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

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.

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 Section 2, 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 function 𝛼∶(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 function 𝛽∶(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 lim_𝑡→∞𝛽(𝑟,𝑡)=0.

Definition 3. For a nonlinear system 𝑥̇=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 ‖𝑥(𝑡)‖≤𝛽(‖𝑥(0)‖,𝑡)+𝛾𝑢_𝑡∞+𝑑, (1) where ‖𝑢_𝑡‖_∞=max_{0≤𝜏≤𝑡}|𝑢(𝜏)| and 𝑢(𝜏) is a truncated function defined as 𝑢_𝑡(𝜏)=𝑢(𝜏)0≤𝜏≤𝑡,0𝜏<0or𝜏>𝑡. (2) When 𝑑=0 in (1), the ISpS property becomes ISS property [21, 26].

Definition 4. A function 𝑉∈ℂ¹ is said to be an ISpS-Lyapunov function for the system 𝑥̇=𝑓(𝑥,𝑢) if(1)there exist class 𝐾_∞ functions 𝛼₁, 𝛼₂, such that 𝛼₁(‖𝑥‖)≤𝑉(𝑥)≤𝛼₂(‖𝑥‖)∀𝑥∈ℝ^𝑛, (3)(2)there exist class 𝐾 function 𝛼₃, 𝛼₄ and a constant 𝑑>0, such that 𝜕𝑉(𝑥)𝑓()𝜕𝑥𝑥,𝑢≤−𝛼₃(‖𝑥‖)+𝛼₄(‖𝑢‖)+𝑑. (4)

When 𝑑=0 in (4), 𝑉 is said to be an ISS-Lyapunov function [26]. Then nonlinear 𝐿_∞ gain 𝛾 in (1) can be chosen as 𝛾(𝑠)=𝛼₁(𝑠)⁻¹⋅𝛼₂(𝑠)⋅𝛼₃(𝑠)⁻¹⋅𝛼₄(𝑠)∀𝑠>0. (5)

Proposition 5. The nonlinear system 𝑥̇=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]: 𝑧^𝜔∶̇()(),𝑥=𝑓𝑡,𝑥,𝜔𝑧=𝐻𝑡,𝑥,𝜔_𝜔𝑧∶̇𝑧𝑧.𝑦=𝑔𝑡,𝑦,𝜔=𝐾𝑡,𝑦, (6) In particular, if there exist class 𝐾𝐿 functions 𝛽_𝜔, 𝛽_𝑧, class 𝐾 functions 𝛾_𝑧, 𝛾_𝜔, 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: ‖𝐻(𝑋(𝑥;𝜔,𝑡))‖≤𝛽_𝜔(‖𝑥(0)‖,𝑡)+𝛾_𝑧‖𝜔‖_∞+𝑑₁,𝐾𝑌𝑦;𝑧,𝑡≤𝛽_𝑧𝑦(0),𝑡+𝛾_𝜔‖𝑧‖_∞+𝑑₂. (7) If 𝛾_𝑧𝛾_𝜔(𝑠)<𝑠or𝛾_𝜔𝛾_𝑧(𝑠)<𝑠,∀𝑠>0, (8) 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: rule𝑅_𝑖∶IF𝑥₁is𝑀^𝑖₁and𝑥₂is𝑀^𝑖₂and,…,and𝑥_𝑛is𝑀^𝑖_𝑛,then𝑦_𝑖=𝑎_𝑖0+𝑎_𝑖1𝑥₁+𝑎_𝑖2𝑥₂+⋯+𝑎_𝑖𝑛𝑥_𝑛, (9) where 𝑀^𝑖_𝑗 are fuzzy subsets, 𝑎_𝑖0,𝑎_𝑖1,𝑎_𝑖2,…,𝑎_𝑖𝑛 denote designed constant parameters in fuzzy rule conclusion step, 𝑦_𝑖 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 𝑦_𝑓=^𝑞_𝑖=1𝑦_𝑓𝑖𝜉_𝑓𝑖(𝑥)=𝜉_𝑓(𝑥)𝐴_𝑓𝑥, (10) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇, 𝑥=[1,𝑥^𝑇]^𝑇, 𝐴_𝑓=𝑎₁₀𝑎₁₁⋯𝑎_1𝑛𝑎₂₀𝑎₂₁⋯𝑎_2𝑛𝑎⋮⋮⋮⋮_𝑞0𝑎_𝑞1⋯𝑎_𝑞𝑛, (11)𝜉_𝑓(𝑥) and 𝜉_𝑓𝑖(𝑥) are fuzzy basis vector and fuzzy basis function, respectively, and 𝜉_𝑓(𝑥)=[𝜉_𝑓1(𝑥),𝜉_𝑓2(𝑥),…,𝜉_𝑓𝑞(𝑥)], 𝜉_𝑓𝑖(𝑥)=∏^𝑛_𝑗=1𝜇_𝑀𝑖𝑗(𝑥_𝑗)/∑^𝑞_𝑖=1(∏^𝑛_𝑗=1𝜇_𝑀𝑖𝑗(𝑥_𝑗)). Usually, the membership function of fuzzy set 𝑀^𝑖_𝑗 is given by [26] 𝜇^𝑖_𝑥𝑗𝑥_𝑗−𝑔=exp^𝑖𝑗𝑥_𝑗−𝑏^𝑖_𝑗²2ℎ^𝑖_𝑗², (12) where 𝑔^𝑖_𝑗∈ℝ, 𝑏^𝑖_𝑗∈ℝ and ℎ^𝑖_𝑗>0 are adjustable parameters of 𝜇_𝑀𝑖𝑗(𝑥_𝑗), 𝑖=1,2,…,𝑞, 𝑗=1,2…,𝑛.

