# Adaptive Fuzzy Robust Control for a Class of Nonlinear Systems via Small Gain Theorem

Authors: Xingjian Wang; Shaoping Wang
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

Practical nonlinear systems can usually be represented by partly linearizable models with unknown nonlinearities and external disturbances. Based on this consideration, we propose a novel adaptive fuzzy robust control (AFRC) algorithm for such systems. The AFRC effectively combines techniques of adaptive control and fuzzy control, and it improves the performance by retaining the advantages of both methods. The linearizable part will be linearly parameterized with unknown but constant parameters, and the discontinuous-projection-based adaptive control law is used to compensate these parts. The Takagi-Sugeno fuzzy logic systems are used to approximate unknown nonlinearities. Robust control law ensures the robustness of closed-loop control system. A systematic design procedure of the AFRC algorithm by combining the backstepping technique and small-gain approach is presented. Then the closed-loop stability is studied by using small gain theorem, and the result indicates that the closed-loop system is semiglobally uniformly ultimately bounded.


---

## Body

## 1. Introduction

High performance control algorithm is often required in modern industrial control systems, such as robot manipulators, precision manufacturing equipment, and automatic inspection machines. During the past two decades, mathematically rigorous design methods for adaptive control of nonlinear systems have developed rapidly, and many remarkable results have been obtained [1, 2]. As a breakthrough in the research field of nonlinear control systems, a recursive design technique named adaptive backstepping approach was proposed to obtain global stability and asymptotic tracking for nonlinear systems [3, 4]. Many researchers have made a lot of efforts in this field and achieved many significant results. For example, the adaptive robust control (ARC) approach for uncertain nonlinear systems proposed by Yao and Tomizuka belongs to this kind of work [5–7].However, the common feature of the adaptive control algorithms discussed before is to deal with the nonlinearities of the control systems in the linearly parameterized forms; that is, these nonlinearities are assumed to be in the form of known nonlinear functions with unknown but constant parameters [8]. But in fact, in practical industrial control environment, uncertainties or nonlinearities of one system cannot always be totally repeatable, structured, or modeled. In order to deal with this kind of nonlinearities, many researchers focused on approximator-based approaches by using backstepping technique and Lyapunov stability theory [9–13]. Some advanced approximators such as fuzzy logic and neural network were used in control methods for nonlinear systems with unstructured uncertainties or nonlinearities [14–16]. Recently, Tong and his colleagues made many achievement on adaptive fuzzy controller design for large-scale nonlinear systems [17–20]. In these methods, all system dynamics are estimated by approximator, but the control performance is not necessarily better.Actually, as the development of modern control theorem and system identification technique, part of the dynamics of control system can be modeled quite accurately, while some nonlinearities are still unknown or not replicable. In order to cope with the control problem of such class of nonlinear systems, we present a novel adaptive fuzzy robust control (AFRC) algorithm. Adaptive control law is used to compensate the linearizable part, and discontinuous-projection-based parameter adaptive law is employed to estimate unknown system parameters. The Takagi-Sugeno fuzzy logic systems are used to approximate unknown nonlinearities. Robust control law ensures the robustness of closed-loop control system. A mathematically design procedure of AFRC is developed by combining the adaptive backstepping technique and small-gain theorem [21, 22]. This proposed AFRC can combine technical characters of adaptive control and fuzzy control and improve the control performance for the class of nonlinear systems we considered.The reminder of this paper is organized as follows. In Section2, we will review some definitions about small gain theorem and Takagi-Sugeno fuzzy logic system, and the control problem is also formulated in this section. The systematic procedure for adaptive fuzzy robust control law is presented in Section 3. In Section 4, the closed-loop stability is studied by using small gain theorem. In Section 5, an illustrative example is presented to demonstrate the effectiveness of the proposed controller. Conclusions are given in Section 6.


## 2. Problem Formulation

### 2.1. Preliminaries

In this section, some definitions about input-to-state stable (ISS) and small gain theorem [21–23] will be introduced, which have been widely used in nonlinear control problems for stability analysis [24]. First, the class K, K∞, and KL functions [25] will be reviewed. Then the structure of fuzzy logic system will be briefly introduced.


#### 2.1.1. ISS and Small Gain Theorem

Definition 1.
A functionα:(0,∞)→(0,∞) is said to belong to class K if it is continuous, strictly increasing, and α(0)=0. It is said to belong to class K∞ if additionally α is unbounded.Definition 2.
A functionβ:(0,∞)×(0,∞)→(0,∞) is of class KL if  β(·,t) is said to belong to class K for each fixed t≥0, and β(r,·) is strictly decreasing for each fixed r>0 and limt→∞β(r,t)=0.Definition 3.
For a nonlinear systemx˙=f(x,u), it is said to be input-to-state practically stable (ISpS) if there exist a class KL function β and a class K function γ, such that for any initial condition x(0), each bounded control input u(t) defined for all t>0 and a constant d>0, the associated solution x(t)∈[0,∞) satisfies
(1)∥x(t)∥≤β(∥x(0)∥,t)+γ(∥ut∥∞)+d,
where ∥ut∥∞=max0≤τ≤t|u(τ)| and u(τ) is a truncated function defined as
(2)ut(τ)={u(τ)0≤τ≤t,0τ<0orτ>t.
When d=0 in (1), the ISpS property becomes ISS property [21, 26].Definition 4.
A functionV∈ℂ1 is said to be an ISpS-Lyapunov function for the system x˙=f(x,u) if(1)
there exist classK∞ functions α1, α2, such that
(3)α1(∥x∥)≤V(x)≤α2(∥x∥)∀x∈ℝn,(2)
there exist classK function α3, α4 and a constant d>0, such that
(4)∂V(x)∂xf(x,u)≤-α3(∥x∥)+α4(∥u∥)+d.Whend=0 in (4), V is said to be an ISS-Lyapunov function [26]. Then nonlinear L∞ gain γ in (1) can be chosen as
(5)γ(s)=α1(s)-1·α2(s)·α3(s)-1·α4(s)∀s>0.Proposition 5.
The nonlinear systemx˙=f(x,u) is ISpS if and only if there exists an ISpS-Lyapunov function [24].Theorem 6 (small gain theorem).
Consider a system in composite feedback form of two ISpS systems [23, 24]:
(6)∑z~ω:{x˙=f(t,x,ω)z~=H(t,x,ω),∑ωz~:{y˙=g(t,y,z~)ω=K(t,y,z~).
In particular, if there exist class KL functions βω, βz~, class K functions γz~, γω, and two constants d1≥0, d2≥0, for all t≥0, the solutions X(x;ω,t) and Y(y;z~,t) are satisfied as follows:
(7)∥H(X(x;ω,t))∥≤βω(∥x(0)∥,t)+γz~(∥ω∥∞)+d1,∥K(Y(y;z~,t))∥≤βz~(∥y(0)∥,t)+γω(∥z~∥∞)+d2.
If
(8)γz~(γω(s))<s(orγω(γz~(s))<s),∀s>0,
then the solution of the composite systems (6)  is ISpS.


#### 2.1.2. Fuzzy Logic System

In the past several years, fuzzy logic system has been extensively studied [24]. In this paper, it will be used as a approximator to estimate these system dynamics which cannot be linearly parameterized in adaptive fuzzy torque controller design.Takagi-Sugeno (T-S) fuzzy system [27] is applied in this study, which can be described as the following form which consists of q rules:
(9)ruleRi:IFx1isM1iandx2isM2iand,…,andxnisMni,thenyi=ai0+ai1x1+ai2x2+⋯+ainxn,
where Mji are fuzzy subsets, ai0,ai1,ai2,…,ain denote designed constant parameters in fuzzy rule conclusion step, yi is fuzzy output, and q is the total number of fuzzy rules.If we choose product inference engine, singleton fuzzifier, and center average defuzzifier (which is very commonplace choice), the output of T-S fuzzy system can be given as(10)yf=∑i=1qyfiξfi(x)=ξf(x)Afx-,
where x=[x1,x2,…,xn]T, x-=[1,xT]T,
(11)Af=[a10a11⋯a1na20a21⋯a2n⋮⋮⋮⋮aq0aq1⋯aqn],ξf(x) and ξfi(x) are fuzzy basis vector and fuzzy basis function, respectively, and ξf(x)=[ξf1(x),ξf2(x),…,ξfq(x)], ξfi(x)=∏j=1nμMji(xj)/∑i=1q(∏j=1nμMji(xj)). Usually, the membership function of fuzzy set Mji is given by [26]
(12)μxji(xj)=exp[-(gjixj-bji)22(hji)2],
where gji∈ℝ, bji∈ℝ and hji>0 are adjustable parameters of μMji(xj), i=1,2,…,q, j=1,2…,n.Lemma 7.
For any continuous functionf(x) defined in a compact set x∈Ωx and for all b>0, there exists an ideal fuzzy logic, such that [28]
(13)|y-f(x)|≤b∀x∈Ωx.


