiong Lin; Jiakang Zhou; Mingming Lu; Hao Wang; Allen Yi
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In order to solve the precision and stability control problems of nonlinear uncertain systems applied in machining systems, in this paper, a robust adaptive fuzzy control technique based on Dynamic Surface Control (DSC) method is proposed for the generalized single-input single-output (SISO) uncertain nonlinear system. A first-order low-pass filter is introduced in each step of the traditional robust control method to overcome the “calculation expansion” problem, and Takagi–Sugeno (T-S) fuzzy logic system is applied to approximate an uncertain nonlinear function of unknown structure in the system. The designed robust adaptive fuzzy controller is applied to the 3D elliptical vibration cutting (3D EVC) device system model, and the effectiveness of the controller design is verified by analysis of position tracking, speed tracking, and tracking error. The results of studies show that the robust adaptive fuzzy controller can effectively suppress the jitter problem of the three-dimensional elliptical vibration cutting device so that the control object can be stabilized quickly even if it has a little jitter at the beginning. It can be smoothed to move along the ideal displacement and velocity signals. It is verified that the designed controller has strong robust adaptability.


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## Body

## 1. Introduction

Adaptive control technology provides a very effective mean to solve the uncertainty in the system [1–3]. Since the 1970s, adaptive control has attracted great attention in the field of control theory [2, 4, 5]. In actual engineering control, the model is always nonlinear and contains some uncertainty. Uncertain nonlinear systems are susceptible to two types of uncertainties: known structural uncertainties, i.e., parameter uncertainties [6] and uncertainties in structural unknowns, i.e., modeling errors and external disturbances [7]. Considering nonlinear systems with known uncertainties in structure, in the past research, significant results have been achieved in adaptive techniques for feedback linearization of nonlinear systems [8–10]. For nonlinear systems with structurally unknown uncertainties, robust control strategies can be applied. Since some controlled systems cannot know the bounds of the unknown uncertainty of their structure, adaptive control and robust control methods cannot be used for controller design. However, for such problems, neural networks and fuzzy control can approximate uncertain continuous functions of the unknown structure in the system [11, 12]. Ge et al. researched a robust adaptive neural network control method for a perturbed strictly feedback nonlinear system, which can guarantee the final boundedness in the case of unknown structural uncertainty [13]; Cao et al. proposed a new system control strategy using the global approximation property of the fuzzy system to approximate the unknown function of the designed system and ensure the stability of the whole system [14].However, in the system described above, it is satisfied that the unknown nonlinearity and the control input appear in the same state space model equation. In order to overcome the limitations of such systems, relevant research scholars have proposed adaptive post-push technology [4, 6, 15, 16], furthermore successfully solved the constraint of the matching conditions imposed on the nonlinearity of the system. However, adaptive post-pushing technology has some disadvantages such as overparameterization. In order to solve this shortcoming, Yang et al. designed the controller with adaptive post-pushing technology and robust control technology, which made the closed-loop system globally consistent and bounded under the influence of unknown nonlinearity and parameter uncertainty factors [17]. In addition, robust adaptive neural networks and fuzzy controllers are studied for systems with matching conditions imposed by the abovementioned nonlinearities [18, 19]. Especially for high-order systems and multivariable systems, when neural network basis functions or fuzzy rule numbers are used to approximate uncertain nonlinear functions, many parameters need to be adjusted, resulting in a “dimension disaster” problem. As a result, the computational burden becomes too large and the “calculation expansion” problem arises at the same time, which is not conducive to engineering applications [19, 20]. Yang et al. investigated an adaptive control technique based on the T-S fuzzy system approximator, which adjusted the adaptive parameters of each system to two and solved the problem of “dimensionality disaster” [21]. Hedrick et al. proposed a “Dynamic Surface Control” (DSC) method that overcame the “computational expansion” problem by introducing a first-order low-pass filter at each step of the pushback technique [22]. Sui et al. proposed a new stochastic finite-time stability theorem combined with the backstepping technique and an adaptive fuzzy stochastic finite-time control method. Simulation results proved the effectiveness of the method [23]. Mu et al. proposed finite-time switching mode manifolds and corresponding nonsingular controllers, and the reduced system is obtained from terminal sliding mode control (TSMC), explained that TSMC is a finite-time control by the homogeneous theory [24].Backstepping technology is a systematic controller synthesis method for uncertain systems. It is a regression design method that combines the selection of the Lyapunov function with the design of the controller. The reason why backstepping technology is widely used in many industrial control fields is mainly because the method eliminates the constraint that the system uncertainty and meets the matching conditions, thereby solving the control problem of relatively complex nonlinear systems. With the continuous development of intelligent control technologies such as neural networks and fuzzy systems, backstepping technology has gained great potential for development. In particular, the introduction of neural networks and adaptive technologies has greatly promoted the application of the backstepping method. Sui et al. proposed a finite-time adaptive fuzzy decentralized control method by combining backstepping recursive design and Lyapunov function theory. The experimental result has proved that the developed control strategy can make the output of the system have good tracking performance in a limited time [25]. Soon after, Sui et al. presented a finite-time switching control method based on the backstepping recursive technique and the common Lyapunov function method and applied it to a mass spring-damper system to prove the effectiveness [26].In this paper, a robust adaptive fuzzy control technique based on “Dynamic Surface Control” (DSC) method is proposed for generalized single-input single-output (SISO) uncertain nonlinear systems. The T-S fuzzy logic system is used to approximate the uncertain nonlinear function with an unknown structure in the system. The main contributions of the paper include the following:(1)
In the case of unknown structural uncertainty, the resulting closed-loop system is ultimately bounded.(2)
Only one function needs to have a T-S fuzzy logic system approximation and each subsystem needs only one parameter to be adjusted, which overcomes the problem of “dimensionality disaster.” In turn, the calculation amount of the control algorithm is greatly reduced, and the problem of “calculation expansion” is solved.The chapters of this paper are distributed as follows. Section1 mainly introduces the expressions and assumptions of the questions and provides some preliminary knowledge. A design of robust adaptive fuzzy controller based on “Dynamic Surface Control” (DSC) method and Lyapunov stability analysis are described in Section 2. Section 3 uses MATLAB to carry out simulation research to prove the effectiveness of the proposed method. Finally, the conclusion is reflected in Section 4.


## 2. Problem Formulation and Preparation

### 2.1. Problem Preparation

#### 2.1.1. Fuzzy Control Theory

Fuzzy Logic Control, referred to as Fuzzy Control, is a computer digital control technology based on fuzzy set theory, fuzzy linguistic variables, and fuzzy logic reasoning. Professor L. A. Zadeh of the University of California, United States, published the famous “Fuzzy Sets” paper [27] and proposed the concepts of fuzzy sets and fuzzy algorithms.The fuzzy control system is mainly composed of four parts: fuzzification, rule base, fuzzy reasoning machine, and defuzzification:(1)
Fuzzification: the main function is to select the input amount of the fuzzy controller and convert it into a fuzzy amount that can be recognized by the system. It consists of three steps. First, the input volume is processed to meet the needs of fuzzy control. Second, the input amount is scaled. Third, the fuzzy language value of each input and the corresponding membership function are determined.(2)
Rule base: it consists of a fuzzy “If-Then” rule set. The fuzzy rule base contains many control rules and is a key step in the transition from actual control experience to fuzzy controllers.(3)
Fuzzy reasoning machine: based on combined reasoning or independent reasoning for fuzzy rule bases, a variety of fuzzy reasoning machines are proposed, which mainly implement knowledge-based reasoning decisions.(4)
Defuzzification: the main function is to convert the control amount obtained by reasoning into the control output.There are two types of fuzzy control systems: Mamdani and Takagi–Sugeno (T-S).Consider a fuzzy system approximating a continuous multidimensional functiony=fx, where x=x1,x2,…,xnT∈U is the input vector and y∈V⊂R is the output vector. The fuzzy logic system constitutes a mapping from subspace U to subspace V, as shown in Figure 1. The system usually consists of the following P P>1 fuzzy rules:Figure 1
Structure diagram of the fuzzy logic system.Ifx1=Ah1i,x2=Ah2i,…,xn=Ahni, then yi=Bh1,h2,…,hni, i=1,2,…,P, where Ah1,h2,…,hni and Bh1,h2,…,hni, respectively, represent the input fuzzy set and the output fuzzy set.IfBh1,h2,…,hni is a single fuzzy set, then it is a Mamdani-type fuzzy system. If Bh1,h2,…,hni is a function, it is a Takagi–Sugeno (T-S)-type fuzzy system [28]. Because the T-S-type fuzzy system is used in this paper, the Mamdani-type fuzzy system will not be introduced too much. After deblurring the abovementioned T-S fuzzy system with product fuzzy reasoning, the output of the T-S fuzzy system becomes(1)Fx=∑i=1Kyiξix,where ξix represents a fuzzy basis function, yi=a1ix1+a2ix2+⋯+anixn; then, formula (1) can be changed to(2)Fx=ξxAxx,where(3)ξx=ξ1x,ξ2x,…,ξKx,Ax=a11a12…a1na21a22…a2n⋮⋮⋮aK1aK2…aKn.The proposed T-S-type fuzzy system provides convenient conditions for the analytical design of the fuzzy control system. Because complex nonlinear systems will be decomposed into a series of linear subsystems, the originally difficulty to parse and describe the system will not be complicated by the followers of each rule in the T-S fuzzy system.Assuming that the output universe is a compact set withinRr, then for any given real continuous functions U and fx on ∀ε>0, there exists a fuzzy system Fx of form (2), such that(4)fx−FxxϵUsup≤ε.For the T-S-type fuzzy system,Fx=f^x,Ax, and the detailed proof process is given in [29, 30].