Lemma 7. For any continuous function 𝑓(𝑥) defined in a compact set 𝑥∈Ω_𝑥 and for all 𝑏>0, there exists an ideal fuzzy logic, such that [28] 𝑦−𝑓(𝑥)≤𝑏∀𝑥∈Ω_𝑥. (13)

2.2. Problem Formulation

We consider a class of SISO nonlinear systems given as 𝑥̇₁=𝑥₂+𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑓₁𝑥₁+Δ₁(),𝑥̇𝑥,𝑡₂=𝑥₃+𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑓₂𝑥₂+Δ₂(),⋮𝑥̇𝑥,𝑡_𝑖=𝑥_𝑖+1+𝜃^𝑇𝜑_𝑖𝑥_𝑖,𝑡+𝑓_𝑖𝑥_𝑖+Δ_𝑖(),⋮𝑥̇𝑥,𝑡_𝑛=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑓_𝑛(𝑥)+Δ_𝑛(𝑥,𝑡),𝑦=𝑥₁, (14) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈ℝ^𝑛 is system state vector, 𝑢∈ℝ is system input, and 𝑦∈ℝ is system output. Denote 𝑥_𝑖=[𝑥₁,𝑥₂,…,𝑥_𝑖]^𝑇, 1≤𝑖≤𝑛. 𝜃^𝑇𝜑_𝑖(𝑥_𝑖,𝑡) is the linearly parameterized system model where 𝜃=[𝜃₁,𝜃₂,…,𝜃_𝑝]^𝑇∈ℝ^𝑝 is the vector of unknown but constant parameters and 𝜑_𝑖(𝑥_𝑖,𝑡) is the vector of known system function. 𝑓_𝑖(𝑥_𝑖) is unknown but continuous system function which cannot be linearly parameterized. Δ_𝑖(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., 𝑥_𝑑⁽¹⁾,𝑥_𝑑⁽²⁾,…,𝑥_𝑑^(𝑛)). Control objective is to design a bounded control law for the input 𝑢, such that output 𝑦=𝑥₁ tracks 𝑥_𝑑(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 𝜃∈Ω_𝜃≜𝜃∶𝜃_min≤𝜃≤𝜃_max,Δ_𝑖(𝑥,𝑡)≤𝐷_𝑖𝑥_𝑖,𝑡, (15)(16) where 𝜃_min=[𝜃_1min,𝜃_2min,…,𝜃_𝑝min]^𝑇∈ℝ^𝑝 and 𝜃_max=[𝜃_1max,𝜃_2max,…,𝜃_𝑝max]^𝑇∈ℝ^𝑝 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<𝜃_max means that 𝜃_𝑗<𝜃_𝑗max, for all 𝑗).

Remark 9. When Δ_𝑖(𝑥,𝑡)=0, if we treat 𝑥_𝑖+1 as the control input of the 𝑥̇_𝑖 dynamics, the 𝑥̇_𝑖 dynamics would only depend on the states of its previous dynamics, that is, 𝑥₁,…,𝑥_𝑖. 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 𝐷_𝑖(𝑥_𝑖,𝑡) is required to be function of 𝑥_𝑗, 𝑗≤𝑖 and 𝑡 only, but Δ_𝑖(𝑥,𝑡) may contain some bounded functions of 𝑥_𝑗, 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. Let 𝜃 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 ̇𝜃𝜃=Proj(Γ𝜏), (17) 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 𝜃Proj(•) can be defined as 𝜃Proj(•)=0,if𝜃=𝜃_maxand•>00,if𝜃=𝜃_minand•<0•,otherwise. (18)

It can be shown that for any adaptation function 𝜏, the projection mapping used in (18) has the following properties [5]: (P1)𝜃∈Ω_𝜃≜𝜃∶𝜃_min≤𝜃≤𝜃_max,(P2)𝜃^𝑇Γ⁻¹𝜃Proj(Γ𝜏)−𝜏≤0,∀𝜏. (19)

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 contains 𝑛 steps. At each step, an intermediate control signal 𝑢_𝑖 will be developed by using an appropriate Lyapunov function 𝑉_𝑖.

Step 1. Consider the first subsystem of (14), which is given as 𝑥̇₁=𝑥₂+𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑓₁𝑥₁+Δ₁(𝑥,𝑡). (20)

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 𝑥_𝑑(𝑡). Define the tracking error variables as 𝑧₁=𝑥₁−𝑥_𝑑. (21)

The time derivative of 𝑧₁ can be given as 𝑧̇₁=𝑥̇₁−𝑥_𝑑⁽¹⁾=𝑥₂+𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑓₁𝑥₁+Δ₁(𝑥,𝑡)−𝑥_𝑑⁽¹⁾. (22)

Since 𝑓₁(𝑥₁) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, 𝑓₁(𝑥₁) can be expressed as 𝑓₁𝑥₁=𝜉₁𝐴₁1,𝑥₁^𝑇+𝛿₁=𝜉₁𝐴₁₁𝑧₁+𝜉₁𝐴₁₂1,𝑥_𝑑^𝑇+𝛿₁, (23) where𝐴₁=𝑎¹₁₀𝑎¹₁₁𝑎¹₂₀𝑎¹₂₁𝑎⋮⋮_1𝑞0𝑎¹_𝑞1,𝐴₁₁=𝑎¹₁₁𝑎¹₂₁⋮𝑎¹_𝑞1,𝐴₁₂=𝑎¹₁₀𝑎¹₁₁𝑎¹₂₀𝑎¹₂₁𝑎⋮⋮_1𝑞0𝑎¹_𝑞1, (24)𝜉₁=[𝜉₁₁,𝜉₁₂,…,𝜉_1𝑞], 𝛿₁ represents fuzzy approximating error, and |𝛿₁|≤𝑏₁; here 𝑏₁ is unknown but bounded positive constant.