### 2.2. Problem Formulation

We consider a class of SISO nonlinear systems given as(14)x˙1=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t),x˙2=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t),⋮x˙i=xi+1+θTφi(x-i,t)+fi(x-i)+Δi(x,t),⋮x˙n=u+θTφn(x,t)+fn(x)+Δn(x,t),y=x1,
where x=[x1,x2,…,xn]T∈ℝn is system state vector, u∈ℝ is system input, and y∈ℝ is system output. Denote x-i=[x1,x2,…,xi]T, 1≤i≤n. θTφi(x-i,t) is the linearly parameterized system model where θ=[θ1,θ2,…,θp]T∈ℝp is the vector of unknown but constant parameters and φi(x-i,t) is the vector of known system function. fi(x-i) is unknown but continuous system function which cannot be linearly parameterized. Δi(x,t) is external unknown disturbance.x
d
(
t
) is the desired trajectory and is assumed to be bounded with bounded derivatives up to nth order (i.e., xd(1),xd(2),…,xd(n)). Control objective is to design a bounded control law for the input u, such that output y=x1 tracks xd(t) as closely as possible.Assumption 8.
The extents of parametric uncertainties are known, and unknown disturbances are bounded by known function; that is, parametric uncertainties and unknown disturbances are assumed to satisfy(15)θ∈Ωθ≜{θ:θmin≤θ≤θmax},(16)|Δi(x,t)|≤Di(x-i,t),
where θmin=[θ1min,θ2min,…,θpmin]T∈ℝp and θmax=[θ1max,θ2max,…,θpmax]T∈ℝp are known lower and upper bound constant vectors of θ. In this paper, the operation < for two vectors is performed in terms of corresponding elements (e.g., θ<θmax means that θj<θjmax, for all j).Remark 9.
WhenΔi(x,t)=0, if we treat xi+1 as the control input of the x˙i dynamics, the x˙i dynamics would only depend on the states of its previous dynamics, that is, x1,…,xi. In other words, only the feedback signals determine the dynamics. Such a form is called strict feedback form.  Equation (14) is called a semistrict feedback form in which the bounding function Di(x-i,t) is required to be function of xj, j≤i and t only, but Δi(x,t) may contain some bounded functions of xj, j>i, thus violating the strict feedback property [6, 7].


## 2.1. Preliminaries

In this section, some definitions about input-to-state stable (ISS) and small gain theorem [21–23] will be introduced, which have been widely used in nonlinear control problems for stability analysis [24]. First, the class K, K∞, and KL functions [25] will be reviewed. Then the structure of fuzzy logic system will be briefly introduced.


### 2.1.1. ISS and Small Gain Theorem

Definition 1.
A functionα:(0,∞)→(0,∞) is said to belong to class K if it is continuous, strictly increasing, and α(0)=0. It is said to belong to class K∞ if additionally α is unbounded.Definition 2.
A functionβ:(0,∞)×(0,∞)→(0,∞) is of class KL if  β(·,t) is said to belong to class K for each fixed t≥0, and β(r,·) is strictly decreasing for each fixed r>0 and limt→∞β(r,t)=0.Definition 3.
For a nonlinear systemx˙=f(x,u), it is said to be input-to-state practically stable (ISpS) if there exist a class KL function β and a class K function γ, such that for any initial condition x(0), each bounded control input u(t) defined for all t>0 and a constant d>0, the associated solution x(t)∈[0,∞) satisfies
(1)∥x(t)∥≤β(∥x(0)∥,t)+γ(∥ut∥∞)+d,
where ∥ut∥∞=max0≤τ≤t|u(τ)| and u(τ) is a truncated function defined as
(2)ut(τ)={u(τ)0≤τ≤t,0τ<0orτ>t.
When d=0 in (1), the ISpS property becomes ISS property [21, 26].Definition 4.
A functionV∈ℂ1 is said to be an ISpS-Lyapunov function for the system x˙=f(x,u) if(1)
there exist classK∞ functions α1, α2, such that
(3)α1(∥x∥)≤V(x)≤α2(∥x∥)∀x∈ℝn,(2)
there exist classK function α3, α4 and a constant d>0, such that
(4)∂V(x)∂xf(x,u)≤-α3(∥x∥)+α4(∥u∥)+d.Whend=0 in (4), V is said to be an ISS-Lyapunov function [26]. Then nonlinear L∞ gain γ in (1) can be chosen as
(5)γ(s)=α1(s)-1·α2(s)·α3(s)-1·α4(s)∀s>0.Proposition 5.
The nonlinear systemx˙=f(x,u) is ISpS if and only if there exists an ISpS-Lyapunov function [24].Theorem 6 (small gain theorem).
Consider a system in composite feedback form of two ISpS systems [23, 24]:
(6)∑z~ω:{x˙=f(t,x,ω)z~=H(t,x,ω),∑ωz~:{y˙=g(t,y,z~)ω=K(t,y,z~).
In particular, if there exist class KL functions βω, βz~, class K functions γz~, γω, and two constants d1≥0, d2≥0, for all t≥0, the solutions X(x;ω,t) and Y(y;z~,t) are satisfied as follows:
(7)∥H(X(x;ω,t))∥≤βω(∥x(0)∥,t)+γz~(∥ω∥∞)+d1,∥K(Y(y;z~,t))∥≤βz~(∥y(0)∥,t)+γω(∥z~∥∞)+d2.
If
(8)γz~(γω(s))<s(orγω(γz~(s))<s),∀s>0,
then the solution of the composite systems (6)  is ISpS.


### 2.1.2. Fuzzy Logic System

In the past several years, fuzzy logic system has been extensively studied [24]. In this paper, it will be used as a approximator to estimate these system dynamics which cannot be linearly parameterized in adaptive fuzzy torque controller design.Takagi-Sugeno (T-S) fuzzy system [27] is applied in this study, which can be described as the following form which consists of q rules:
(9)ruleRi:IFx1isM1iandx2isM2iand,…,andxnisMni,thenyi=ai0+ai1x1+ai2x2+⋯+ainxn,
where Mji are fuzzy subsets, ai0,ai1,ai2,…,ain denote designed constant parameters in fuzzy rule conclusion step, yi is fuzzy output, and q is the total number of fuzzy rules.If we choose product inference engine, singleton fuzzifier, and center average defuzzifier (which is very commonplace choice), the output of T-S fuzzy system can be given as(10)yf=∑i=1qyfiξfi(x)=ξf(x)Afx-,
where x=[x1,x2,…,xn]T, x-=[1,xT]T,
(11)Af=[a10a11⋯a1na20a21⋯a2n⋮⋮⋮⋮aq0aq1⋯aqn],ξf(x) and ξfi(x) are fuzzy basis vector and fuzzy basis function, respectively, and ξf(x)=[ξf1(x),ξf2(x),…,ξfq(x)], ξfi(x)=∏j=1nμMji(xj)/∑i=1q(∏j=1nμMji(xj)). Usually, the membership function of fuzzy set Mji is given by [26]
(12)μxji(xj)=exp[-(gjixj-bji)22(hji)2],
where gji∈ℝ, bji∈ℝ and hji>0 are adjustable parameters of μMji(xj), i=1,2,…,q, j=1,2…,n.Lemma 7.
For any continuous functionf(x) defined in a compact set x∈Ωx and for all b>0, there exists an ideal fuzzy logic, such that [28]
(13)|y-f(x)|≤b∀x∈Ωx.