#### 2.1.2. Dynamic Surface Control

Dynamic surface control is developed on the basis of Backstepping Control. Backstepping method has obvious advantages in implementing robust control or adaptive control of uncertain nonlinear systems. However, the backstepping method itself does not have a good solution to the term expansion caused by the derivation of virtual control and the problem caused by the term expansion. This disadvantage is particularly prominent in higher-order systems. It is to overcome the computational complexity of the traditional inverse design. The main advantages of the dynamic surface control method are as follows: (1) it can eliminate the expansion of differential terms and make the controller and parameter design simple; (2) it can reduce the number of neural network and fuzzy system input variables used for modeling; and (3) there is no need to approximate the bounded error in the stability analysis, thereby avoiding the cyclical argument.The following briefly introduces the design of dynamic surface control methods.Take a second-order nonlinear system as an example:(5)x˙1=x2+af1x1,x˙2=u+bf2x¯2,y=x1,where x¯2=x1,x2T represents the system state variable and u represents the system input.Define the first error surfaceS1=x1−yr, where yr represents the system reference input signal, and derive the error surface S1 to obtain the following formula:(6)S˙1=x2+x2d−x2d+af1x1−y˙r,where x2d represents the intermediate virtual control rate.In order to overcome the expansion of the differential term, a new low-pass filter is introduced to obtain a new variablez2:(7)τ2z2˙+z2=x2d,where τ2 represents the time constant of the first-order low-pass filter.By introducing a first-order filter in a nonlinear system, the problem of term expansion caused by the derivation of virtual control in high-order systems is avoided. The design and stability analysis of the dynamic surface control method proves that the literature [31] gives specific steps.


#### 2.1.3. Statement of Problem

Consider a class of nonlinearly indeterminate single-input single-output (SISO) systems that are subject to external disturbances:(8)x˙i=gix¯ixi+1+fix¯i+dix,t,x˙n=gnxu+fnx+dnx,t,y=x1,where x¯i=x1,…,xiT, x=x1,…,xnT is the system state vector; u,y∈R is the input and output of the control system; fi⋅ is the unknown smooth nonlinear function; gi⋅ is the unknown smooth virtual control nonlinear gain function; and dix,t1≤i≤n is the uncertain external disturbance, and bounded.Define the tracking errorz, track a given reference signal yrt, and assume that the given reference signal yrt is bounded:(9)z=yt−yrt.Make assumptions about the abovementioned system.Assumption 1.
The unknown smooth virtual control nonlinear gain functiongi⋅ satisfies 0≤amin≤gix¯i,t≤bmax, where amin and bmax represent the upper and lower bounds of the parameter and are some unknown constant.Assumption 2.
It is uncertain that the external disturbancedix,t1≤i≤n is bounded, and dix,t≤di, di is an unknown normal number.Assumption 3.
The weight of the fuzzy systemθi∗ and the approximation error εi are bounded.


### 2.2. Design of Robust Adaptive Fuzzy Controller

#### 2.2.1. Controller Design

Consider the first subsystem of the system shown in equation (8), defining the first tracking error z1=x1−yr; then,(10)z˙1=g1x¯1,tx2+f1x1,t+d1−y˙r.Using a T-S-type fuzzy system to approximatef1x1,ω, then(11)f1x1,ω=ξ1x1A1x1+ε1=ξ1x1A1z1+ξ1x1A1yr+ε1=qθ1ξ1ω1+ξ1x1A1yr+ε1,where ε1 represents the approximation error, qθ1 is an unknown constant, and qθ1=A1, A1=qθ1A1m, and ω1=A1z1; then,(12)z˙1=g1x¯1,tx2+qθ1ξ1xω1+l1−y˙r,where l1=ξ1x1A1yr+ε1+d1, l1≤ξ1x1A1yr+ε1+d1≤aminθ1φ1x1, θ1=amin−1maxA1yr,ε1+d1, and φ1x1=1+ξ1.Therefore, the intermediate stabilization functionψ2 and the parameter adaptation rate are(13)ψ2=−k1z1+y˙r−λ^1ρ1x1z1,(14)λ^˙1=μ1ρ1x1z12−σ1λ^1−λ10,(15)ρ1x1=14ι12ξ1ξ1T+14κ12φ12,where k1, μ1, ι1, and κ1 are normal numbers; λ1^ is an estimated value of λ1; λ1=amin−1maxqθ12,θ12; and λ10 is the initial value.At this point, the estimated values2 of ψ2 is obtained:(16)s2=ψ2−τ2s˙2,where τ is the time constant.Considering thei−th2≤i≤n−1 subsystems of the system shown in equation (8) and defining the i−th tracking error: zi=xi−si; then,(17)z˙i=gix¯i,txi+1+fix¯i,t+di−s˙i.Using the T-S fuzzy system to approximate the uncertain functionfix¯i,t, then(18)fix¯i,t=ξix¯iAix¯iT+εi=ξix¯iAiz1+yrz2+s2⋯zi+si+εi=qθiξiωi+di′,where di'=ξiAi1yr+ξi∑j=2iAijsj+εi, qθi=Ai1, Ai1=qθiAim, and ωi=Aimz¯i; then,(19)z˙i=gix¯i,txi+1+qθiξixωi+li−s˙i,where li=di+di′, li≤di+ξiAi1yr+ξi∑j=2iAij+εi≤aminθiφixi, θi=amin−1maxAi1yr,∑j=2iAij,εi+di, and φixi=1+ξi.Therefore, the intermediate stabilization functionψi+1 and the parameter adaptation rate are(20)ψi+1=−kizi+s˙i−λ^iρix¯izi,(21)λ^˙i=μiρix¯izi2−σiλ^i−λi0,(22)ρix¯i=14ιi2ξiξiT+14κi2φi2,where ki, μi, ιi, and κi are normal numbers; λ^i is an estimated value of λi; λi=amin−1maxqθi2,θi2; and λi0 is the initial value.At this point, the estimated valuesi+1 of ψi+1 is obtained:(23)si+1=ψi+1−τi+1s˙i+1.Define then−th tracking error: zn=xn−sn; similarly, there is(24)z˙n=gnx,tu+fnx¯n,t+dn−s˙n,fix¯n,t=ξnx¯nAnx¯nT+εn=ξnx¯nAnz1+yrz2+s2⋯zn+sn+εn=qθnξnωn+dn′,where dn′=ξnAn1yr+ξn∑j=2nAnjsj+εn, qθn=An1, An1=qθnAnm, and ωn=Anmz¯n; then,(25)z˙n=gnx,tu+qθnξnxωn+ln−s˙n,where ln=dn+dn′, ln≤dn+ξnAn1yr+ξn∑j=2nAnj+εn≤aminθnφnx, θn=amin−1maxAn1yr,∑j=2nAin,εn+dn, and φnx=1+ξn.Therefore, select the control rate and parameter adaptation rate:(26)u=−knzn+s˙n−λ^nρnxzn,(27)λ^˙n=μnρnxzn2−σnλ^n−λn0,(28)ρnx=14ιn2ξnξnT+14tn2φn2,where kn, μn, ιn, and κn are normal numbers; λ^n is an estimated value of λn; λn=amin−1maxqθn2,θn2; and λn0 is the initial value.