Let 𝑐₁=𝑛‖𝐴₁₁‖, 𝐴^∗₁=𝑐₁⁻¹𝐴₁₁. It is clear that ‖𝐴^∗₁‖≤1/𝑛; then let 𝜔₁=𝐴^∗₁𝑧₁, and (23) can be rewritten as 𝑓₁𝑥₁=𝑐₁𝜉₁𝜔₁+𝜉₁𝐴₁₂1,𝑥_𝑑^𝑇+𝛿₁. (25)

Define the 2th tracking error variable as 𝑧₂=𝑥₂−𝑢₁. (26)

Then substitute (25) into (22), 𝑧̇₁=𝑧₂+𝑢₁+𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑐₁𝜉₁𝜔₁+𝑑₁, (27) where 𝑑₁=𝜉₁𝐴₁₂1,𝑥_𝑑^𝑇+𝛿₁+Δ₁(𝑥,𝑡)−𝑥_𝑑⁽¹⁾,𝑑₁≤𝐵₁𝑞₁, (28) where 𝐵₁=max{‖𝐴₁₂‖,𝑏₁,1} is a positive constant and 𝑞₁=‖𝜉₁‖⋅‖[1,𝑥_𝑑]^𝑇‖+1+𝐷₁(𝑥₁,𝑡)+|𝑥_𝑑⁽¹⁾|.

The intermediate control law 𝑢₁ consists of three parts as 𝑢₁=𝑢_1𝑎+𝑢_1𝑓+𝑢_1𝑟. (29)

At first, we design the adaptive control item as 𝑢_1𝑎=−𝑘₁𝑧₁−𝜃^𝑇𝜑₁𝑥₁,𝑡. (30)

Substituting (29) and (30) into (22) leads to 𝑧̇₁=𝑧₂+𝑢_1𝑓+𝑢_1𝑟−𝑘₁𝑧₁−𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑓₁𝑥₁+Δ₁(𝑥,𝑡). (31)

Then, we design the fuzzy control law 𝑢_1𝑓 as 𝜇_1𝑓=−𝜆₁𝜓₁𝑧₁,̇𝜆₁=𝜌₁𝜓₁𝑧²₁−𝜂₁𝜆₁−𝜆₁. (32) where 𝜓₁=‖𝜉₁‖²/(4𝛾²₁), 𝜆₁=𝑐²₁, and 𝜆₁ is an online estimate of parameter 𝜆₁ with estimate error 𝜆₁=𝜆₁−𝜆₁. 𝛾₁>0, 𝜌₁>0, and 𝜂₁>0 are positive design parameters.

Define a positive semidefinite (p.s.d.) function 𝑉₁ as 𝑉₁=12𝑧²₁+12𝜌₁⁻¹𝜆₁². (33)

Considering 𝜆̇₁=−𝜆̇₁, time derivative of 𝑉₁ can be given as 𝑉̇₁=𝑧₁𝑧̇₁−𝜌₁⁻¹𝜆₁𝜆̇₁=𝑧₁𝑧₂+𝑧₁𝑢_1𝑓+𝑐₁𝜉₁𝜔₁𝑧₁−𝜌₁⁻¹𝜆₁̇𝜆₁−𝑘₁𝑧²₁+𝑧₁𝑢_1𝑟−𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑑₁. (34)

Applying Young’s inequality, the following inequality can be obtained: 𝑐₁𝜉₁𝜔₁𝑧₁≤𝑐₁𝜉₁𝜔₁𝑧₁≤𝑐²₁𝜉₁²𝑧²₁4𝛾²₁+2𝛾²₁𝜔₁²2=𝑐²₁𝜉₁²4𝛾²₁𝑧²₁+𝛾²₁𝜔₁². (35)

Considering that 𝜆₁=𝑐²₁ and 𝜓₁=‖𝜉₁‖²/(4𝛾²₁), from (35), the following inequality is holding: 𝑐₁𝜉₁𝜔₁𝑧₁≤𝜆₁𝜓₁𝑧²₁+𝛾²₁𝜔₁²=𝜆₁𝜓₁𝑧²₁+𝜆₁𝜓₁𝑧²₁+𝛾²₁𝜔₁². (36)

Substituting (32) and (36) into (34) leads to 𝑉̇₁≤𝑧₁𝑧₂−𝑘₁𝑧²₁−𝜂₁𝜆²₁+𝛾²₁𝜔₁²+𝑧₁𝑢_1𝑟−𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑑₁. (37)

Now we can get the following robust performance conditions for 𝑢_1𝑟: condition(i)𝑧₁𝑢_1𝑟−𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑑₁≤𝜀₁,condition(ii)𝑧₁𝑢_1𝑟≤0. (38)

Remark 10. One smooth example of robust control function 𝑢_1𝑟 satisfying (38) can be found in the following way. Let ℎ₁ be any smooth function satisfying ℎ₁≥||𝜃_𝑀||||𝜑₁(𝑥₁,𝑡)||+𝐵₁𝑞₁ where 𝜃_𝑀=𝜃_max−𝜃_min. 𝜀₁>0 is positive design parameter.

Then we can design a robust control law 𝑢_1𝑟 as 𝑢_1𝑟1=−4𝜀₁ℎ²₁𝑧₁. (39)

Thus, (37) can be given as 𝑉̇₁≤𝑧₁𝑧₂−𝑘₁𝑧²₁−𝜂₁𝜆²₁+𝛾²₁𝜔₁²+𝜀₁. (40)

Step 2. Consider the second subsystem of (14), which is given as 𝑥̇₂=𝑥₃+𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑓₂𝑥₂+Δ₂(𝑥,𝑡). (41)

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 of 𝑧₂ can be given as 𝑧̇₂=𝑥̇₂−𝑢̇₁=𝑥₃+𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑓₂𝑥₂+Δ₂(𝑥,𝑡)−𝑢̇₁. (42) And the time derivative of 𝑢₁ is 𝑢̇₁=𝜕𝑢₁𝜕𝑥₁𝑥̇₁+𝜕𝑢₁𝜕𝜃̇𝜃+𝜕𝑢₁𝜕𝜆₁̇𝜆₁+𝜕𝑢₁𝜕𝑥_𝑑𝑥_𝑑⁽¹⁾. (43)

Then substituting (43) into (42) leads to 𝑧̇₂=𝑥₃+𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑓^′₂𝑥₂+Δ₂(𝑥,𝑡)+𝜕𝑢₁𝜕𝑥_𝑑𝑥_𝑑⁽¹⁾, (44) where 𝑓^′₂𝑥₂=𝑓₂𝑥₂−𝜕𝑢₁𝜕𝑥₁𝑥̇₁−𝜕𝑢₁𝜕𝜃̇𝜃−𝜕𝑢₁𝜕𝜆₁̇𝜆₁. (45)