## 2.1.1. ISS and Small Gain Theorem

Definition 1.
A functionα:(0,∞)→(0,∞) is said to belong to class K if it is continuous, strictly increasing, and α(0)=0. It is said to belong to class K∞ if additionally α is unbounded.Definition 2.
A functionβ:(0,∞)×(0,∞)→(0,∞) is of class KL if  β(·,t) is said to belong to class K for each fixed t≥0, and β(r,·) is strictly decreasing for each fixed r>0 and limt→∞β(r,t)=0.Definition 3.
For a nonlinear systemx˙=f(x,u), it is said to be input-to-state practically stable (ISpS) if there exist a class KL function β and a class K function γ, such that for any initial condition x(0), each bounded control input u(t) defined for all t>0 and a constant d>0, the associated solution x(t)∈[0,∞) satisfies
(1)∥x(t)∥≤β(∥x(0)∥,t)+γ(∥ut∥∞)+d,
where ∥ut∥∞=max0≤τ≤t|u(τ)| and u(τ) is a truncated function defined as
(2)ut(τ)={u(τ)0≤τ≤t,0τ<0orτ>t.
When d=0 in (1), the ISpS property becomes ISS property [21, 26].Definition 4.
A functionV∈ℂ1 is said to be an ISpS-Lyapunov function for the system x˙=f(x,u) if(1)
there exist classK∞ functions α1, α2, such that
(3)α1(∥x∥)≤V(x)≤α2(∥x∥)∀x∈ℝn,(2)
there exist classK function α3, α4 and a constant d>0, such that
(4)∂V(x)∂xf(x,u)≤-α3(∥x∥)+α4(∥u∥)+d.Whend=0 in (4), V is said to be an ISS-Lyapunov function [26]. Then nonlinear L∞ gain γ in (1) can be chosen as
(5)γ(s)=α1(s)-1·α2(s)·α3(s)-1·α4(s)∀s>0.Proposition 5.
The nonlinear systemx˙=f(x,u) is ISpS if and only if there exists an ISpS-Lyapunov function [24].Theorem 6 (small gain theorem).
Consider a system in composite feedback form of two ISpS systems [23, 24]:
(6)∑z~ω:{x˙=f(t,x,ω)z~=H(t,x,ω),∑ωz~:{y˙=g(t,y,z~)ω=K(t,y,z~).
In particular, if there exist class KL functions βω, βz~, class K functions γz~, γω, and two constants d1≥0, d2≥0, for all t≥0, the solutions X(x;ω,t) and Y(y;z~,t) are satisfied as follows:
(7)∥H(X(x;ω,t))∥≤βω(∥x(0)∥,t)+γz~(∥ω∥∞)+d1,∥K(Y(y;z~,t))∥≤βz~(∥y(0)∥,t)+γω(∥z~∥∞)+d2.
If
(8)γz~(γω(s))<s(orγω(γz~(s))<s),∀s>0,
then the solution of the composite systems (6)  is ISpS.


## 2.1.2. Fuzzy Logic System

In the past several years, fuzzy logic system has been extensively studied [24]. In this paper, it will be used as a approximator to estimate these system dynamics which cannot be linearly parameterized in adaptive fuzzy torque controller design.Takagi-Sugeno (T-S) fuzzy system [27] is applied in this study, which can be described as the following form which consists of q rules:
(9)ruleRi:IFx1isM1iandx2isM2iand,…,andxnisMni,thenyi=ai0+ai1x1+ai2x2+⋯+ainxn,
where Mji are fuzzy subsets, ai0,ai1,ai2,…,ain denote designed constant parameters in fuzzy rule conclusion step, yi is fuzzy output, and q is the total number of fuzzy rules.If we choose product inference engine, singleton fuzzifier, and center average defuzzifier (which is very commonplace choice), the output of T-S fuzzy system can be given as(10)yf=∑i=1qyfiξfi(x)=ξf(x)Afx-,
where x=[x1,x2,…,xn]T, x-=[1,xT]T,
(11)Af=[a10a11⋯a1na20a21⋯a2n⋮⋮⋮⋮aq0aq1⋯aqn],ξf(x) and ξfi(x) are fuzzy basis vector and fuzzy basis function, respectively, and ξf(x)=[ξf1(x),ξf2(x),…,ξfq(x)], ξfi(x)=∏j=1nμMji(xj)/∑i=1q(∏j=1nμMji(xj)). Usually, the membership function of fuzzy set Mji is given by [26]
(12)μxji(xj)=exp[-(gjixj-bji)22(hji)2],
where gji∈ℝ, bji∈ℝ and hji>0 are adjustable parameters of μMji(xj), i=1,2,…,q, j=1,2…,n.Lemma 7.
For any continuous functionf(x) defined in a compact set x∈Ωx and for all b>0, there exists an ideal fuzzy logic, such that [28]
(13)|y-f(x)|≤b∀x∈Ωx.


## 2.2. Problem Formulation

We consider a class of SISO nonlinear systems given as(14)x˙1=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t),x˙2=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t),⋮x˙i=xi+1+θTφi(x-i,t)+fi(x-i)+Δi(x,t),⋮x˙n=u+θTφn(x,t)+fn(x)+Δn(x,t),y=x1,
where x=[x1,x2,…,xn]T∈ℝn is system state vector, u∈ℝ is system input, and y∈ℝ is system output. Denote x-i=[x1,x2,…,xi]T, 1≤i≤n. θTφi(x-i,t) is the linearly parameterized system model where θ=[θ1,θ2,…,θp]T∈ℝp is the vector of unknown but constant parameters and φi(x-i,t) is the vector of known system function. fi(x-i) is unknown but continuous system function which cannot be linearly parameterized. Δi(x,t) is external unknown disturbance.x
d
(
t
) is the desired trajectory and is assumed to be bounded with bounded derivatives up to nth order (i.e., xd(1),xd(2),…,xd(n)). Control objective is to design a bounded control law for the input u, such that output y=x1 tracks xd(t) as closely as possible.Assumption 8.
The extents of parametric uncertainties are known, and unknown disturbances are bounded by known function; that is, parametric uncertainties and unknown disturbances are assumed to satisfy(15)θ∈Ωθ≜{θ:θmin≤θ≤θmax},(16)|Δi(x,t)|≤Di(x-i,t),
where θmin=[θ1min,θ2min,…,θpmin]T∈ℝp and θmax=[θ1max,θ2max,…,θpmax]T∈ℝp are known lower and upper bound constant vectors of θ. In this paper, the operation < for two vectors is performed in terms of corresponding elements (e.g., θ<θmax means that θj<θjmax, for all j).Remark 9.
WhenΔi(x,t)=0, if we treat xi+1 as the control input of the x˙i dynamics, the x˙i dynamics would only depend on the states of its previous dynamics, that is, x1,…,xi. In other words, only the feedback signals determine the dynamics. Such a form is called strict feedback form.  Equation (14) is called a semistrict feedback form in which the bounding function Di(x-i,t) is required to be function of xj, j≤i and t only, but Δi(x,t) may contain some bounded functions of xj, j>i, thus violating the strict feedback property [6, 7].


## 3. Adaptive Fuzzy Robust Controller Design

### 3.1. Discontinuous Projection-Based Adaptive Law

The discontinuous projection-type parameter adaptation law is used to update parameters in AFRC controller. Letθ^ denote the estimate of θ, and let θ~ denote estimation error (i.e., θ~=θ^-θ). Specifically, viewing (15), parameter estimate θ^ is updated through a parameter adaptation law having the form of
(17)θ^˙=Projθ^(Γτ),
where Γ>0 is a diagonal matrix of adaptation rate and τ is an adaptation function to be designed according to system model. The discontinuous projection mapping Projθ^(•) can be defined as
(18)Projθ^(•)={0,ifθ^=θmaxand•>00,ifθ^=θminand•<0•,otherwise.It can be shown that for any adaptation functionτ, the projection mapping used in (18) has the following properties [5]:
(19)(P1)θ^∈Ωθ≜{θ^:θmin≤θ^≤θmax},(P2)θ~T[Γ-1Projθ^(Γτ)-τ]≤0,∀τ.