#### 2.2.2. Lyapunov Stability Analysis

Define a new tracking error:(29)yi+1=si+1−ψi+1,i=1,2,…,n−1.From equation (23), s˙i=−si+ψi/τi=−yi/τi; then,(30)y˙2=s˙2−ψ˙2=−y2τ2+−∂ψ2∂z1z˙1−∂ψ2∂x1x˙1−∂ψ2∂θ1^θn^˙−∂ψ2∂λ1^λ1^˙+yr¨=−y2τ2+Ο2z1,z2,y2,θ1^,λ1^,yr,yr˙,yr¨,where Ο2⋅ is the remainder. Similarly, there is(31)y˙i+1=s˙i+1−ψ˙i+1=−yi+1τi+1+Οi+1z1,…,zi+1,y2,…yi,θ^1,…θ^i,λ^1,…,λ^i,yr,y˙r,y¨r.According to thei−th tracking error equation and equation (29), xi+1=zi+1+si+1 and si+1=yi+1+ψi+1, and the tracking error can be expressed as follows:(32)z˙1=g1x¯1,tz1+g1x¯1,ty1+g1x¯1,tψ1+qθ1ξ1xω1+l1−y˙r,z˙i=gix¯i,tzi+1+gix¯i,tyi+1+gix¯i,tψi+1+qθiξixωi+li−s˙i,z˙n=gnx,tu+qθnξnxωn+ln−s˙n.Then, bring equations (13)∼(15), (19)∼(21), and (26)∼(28) into the abovementioned equation and obtain the following expression:(33)z˙1=g1z1+g1y1−g1k1z1+g1y˙r−g1λ^114ι12ξ1ξ1T+14κ12φ12z1+qθ1ξ1xω1+l1−y˙r,z˙i=gizi+1+giyi+1−gikizi+gis˙i−giλ^i14ιi2ξiξiT+14κi2φi2zi+qθiξixωi+li−s˙i,z˙n=−gnknzn+gns˙n−gnλ^n14ιn2ξnξnT+14tn2φn2zn+qθnξnxωn+ln−s˙n.Define the closed-loop system Lyapunov function as(34)V=12∑i=1nzi2+λ^iTaminμi−1λ˜i+12∑i=1n−1yi+12,where λ^i=λi−λ^i.In the following, it will be proved that the given arbitraryζ>0, the existence of li, and ιi, κi, σi, and μi make the solution of the closed-loop system consistent and bounded.The derivative of the closed-loop system Lyapunov function with respect to time is(35)V˙=∑i=1nziz˙i−λ^iTaminμi−1λ^˙i+∑i=1n−1yi+1y˙i+1.Deduced analysis can be drawn(36)V˙≤∑i=1n−1gizi+1zi+giyi+1zi−aminkizi2−aminλ^i14ιi2ξiξiTzi2−giλ^i14κi2φi2zi2+qθiξixi¯ωizi+lizi−λ^iTaminμi−1λ^˙i+g1y˙rz1−y˙rz1+∑i=2ngis˙izi−s˙izi−aminknzn2−gnλ^n14tn2ξnξnTzn2−aminλ^n14κn2φn2zn2+qθnξnxωnzn+lnzn−λ^nTaminμn−1λ^˙n+∑i=1n−1−yi+12τi+1+yi+1Οi+1.However,(37)qθiξiωizi=qθiξiωizi−ιi2ωiTωi+ιi2ωiTωi=−ιi2ωi−qθi22ιi2ξizi2+qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤aminλ^i14ιi2ξiξiTzi2+aminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi,lizi≤aminθ^iφix¯izi+aminθ˜iφix¯izi≤giθ^iφix¯izi+aminθ˜iφix¯izi.At the same time, it can be known froms˙i=−si+ψi/τi=−yi/τi:(38)g1y˙rz1−y˙rz1≤1+bmaxτizi2+1+bmaxy˙r2≤1+bmax4zi2+1+bmaxΟ02,gis˙izi−s˙izi≤gis˙izi+s˙izi≤gi+1yiτizi≤1+bmaxτizi2+1+bmax4τiyi2.Then, equation (36) can be changed to(39)V˙≤∑i=1n−1gizi+1zi+giyi+1zi+∑i=1naminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi+giθ^iφix¯izi+aminθ˜iφix¯izi−giλ^i14κi2φi2zi2gn−gnλ^n14tn2ξnξnTzn2+1+bmax4−amink1z12+1+bmaxΟ02+∑i=2n−−1+bmaxτi+aminkizi2+∑i=2n1+bmax4τiyi2+∑i=1n−1−yi+12τi+1+yi+1Οi+1.Since,giθ^iφix¯izi+aminθ˜iφix¯izi−giλi^1/4κi2φi2zi2≤θiφixi¯zi−giλi^1/4κi2φi2zi2≤θi2/4tn2φi2x¯zi2+κi2−giλ^i1/4κi2φi2zi2≤giκi≤bmaxκi, gizi+1zi≤zi2+bmax/4zi+12 and giyi+1zi≤zi2+bmax/4yi+12. Bring equations (21) and (22) into the abovementioned equation; then,(40)V˙≤1+bmax4−amink1−2z12+∑i=2n−−1+bmaxτi+aminki−2zi2+1+bmax4−aminknzn2−∑i=1nσi2λmaxaminμi−1λ˜iTμi−1λ˜i+∑i=1n−1bmax4yi+12−3−bmax4τi+1+yi+1Οi+1+∑i=1nδi′+ι2ω2,where δi′=1+bmaxΟ02+bmaxκi+σi/2λi∗−λi02, ι=ι12+ι22+⋯+ιn21/2, and ω=ω1,ω2,…,ωnT.Sinceωi=Aimz¯iT, Aim≤1; however,(41)ω=ω1ω2⋯ωn=A1m0⋯0A2m1A2m2⋯0⋮⋮⋮Anm1Anm2⋯Anmnz1z2⋯zn=Az.This can be obtained as(42)ω≤Az≤z.First inspection the properties of equation (8):LetΟi+1 be the maximum value of Μi+1 on the set Ω, and let 1/τi+1=3−bmax/4τi+1−1bmax/4+Μi+12/2ψ+ψ0, and since yi+1Οi+1≤yi+12Μi+12/2ψ+ψ/2, then(43)bmax4yi+12−3−bmax4τi+1+yi+1Οi+1≤−bmax4+Μi+122ψ+ψ0yi+12+bmax4yi+12+yi+12Μi+12Οi+122ψΜi+12+ψ2=−ψ0yi+10−1−Οi+12Μi+12yi+12Μi+122ψ+ψ2.Let beσi/2λmaxaminμi−1=ψ0, k1=bmax−12+1+bmax/4+ψ0, ki=bmax−12+1+bmax/τi+bmax/4+ψ0, kn=bmax−11+bmax/τi+ψ0, and ψ0 is a normal number. Then, equation (36) becomes(44)V˙≤−ψ0∑i=1nzi2−ψ0∑i=1nλ˜iTaminμi−1λ˜i−ψ0∑i=1n−1yi+12+∑i=1nιi2ωiTωi+χ≤−2ψ0V+χ,where z=z1,z2,…,znT, χ=∑i=1nδi′+∑i=1n−1ψ/2, then(45)Vt≤χ2ψ0+Vt0e−2ψ0t−t0,∀t≥t0≥0.Obviously, for anyζ1>0, exist Τ>0 so that for all t≥t0+Τ, z1≤ζ1 is established. Then, all solutions of the closed-loop system are consistent and bounded, that is, by adjusting the controller parameters so that the tracking error z1=yt−yrt is as small as possible to achieve the desired tracking accuracy.


## 2.1. Problem Preparation

### 2.1.1. Fuzzy Control Theory

Fuzzy Logic Control, referred to as Fuzzy Control, is a computer digital control technology based on fuzzy set theory, fuzzy linguistic variables, and fuzzy logic reasoning. Professor L. A. Zadeh of the University of California, United States, published the famous “Fuzzy Sets” paper [27] and proposed the concepts of fuzzy sets and fuzzy algorithms.The fuzzy control system is mainly composed of four parts: fuzzification, rule base, fuzzy reasoning machine, and defuzzification:(1)
Fuzzification: the main function is to select the input amount of the fuzzy controller and convert it into a fuzzy amount that can be recognized by the system. It consists of three steps. First, the input volume is processed to meet the needs of fuzzy control. Second, the input amount is scaled. Third, the fuzzy language value of each input and the corresponding membership function are determined.(2)
Rule base: it consists of a fuzzy “If-Then” rule set. The fuzzy rule base contains many control rules and is a key step in the transition from actual control experience to fuzzy controllers.(3)
Fuzzy reasoning machine: based on combined reasoning or independent reasoning for fuzzy rule bases, a variety of fuzzy reasoning machines are proposed, which mainly implement knowledge-based reasoning decisions.(4)
Defuzzification: the main function is to convert the control amount obtained by reasoning into the control output.There are two types of fuzzy control systems: Mamdani and Takagi–Sugeno (T-S).Consider a fuzzy system approximating a continuous multidimensional functiony=fx, where x=x1,x2,…,xnT∈U is the input vector and y∈V⊂R is the output vector. The fuzzy logic system constitutes a mapping from subspace U to subspace V, as shown in Figure 1. The system usually consists of the following P P>1 fuzzy rules:Figure 1
Structure diagram of the fuzzy logic system.Ifx1=Ah1i,x2=Ah2i,…,xn=Ahni, then yi=Bh1,h2,…,hni, i=1,2,…,P, where Ah1,h2,…,hni and Bh1,h2,…,hni, respectively, represent the input fuzzy set and the output fuzzy set.IfBh1,h2,…,hni is a single fuzzy set, then it is a Mamdani-type fuzzy system. If Bh1,h2,…,hni is a function, it is a Takagi–Sugeno (T-S)-type fuzzy system [28]. Because the T-S-type fuzzy system is used in this paper, the Mamdani-type fuzzy system will not be introduced too much. After deblurring the abovementioned T-S fuzzy system with product fuzzy reasoning, the output of the T-S fuzzy system becomes(1)Fx=∑i=1Kyiξix,where ξix represents a fuzzy basis function, yi=a1ix1+a2ix2+⋯+anixn; then, formula (1) can be changed to(2)Fx=ξxAxx,where(3)ξx=ξ1x,ξ2x,…,ξKx,Ax=a11a12…a1na21a22…a2n⋮⋮⋮aK1aK2…aKn.The proposed T-S-type fuzzy system provides convenient conditions for the analytical design of the fuzzy control system. Because complex nonlinear systems will be decomposed into a series of linear subsystems, the originally difficulty to parse and describe the system will not be complicated by the followers of each rule in the T-S fuzzy system.Assuming that the output universe is a compact set withinRr, then for any given real continuous functions U and fx on ∀ε>0, there exists a fuzzy system Fx of form (2), such that(4)fx−FxxϵUsup≤ε.For the T-S-type fuzzy system,Fx=f^x,Ax, and the detailed proof process is given in [29, 30].