Since 𝑓^′₂(𝑥₂) is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, 𝑓^′₂(𝑥₂) can be expressed as 𝑓^′₂𝑥₂=𝜉₂𝐴₂1,𝑧₁,𝑥₂,𝑥_𝑑,𝑥_𝑑^(1)𝑇+𝛿₂=𝜉₂𝐴₂₁𝑧₁,𝑧₂^𝑇+𝜉₂𝐴₂₂1,𝑢₁,𝑥_𝑑,𝑥_𝑑^(1)𝑇+𝛿₂, (46) where 𝐴₂=𝑎¹₁₀𝑎¹₁₁⋯𝑎¹₁₄𝑎¹₂₀𝑎¹₂₁⋯𝑎¹₂₄𝑎⋮⋮⋱⋮_1𝑞0𝑎¹_𝑞1⋯𝑎¹_𝑞4,𝐴₂₁=𝑎¹₁₁𝑎¹₁₂𝑎¹₂₁𝑎¹₂₂𝑎⋮⋮_1𝑞1𝑎¹_𝑞2,𝐴₂₂=𝑎¹₁₀𝑎¹₁₂𝑎¹₁₃𝑎¹₁₄𝑎¹₂₀𝑎¹₂₂𝑎¹₂₃𝑎¹₂₄𝑎⋮⋮⋮⋮_1𝑞0𝑎¹_𝑞2𝑎¹_𝑞3𝑎¹_𝑞4, (47)𝜉₂=[𝜉₂₁,𝜉₂₂,…,𝜉_2𝑞], 𝛿₂ represents fuzzy approximating error, and |𝛿₂|≤𝑏₂; here 𝑏₂ is unknown but bounded positive constant.

Let 𝑐₂=𝑛‖𝐴₂₁‖, 𝐴^∗₂=𝑐₂⁻¹𝐴₂₁. It is clear that ‖𝐴^∗₂‖≤1/𝑛; then let 𝜔₂=𝐴^∗₂𝑧₂, and (23) can be rewritten as 𝑓^′₂𝑥₂=𝑐₂𝜉₂𝜔₂+𝜉₂𝐴₂₂1,𝑢₁,𝑥_𝑑,𝑥_𝑑^(1)𝑇+𝛿₂. (48)

Define the 3th tracking error variable as 𝑧₃=𝑥₃−𝑢₂. (49)

Substituting (48) and (49) into (44), we have 𝑧̇₂=𝑧₃+𝑢₂+𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑐₂𝜉₂𝜔₂+𝑑₂, (50) where 𝑑₂=𝜉₂𝐴₂₂1,𝑢₁,𝑥_𝑑,𝑥_𝑑^(1)𝑇+𝛿₂+Δ₂(𝑥,𝑡)+𝜕𝑢₁𝜕𝑥_𝑑𝑥_𝑑⁽¹⁾,𝑑₂≤𝐵₂𝑞₂, (51) where 𝐵₂=max{‖𝐴₂₂‖,𝑏₂,1} is a positive constant and 𝑞₂=‖𝜉₂‖⋅‖[1,𝑢₁,𝑥_𝑑,𝑥_𝑑⁽¹⁾]^𝑇‖+1+𝐷₂(𝑥₂,𝑡)+|(𝜕𝑢₁/𝜕𝑥_𝑑)𝑥_𝑑⁽¹⁾|.

The control law 𝑢₂ consists of three parts as 𝑢₂=𝑢_2𝑎+𝑢_2𝑓+𝑢_2𝑟. (52)

At first, we design the adaptive control item as 𝑢_2𝑎=−𝑘₂𝑧₂−𝜃^𝑇𝜑₂𝑥₂,𝑡. (53)

Substituting (52) and (53) into (50) leads to 𝑧̇₂=𝑧₃+𝑢_2𝑓+𝑢_2𝑟−𝑘₂𝑧₂−𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑓^′₂𝑥₂+Δ₂(𝑥,𝑡). (54)

Then, we design the fuzzy control law 𝑢_2𝑓 as 𝑢_2𝑓=−𝜆₂𝜓₂𝑧₂,̇𝜆₂=𝜌₂𝜓₂𝑧²₂−𝜂₂𝜆₂−𝜆₂, (55) where 𝜓₂=‖𝜉₂‖²/(4𝛾²₂), 𝜆₂ is an online estimate of parameter 𝜆₂ with estimate error 𝜆₂=𝜆₂−𝜆₂. 𝛾₂>0, 𝜌₂>0, and 𝜂₂>0 are positive design parameters.

Define a positive semidefinite (p.s.d.) function 𝑉₂ as 𝑉₂=12𝑧²₂+12𝜌₂⁻¹𝜆²₂. (56)

Considering 𝜆̇₂=−𝜆̇₂, time derivative of 𝑉₂ can be given as 𝑉̇₂=𝑧₂𝑧̇₂−𝜌₂⁻¹𝜆₂𝜆̇₂=𝑧₂𝑧₃+𝑧₂𝑢_2𝑓+𝑐₂𝜉₂𝜔₂𝑧₂−𝜌₂⁻¹𝜆₂̇𝜆₂−𝑘₂𝑧²₂+𝑧₂𝑢_2𝑟−𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑑₂. (57)

Applying Young’s inequality, the following inequality can be obtained: 𝑐₂𝜉₂𝜔₂𝑧₂≤𝑐₂𝜉₂𝜔₂𝑧₂≤𝑐²₂𝜉₂²𝑧²₂4𝛾²₂+2𝛾²₂𝜔₂²2=𝑐²₂𝜉₂²𝑧²₂4𝛾²₂+𝛾²₂𝜔₂². (58)