### 3.2. Backstepping Design of Adaptive Fuzzy Robust Control

In this section, we give out adaptive fuzzy robust controller design procedure by using backstepping techniques, and it containsn steps. At each step, an intermediate control signal ui will be developed by using an appropriate Lyapunov function Vi.Step 1.
Consider the first subsystem of (14), which is given as
(20)x˙1=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t).In (20), by viewing x2 as the control input of this subsystem, we can design an intermediate control law u1 for x2, such that x1 tracks its desired trajectory xd(t). Define the tracking error variables as
(21)z1=x1-xd.The time derivative ofz1 can be given as
(22)z˙1=x˙1-xd(1)=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t)-xd(1).Sincef1(x-1) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, f1(x-1) can be expressed as
(23)f1(x-1)=ξ1A1[1,x1]T+δ1=ξ1A11z1+ξ1A12[1,xd]T+δ1,
where(24)A1=[a101a111a201a211⋮⋮aq01aq11],A11=[a111a211⋮aq11],A12=[a101a111a201a211⋮⋮aq01aq11],ξ1=[ξ11,ξ12,…,ξ1q], δ1 represents fuzzy approximating error, and |δ1|≤b1; here b1 is unknown but bounded positive constant.Letc1=n∥A11∥, A1*=c1-1A11. It is clear that ∥A1*∥≤1/n; then let ω1=A1*z1, and (23) can be rewritten as
(25)f1(x-1)=c1ξ1ω1+ξ1A12[1,xd]T+δ1.Define the 2th tracking error variable as(26)z2=x2-u1.Then substitute (25) into (22),
(27)z˙1=(z2+u1)+θTφ1(x-1,t)+c1ξ1ω1+d1,
where
(28)d1=ξ1A12[1,xd]T+δ1+Δ1(x,t)-xd(1),|d1|≤B1q1,
where B1=max{∥A12∥,b1,1} is a positive constant and q1=∥ξ1∥·∥[1,xd]T∥+1+D1(x-1,t)+|xd(1)|.The intermediate control lawu1 consists of three parts as
(29)u1=u1a+u1f+u1r.At first, we design the adaptive control item as(30)u1a=-k1z1-θ^Tφ1(x-1,t).Substituting (29) and (30) into (22) leads to
(31)z˙1=(z2+u1f+u1r)-k1z1-θ~Tφ1(x-1,t)+f1(x-1)+Δ1(x,t).Then, we design the fuzzy control lawu1f as
(32)μ1f=-λ^1ψ1z1,λ^˙1=ρ1[ψ1z12-η1(λ^1-λ1)].
where ψ1=∥ξ1∥2/(4γ12), λ1=c12, and λ^1 is an online estimate of parameter λ1 with estimate error λ~1=λ^1-λ1.  γ1>0, ρ1>0, and η1>0 are positive design parameters.Define a positive semidefinite (p.s.d.) functionV1 as
(33)V1=12z12+12ρ1-1λ~12.Consideringλ~˙1=-λ^˙1, time derivative of V1 can be given as
(34)V˙1=z1z˙1-ρ1-1λ~1λ^˙1=z1z2+z1u1f+c1ξ1ω1z1-ρ1-1λ~1λ^˙1-k1z12+z1{u1r-θ~Tφ1(x-1,t)+d1}.Applying Young’s inequality, the following inequality can be obtained:(35)c1ξ1ω1z1≤|c1ξ1ω1z1|≤c12∥ξ1∥2z124γ12+2γ12∥ω1∥22=c12∥ξ1∥24γ12z12+γ12∥ω1∥2.Considering thatλ1=c12 and ψ1=∥ξ1∥2/(4γ12), from (35), the following inequality is holding:
(36)c1ξ1ω1z1≤λ1ψ1z12+γ12∥ω1∥2=λ^1ψ1z12+λ~1ψ1z12+γ12∥ω1∥2.Substituting (32) and (36) into (34) leads to
(37)V˙1≤z1z2-k1z12-η1λ~12+γ12∥ω1∥2+z1{u1r-θ~Tφ1(x-1,t)+d1}.Now we can get the following robust performance conditions foru1r:
(38)condition(i)z1{u1r-θ~Tφ1(x-1,t)+d1}≤ε1,condition(ii)z1u1r≤0.Remark 10.
One smooth example of robust control functionu1r satisfying (38) can be found in the following way. Let h1 be any smooth function satisfying h1≥||θM||||φ1(x-1,t)||+B1q1 where θM=θmax-θmin. ε1>0 is positive design parameter.Then we can design a robust control lawu1r as
(39)u1r=-14ε1h12z1.Thus, (37) can be given as
(40)V˙1≤z1z2-k1z12-η1λ~12+γ12∥ω1∥2+ε1.Step 2.
Consider the second subsystem of  (14), which is given as
(41)x˙2=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t).In (41), by viewing x3 as the control input of this subsystem, we can design an intermediate control law u2 for x3, such that x2 tracks the intermediate control law u1.The time derivative ofz2 can be given as
(42)z˙2=x˙2-u˙1=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t)-u˙1.
And the time derivative of u1 is
(43)u˙1=∂u1∂x1x˙1+∂u1∂θ^θ^˙+∂u1∂λ^1λ^˙1+∂u1∂xdxd(1).Then substituting (43) into (42) leads to
(44)z˙2=x3+θTφ2(x-2,t)+f2′(x-2)+Δ2(x,t)+∂u1∂xdxd(1),
where
(45)f2′(x-2)=f2(x-2)-∂u1∂x1x˙1-∂u1∂θ^θ^˙-∂u1∂λ^1λ^˙1.Sincef2′(x-2) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, f2′(x-2) can be expressed as
(46)f2′(x-2)=ξ2A2[1,z1,x2,xd,xd(1)]T+δ2=ξ2A21[z1,z2]T+ξ2A22[1,u1,xd,xd(1)]T+δ2,
where
(47)A2=[a101a111⋯a141a201a211⋯a241⋮⋮⋱⋮aq01aq11⋯aq41],A21=[a111a121a211a221⋮⋮aq11aq21],A22=[a101a121a131a141a201a221a231a241⋮⋮⋮⋮aq01aq21aq31aq41],ξ2=[ξ21,ξ22,…,ξ2q], δ2 represents fuzzy approximating error, and |δ2|≤b2; here b2 is unknown but bounded positive constant.Letc2=n∥A21∥, A2*=c2-1A21. It is clear that ∥A2*∥≤1/n; then let ω2=A2*z2, and (23) can be rewritten as
(48)f2′(x-2)=c2ξ2ω2+ξ2A22[1,u1,xd,xd(1)]T+δ2.Define the 3th tracking error variable as(49)z3=x3-u2.Substituting (48) and (49) into (44), we have
(50)z˙2=(z3+u2)+θTφ2(x-2,t)+c2ξ2ω2+d2,
where
(51)d2=ξ2A22[1,u1,xd,xd(1)]T+δ2+Δ2(x,t)+∂u1∂xdxd(1),|d2|≤B2q2,
where B2=max{∥A22∥,b2,1} is a positive constant and q2=∥ξ2∥·∥[1,u1,xd,xd(1)]T∥+1+D2(x-2,t)+|(∂u1/∂xd)xd(1)|.The control lawu2 consists of three parts as