### 2.1.2. Dynamic Surface Control

Dynamic surface control is developed on the basis of Backstepping Control. Backstepping method has obvious advantages in implementing robust control or adaptive control of uncertain nonlinear systems. However, the backstepping method itself does not have a good solution to the term expansion caused by the derivation of virtual control and the problem caused by the term expansion. This disadvantage is particularly prominent in higher-order systems. It is to overcome the computational complexity of the traditional inverse design. The main advantages of the dynamic surface control method are as follows: (1) it can eliminate the expansion of differential terms and make the controller and parameter design simple; (2) it can reduce the number of neural network and fuzzy system input variables used for modeling; and (3) there is no need to approximate the bounded error in the stability analysis, thereby avoiding the cyclical argument.The following briefly introduces the design of dynamic surface control methods.Take a second-order nonlinear system as an example:(5)x˙1=x2+af1x1,x˙2=u+bf2x¯2,y=x1,where x¯2=x1,x2T represents the system state variable and u represents the system input.Define the first error surfaceS1=x1−yr, where yr represents the system reference input signal, and derive the error surface S1 to obtain the following formula:(6)S˙1=x2+x2d−x2d+af1x1−y˙r,where x2d represents the intermediate virtual control rate.In order to overcome the expansion of the differential term, a new low-pass filter is introduced to obtain a new variablez2:(7)τ2z2˙+z2=x2d,where τ2 represents the time constant of the first-order low-pass filter.By introducing a first-order filter in a nonlinear system, the problem of term expansion caused by the derivation of virtual control in high-order systems is avoided. The design and stability analysis of the dynamic surface control method proves that the literature [31] gives specific steps.


### 2.1.3. Statement of Problem

Consider a class of nonlinearly indeterminate single-input single-output (SISO) systems that are subject to external disturbances:(8)x˙i=gix¯ixi+1+fix¯i+dix,t,x˙n=gnxu+fnx+dnx,t,y=x1,where x¯i=x1,…,xiT, x=x1,…,xnT is the system state vector; u,y∈R is the input and output of the control system; fi⋅ is the unknown smooth nonlinear function; gi⋅ is the unknown smooth virtual control nonlinear gain function; and dix,t1≤i≤n is the uncertain external disturbance, and bounded.Define the tracking errorz, track a given reference signal yrt, and assume that the given reference signal yrt is bounded:(9)z=yt−yrt.Make assumptions about the abovementioned system.Assumption 1.
The unknown smooth virtual control nonlinear gain functiongi⋅ satisfies 0≤amin≤gix¯i,t≤bmax, where amin and bmax represent the upper and lower bounds of the parameter and are some unknown constant.Assumption 2.
It is uncertain that the external disturbancedix,t1≤i≤n is bounded, and dix,t≤di, di is an unknown normal number.Assumption 3.
The weight of the fuzzy systemθi∗ and the approximation error εi are bounded.


## 2.1.1. Fuzzy Control Theory

Fuzzy Logic Control, referred to as Fuzzy Control, is a computer digital control technology based on fuzzy set theory, fuzzy linguistic variables, and fuzzy logic reasoning. Professor L. A. Zadeh of the University of California, United States, published the famous “Fuzzy Sets” paper [27] and proposed the concepts of fuzzy sets and fuzzy algorithms.The fuzzy control system is mainly composed of four parts: fuzzification, rule base, fuzzy reasoning machine, and defuzzification:(1)
Fuzzification: the main function is to select the input amount of the fuzzy controller and convert it into a fuzzy amount that can be recognized by the system. It consists of three steps. First, the input volume is processed to meet the needs of fuzzy control. Second, the input amount is scaled. Third, the fuzzy language value of each input and the corresponding membership function are determined.(2)
Rule base: it consists of a fuzzy “If-Then” rule set. The fuzzy rule base contains many control rules and is a key step in the transition from actual control experience to fuzzy controllers.(3)
Fuzzy reasoning machine: based on combined reasoning or independent reasoning for fuzzy rule bases, a variety of fuzzy reasoning machines are proposed, which mainly implement knowledge-based reasoning decisions.(4)
Defuzzification: the main function is to convert the control amount obtained by reasoning into the control output.There are two types of fuzzy control systems: Mamdani and Takagi–Sugeno (T-S).Consider a fuzzy system approximating a continuous multidimensional functiony=fx, where x=x1,x2,…,xnT∈U is the input vector and y∈V⊂R is the output vector. The fuzzy logic system constitutes a mapping from subspace U to subspace V, as shown in Figure 1. The system usually consists of the following P P>1 fuzzy rules:Figure 1
Structure diagram of the fuzzy logic system.Ifx1=Ah1i,x2=Ah2i,…,xn=Ahni, then yi=Bh1,h2,…,hni, i=1,2,…,P, where Ah1,h2,…,hni and Bh1,h2,…,hni, respectively, represent the input fuzzy set and the output fuzzy set.IfBh1,h2,…,hni is a single fuzzy set, then it is a Mamdani-type fuzzy system. If Bh1,h2,…,hni is a function, it is a Takagi–Sugeno (T-S)-type fuzzy system [28]. Because the T-S-type fuzzy system is used in this paper, the Mamdani-type fuzzy system will not be introduced too much. After deblurring the abovementioned T-S fuzzy system with product fuzzy reasoning, the output of the T-S fuzzy system becomes(1)Fx=∑i=1Kyiξix,where ξix represents a fuzzy basis function, yi=a1ix1+a2ix2+⋯+anixn; then, formula (1) can be changed to(2)Fx=ξxAxx,where(3)ξx=ξ1x,ξ2x,…,ξKx,Ax=a11a12…a1na21a22…a2n⋮⋮⋮aK1aK2…aKn.The proposed T-S-type fuzzy system provides convenient conditions for the analytical design of the fuzzy control system. Because complex nonlinear systems will be decomposed into a series of linear subsystems, the originally difficulty to parse and describe the system will not be complicated by the followers of each rule in the T-S fuzzy system.Assuming that the output universe is a compact set withinRr, then for any given real continuous functions U and fx on ∀ε>0, there exists a fuzzy system Fx of form (2), such that(4)fx−FxxϵUsup≤ε.For the T-S-type fuzzy system,Fx=f^x,Ax, and the detailed proof process is given in [29, 30].


## 2.1.2. Dynamic Surface Control

Dynamic surface control is developed on the basis of Backstepping Control. Backstepping method has obvious advantages in implementing robust control or adaptive control of uncertain nonlinear systems. However, the backstepping method itself does not have a good solution to the term expansion caused by the derivation of virtual control and the problem caused by the term expansion. This disadvantage is particularly prominent in higher-order systems. It is to overcome the computational complexity of the traditional inverse design. The main advantages of the dynamic surface control method are as follows: (1) it can eliminate the expansion of differential terms and make the controller and parameter design simple; (2) it can reduce the number of neural network and fuzzy system input variables used for modeling; and (3) there is no need to approximate the bounded error in the stability analysis, thereby avoiding the cyclical argument.The following briefly introduces the design of dynamic surface control methods.Take a second-order nonlinear system as an example:(5)x˙1=x2+af1x1,x˙2=u+bf2x¯2,y=x1,where x¯2=x1,x2T represents the system state variable and u represents the system input.Define the first error surfaceS1=x1−yr, where yr represents the system reference input signal, and derive the error surface S1 to obtain the following formula:(6)S˙1=x2+x2d−x2d+af1x1−y˙r,where x2d represents the intermediate virtual control rate.In order to overcome the expansion of the differential term, a new low-pass filter is introduced to obtain a new variablez2:(7)τ2z2˙+z2=x2d,where τ2 represents the time constant of the first-order low-pass filter.By introducing a first-order filter in a nonlinear system, the problem of term expansion caused by the derivation of virtual control in high-order systems is avoided. The design and stability analysis of the dynamic surface control method proves that the literature [31] gives specific steps.