Considering that 𝜆₂=𝑐²₂ and 𝜓₂=‖𝜉₂‖²/(4𝛾²₂), from (58), the following inequality is holding: 𝑐₂𝜉₂𝜔₂𝑧₂≤𝜆₂𝜓₂𝑧²₂+𝛾²₂𝜔₂²=𝜆₂𝜓₂𝑧²₂+𝜆₂𝜓₂𝑧²₂+𝛾²₂𝜔₂². (59)

Substituting (55) and (59) into (57) leads to 𝑉̇₂≤𝑧₂𝑧₃−𝑘₂𝑧²₂−𝜂₂𝜆²₂+𝛾²₂𝜔₂²+𝑧₂𝑢_2𝑟−𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑑₂. (60)

Now we can get the following robust performance conditions for 𝑢_2𝑟: condition(i)𝑧₂𝑢_2𝑟−𝜃^𝑇𝜑₂𝑥₂,𝑡+𝑑₂≤𝜀₂,condition(ii)𝑧₂𝑢_2𝑟≤0. (61)

According to Remark 10, we can design a robust control law 𝑢_2𝑟 as 𝑢_2𝑟1=−4𝜀₂ℎ²₂𝑧₂, (62) where ℎ₂ can be any smooth function satisfying ℎ₂≥||𝜃_𝑀||||𝜑₂(𝑥₂,𝑡)||+𝐵₂𝑞₂ and 𝜀₂>0 is positive design parameter.

Thus, (60) can be given as 𝑉̇₂≤𝑧₂𝑧₃−𝑘₂𝑧²₂−𝜂₂𝜆²₂+𝛾²₂𝜔₂²+𝜀₂. (63)

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 𝑥_𝑖=𝑥_𝑖+1+𝜃^𝑇𝜑_𝑖𝑥_𝑖,𝑡+𝑓_𝑖𝑥_𝑖+Δ_𝑖(𝑥,𝑡). (64)

We can design a control law 𝑢_𝑖 for subsystem (64) as 𝑢_𝑖=𝑢_𝑖𝑎+𝑢_𝑖𝑓+𝑢_𝑖𝑟,𝑢_𝑖𝑎=−𝑘_𝑖𝑧_𝑖−𝜃^𝑇𝜑_𝑖𝑥_𝑖,𝜇,𝑡_𝑖𝑓=−𝜆_𝑖𝜓_𝑖𝑧_𝑖,̇𝜆_𝑖=𝜌_𝑖𝜓_𝑖𝑧²_𝑖−𝜂_𝑖𝜆_𝑖−𝜆_𝑖,𝑢_𝑖𝑟1=−4𝜀_𝑖ℎ²_𝑖𝑧_𝑖. (65)

Define a p.s.d. function 𝑉_𝑖 as 𝑉_𝑖=𝑉_𝑖−1+12𝑧²_𝑖+12𝜌_𝑖⁻¹𝜆²_𝑖. (66)

By using the same technique, the time derivative of 𝑉_𝑖 can be given as 𝑉_𝑖≤𝑉_𝑖−1+𝑧_𝑖𝑧_𝑖+1−𝑘_𝑖𝑧²_𝑖−𝜂_𝑖𝜆²_𝑖+𝛾²_𝑖𝜔_𝑖²+𝜀_𝑖. (67)

Step n. Consider the 𝑛th subsystem of (14), which is given as 𝑥̇_𝑛=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑓_𝑛(𝑥)+Δ_𝑛(𝑥,𝑡). (68)

Define the tracking error variable as 𝑧_𝑛=𝑥_𝑛−𝑢_𝑛−1. (69)

The time derivative of 𝑧_𝑛 can be given as 𝑧̇_𝑛=𝑥̇_𝑛−𝑢̇_𝑛−1=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑓_𝑛(𝑥)+Δ_𝑛(𝑥,𝑡)−𝑢̇_𝑛−1. (70)

The time derivative of 𝑢_𝑛−1 is 𝑢̇_𝑛−1=^𝑛−1_𝑗=1𝜕𝑢_𝑛−1𝜕𝑥_𝑗𝑥̇_𝑗+𝜕𝑢_𝑛−1𝜕𝜃̇𝜃+𝜕𝑢_𝑛−1𝜕𝜆_𝑛−1̇𝜆_𝑛−1+^𝑛−1_𝑗=1𝜕𝑢_𝑛−1𝜕𝑥_𝑑^(𝑗−1)𝑥_𝑑^(𝑗). (71)

Substituting (71) into (70) leads to 𝑧̇_𝑛=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑓^′_𝑛𝑥_𝑛+Δ_𝑛(𝑥,𝑡)+^𝑛−1_𝑗=1𝜕𝑢_𝑛−1𝜕𝑥_𝑑^(𝑗−1)𝑥_𝑑^(𝑗), (72) where 𝑓^′_𝑛𝑥_𝑛=^𝑛−1_𝑗=1𝜕𝑢_𝑛−1𝜕𝑥_𝑗𝑥̇_𝑗+𝜕𝑢_𝑛−1𝜕𝜃̇𝜃+𝜕𝑢_𝑛−1𝜕𝜆_𝑛−1̇𝜆_𝑛−1+𝑓_𝑛𝑥_𝑛. (73)