(52)u2=u2a+u2f+u2r.At first, we design the adaptive control item as(53)u2a=-k2z2-θ^Tφ2(x-2,t).Substituting (52) and (53) into (50) leads to
(54)z˙2=(z3+u2f+u2r)-k2z2-θ~Tφ2(x-2,t)+f2′(x-2)+Δ2(x,t).Then, we design the fuzzy control lawu2f as
(55)u2f=-λ^2ψ2z2,λ^˙2=ρ2[ψ2z22-η2(λ^2-λ2)],
where ψ2=∥ξ2∥2/(4γ22),  λ^2 is an online estimate of parameter λ2 with estimate error λ~2=λ^2-λ2.  γ2>0, ρ2>0, and η2>0 are positive design parameters.Define a positive semidefinite (p.s.d.) functionV2 as
(56)V2=12z22+12ρ2-1λ~22.Consideringλ~˙2=-λ^˙2, time derivative of V2 can be given as
(57)V˙2=z2z˙2-ρ2-1λ~2λ^˙2=z2z3+z2u2f+c2ξ2ω2z2-ρ2-1λ~2λ^˙2-k2z22+z2{u2r-θ~Tφ2(x-2,t)+d2}.Applying Young’s inequality, the following inequality can be obtained:(58)c2ξ2ω2z2≤|c2ξ2ω2z2|≤c22∥ξ2∥2z224γ22+2γ22∥ω2∥22=c22∥ξ2∥2z224γ22+γ22∥ω2∥2.Considering thatλ2=c22 and ψ2=∥ξ2∥2/(4γ22), from (58), the following inequality is holding:
(59)c2ξ2ω2z2≤λ2ψ2z22+γ22∥ω2∥2=λ^2ψ2z22+λ~2ψ2z22+γ22∥ω2∥2.Substituting (55) and (59) into (57) leads to
(60)V˙2≤z2z3-k2z22-η2λ~22+γ22∥ω2∥2+z2{u2r-θ~Tφ2(x-2,t)+d2}.Now we can get the following robust performance conditions foru2r:
(61)condition(i)z2{u2r-θ~Tφ2(x-2,t)+d2}≤ε2,condition(ii)z2u2r≤0.According to Remark10, we can design a robust control law u2r as
(62)u2r=-14ε2h22z2,
where h2 can be any smooth function satisfying h2≥||θM||||φ2(x-2,t)||+B2q2 and ε2>0 is positive design parameter.Thus, (60) can be given as
(63)V˙2≤z2z3-k2z22-η2λ~22+γ22∥ω2∥2+ε2.Step i. We can use the similar procedure for each Step i(3≤i≤n-1). Consider the ith subsystem of (14), which is given as
(64)xi=xi+1+θTφi(x-i,t)+fi(x-i)+Δi(x,t).We can design a control lawui for subsystem (64) as
(65)ui=uia+uif+uir,uia=-kizi-θ^Tφi(x-i,t),μif=-λ^iψizi,λ^˙i=ρi[ψizi2-ηi(λ^i-λi)],uir=-14εihi2zi.Define a p.s.d. functionVi as
(66)Vi=Vi-1+12zi2+12ρi-1λ~i2.By using the same technique, the time derivative ofVi can be given as
(67)Vi≤Vi-1+zizi+1-kizi2-ηiλ~i2+γi2∥ωi∥2+εi.Step n. Consider the nth subsystem of (14), which is given as
(68)x˙n=u+θTφn(x,t)+fn(x)+Δn(x,t).Define the tracking error variable as(69)zn=xn-un-1.The time derivative ofzn can be given as
(70)z˙n=x˙n-u˙n-1=u+θTφn(x,t)+fn(x)+Δn(x,t)-u˙n-1.The time derivative ofun-1 is
(71)u˙n-1=∑j=1n-1∂un-1∂xjx˙j+∂un-1∂θ^θ^˙+∂un-1∂λ^n-1λ^˙n-1+∑j=1n-1∂un-1∂xd(j-1)xd(j).Substituting (71) into (70) leads to
(72)z˙n=u+θTφn(x,t)+fn′(x-n)+Δn(x,t)+∑j=1n-1∂un-1∂xd(j-1)xd(j),
where
(73)fn′(x-n)=∑j=1n-1∂un-1∂xjx˙j+∂un-1∂θ^θ^˙+∂un-1∂λ^n-1λ^˙n-1+fn(x-n).According to fuzzy logic theorem as mentioned in Remark10, (73) can be expressed as
(74)fn′(x-n)=ξnAn[1,z1,…,zn-1,xn,xd,…,xd(n-1)]T+δn=ξnAn1[z1,…,zn]T+ξnAn2[1,un-1,xd,…,xd(n-1)]T+δn,
where
(75)An=[a10na11n⋯a1(2n)na20na21n⋯a2(2n)n⋮⋮⋱⋮aq0naq1n⋯aq(2n)n],An1=[a11na12n⋯a1nna21na22n⋯a2nn⋮⋮⋱⋮aq1naq2n⋯aqnn],An2=[a10na1nna1(n+1)n⋯a1(2n)na20na2nna2(n+1)n⋯a2(2n)n⋮⋮⋮⋱⋮aq0naqnnaq(n+1)n⋯aq(2n)n],ξn=[ξn1,ξn2,…,ξnq], δn represents fuzzy approximating error, and |δn|≤bn, and here bn is unknown but bounded positive constant.Letcn=n∥An1∥, An*=cn-1An1. It is clear that ∥An*∥≤1/n; then let ωn=An*[z1,…,zn]T, and (74) can be rewritten as
(76)fn′(x-n)=cnξnωn+ξnAn2[u1,xd,…,xd(n-1)]T+δn.Substitute (76) into (72), we have
(77)z˙n=u+θTφn(x,t)+cnξnωn+dn,
where
(78)dn=ξnAn2[1,un-1,xd,…,xd(n-1)]T+δn+Δn(x,t)+∑j=1n-1∂un-1∂xd(j-1)xd(j),|dn|≤Bnqn,
where Bn=max{∥An2∥,bn,1} is a positive constant and q2=∥ξn∥·∥[1,un-1,xd,…,xd(n-1)]T∥+1+Dn(x-n,t)+∥∑j=1n-1((∂un-1/∂xd(j-1))xd(j))∥.The control lawu consists of three parts giving as
(79)u=un=una+unf+unr.At first, we design the adaptive control item as(80)una=-knzn-θ^Tφn(x-n,t).Substituting (79) and (80) into (77) leads to
(81)z˙n=unf+unr-knzn-θ~Tφn(x,t)+cnξnωn+dn.Design the fuzzy control lawunf as
(82)μnf=-λ^nψnzn,λ^˙n=ρn[ψnzn2-ηn(λ^n-λn)],
where ψn=∥ξn∥2/(4γn2), λn=cn2, and  λ^n is an online estimate of parameter λn with estimate error λ~n=λ^n-λn.  γn>0, ρn>0 and ηn>0 are positive design parameters.Define a p.s.d. functionVn as
(83)Vn=Vn-1+12zn2+12ρn-1λ~n2.Consideringλ~˙n=-λ^˙n, the time derivative of Vn can be given as
(84)V˙n=V˙n-1+znz˙n-γn-1λ~nλ^˙n=V˙n-1+znunf+cnξnωnzn-ρn-1λ~nλ^˙n-knzn2+zn{unr-θ~Tφn(x-n,t)+dn}.Applying Young’s inequality, the following inequality can be obtained:(85)cnξnωnzn≤|cnξnωnzn|≤cn2∥ξn∥2zn24γn2+2γn2∥ωn∥22=cn2∥ξn∥2zn24γn2+γn2∥ωn∥2.Considering thatλn=cn2 and ψn=∥ξn∥2/(4γn2), from (85), the following inequality is holding:
(86)cnξnωnzn≤λnψnzn2+γn2∥ωn∥2=λ^nψnzn2+λ~nψnzn2+γn2∥ωn∥2.Substituting (82) and (86) into (84) leads to
(87)V˙n≤V˙n-1-knzn2-ηnλ~n2+γn2∥ωn∥2+zn{unr-θ~Tφn(x-n,t)+dn}.Now we can get the following robust performance conditions forunr:
(88)condition(i)zn{unr-θ~Tφn(x-n,t)+dn}≤εn,condition(ii)znunr≤0.Then we can design a robust control lawunr as
(89)unr=-14εnhn2zn,
where hn can be any smooth function satisfying hn≥||θM||||φn(x-n,t)||+Bnqn and εn is positive parameter.Thus, (87) can be given as
(90)V˙n≤V˙n-1-knzn2-ηnλ~n2+γn2∥ωn∥2+εn.