## 2.1.3. Statement of Problem

Consider a class of nonlinearly indeterminate single-input single-output (SISO) systems that are subject to external disturbances:(8)x˙i=gix¯ixi+1+fix¯i+dix,t,x˙n=gnxu+fnx+dnx,t,y=x1,where x¯i=x1,…,xiT, x=x1,…,xnT is the system state vector; u,y∈R is the input and output of the control system; fi⋅ is the unknown smooth nonlinear function; gi⋅ is the unknown smooth virtual control nonlinear gain function; and dix,t1≤i≤n is the uncertain external disturbance, and bounded.Define the tracking errorz, track a given reference signal yrt, and assume that the given reference signal yrt is bounded:(9)z=yt−yrt.Make assumptions about the abovementioned system.Assumption 1.
The unknown smooth virtual control nonlinear gain functiongi⋅ satisfies 0≤amin≤gix¯i,t≤bmax, where amin and bmax represent the upper and lower bounds of the parameter and are some unknown constant.Assumption 2.
It is uncertain that the external disturbancedix,t1≤i≤n is bounded, and dix,t≤di, di is an unknown normal number.Assumption 3.
The weight of the fuzzy systemθi∗ and the approximation error εi are bounded.


## 2.2. Design of Robust Adaptive Fuzzy Controller

### 2.2.1. Controller Design

Consider the first subsystem of the system shown in equation (8), defining the first tracking error z1=x1−yr; then,(10)z˙1=g1x¯1,tx2+f1x1,t+d1−y˙r.Using a T-S-type fuzzy system to approximatef1x1,ω, then(11)f1x1,ω=ξ1x1A1x1+ε1=ξ1x1A1z1+ξ1x1A1yr+ε1=qθ1ξ1ω1+ξ1x1A1yr+ε1,where ε1 represents the approximation error, qθ1 is an unknown constant, and qθ1=A1, A1=qθ1A1m, and ω1=A1z1; then,(12)z˙1=g1x¯1,tx2+qθ1ξ1xω1+l1−y˙r,where l1=ξ1x1A1yr+ε1+d1, l1≤ξ1x1A1yr+ε1+d1≤aminθ1φ1x1, θ1=amin−1maxA1yr,ε1+d1, and φ1x1=1+ξ1.Therefore, the intermediate stabilization functionψ2 and the parameter adaptation rate are(13)ψ2=−k1z1+y˙r−λ^1ρ1x1z1,(14)λ^˙1=μ1ρ1x1z12−σ1λ^1−λ10,(15)ρ1x1=14ι12ξ1ξ1T+14κ12φ12,where k1, μ1, ι1, and κ1 are normal numbers; λ1^ is an estimated value of λ1; λ1=amin−1maxqθ12,θ12; and λ10 is the initial value.At this point, the estimated values2 of ψ2 is obtained:(16)s2=ψ2−τ2s˙2,where τ is the time constant.Considering thei−th2≤i≤n−1 subsystems of the system shown in equation (8) and defining the i−th tracking error: zi=xi−si; then,(17)z˙i=gix¯i,txi+1+fix¯i,t+di−s˙i.Using the T-S fuzzy system to approximate the uncertain functionfix¯i,t, then(18)fix¯i,t=ξix¯iAix¯iT+εi=ξix¯iAiz1+yrz2+s2⋯zi+si+εi=qθiξiωi+di′,where di'=ξiAi1yr+ξi∑j=2iAijsj+εi, qθi=Ai1, Ai1=qθiAim, and ωi=Aimz¯i; then,(19)z˙i=gix¯i,txi+1+qθiξixωi+li−s˙i,where li=di+di′, li≤di+ξiAi1yr+ξi∑j=2iAij+εi≤aminθiφixi, θi=amin−1maxAi1yr,∑j=2iAij,εi+di, and φixi=1+ξi.Therefore, the intermediate stabilization functionψi+1 and the parameter adaptation rate are(20)ψi+1=−kizi+s˙i−λ^iρix¯izi,(21)λ^˙i=μiρix¯izi2−σiλ^i−λi0,(22)ρix¯i=14ιi2ξiξiT+14κi2φi2,where ki, μi, ιi, and κi are normal numbers; λ^i is an estimated value of λi; λi=amin−1maxqθi2,θi2; and λi0 is the initial value.At this point, the estimated valuesi+1 of ψi+1 is obtained:(23)si+1=ψi+1−τi+1s˙i+1.Define then−th tracking error: zn=xn−sn; similarly, there is(24)z˙n=gnx,tu+fnx¯n,t+dn−s˙n,fix¯n,t=ξnx¯nAnx¯nT+εn=ξnx¯nAnz1+yrz2+s2⋯zn+sn+εn=qθnξnωn+dn′,where dn′=ξnAn1yr+ξn∑j=2nAnjsj+εn, qθn=An1, An1=qθnAnm, and ωn=Anmz¯n; then,(25)z˙n=gnx,tu+qθnξnxωn+ln−s˙n,where ln=dn+dn′, ln≤dn+ξnAn1yr+ξn∑j=2nAnj+εn≤aminθnφnx, θn=amin−1maxAn1yr,∑j=2nAin,εn+dn, and φnx=1+ξn.Therefore, select the control rate and parameter adaptation rate:(26)u=−knzn+s˙n−λ^nρnxzn,(27)λ^˙n=μnρnxzn2−σnλ^n−λn0,(28)ρnx=14ιn2ξnξnT+14tn2φn2,where kn, μn, ιn, and κn are normal numbers; λ^n is an estimated value of λn; λn=amin−1maxqθn2,θn2; and λn0 is the initial value.


### 2.2.2. Lyapunov Stability Analysis

Define a new tracking error:(29)yi+1=si+1−ψi+1,i=1,2,…,n−1.From equation (23), s˙i=−si+ψi/τi=−yi/τi; then,(30)y˙2=s˙2−ψ˙2=−y2τ2+−∂ψ2∂z1z˙1−∂ψ2∂x1x˙1−∂ψ2∂θ1^θn^˙−∂ψ2∂λ1^λ1^˙+yr¨=−y2τ2+Ο2z1,z2,y2,θ1^,λ1^,yr,yr˙,yr¨,where Ο2⋅ is the remainder. Similarly, there is(31)y˙i+1=s˙i+1−ψ˙i+1=−yi+1τi+1+Οi+1z1,…,zi+1,y2,…yi,θ^1,…θ^i,λ^1,…,λ^i,yr,y˙r,y¨r.According to thei−th tracking error equation and equation (29), xi+1=zi+1+si+1 and si+1=yi+1+ψi+1, and the tracking error can be expressed as follows:(32)z˙1=g1x¯1,tz1+g1x¯1,ty1+g1x¯1,tψ1+qθ1ξ1xω1+l1−y˙r,z˙i=gix¯i,tzi+1+gix¯i,tyi+1+gix¯i,tψi+1+qθiξixωi+li−s˙i,z˙n=gnx,tu+qθnξnxωn+ln−s˙n.Then, bring equations (13)∼(15), (19)∼(21), and (26)∼(28) into the abovementioned equation and obtain the following expression:(33)z˙1=g1z1+g1y1−g1k1z1+g1y˙r−g1λ^114ι12ξ1ξ1T+14κ12φ12z1+qθ1ξ1xω1+l1−y˙r,z˙i=gizi+1+giyi+1−gikizi+gis˙i−giλ^i14ιi2ξiξiT+14κi2φi2zi+qθiξixωi+li−s˙i,z˙n=−gnknzn+gns˙n−gnλ^n14ιn2ξnξnT+14tn2φn2zn+qθnξnxωn+ln−s˙n.Define the closed-loop system Lyapunov function as(34)V=12∑i=1nzi2+λ^iTaminμi−1λ˜i+12∑i=1n−1yi+12,where λ^i=λi−λ^i.In the following, it will be proved that the given arbitraryζ>0, the existence of li, and ιi, κi, σi, and μi make the solution of the closed-loop system consistent and bounded.The derivative of the closed-loop system Lyapunov function with respect to time is(35)V˙=∑i=1nziz˙i−λ^iTaminμi−1λ^˙i+∑i=1n−1yi+1y˙i+1.Deduced analysis can be drawn(36)V˙≤∑i=1n−1gizi+1zi+giyi+1zi−aminkizi2−aminλ^i14ιi2ξiξiTzi2−giλ^i14κi2φi2zi2+qθiξixi¯ωizi+lizi−λ^iTaminμi−1λ^˙i+g1y˙rz1−y˙rz1+∑i=2ngis˙izi−s˙izi−aminknzn2−gnλ^n14tn2ξnξnTzn2−aminλ^n14κn2φn2zn2+qθnξnxωnzn+lnzn−λ^nTaminμn−1λ^˙n+∑i=1n−1−yi+12τi+1+yi+1Οi+1.However,(37)qθiξiωizi=qθiξiωizi−ιi2ωiTωi+ιi2ωiTωi=−ιi2ωi−qθi22ιi2ξizi2+qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤aminλ^i14ιi2ξiξiTzi2+aminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi,lizi≤aminθ^iφix¯izi+aminθ˜iφix¯izi≤giθ^iφix¯izi+aminθ˜iφix¯izi.At the same time, it can be known froms˙i=−si+ψi/τi=−yi/τi:(38)g1y˙rz1−y˙rz1≤1+bmaxτizi2+1+bmaxy˙r2≤1+bmax4zi2+1+bmaxΟ02,gis˙izi−s˙izi≤gis˙izi+s˙izi≤gi+1yiτizi≤1+bmaxτizi2+1+bmax4τiyi2.Then, equation (36) can be changed to(39)V˙≤∑i=1n−1gizi+1zi+giyi+1zi+∑i=1naminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi+giθ^iφix¯izi+aminθ˜iφix¯izi−giλ^i14κi2φi2zi2gn−gnλ^n14tn2ξnξnTzn2+1+bmax4−amink1z12+1+bmaxΟ02+∑i=2n−−1+bmaxτi+aminkizi2+∑i=2n1+bmax4τiyi2+∑i=1n−1−yi+12τi+1+yi+1Οi+1.Since,giθ^iφix¯izi+aminθ˜iφix¯izi−giλi^1/4κi2φi2zi2≤θiφixi¯zi−giλi^1/4κi2φi2zi2≤θi2/4tn2φi2x¯zi2+κi2−giλ^i1/4κi2φi2zi2≤giκi≤bmaxκi, gizi+1zi≤zi2+bmax/4zi+12 and giyi+1zi≤zi2+bmax/4yi+12. Bring equations (21) and (22) into the abovementioned equation; then,(40)V˙≤1+bmax4−amink1−2z12+∑i=2n−−1+bmaxτi+aminki−2zi2+1+bmax4−aminknzn2−∑i=1nσi2λmaxaminμi−1λ˜iTμi−1λ˜i+∑i=1n−1bmax4yi+12−3−bmax4τi+1+yi+1Οi+1+∑i=1nδi′+ι2ω2,where δi′=1+bmaxΟ02+bmaxκi+σi/2λi∗−λi02, ι=ι12+ι22+⋯+ιn21/2, and ω=ω1,ω2,…,ωnT.Sinceωi=Aimz¯iT, Aim≤1; however,(41)ω=ω1ω2⋯ωn=A1m0⋯0A2m1A2m2⋯0⋮⋮⋮Anm1Anm2⋯Anmnz1z2⋯zn=Az.This can be obtained as(42)ω≤Az≤z.First inspection the properties of equation (8):LetΟi+1 be the maximum value of Μi+1 on the set Ω, and let 1/τi+1=3−bmax/4τi+1−1bmax/4+Μi+12/2ψ+ψ0, and since yi+1Οi+1≤yi+12Μi+12/2ψ+ψ/2, then(43)bmax4yi+12−3−bmax4τi+1+yi+1Οi+1≤−bmax4+Μi+122ψ+ψ0yi+12+bmax4yi+12+yi+12Μi+12Οi+122ψΜi+12+ψ2=−ψ0yi+10−1−Οi+12Μi+12yi+12Μi+122ψ+ψ2.Let beσi/2λmaxaminμi−1=ψ0, k1=bmax−12+1+bmax/4+ψ0, ki=bmax−12+1+bmax/τi+bmax/4+ψ0, kn=bmax−11+bmax/τi+ψ0, and ψ0 is a normal number. Then, equation (36) becomes(44)V˙≤−ψ0∑i=1nzi2−ψ0∑i=1nλ˜iTaminμi−1λ˜i−ψ0∑i=1n−1yi+12+∑i=1nιi2ωiTωi+χ≤−2ψ0V+χ,where z=z1,z2,…,znT, χ=∑i=1nδi′+∑i=1n−1ψ/2, then(45)Vt≤χ2ψ0+Vt0e−2ψ0t−t0,∀t≥t0≥0.Obviously, for anyζ1>0, exist Τ>0 so that for all t≥t0+Τ, z1≤ζ1 is established. Then, all solutions of the closed-loop system are consistent and bounded, that is, by adjusting the controller parameters so that the tracking error z1=yt−yrt is as small as possible to achieve the desired tracking accuracy.