According to fuzzy logic theorem as mentioned in Remark 10, (73) can be expressed as 𝑓^′_𝑛𝑥_𝑛=𝜉_𝑛𝐴_𝑛1,𝑧₁,…,𝑧_𝑛−1,𝑥_𝑛,𝑥_𝑑,…,𝑥_𝑑^{(𝑛−1)𝑇}+𝛿_𝑛=𝜉_𝑛𝐴_𝑛1𝑧₁,…,𝑧_𝑛^𝑇+𝜉_𝑛𝐴_𝑛21,𝑢_𝑛−1,𝑥_𝑑,…,𝑥_𝑑^{(𝑛−1)𝑇}+𝛿_𝑛, (74) where 𝐴_𝑛=𝑎^𝑛₁₀𝑎^𝑛₁₁⋯𝑎^𝑛_1(2𝑛)𝑎^𝑛₂₀𝑎^𝑛₂₁⋯𝑎^𝑛_2(2𝑛)𝑎⋮⋮⋱⋮_𝑛𝑞0𝑎^𝑛_𝑞1⋯𝑎^𝑛_𝑞(2𝑛),𝐴_𝑛1=𝑎^𝑛₁₁𝑎^𝑛₁₂⋯𝑎^𝑛_1𝑛𝑎^𝑛₂₁𝑎^𝑛₂₂⋯𝑎^𝑛_2𝑛𝑎⋮⋮⋱⋮_𝑛𝑞1𝑎^𝑛_𝑞2⋯𝑎^𝑛_𝑞𝑛,𝐴_𝑛2=𝑎^𝑛₁₀𝑎^𝑛_1𝑛𝑎^𝑛_1(𝑛+1)⋯𝑎^𝑛_1(2𝑛)𝑎^𝑛₂₀𝑎^𝑛_2𝑛𝑎^𝑛_2(𝑛+1)⋯𝑎^𝑛_2(2𝑛)𝑎⋮⋮⋮⋱⋮_𝑛𝑞0𝑎^𝑛_𝑞𝑛𝑎^𝑛_𝑞(𝑛+1)⋯𝑎^𝑛_𝑞(2𝑛), (75)𝜉_𝑛=[𝜉_𝑛1,𝜉_𝑛2,…,𝜉_𝑛𝑞], 𝛿_𝑛 represents fuzzy approximating error, and |𝛿_𝑛|≤𝑏_𝑛, and here 𝑏_𝑛 is unknown but bounded positive constant.

Let 𝑐_𝑛=𝑛‖𝐴_𝑛1‖, 𝐴^∗_𝑛=𝑐_𝑛⁻¹𝐴_𝑛1. It is clear that ‖𝐴^∗_𝑛‖≤1/𝑛; then let 𝜔_𝑛=𝐴^∗_𝑛[𝑧₁,…,𝑧_𝑛]^𝑇, and (74) can be rewritten as 𝑓^′_𝑛𝑥_𝑛=𝑐_𝑛𝜉_𝑛𝜔_𝑛+𝜉_𝑛𝐴_𝑛2𝑢₁,𝑥_𝑑,…,𝑥_𝑑^{(𝑛−1)𝑇}+𝛿_𝑛. (76)

Substitute (76) into (72), we have 𝑧̇_𝑛=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑐_𝑛𝜉_𝑛𝜔_𝑛+𝑑_𝑛, (77) where 𝑑_𝑛=𝜉_𝑛𝐴_𝑛21,𝑢_𝑛−1,𝑥_𝑑,…,𝑥_𝑑^{(𝑛−1)𝑇}+𝛿_𝑛+Δ_𝑛()+𝑥,𝑡_{𝑛−1𝑗=1}𝜕𝑢_𝑛−1𝜕𝑥_𝑑^(𝑗−1)𝑥_𝑑^(𝑗),𝑑_𝑛≤𝐵_𝑛𝑞_𝑛, (78) where 𝐵_𝑛=max{‖𝐴_𝑛2‖,𝑏_𝑛,1} is a positive constant and 𝑞₂=‖𝜉_𝑛‖⋅‖[1,𝑢_𝑛−1,𝑥_𝑑,…,𝑥_𝑑^(𝑛−1)]^𝑇‖+1+𝐷_𝑛(𝑥_𝑛,𝑡)+‖∑^𝑛−1_𝑗=1((𝜕𝑢_𝑛−1/𝜕𝑥_𝑑^(𝑗−1))𝑥_𝑑^(𝑗))‖.

The control law 𝑢 consists of three parts giving as 𝑢=𝑢_𝑛=𝑢_𝑛𝑎+𝑢_𝑛𝑓+𝑢_𝑛𝑟. (79)

At first, we design the adaptive control item as 𝑢_𝑛𝑎=−𝑘_𝑛𝑧_𝑛−𝜃^𝑇𝜑_𝑛𝑥_𝑛,𝑡. (80)

Substituting (79) and (80) into (77) leads to 𝑧̇_𝑛=𝑢_𝑛𝑓+𝑢_𝑛𝑟−𝑘_𝑛𝑧_𝑛−𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑐_𝑛𝜉_𝑛𝜔_𝑛+𝑑_𝑛. (81)

Design the fuzzy control law 𝑢_𝑛𝑓 as 𝜇_𝑛𝑓=−𝜆_𝑛𝜓_𝑛𝑧_𝑛,̇𝜆_𝑛=𝜌_𝑛𝜓_𝑛𝑧²_𝑛−𝜂_𝑛𝜆_𝑛−𝜆_𝑛, (82) where 𝜓_𝑛=‖𝜉_𝑛‖²/(4𝛾²_𝑛), 𝜆_𝑛=𝑐²_𝑛, and 𝜆_𝑛 is an online estimate of parameter 𝜆_𝑛 with estimate error 𝜆_𝑛=𝜆_𝑛−𝜆_𝑛. 𝛾_𝑛>0, 𝜌_𝑛>0 and 𝜂_𝑛>0 are positive design parameters.

Define a p.s.d. function 𝑉_𝑛 as 𝑉_𝑛=𝑉_𝑛−1+12𝑧²_𝑛+12𝜌_𝑛⁻¹𝜆²_𝑛. (83)

Considering 𝜆̇_𝑛=−𝜆̇_𝑛, the time derivative of 𝑉_𝑛 can be given as 𝑉̇_𝑛=𝑉̇_𝑛−1+𝑧_𝑛𝑧̇_𝑛−𝛾_𝑛⁻¹𝜆_𝑛𝜆̇_𝑛=𝑉̇_𝑛−1+𝑧_𝑛𝑢_𝑛𝑓+𝑐_𝑛𝜉_𝑛𝜔_𝑛𝑧_𝑛−𝜌_𝑛⁻¹𝜆_𝑛̇𝜆_𝑛−𝑘_𝑛𝑧²_𝑛+𝑧_𝑛𝑢_𝑛𝑟−𝜃^𝑇𝜑_𝑛𝑥_𝑛,𝑡+𝑑_𝑛. (84)