## 3.1. Discontinuous Projection-Based Adaptive Law

The discontinuous projection-type parameter adaptation law is used to update parameters in AFRC controller. Letθ^ denote the estimate of θ, and let θ~ denote estimation error (i.e., θ~=θ^-θ). Specifically, viewing (15), parameter estimate θ^ is updated through a parameter adaptation law having the form of
(17)θ^˙=Projθ^(Γτ),
where Γ>0 is a diagonal matrix of adaptation rate and τ is an adaptation function to be designed according to system model. The discontinuous projection mapping Projθ^(•) can be defined as
(18)Projθ^(•)={0,ifθ^=θmaxand•>00,ifθ^=θminand•<0•,otherwise.It can be shown that for any adaptation functionτ, the projection mapping used in (18) has the following properties [5]:
(19)(P1)θ^∈Ωθ≜{θ^:θmin≤θ^≤θmax},(P2)θ~T[Γ-1Projθ^(Γτ)-τ]≤0,∀τ.


## 3.2. Backstepping Design of Adaptive Fuzzy Robust Control

In this section, we give out adaptive fuzzy robust controller design procedure by using backstepping techniques, and it containsn steps. At each step, an intermediate control signal ui will be developed by using an appropriate Lyapunov function Vi.Step 1.
Consider the first subsystem of (14), which is given as
(20)x˙1=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t).In (20), by viewing x2 as the control input of this subsystem, we can design an intermediate control law u1 for x2, such that x1 tracks its desired trajectory xd(t). Define the tracking error variables as
(21)z1=x1-xd.The time derivative ofz1 can be given as
(22)z˙1=x˙1-xd(1)=x2+θTφ1(x-1,t)+f1(x-1)+Δ1(x,t)-xd(1).Sincef1(x-1) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, f1(x-1) can be expressed as
(23)f1(x-1)=ξ1A1[1,x1]T+δ1=ξ1A11z1+ξ1A12[1,xd]T+δ1,
where(24)A1=[a101a111a201a211⋮⋮aq01aq11],A11=[a111a211⋮aq11],A12=[a101a111a201a211⋮⋮aq01aq11],ξ1=[ξ11,ξ12,…,ξ1q], δ1 represents fuzzy approximating error, and |δ1|≤b1; here b1 is unknown but bounded positive constant.Letc1=n∥A11∥, A1*=c1-1A11. It is clear that ∥A1*∥≤1/n; then let ω1=A1*z1, and (23) can be rewritten as
(25)f1(x-1)=c1ξ1ω1+ξ1A12[1,xd]T+δ1.Define the 2th tracking error variable as(26)z2=x2-u1.Then substitute (25) into (22),
(27)z˙1=(z2+u1)+θTφ1(x-1,t)+c1ξ1ω1+d1,
where
(28)d1=ξ1A12[1,xd]T+δ1+Δ1(x,t)-xd(1),|d1|≤B1q1,
where B1=max{∥A12∥,b1,1} is a positive constant and q1=∥ξ1∥·∥[1,xd]T∥+1+D1(x-1,t)+|xd(1)|.The intermediate control lawu1 consists of three parts as
(29)u1=u1a+u1f+u1r.At first, we design the adaptive control item as(30)u1a=-k1z1-θ^Tφ1(x-1,t).Substituting (29) and (30) into (22) leads to
(31)z˙1=(z2+u1f+u1r)-k1z1-θ~Tφ1(x-1,t)+f1(x-1)+Δ1(x,t).Then, we design the fuzzy control lawu1f as
(32)μ1f=-λ^1ψ1z1,λ^˙1=ρ1[ψ1z12-η1(λ^1-λ1)].
where ψ1=∥ξ1∥2/(4γ12), λ1=c12, and λ^1 is an online estimate of parameter λ1 with estimate error λ~1=λ^1-λ1.  γ1>0, ρ1>0, and η1>0 are positive design parameters.Define a positive semidefinite (p.s.d.) functionV1 as
(33)V1=12z12+12ρ1-1λ~12.Consideringλ~˙1=-λ^˙1, time derivative of V1 can be given as
(34)V˙1=z1z˙1-ρ1-1λ~1λ^˙1=z1z2+z1u1f+c1ξ1ω1z1-ρ1-1λ~1λ^˙1-k1z12+z1{u1r-θ~Tφ1(x-1,t)+d1}.Applying Young’s inequality, the following inequality can be obtained:(35)c1ξ1ω1z1≤|c1ξ1ω1z1|≤c12∥ξ1∥2z124γ12+2γ12∥ω1∥22=c12∥ξ1∥24γ12z12+γ12∥ω1∥2.Considering thatλ1=c12 and ψ1=∥ξ1∥2/(4γ12), from (35), the following inequality is holding:
(36)c1ξ1ω1z1≤λ1ψ1z12+γ12∥ω1∥2=λ^1ψ1z12+λ~1ψ1z12+γ12∥ω1∥2.Substituting (32) and (36) into (34) leads to
(37)V˙1≤z1z2-k1z12-η1λ~12+γ12∥ω1∥2+z1{u1r-θ~Tφ1(x-1,t)+d1}.Now we can get the following robust performance conditions foru1r:
(38)condition(i)z1{u1r-θ~Tφ1(x-1,t)+d1}≤ε1,condition(ii)z1u1r≤0.Remark 10.
One smooth example of robust control functionu1r satisfying (38) can be found in the following way. Let h1 be any smooth function satisfying h1≥||θM||||φ1(x-1,t)||+B1q1 where θM=θmax-θmin. ε1>0 is positive design parameter.Then we can design a robust control lawu1r as
(39)u1r=-14ε1h12z1.Thus, (37) can be given as
(40)V˙1≤z1z2-k1z12-η1λ~12+γ12∥ω1∥2+ε1.Step 2.
Consider the second subsystem of  (14), which is given as
(41)x˙2=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t).In (41), by viewing x3 as the control input of this subsystem, we can design an intermediate control law u2 for x3, such that x2 tracks the intermediate control law u1.The time derivative ofz2 can be given as
(42)z˙2=x˙2-u˙1=x3+θTφ2(x-2,t)+f2(x-2)+Δ2(x,t)-u˙1.
And the time derivative of u1 is
(43)u˙1=∂u1∂x1x˙1+∂u1∂θ^θ^˙+∂u1∂λ^1λ^˙1+∂u1∂xdxd(1).Then substituting (43) into (42) leads to
(44)z˙2=x3+θTφ2(x-2,t)+f2′(x-2)+Δ2(x,t)+∂u1∂xdxd(1),
where
(45)f2′(x-2)=f2(x-2)-∂u1∂x1x˙1-∂u1∂θ^θ^˙-∂u1∂λ^1λ^˙1.Sincef2′(x-2) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, f2′(x-2) can be expressed as
(46)f2′(x-2)=ξ2A2[1,z1,x2,xd,xd(1)]T+δ2=ξ2A21[z1,z2]T+ξ2A22[1,u1,xd,xd(1)]T+δ2,
where
(47)A2=[a101a111⋯a141a201a211⋯a241⋮⋮⋱⋮aq01aq11⋯aq41],A21=[a111a121a211a221⋮⋮aq11aq21],A22=[a101a121a131a141a201a221a231a241⋮⋮⋮⋮aq01aq21aq31aq41],ξ2=[ξ21,ξ22,…,ξ2q], δ2 represents fuzzy approximating error, and |δ2|≤b2; here b2 is unknown but bounded positive constant.Letc2=n∥A21∥, A2*=c2-1A21. It is clear that ∥A2*∥≤1/n; then let ω2=A2*z2, and (23) can be rewritten as
(48)f2′(x-2)=c2ξ2ω2+ξ2A22[1,u1,xd,xd(1)]T+δ2.Define the 3th tracking error variable as(49)z3=x3-u2.Substituting (48) and (49) into (44), we have