## 2.2.1. Controller Design

Consider the first subsystem of the system shown in equation (8), defining the first tracking error z1=x1−yr; then,(10)z˙1=g1x¯1,tx2+f1x1,t+d1−y˙r.Using a T-S-type fuzzy system to approximatef1x1,ω, then(11)f1x1,ω=ξ1x1A1x1+ε1=ξ1x1A1z1+ξ1x1A1yr+ε1=qθ1ξ1ω1+ξ1x1A1yr+ε1,where ε1 represents the approximation error, qθ1 is an unknown constant, and qθ1=A1, A1=qθ1A1m, and ω1=A1z1; then,(12)z˙1=g1x¯1,tx2+qθ1ξ1xω1+l1−y˙r,where l1=ξ1x1A1yr+ε1+d1, l1≤ξ1x1A1yr+ε1+d1≤aminθ1φ1x1, θ1=amin−1maxA1yr,ε1+d1, and φ1x1=1+ξ1.Therefore, the intermediate stabilization functionψ2 and the parameter adaptation rate are(13)ψ2=−k1z1+y˙r−λ^1ρ1x1z1,(14)λ^˙1=μ1ρ1x1z12−σ1λ^1−λ10,(15)ρ1x1=14ι12ξ1ξ1T+14κ12φ12,where k1, μ1, ι1, and κ1 are normal numbers; λ1^ is an estimated value of λ1; λ1=amin−1maxqθ12,θ12; and λ10 is the initial value.At this point, the estimated values2 of ψ2 is obtained:(16)s2=ψ2−τ2s˙2,where τ is the time constant.Considering thei−th2≤i≤n−1 subsystems of the system shown in equation (8) and defining the i−th tracking error: zi=xi−si; then,(17)z˙i=gix¯i,txi+1+fix¯i,t+di−s˙i.Using the T-S fuzzy system to approximate the uncertain functionfix¯i,t, then(18)fix¯i,t=ξix¯iAix¯iT+εi=ξix¯iAiz1+yrz2+s2⋯zi+si+εi=qθiξiωi+di′,where di'=ξiAi1yr+ξi∑j=2iAijsj+εi, qθi=Ai1, Ai1=qθiAim, and ωi=Aimz¯i; then,(19)z˙i=gix¯i,txi+1+qθiξixωi+li−s˙i,where li=di+di′, li≤di+ξiAi1yr+ξi∑j=2iAij+εi≤aminθiφixi, θi=amin−1maxAi1yr,∑j=2iAij,εi+di, and φixi=1+ξi.Therefore, the intermediate stabilization functionψi+1 and the parameter adaptation rate are(20)ψi+1=−kizi+s˙i−λ^iρix¯izi,(21)λ^˙i=μiρix¯izi2−σiλ^i−λi0,(22)ρix¯i=14ιi2ξiξiT+14κi2φi2,where ki, μi, ιi, and κi are normal numbers; λ^i is an estimated value of λi; λi=amin−1maxqθi2,θi2; and λi0 is the initial value.At this point, the estimated valuesi+1 of ψi+1 is obtained:(23)si+1=ψi+1−τi+1s˙i+1.Define then−th tracking error: zn=xn−sn; similarly, there is(24)z˙n=gnx,tu+fnx¯n,t+dn−s˙n,fix¯n,t=ξnx¯nAnx¯nT+εn=ξnx¯nAnz1+yrz2+s2⋯zn+sn+εn=qθnξnωn+dn′,where dn′=ξnAn1yr+ξn∑j=2nAnjsj+εn, qθn=An1, An1=qθnAnm, and ωn=Anmz¯n; then,(25)z˙n=gnx,tu+qθnξnxωn+ln−s˙n,where ln=dn+dn′, ln≤dn+ξnAn1yr+ξn∑j=2nAnj+εn≤aminθnφnx, θn=amin−1maxAn1yr,∑j=2nAin,εn+dn, and φnx=1+ξn.Therefore, select the control rate and parameter adaptation rate:(26)u=−knzn+s˙n−λ^nρnxzn,(27)λ^˙n=μnρnxzn2−σnλ^n−λn0,(28)ρnx=14ιn2ξnξnT+14tn2φn2,where kn, μn, ιn, and κn are normal numbers; λ^n is an estimated value of λn; λn=amin−1maxqθn2,θn2; and λn0 is the initial value.