Applying Young’s inequality, the following inequality can be obtained: 𝑐_𝑛𝜉_𝑛𝜔_𝑛𝑧_𝑛≤𝑐_𝑛𝜉_𝑛𝜔_𝑛𝑧_𝑛≤𝑐²_𝑛𝜉_𝑛²𝑧²_𝑛4𝛾²_𝑛+2𝛾²_𝑛𝜔_𝑛²2=𝑐²_𝑛𝜉_𝑛²𝑧²_𝑛4𝛾²_𝑛+𝛾²_𝑛𝜔_𝑛². (85)

Considering that 𝜆_𝑛=𝑐²_𝑛 and 𝜓_𝑛=‖𝜉_𝑛‖²/(4𝛾²_𝑛), from (85), the following inequality is holding: 𝑐_𝑛𝜉_𝑛𝜔_𝑛𝑧_𝑛≤𝜆_𝑛𝜓_𝑛𝑧²_𝑛+𝛾²_𝑛𝜔_𝑛²=𝜆_𝑛𝜓_𝑛𝑧²_𝑛+𝜆_𝑛𝜓_𝑛𝑧²_𝑛+𝛾²_𝑛𝜔_𝑛². (86)

Substituting (82) and (86) into (84) leads to 𝑉̇_𝑛≤𝑉̇_𝑛−1−𝑘_𝑛𝑧²_𝑛−𝜂_𝑛𝜆²_𝑛+𝛾²_𝑛𝜔_𝑛²+𝑧_𝑛𝑢_𝑛𝑟−𝜃^𝑇𝜑_𝑛𝑥_𝑛,𝑡+𝑑_𝑛. (87)

Now we can get the following robust performance conditions for 𝑢_𝑛𝑟: condition(i)𝑧_𝑛𝑢_𝑛𝑟−𝜃^𝑇𝜑_𝑛𝑥_𝑛,𝑡+𝑑_𝑛≤𝜀_𝑛,condition(ii)𝑧_𝑛𝑢_𝑛𝑟≤0. (88)

Then we can design a robust control law 𝑢_𝑛𝑟 as 𝑢_𝑛𝑟1=−4𝜀_𝑛ℎ²_𝑛𝑧_𝑛, (89) where ℎ_𝑛 can be any smooth function satisfying ℎ_𝑛≥||𝜃_𝑀||||𝜑_𝑛(𝑥_𝑛,𝑡)||+𝐵_𝑛𝑞_𝑛 and 𝜀_𝑛 is positive parameter.

Thus, (87) can be given as 𝑉̇_𝑛≤𝑉̇_𝑛−1−𝑘_𝑛𝑧²_𝑛−𝜂_𝑛𝜆²_𝑛+𝛾²_𝑛𝜔_𝑛²+𝜀_𝑛. (90)

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=𝑢_𝑛𝑎+𝑢_𝑛𝑓+𝑢_𝑛𝑟 with the intermediate control law 𝑢_𝑖, 𝑖=1,2,…,𝑛−1, and adaptive laws for 𝜃_𝑖 and 𝜆_𝑖, 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, Σ_𝑧𝜔 and Σ_𝜔𝑧 as follows: 𝑧̇₁=𝑧₂+𝑢₁+𝜃^𝑇𝜑₁𝑥₁,𝑡+𝑐₁𝜉₁𝜔₁+𝑑₁,𝑧̇_𝑖=𝑧_𝑖+1+𝑢_𝑖+𝜃^𝑇𝜑_𝑖𝑥_𝑖,𝑡+𝑐_𝑖𝜉_𝑖𝜔_𝑖+𝑑_𝑖,𝑧̇_𝑛=𝑢+𝜃^𝑇𝜑_𝑛(𝑥,𝑡)+𝑐_𝑛𝜉_𝑛𝜔_𝑛+𝑑_𝑛,̇𝜆₁=𝜌₁𝜓₁𝑧²₁−𝜂₁𝜆₁−𝜆₁,̇𝜆_𝑖=𝜌_𝑖𝜓_𝑖𝑧²_𝑖−𝜂_𝑖𝜆_𝑖−𝜆_𝑖,̇𝜆_𝑛=𝜌_𝑛𝜓_𝑛𝑧²_𝑛−𝜂_𝑛𝜆_𝑛−𝜆_𝑛,𝑍=𝐻(𝑍)=𝑍, (91) where 𝜔=[𝜔₁,…,𝜔_𝑛]^𝑇 is given as input of subsystem Σ_𝑧𝜔 and 𝑍 is output, 𝑍=[𝑧^𝑇,𝜆^𝑇]^𝑇, 𝑧=[𝑧₁,…,𝑧_𝑛]^𝑇, and 𝜆=[𝜆₁,…,𝜆_𝑛]^𝑇.
Design a Lyapunov function as 𝑉=𝑉_𝑛. (92)
Choosing 𝛾=(𝛾²₁+⋯+𝛾²_𝑛)^1/2, noting the time derivative of 𝑉 can be obtained as 𝑉̇≤−‖𝑍‖²+𝛾²‖𝜔‖²+𝜀, (93) where 𝜀=^𝑛_𝑖=1𝜀_𝑖. (94)
According to small gain theorem [8, 13, 21], the subsystem Σ_𝑧𝜔 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 Σ_𝑧𝜔 can be obtained as 𝛾_̃‌z(𝑠)=𝛼₁⁻¹⋅𝛼₂⋅𝛼₃⁻¹⋅𝛼₄(𝑠)∀𝑠>0. (95)
Consider subsystem Σ_𝜔𝑧 as 𝜔₁=𝐴^∗₁𝑧₁,𝜔_𝑖=𝐴^∗_𝑖𝑧_𝑖=𝐴^∗_𝑖𝑧₁,…,𝑧_𝑖^𝑇,𝜔_𝑛=𝐴^∗_𝑛𝑧_𝑛=𝐴^∗_𝑛𝑧₁,…,𝑧_𝑛^𝑇,⟹𝜔=𝐴𝑧=𝐴Ζ=𝐴Ζ, (96) where 𝑍 is the input of subsystem, output is 𝜔=[𝜔₁,…,𝜔_𝑛]^𝑇. 𝑧=[𝑧₁,…,𝑧_𝑛]^𝑇, 𝜆=[𝜆₁,…,𝜆_𝑛]^𝑇, Ζ=𝑍=[𝑧^𝑇,𝜆^𝑇]^𝑇, and 𝐴=[𝐴,0]. Considering 𝐴^∗_𝑖=[𝐴_𝑖^∗1𝐴_𝑖^∗2⋯𝐴_𝑖^∗𝑖] and ‖𝐴^∗_𝑖‖≤1/𝑛, 𝑖=1,2,…,𝑛, we have 𝐴=‖𝐴‖=𝐴^∗₁𝐴0⋯0_2∗1𝐴₂^∗2𝐴⋯0⋮⋮⋱⋮_𝑛∗1𝐴_𝑛^∗2⋯𝐴_𝑛^∗𝑛=𝐴^∗₁²+𝐴^∗₂²+⋯+𝐴^∗_𝑛^21/2≤1. (97) Let 𝛾^′=‖𝐴‖, ‖𝜔‖≤𝐴‖Ζ‖=𝛾^′Ζ. (98)
Therefore 𝛾^′≤1, the gain of subsystem Σ_𝜔𝑧𝛾_𝜔=𝛾^′≤1.
According to small gain theorem, if 𝛾_𝑧(𝛾_𝜔(𝑠))<𝑠, then the closed-loop system (91) is ISpS. Then, 𝛾_𝑧𝛾_𝜔(𝑠)<𝑠⟹𝛾𝛾^′<1. (99)
Because 0<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 𝑧(𝑡)^𝑇,𝜆(𝑡)^𝑇𝑇≤𝛽𝑧(0)^𝑇,𝜆(0)^𝑇𝑇,𝑡+𝛿₀. (100)
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 Σ_𝑧𝜔 is ISS, and the closed-loop system is also ISS. Therefore, there exists a class 𝐾𝐿 function, such that 𝑧(𝑡)^𝑇,𝜆(𝑡)^𝑇𝑇≤𝛽𝑧(0)^𝑇,𝜆(0)^𝑇𝑇,𝑡. (101)
That means lim_𝑡→∞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: 𝑑̇=−𝑑+0.25𝑥₁,𝑥̇₁=𝑥₂+𝜃₁𝑥₁+𝑓₁𝑥₁+Δ₁(),𝑥̇𝑥,𝑡₂=𝑢+𝜃₂𝑥₂+𝑓₂𝑥₂+Δ₂(𝑥,𝑡),𝑦=𝑥₁, (102) 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(𝑥₁) and 𝑓₂(𝑥₂) 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(𝑥₁)=−0.6𝑥²₁, 𝑓₂(𝑥₂)=−0.02𝑥₁𝑥₂, Δ₁(𝑥,𝑡)=−2.0𝑑², and Δ₂(𝑥,𝑡)=−3.0𝑑−0.1𝑑̇.