(50)z˙2=(z3+u2)+θTφ2(x-2,t)+c2ξ2ω2+d2,
where
(51)d2=ξ2A22[1,u1,xd,xd(1)]T+δ2+Δ2(x,t)+∂u1∂xdxd(1),|d2|≤B2q2,
where B2=max{∥A22∥,b2,1} is a positive constant and q2=∥ξ2∥·∥[1,u1,xd,xd(1)]T∥+1+D2(x-2,t)+|(∂u1/∂xd)xd(1)|.The control lawu2 consists of three parts as
(52)u2=u2a+u2f+u2r.At first, we design the adaptive control item as(53)u2a=-k2z2-θ^Tφ2(x-2,t).Substituting (52) and (53) into (50) leads to
(54)z˙2=(z3+u2f+u2r)-k2z2-θ~Tφ2(x-2,t)+f2′(x-2)+Δ2(x,t).Then, we design the fuzzy control lawu2f as
(55)u2f=-λ^2ψ2z2,λ^˙2=ρ2[ψ2z22-η2(λ^2-λ2)],
where ψ2=∥ξ2∥2/(4γ22),  λ^2 is an online estimate of parameter λ2 with estimate error λ~2=λ^2-λ2.  γ2>0, ρ2>0, and η2>0 are positive design parameters.Define a positive semidefinite (p.s.d.) functionV2 as
(56)V2=12z22+12ρ2-1λ~22.Consideringλ~˙2=-λ^˙2, time derivative of V2 can be given as
(57)V˙2=z2z˙2-ρ2-1λ~2λ^˙2=z2z3+z2u2f+c2ξ2ω2z2-ρ2-1λ~2λ^˙2-k2z22+z2{u2r-θ~Tφ2(x-2,t)+d2}.Applying Young’s inequality, the following inequality can be obtained:(58)c2ξ2ω2z2≤|c2ξ2ω2z2|≤c22∥ξ2∥2z224γ22+2γ22∥ω2∥22=c22∥ξ2∥2z224γ22+γ22∥ω2∥2.Considering thatλ2=c22 and ψ2=∥ξ2∥2/(4γ22), from (58), the following inequality is holding:
(59)c2ξ2ω2z2≤λ2ψ2z22+γ22∥ω2∥2=λ^2ψ2z22+λ~2ψ2z22+γ22∥ω2∥2.Substituting (55) and (59) into (57) leads to
(60)V˙2≤z2z3-k2z22-η2λ~22+γ22∥ω2∥2+z2{u2r-θ~Tφ2(x-2,t)+d2}.Now we can get the following robust performance conditions foru2r:
(61)condition(i)z2{u2r-θ~Tφ2(x-2,t)+d2}≤ε2,condition(ii)z2u2r≤0.According to Remark10, we can design a robust control law u2r as
(62)u2r=-14ε2h22z2,
where h2 can be any smooth function satisfying h2≥||θM||||φ2(x-2,t)||+B2q2 and ε2>0 is positive design parameter.Thus, (60) can be given as
(63)V˙2≤z2z3-k2z22-η2λ~22+γ22∥ω2∥2+ε2.Step i. We can use the similar procedure for each Step i(3≤i≤n-1). Consider the ith subsystem of (14), which is given as
(64)xi=xi+1+θTφi(x-i,t)+fi(x-i)+Δi(x,t).We can design a control lawui for subsystem (64) as
(65)ui=uia+uif+uir,uia=-kizi-θ^Tφi(x-i,t),μif=-λ^iψizi,λ^˙i=ρi[ψizi2-ηi(λ^i-λi)],uir=-14εihi2zi.Define a p.s.d. functionVi as
(66)Vi=Vi-1+12zi2+12ρi-1λ~i2.By using the same technique, the time derivative ofVi can be given as
(67)Vi≤Vi-1+zizi+1-kizi2-ηiλ~i2+γi2∥ωi∥2+εi.Step n. Consider the nth subsystem of (14), which is given as
(68)x˙n=u+θTφn(x,t)+fn(x)+Δn(x,t).Define the tracking error variable as(69)zn=xn-un-1.The time derivative ofzn can be given as
(70)z˙n=x˙n-u˙n-1=u+θTφn(x,t)+fn(x)+Δn(x,t)-u˙n-1.The time derivative ofun-1 is
(71)u˙n-1=∑j=1n-1∂un-1∂xjx˙j+∂un-1∂θ^θ^˙+∂un-1∂λ^n-1λ^˙n-1+∑j=1n-1∂un-1∂xd(j-1)xd(j).Substituting (71) into (70) leads to
(72)z˙n=u+θTφn(x,t)+fn′(x-n)+Δn(x,t)+∑j=1n-1∂un-1∂xd(j-1)xd(j),
where
(73)fn′(x-n)=∑j=1n-1∂un-1∂xjx˙j+∂un-1∂θ^θ^˙+∂un-1∂λ^n-1λ^˙n-1+fn(x-n).According to fuzzy logic theorem as mentioned in Remark10, (73) can be expressed as
(74)fn′(x-n)=ξnAn[1,z1,…,zn-1,xn,xd,…,xd(n-1)]T+δn=ξnAn1[z1,…,zn]T+ξnAn2[1,un-1,xd,…,xd(n-1)]T+δn,
where
(75)An=[a10na11n⋯a1(2n)na20na21n⋯a2(2n)n⋮⋮⋱⋮aq0naq1n⋯aq(2n)n],An1=[a11na12n⋯a1nna21na22n⋯a2nn⋮⋮⋱⋮aq1naq2n⋯aqnn],An2=[a10na1nna1(n+1)n⋯a1(2n)na20na2nna2(n+1)n⋯a2(2n)n⋮⋮⋮⋱⋮aq0naqnnaq(n+1)n⋯aq(2n)n],ξn=[ξn1,ξn2,…,ξnq], δn represents fuzzy approximating error, and |δn|≤bn, and here bn is unknown but bounded positive constant.Letcn=n∥An1∥, An*=cn-1An1. It is clear that ∥An*∥≤1/n; then let ωn=An*[z1,…,zn]T, and (74) can be rewritten as
(76)fn′(x-n)=cnξnωn+ξnAn2[u1,xd,…,xd(n-1)]T+δn.Substitute (76) into (72), we have
(77)z˙n=u+θTφn(x,t)+cnξnωn+dn,
where
(78)dn=ξnAn2[1,un-1,xd,…,xd(n-1)]T+δn+Δn(x,t)+∑j=1n-1∂un-1∂xd(j-1)xd(j),|dn|≤Bnqn,
where Bn=max{∥An2∥,bn,1} is a positive constant and q2=∥ξn∥·∥[1,un-1,xd,…,xd(n-1)]T∥+1+Dn(x-n,t)+∥∑j=1n-1((∂un-1/∂xd(j-1))xd(j))∥.The control lawu consists of three parts giving as
(79)u=un=una+unf+unr.At first, we design the adaptive control item as(80)una=-knzn-θ^Tφn(x-n,t).Substituting (79) and (80) into (77) leads to
(81)z˙n=unf+unr-knzn-θ~Tφn(x,t)+cnξnωn+dn.Design the fuzzy control lawunf as
(82)μnf=-λ^nψnzn,λ^˙n=ρn[ψnzn2-ηn(λ^n-λn)],
where ψn=∥ξn∥2/(4γn2), λn=cn2, and  λ^n is an online estimate of parameter λn with estimate error λ~n=λ^n-λn.  γn>0, ρn>0 and ηn>0 are positive design parameters.Define a p.s.d. functionVn as
(83)Vn=Vn-1+12zn2+12ρn-1λ~n2.Consideringλ~˙n=-λ^˙n, the time derivative of Vn can be given as
(84)V˙n=V˙n-1+znz˙n-γn-1λ~nλ^˙n=V˙n-1+znunf+cnξnωnzn-ρn-1λ~nλ^˙n-knzn2+zn{unr-θ~Tφn(x-n,t)+dn}.Applying Young’s inequality, the following inequality can be obtained:(85)cnξnωnzn≤|cnξnωnzn|≤cn2∥ξn∥2zn24γn2+2γn2∥ωn∥22=cn2∥ξn∥2zn24γn2+γn2∥ωn∥2.Considering thatλn=cn2 and ψn=∥ξn∥2/(4γn2), from (85), the following inequality is holding:
(86)cnξnωnzn≤λnψnzn2+γn2∥ωn∥2=λ^nψnzn2+λ~nψnzn2+γn2∥ωn∥2.Substituting (82) and (86) into (84) leads to
(87)V˙n≤V˙n-1-knzn2-ηnλ~n2+γn2∥ωn∥2+zn{unr-θ~Tφn(x-n,t)+dn}.Now we can get the following robust performance conditions forunr:
(88)condition(i)zn{unr-θ~Tφn(x-n,t)+dn}≤εn,condition(ii)znunr≤0.Then we can design a robust control lawunr as
(89)unr=-14εnhn2zn,
where hn can be any smooth function satisfying hn≥||θM||||φn(x-n,t)||+Bnqn and εn is positive parameter.Thus, (87) can be given as
(90)V˙n≤V˙n-1-knzn2-ηnλ~n2+γn2∥ωn∥2+εn.