## 2.2.2. Lyapunov Stability Analysis

Define a new tracking error:(29)yi+1=si+1−ψi+1,i=1,2,…,n−1.From equation (23), s˙i=−si+ψi/τi=−yi/τi; then,(30)y˙2=s˙2−ψ˙2=−y2τ2+−∂ψ2∂z1z˙1−∂ψ2∂x1x˙1−∂ψ2∂θ1^θn^˙−∂ψ2∂λ1^λ1^˙+yr¨=−y2τ2+Ο2z1,z2,y2,θ1^,λ1^,yr,yr˙,yr¨,where Ο2⋅ is the remainder. Similarly, there is(31)y˙i+1=s˙i+1−ψ˙i+1=−yi+1τi+1+Οi+1z1,…,zi+1,y2,…yi,θ^1,…θ^i,λ^1,…,λ^i,yr,y˙r,y¨r.According to thei−th tracking error equation and equation (29), xi+1=zi+1+si+1 and si+1=yi+1+ψi+1, and the tracking error can be expressed as follows:(32)z˙1=g1x¯1,tz1+g1x¯1,ty1+g1x¯1,tψ1+qθ1ξ1xω1+l1−y˙r,z˙i=gix¯i,tzi+1+gix¯i,tyi+1+gix¯i,tψi+1+qθiξixωi+li−s˙i,z˙n=gnx,tu+qθnξnxωn+ln−s˙n.Then, bring equations (13)∼(15), (19)∼(21), and (26)∼(28) into the abovementioned equation and obtain the following expression:(33)z˙1=g1z1+g1y1−g1k1z1+g1y˙r−g1λ^114ι12ξ1ξ1T+14κ12φ12z1+qθ1ξ1xω1+l1−y˙r,z˙i=gizi+1+giyi+1−gikizi+gis˙i−giλ^i14ιi2ξiξiT+14κi2φi2zi+qθiξixωi+li−s˙i,z˙n=−gnknzn+gns˙n−gnλ^n14ιn2ξnξnT+14tn2φn2zn+qθnξnxωn+ln−s˙n.Define the closed-loop system Lyapunov function as(34)V=12∑i=1nzi2+λ^iTaminμi−1λ˜i+12∑i=1n−1yi+12,where λ^i=λi−λ^i.In the following, it will be proved that the given arbitraryζ>0, the existence of li, and ιi, κi, σi, and μi make the solution of the closed-loop system consistent and bounded.The derivative of the closed-loop system Lyapunov function with respect to time is(35)V˙=∑i=1nziz˙i−λ^iTaminμi−1λ^˙i+∑i=1n−1yi+1y˙i+1.Deduced analysis can be drawn(36)V˙≤∑i=1n−1gizi+1zi+giyi+1zi−aminkizi2−aminλ^i14ιi2ξiξiTzi2−giλ^i14κi2φi2zi2+qθiξixi¯ωizi+lizi−λ^iTaminμi−1λ^˙i+g1y˙rz1−y˙rz1+∑i=2ngis˙izi−s˙izi−aminknzn2−gnλ^n14tn2ξnξnTzn2−aminλ^n14κn2φn2zn2+qθnξnxωnzn+lnzn−λ^nTaminμn−1λ^˙n+∑i=1n−1−yi+12τi+1+yi+1Οi+1.However,(37)qθiξiωizi=qθiξiωizi−ιi2ωiTωi+ιi2ωiTωi=−ιi2ωi−qθi22ιi2ξizi2+qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤qθi24ιi2ξiξiTzi2+ιi2ωiTωi≤aminλ^i14ιi2ξiξiTzi2+aminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi,lizi≤aminθ^iφix¯izi+aminθ˜iφix¯izi≤giθ^iφix¯izi+aminθ˜iφix¯izi.At the same time, it can be known froms˙i=−si+ψi/τi=−yi/τi:(38)g1y˙rz1−y˙rz1≤1+bmaxτizi2+1+bmaxy˙r2≤1+bmax4zi2+1+bmaxΟ02,gis˙izi−s˙izi≤gis˙izi+s˙izi≤gi+1yiτizi≤1+bmaxτizi2+1+bmax4τiyi2.Then, equation (36) can be changed to(39)V˙≤∑i=1n−1gizi+1zi+giyi+1zi+∑i=1naminλ˜i14ιi2ξiξiTzi2+ιi2ωiTωi+giθ^iφix¯izi+aminθ˜iφix¯izi−giλ^i14κi2φi2zi2gn−gnλ^n14tn2ξnξnTzn2+1+bmax4−amink1z12+1+bmaxΟ02+∑i=2n−−1+bmaxτi+aminkizi2+∑i=2n1+bmax4τiyi2+∑i=1n−1−yi+12τi+1+yi+1Οi+1.Since,giθ^iφix¯izi+aminθ˜iφix¯izi−giλi^1/4κi2φi2zi2≤θiφixi¯zi−giλi^1/4κi2φi2zi2≤θi2/4tn2φi2x¯zi2+κi2−giλ^i1/4κi2φi2zi2≤giκi≤bmaxκi, gizi+1zi≤zi2+bmax/4zi+12 and giyi+1zi≤zi2+bmax/4yi+12. Bring equations (21) and (22) into the abovementioned equation; then,(40)V˙≤1+bmax4−amink1−2z12+∑i=2n−−1+bmaxτi+aminki−2zi2+1+bmax4−aminknzn2−∑i=1nσi2λmaxaminμi−1λ˜iTμi−1λ˜i+∑i=1n−1bmax4yi+12−3−bmax4τi+1+yi+1Οi+1+∑i=1nδi′+ι2ω2,where δi′=1+bmaxΟ02+bmaxκi+σi/2λi∗−λi02, ι=ι12+ι22+⋯+ιn21/2, and ω=ω1,ω2,…,ωnT.Sinceωi=Aimz¯iT, Aim≤1; however,(41)ω=ω1ω2⋯ωn=A1m0⋯0A2m1A2m2⋯0⋮⋮⋮Anm1Anm2⋯Anmnz1z2⋯zn=Az.This can be obtained as(42)ω≤Az≤z.First inspection the properties of equation (8):LetΟi+1 be the maximum value of Μi+1 on the set Ω, and let 1/τi+1=3−bmax/4τi+1−1bmax/4+Μi+12/2ψ+ψ0, and since yi+1Οi+1≤yi+12Μi+12/2ψ+ψ/2, then(43)bmax4yi+12−3−bmax4τi+1+yi+1Οi+1≤−bmax4+Μi+122ψ+ψ0yi+12+bmax4yi+12+yi+12Μi+12Οi+122ψΜi+12+ψ2=−ψ0yi+10−1−Οi+12Μi+12yi+12Μi+122ψ+ψ2.Let beσi/2λmaxaminμi−1=ψ0, k1=bmax−12+1+bmax/4+ψ0, ki=bmax−12+1+bmax/τi+bmax/4+ψ0, kn=bmax−11+bmax/τi+ψ0, and ψ0 is a normal number. Then, equation (36) becomes(44)V˙≤−ψ0∑i=1nzi2−ψ0∑i=1nλ˜iTaminμi−1λ˜i−ψ0∑i=1n−1yi+12+∑i=1nιi2ωiTωi+χ≤−2ψ0V+χ,where z=z1,z2,…,znT, χ=∑i=1nδi′+∑i=1n−1ψ/2, then(45)Vt≤χ2ψ0+Vt0e−2ψ0t−t0,∀t≥t0≥0.Obviously, for anyζ1>0, exist Τ>0 so that for all t≥t0+Τ, z1≤ζ1 is established. Then, all solutions of the closed-loop system are consistent and bounded, that is, by adjusting the controller parameters so that the tracking error z1=yt−yrt is as small as possible to achieve the desired tracking accuracy.


## 3. Simulation Studies of Three-Dimensional Elliptical Vibration Cutting System

### 3.1. System Transformation

Make the following assumptions for system (8).Assumption 4.
System (8) has a strong relative r.
Under the condition of hypothesis 3.1, there is a differential homeomorphic mapu,v=Tx so that equation (8) becomes(46)yr=a1t+b1t+g1u,vu+f1u,v+du,v,v˙=qu,v,where Lgrhx=f1u,v, LgLfr−1hx=g1u,v, v˙=qu,v denotes zero dynamics of the system, a1t and b1t are known functions that are continuously bounded, and a1t=b1t=0 is assumed.Assumption 5.
System (1) is of the minimum phase and pairs and the variable u satisfies the Lipschitz condition:(47)qu,v−q0,v≤Lu,where L is a constant.
Under the assumption of5, if u is bounded, then v must be bounded.
Suppose the system has no zero dynamics, i.e.,v=0.
Letfx=f1u,0, gx=g1u,0, and u=T−1x; then, equation (8) can be changed to(48)yr=gxu+fx+dx,t.
For the three-dimensional elliptical vibration assisted cutting system of the previous research [21], considering this type of single-input single-output (SISO) nonlinear Wiener system, it can be expressed as follows:(49)y3=gx,tut+fx,t+dt,where gx,t and fx,t are nonlinear functions, gx,t>0, and dt is external interference.
Based on the LabVIEW simulation analysis, five fuzzy sets are defined for each variable in the system model to be fuzzy:NL,NM,ZE,PM,PL. These fuzzy sets are described by the following membership functions:(50)ΓNL=exp−x+12,ΓNM=exp−x+0.52,ΓZE=exp−x2,ΓPM=exp−x−0.52,ΓPL=exp−x−12.
According to the control algorithm proposed in Section2, the control rate, intermediate stabilization function, and parameter adaptation rate are selected as follows:(51)ψ2=−5z1+yd,ψ3=−0.05z2+s2−−λ^2ρ2z2,u=−0.02z3+s3−−λ^3ρ3z3,λ^˙2=0.2ρ2z22−0.5λ^2,λ^˙3=0.1ρ3z32−1.5λ^3,ρix¯i=14ιi2ξiξiT+14κi2φi2,i=2,3.