Let the desired trajectory as 𝑥_𝑑(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 𝑥_𝑑 as closely as possible.

Figure 1 

The desired trajectory.

Define fuzzy membership functions for each error dynamics 𝑧₁, 𝑧₂ as 𝜇^𝑖_𝑧1𝑧₁=exp−50𝑧₁−(𝑖−4)²2×0.4247²𝜇,𝑖=1,2,…,7,_𝑖𝑧2𝑧₂=exp−10𝑧₂−(𝑖−4)²2×0.4247²,𝑖=1,2,…,7. (103) The first intermediate control function 𝑢₁ can be given as 𝑢₁=𝑢_1𝑎+𝑢_1𝑓+𝑢_1𝑟=−40𝑧₁−𝜃₁𝑥₁−𝜆₁𝜓₁𝑧₁−14ℎ²₁𝑧₁. (104) Then we design the controller law as 𝑢=𝑢₂=𝑢_2𝑎+𝑢_2𝑓+𝑢_2𝑟=−40𝑧₂−𝜃₂𝑥₂−𝜆₂𝜓₂𝑧₂−1ℎ16²²𝑧₂, (105) where 𝜓_𝑖=‖𝜉_𝑖‖²/(4∗0.5²) and ℎ_𝑖=||𝜃_𝑀||||𝑥_𝑖||+𝐵_𝑖𝑞_𝑖, 𝑖=1,2. The discontinuous projection-type parameter adaptation laws are given as follows: ̇𝜃₁𝜃=Proj₁0.6𝑥₁𝑧₁,̇𝜃₂𝜃=Proj₂0.005𝑥₂𝑧₂ (106) with lower bound 𝜃_min=[−0.6,−0.04]^𝑇 and upper bound 𝜃_max=[−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 parameter 𝜆₁ and 𝜆₂ are given as ̇𝜆₁=200𝜓₁𝑧²₁−0.01𝜆₁,̇𝜆−50₂=100𝜓₂𝑧²₂−0.05𝜆₂−200. (107)

In the simulation, three controllers are compared. (C1): the proposed adaptive fuzzy robust controller as described previously. (C2): by setting 𝑢_𝑖𝑓=0, the controller is equivalent to the adaptive robust control law in [6]. (C3): by setting 𝑢_𝑖𝑎=0, the controller becomes a fuzzy controller.

Simulation results are shown in Figures 2–5. 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 𝜆₁, and the figures indicate that both of them are all bounded.

Figure 2 

The output tracking error 𝑧1 of three compared controllers.

Figure 3 

The error 𝑧2 of three compared controllers.

(a)

(b)

Figure 4 

The online estimations 𝜃𝑖 of system parameters.

(a)

(b)

Figure 5 

The online updated fuzzy parameters 𝜆1.

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.

Acknowledgment

The authors would like to appreciate the support of National Natural Science Foundation of China (51175014) and Program 111 of China.