## 4. The Closed-Loop System Stability Analysis

Consider the system (14) with partly linearizable system dynamics, unknown nonlinearities, and external disturbances, and suppose that the unknown system functions can be approximated by T-S fuzzy systems. If we design the adaptive fuzzy robust control law u=una+unf+unr with the intermediate control law ui, i=1,2,…,n-1, and adaptive laws for θi and λi, then the overall closed-loop control system is semiglobally uniformly ultimately bounded in the sense that all signals in this system is bounded. If control parameters are appropriately selected, the tracking error can be smaller than a prescribed error bound, and it means the tracking error asymptotically converges zero.Proof.
Rewrite the closed-loop system into two composited subsystems, namely,Σz~ω and Σωz~ as follows:
(91)z˙1=(z2+u1)+θTφ1(x-1,t)+c1ξ1ω1+d1,z˙i=(zi+1+ui)+θTφi(x-i,t)+ciξiωi+di,z˙n=u+θTφn(x,t)+cnξnωn+dn,λ^˙1=ρ1[ψ1z12-η1(λ^1-λ1)],λ^˙i=ρi[ψizi2-ηi(λ^i-λi)],λ^˙n=ρn[ψnzn2-ηn(λ^n-λn)],Z~=H(Z)=Z,
where ω=[ω1,…,ωn]T is given as input of subsystem Σz~ω and Z~ is output, Z=[zT,λ~T]T,  z=[z1,…,zn]T, and λ~=[λ~1,…,λ~n]T.
Design a Lyapunov function as(92)V=Vn.
Choosingγ=(γ12+⋯+γn2)1/2, noting the time derivative of V can be obtained as
(93)V˙≤-∥Z∥2+γ2∥ω∥2+ε,
where
(94)ε=∑i=1nεi.
According to small gain theorem [8, 13, 21], the subsystem Σz~ω satisfies ISpS, and there exist class K∞ functions α1(s), α2(s), such that α1(∥Z∥)≤V(Z)≤α2(∥Z∥), and α3(s)=s2, α4(s)=γ2s2, then the gain of subsystem Σz~ω can be obtained as
(95)γz~(s)=α1-1·α2·α3-1·α4(s)∀s>0.
Consider subsystemΣωz~ as
(96)ω1=A1*z1,ωi=Ai*z-i=Ai*[z1,…,zi]T,ωn=An*z-n=An*[z1,…,zn]T,⟹ω=Az=A-Ζ=A-Ζ~,
where Z~ is the input of subsystem, output is ω=[ω1,…,ωn]T. z=[z1,…,zn]T, λ~=[λ~1,…,λ~n]T, Ζ=Z~=[zT,λ~T]T, and A-=[A,0]. Considering Ai*=[Ai*1Ai*2⋯Ai*i] and ∥Ai*∥≤1/n, i=1,2,…,n, we have
(97)∥A-∥=∥A∥=∥[A1*0⋯0A2*1A2*2⋯0⋮⋮⋱⋮An*1An*2⋯An*n]∥=(∥A1*∥2+∥A2*∥2+⋯+∥An*∥2)1/2≤1.
Let γ′=∥A-∥,
(98)∥ω∥≤∥A-∥∥Ζ∥=γ′∥Ζ~∥.
Thereforeγ′≤1, the gain of subsystem Σωz~γω=γ′≤1.
According to small gain theorem, ifγz~(γω(s))<s, then the closed-loop system (91) is ISpS. Then,
(99)γz~(γω(s))<s⟹γγ′<1.
Because0<γ′≤1, if we choose 0<γ<1, the closed-loop system satisfies ISpS conditions. Therefore, there exist a class KL function and a positive condition δ0, such that
(100)∥[z(t)T,λ~(t)T]T∥≤β(∥[z(0)T,λ~(0)T]T∥,t)+δ0.
Therefore,z(t)∈L∞, x(t)∈L∞. There exist σ0>0 and T>0; such that, ∥x∥<σ0, for all t≥T, that is the closed-loop system is uniformly ultimately bounded.
Furthermore, according to (93), if ε=0, the subsystem Σz~ω is ISS, and the closed-loop system is also ISS. Therefore, there exists a class KL function, such that
(101)∥[z(t)T,λ~(t)T]T∥≤β(∥[z(0)T,λ~(0)T]T∥,t).
That meanslimt→∞z1(t)=0; that is, the asymptotic output tracking is achieved.


## 5. Illustrative Example

In order to illustrate the control performance of the proposed AFRC, the following simulation example is considered in this section:(102)d˙=-d+0.25x1,x˙1=x2+θ1x1+f1(x-1)+Δ1(x,t),x˙2=u+θ2x2+f2(x-2)+Δ2(x,t),y=x1,
where x=[x1,x2]T∈R2 is system state vector, u∈R is system input, and y∈R is system output. θ=[θ1,θ2]T∈R2 is the vector of parameters. f1(x-1) and f2(x-2) represent unknown functions. Δ1(x,t) and Δ2(x,t) are nonlinear dynamic disturbances. In the simulation, we set θ1=-0.5, θ2=-0.03, f1(x-1)=-0.6x12, f2(x-2)=-0.02x1x2, Δ1(x,t)=-2.0d2, and Δ2(x,t)=-3.0d-0.1d˙.Let the desired trajectory asxd(t)=0.5sin(2πt)+0.25sin(5πt) which is shown in Figure 1. The control objective is to guarantee that (a) all signals in the closed-loop system are bounded and (b) the output y follows the trajectory xd as closely as possible.Figure 1
The desired trajectory.Define fuzzy membership functions for each error dynamicsz1, z2 as
(103)μz1i(z1)=exp[-(50z1-(i-4))22×0.42472],i=1,2,…,7,μz2i(z2)=exp[-(10z2-(i-4))22×0.42472],i=1,2,…,7.
The first intermediate control function u1 can be given as
(104)u1=u1a+u1f+u1r=-40z1-θ^1x1-λ^1ψ1z1-14h12z1.
Then we design the controller law as
(105)u=u2=u2a+u2f+u2r=-40z2-θ^2x2-λ^2ψ2z2-116h22z2,
where ψi=∥ξi∥2/(4*0.52) and hi=||θM||||xi||+Biqi, i=1,2. The discontinuous projection-type parameter adaptation laws are given as follows:
(106)θ^˙1=Projθ^1(0.6x1z1),θ^˙2=Projθ^2(0.005x2z2)
with lower bound θmin=[-0.6,-0.04]T and upper bound θmax=[-0.4,-0.02]T. The initial values of parameter estimates are chosen as θ(0)=[-0.55,-0.025]T.The online estimation laws of parameterλ1 and λ2 are given as
(107)λ^˙1=200[ψ1z12-0.01(λ^1-50)],λ^˙2=100[ψ2z22-0.05(λ^2-200)].In the simulation, three controllers are compared.(C1): the proposed adaptive fuzzy robust controller as described previously.(C2): by settinguif=0, the controller is equivalent to the adaptive robust control law in [6].(C3): by settinguia=0, the controller becomes a fuzzy controller.Simulation results are shown in Figures2–5. Figure 2 shows the output tracking error z1, and Figure 3 shows the intermediate error z2. According to Figures 2 and 3, it is obvious that all three controllers have very good tracking ability; however, AFRC (C1) has the best performance among the three controllers. Figure 4 shows the online estimations θ^i of system parameters, Figure 5 shows the online updated fuzzy parameters λ^1, and the figures indicate that both of them are all bounded.Figure 2
The output tracking errorz1 of three compared controllers.Figure 3
The errorz2 of three compared controllers.The online estimationsθ^i of system parameters.
(a)
(b)The online updated fuzzy parametersλ^1.
(a)
(b)


## 6. Conclusion

The AFRC algorithm is proposed for a class of nonlinear systems which can be partly linearly parameterized and also have unknown system nonlinearities and external unknown disturbances. Essentially, AFRC is a combination of adaptive control and fuzzy control, and the algorithm is developed by using backstepping technique. By using small-gain approach, the closed-loop system is proved to be semiglobally uniformly ultimately bounded. The comparative simulation results demonstrate the effectiveness of the proposed AFRC algorithm. This proposed that AFRC effectively combines the technical characters of adaptive control and fuzzy control and improves the control performance by retaining the advantages of both control methods.



---

*Source: 201432-2013-05-07.xml*