### 3.2. Description of Three-Dimensional Elliptical Vibration Cutting System

The three-dimensional elliptical vibration assisted cutting device (shown in Figure2) is driven mainly by two parallel, vertically placed piezoelectric stacks in a nonresonant manner. The three piezoelectric stacks are, respectively, distributed on the upper flexible hinge and the lower flexible hinge, and each of the piezoelectric stacks is placed in parallel with a displacement sensor. The 3D EVC system can be viewed as three single-input single-output (SISO) systems using a nonlinear Wiener model to describe the 3D EVC system. The system identification results are detailed in [32].Figure 2
Structure of the 3D EVC system.


### 3.3. Simulation Studies

In order to further illustrate the effectiveness of the proposed method, the three-dimensional elliptical vibration cutting device developed by our group was used to design the nonlinear Wiener control system as an example. The robust adaptive fuzzy control design and simulation research were carried out. A type of single-input and single-output nonlinear control system that can express its system equations is as follows:(52)y3=f1x,t+g1x,tut+d1t,x=y,where f1x,t and g1x,t represent nonlinear functions, g1x,t>0; and d1t are unknown external disturbances.The robust adaptive fuzzy control simulation is carried out by taking the system identification result of theX+ direction subsystem of the designed three-dimensional elliptical vibration cutting device as an example. In the simulation, the same membership function as in equation (50) is taken, and use MATLAB software to generate the membership function graph shown in Figure 3. The external interference is taken as d1t=sint; the initial values of the system variables x1, x2, and x3 are set to 0.2,2π,0; and the reference tracking signal is yd=sinπt. The simulation results of position tracking and speed tracking are shown in Figure 4.Figure 3
Membership function graph.
(a)
(b)
(c)Figure 4
Simulation results: (a) the duration curve of the position tracking signal; (b) the duration curve of the speed tracking signal.
(a)
(b)The tracking error curve of robust adaptive fuzzy control for a three-dimensional elliptical vibratory cutting system is shown in Figure4. It can be seen from Figures 4 and 5 that the nonlinear model identified by the three-dimensional elliptical vibration cutting device can effectively suppress the jitter problem by using the robust adaptive fuzzy controller. The control object has a little jitter at the beginning, but it can be quickly stabilized and smoothed to move along the ideal displacement and velocity signals. Figures 6(a) and 6(b) are simulation results of the estimated parameters λ2^˙ and λ3^˙, respectively. The parameter estimation simulation curve shows that the proposed controller has strong robust adaptability and can freely adjust its own parameters according to the needs of the controlled system.Figure 5
The tracking error curve of robust adaptive fuzzy control for a three-dimensional elliptical vibratory cutting system.Figure 6
Simulation results: (a) parameter estimationλ^˙2; (b) parameter estimation λ^˙3.
(a)
(b)The robust adaptive fuzzy control of theX+ directional subsystem of the abovementioned three-dimensional elliptical vibration cutting system can make the system tend to be stable in a short time, is not affected by external disturbances and has strong robust adaptability.


## 3.1. System Transformation

Make the following assumptions for system (8).Assumption 4.
System (8) has a strong relative r.
Under the condition of hypothesis 3.1, there is a differential homeomorphic mapu,v=Tx so that equation (8) becomes(46)yr=a1t+b1t+g1u,vu+f1u,v+du,v,v˙=qu,v,where Lgrhx=f1u,v, LgLfr−1hx=g1u,v, v˙=qu,v denotes zero dynamics of the system, a1t and b1t are known functions that are continuously bounded, and a1t=b1t=0 is assumed.Assumption 5.
System (1) is of the minimum phase and pairs and the variable u satisfies the Lipschitz condition:(47)qu,v−q0,v≤Lu,where L is a constant.
Under the assumption of5, if u is bounded, then v must be bounded.
Suppose the system has no zero dynamics, i.e.,v=0.
Letfx=f1u,0, gx=g1u,0, and u=T−1x; then, equation (8) can be changed to(48)yr=gxu+fx+dx,t.
For the three-dimensional elliptical vibration assisted cutting system of the previous research [21], considering this type of single-input single-output (SISO) nonlinear Wiener system, it can be expressed as follows:(49)y3=gx,tut+fx,t+dt,where gx,t and fx,t are nonlinear functions, gx,t>0, and dt is external interference.
Based on the LabVIEW simulation analysis, five fuzzy sets are defined for each variable in the system model to be fuzzy:NL,NM,ZE,PM,PL. These fuzzy sets are described by the following membership functions:(50)ΓNL=exp−x+12,ΓNM=exp−x+0.52,ΓZE=exp−x2,ΓPM=exp−x−0.52,ΓPL=exp−x−12.
According to the control algorithm proposed in Section2, the control rate, intermediate stabilization function, and parameter adaptation rate are selected as follows:(51)ψ2=−5z1+yd,ψ3=−0.05z2+s2−−λ^2ρ2z2,u=−0.02z3+s3−−λ^3ρ3z3,λ^˙2=0.2ρ2z22−0.5λ^2,λ^˙3=0.1ρ3z32−1.5λ^3,ρix¯i=14ιi2ξiξiT+14κi2φi2,i=2,3.


## 3.2. Description of Three-Dimensional Elliptical Vibration Cutting System

The three-dimensional elliptical vibration assisted cutting device (shown in Figure2) is driven mainly by two parallel, vertically placed piezoelectric stacks in a nonresonant manner. The three piezoelectric stacks are, respectively, distributed on the upper flexible hinge and the lower flexible hinge, and each of the piezoelectric stacks is placed in parallel with a displacement sensor. The 3D EVC system can be viewed as three single-input single-output (SISO) systems using a nonlinear Wiener model to describe the 3D EVC system. The system identification results are detailed in [32].Figure 2
Structure of the 3D EVC system.


## 3.3. Simulation Studies

In order to further illustrate the effectiveness of the proposed method, the three-dimensional elliptical vibration cutting device developed by our group was used to design the nonlinear Wiener control system as an example. The robust adaptive fuzzy control design and simulation research were carried out. A type of single-input and single-output nonlinear control system that can express its system equations is as follows:(52)y3=f1x,t+g1x,tut+d1t,x=y,where f1x,t and g1x,t represent nonlinear functions, g1x,t>0; and d1t are unknown external disturbances.The robust adaptive fuzzy control simulation is carried out by taking the system identification result of theX+ direction subsystem of the designed three-dimensional elliptical vibration cutting device as an example. In the simulation, the same membership function as in equation (50) is taken, and use MATLAB software to generate the membership function graph shown in Figure 3. The external interference is taken as d1t=sint; the initial values of the system variables x1, x2, and x3 are set to 0.2,2π,0; and the reference tracking signal is yd=sinπt. The simulation results of position tracking and speed tracking are shown in Figure 4.Figure 3
Membership function graph.
(a)
(b)
(c)Figure 4
Simulation results: (a) the duration curve of the position tracking signal; (b) the duration curve of the speed tracking signal.
(a)
(b)The tracking error curve of robust adaptive fuzzy control for a three-dimensional elliptical vibratory cutting system is shown in Figure4. It can be seen from Figures 4 and 5 that the nonlinear model identified by the three-dimensional elliptical vibration cutting device can effectively suppress the jitter problem by using the robust adaptive fuzzy controller. The control object has a little jitter at the beginning, but it can be quickly stabilized and smoothed to move along the ideal displacement and velocity signals. Figures 6(a) and 6(b) are simulation results of the estimated parameters λ2^˙ and λ3^˙, respectively. The parameter estimation simulation curve shows that the proposed controller has strong robust adaptability and can freely adjust its own parameters according to the needs of the controlled system.Figure 5
The tracking error curve of robust adaptive fuzzy control for a three-dimensional elliptical vibratory cutting system.Figure 6
Simulation results: (a) parameter estimationλ^˙2; (b) parameter estimation λ^˙3.
(a)
(b)The robust adaptive fuzzy control of theX+ directional subsystem of the abovementioned three-dimensional elliptical vibration cutting system can make the system tend to be stable in a short time, is not affected by external disturbances and has strong robust adaptability.


## 4. Conclusions

In this paper, the design of a robust adaptive fuzzy controller for a class of single-input single-output (SISO) uncertain nonlinear systems is proposed, and the corresponding design steps are given. The conclusions reached are as follows:(1)
The most notable features of the proposed control algorithm are as follows: in the case of unknown structural uncertainty, the resulting closed-loop system is ultimately bounded. Only one function needs to have a T-S fuzzy logic system approximation, and each subsystem needs only one parameter to be adjusted, which overcomes the problem of “dimensionality disaster.” Thereby, the calculation amount of the control algorithm is greatly reduced and the problem of “calculation expansion” is solved.(2)
The simulation research on the nonlinear control system of the three-dimensional elliptical vibration cutting device developed by our group is carried out. It is verified that the robust adaptive fuzzy control can make the system tend to be stable in a short time, is not affected by external disturbances, and has strong robust adaptability.Future research will focus on extending the parameter optimization problem of the analytical control system to prove that the proposed solution has more effective engineering application value.



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*Source: 6178678-2020-03-27.xml*