Robust Decentralized Adaptive Neural Control for a Class of Nonaffine Nonlinear Large-Scale Systems with Unknown Dead Zones

Abstract

The problem of robust decentralized adaptive neural stabilization control is investigated for a class of nonaffine nonlinear interconnected large-scale systems with unknown dead zones. In the controller design procedure, radical basis function (RBF) neural networks are applied to approximate packaged unknown nonlinearities and then an adaptive neural decentralized controller is systematically derived without requiring any information on the boundedness of dead zone parameters (slopes and break points). It is proven that the developed control scheme can ensure that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded in the sense of mean square. Simulation study is provided to further demonstrate the effectiveness of the developed control scheme.

1. Introduction

During the past several decades, a large number of research results have been obtained on the problems of stability analysis and control design for nonlinear systems because of the extensive existence of nonlinearity in the practical systems. So far, there are many control methods proposed to control design of nonlinear systems, such as adaptive backstepping control [1–4], fault-tolerant control [5–9], 𝐻_∞ control [10–16], and fuzzy control [17–21]. In the control of uncertain complex nonlinear systems, backstepping-based neural networks or fuzzy adaptive control technique is an efficient and practical strategy, and many interesting results have been obtained for a class of uncertain nonlinear strict-feedback systems; for example, see [22–37]. In addition, adaptive neural or fuzzy backstepping control approach can also be extended to control a class of nonaffine pure-feedback nonlinear systems representing a class of more general lower triangular systems, in which no affine appearance of the state variables can be used as virtual control signals. This makes it quite difficult and more meaningful to control the pure-feedback systems. By using small-gain theorem, in [38], an adaptive neural control scheme is presented for a class of completely nonaffine pure-feedback systems. Afterwards, some researchers further consider other types of pure-feedback nonlinear systems, such as pure-feedback systems with time-delay [39] and with dead-zone [40].

Large-scale system is considered as a dynamical system which is composed of some lower-order subsystems with interconnections and often exists in many practical applications such as electric power systems, computer network systems, and aerospace systems. Decentralized adaptive control of unknown nonlinear interconnected systems has attracted much research attention because this problem is important both theoretically and practically. Up to now, many interesting adaptive decentralized control approaches for large-scale nonlinear systems have been proposed in [41, 42] and the references therein. By combining backstepping technique together with adaptive neural or fuzzy control, many research papers have been published in [28–32] for affine nonlinear large-scale interconnected systems and in [43, 44] for nonaffine nonlinear large-scale systems. However, these researches have not considered the effect of the system input signal with dead zone. The existence of dead zone input nonlinearity severely degrades system performance. So, dead zone nonlinearity has to be considered when controller is designed in many industrial processes, such as valves, DC servo motors, and other devices. In [45], an adaptive robust control scheme is proposed for a class of nonlinear systems, in which the dead zone nonlinearity is expressed by a combination of a line and a disturbance-like term and its parameters are tuned by using adaptive technique. Then, some backstepping-based adaptive control schemes are developed for nonlinear systems with unknown dead zone input [46–50].

Based on the above observations, this paper focuses on the problem of adaptive neural decentralized control for a class of nonaffine large-scale nonlinear systems with unknown dead zones. The proposed adaptive neural controller guarantees that all the signals in the closed-loop systems remain semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals eventually converge to small neighborhood around the origin. The main advantages of this research lie in that (i) the dead-zone inverse as well as the prior knowledge of bounds of dead-zone parameters (slopes and break-points) is not required; (ii) only one adaptive parameter is involved in the developed controller for each subsystem. As a result, the computational burden is significantly alleviated. In this way, the proposed control law could be easily implemented in practical applications. Simulation results are provided to further illustrate the effectiveness of the proposed control approach.

The paper is organized as follows. Section 2 provides the problem formulation and preliminaries. The control design and analysis of state feedback controller are given in Section 3, following a simulation example in Section 4. Section 5 concludes this paper.

2. Problem Formulation and Preliminaries

Consider a class of pure-feedback nonlinear interconnected large-scale systems with dead zones and 𝑁 subsystems, and the 𝑖th (𝑖=1,2,…,𝑁) subsystem is in the following form: 𝑥̇_𝑖,𝑗=𝑓_𝑖,𝑗𝑥_𝑖,𝑗,𝑥_𝑖,𝑗+1+𝜑_𝑖,𝑗𝑦,1≤𝑗≤𝑛_𝑖𝑥̇−1,_{𝑖,𝑛𝑖}=𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},𝑢_𝑖+𝜑_{𝑖,𝑛𝑖}𝑦,𝑦_𝑖=𝑥_𝑖,1, (1) where 𝑥_𝑖,𝑗=[𝑥_𝑖,1,𝑥_𝑖,2,…,𝑥_𝑖,𝑗]^𝑇 and 𝑦=[𝑦₁,𝑦₂,…,𝑦_𝑁]^𝑇. 𝑥_𝑖=[𝑥_𝑖,1,𝑥_𝑖,2,…,𝑥_{𝑖,𝑛𝑖}]^𝑇∈𝑅^𝑛𝑖 and 𝑦_𝑖∈𝑅 are the state variables and the output of the 𝑖th subsystem, respectively. 𝑓_𝑖,𝑗(⋅)∶𝑅^𝑗+1→𝑅,(𝑗=1,2,…,𝑛_𝑖) are unknown smooth nonlinear functions; 𝜑_𝑖,𝑗(⋅)∶𝑅^𝑁→𝑅(𝑗=1,2,…,𝑛_𝑖) are unknown interconnections between the 𝑖th subsystem and other subsystems, with 𝑓_𝑖,𝑗(0)=𝜑_𝑖,𝑗(0)=0; 𝑢_𝑖∈𝑅 is the output of an unknown dead zone and defined as 𝑢_𝑖=𝐷𝑣_𝑖=𝑔_𝑖𝑟𝑣_𝑖,𝑣_𝑖≥𝑏_𝑖𝑟,0,𝑏_𝑖𝑙<𝑣_𝑖<𝑏_𝑖𝑟,𝑔_𝑖𝑙𝑣_𝑖,𝑣_𝑖≤𝑏_𝑖𝑙, (2) where 𝑏_𝑖𝑙<0 and 𝑏_𝑖𝑟>0 are the unknown parameters and 𝑣_𝑖(𝑡)∈𝑅 is the input of the dead zone.

For the unknown dead zone input, the following assumption is required.

Assumption 1 (see [40]). The functions, 𝑔_𝑖𝑙(𝑣_𝑖) and 𝑔_𝑖𝑟(𝑣_𝑖), are smooth and there exist unknown positive constants, 𝑘_𝑖𝑙0,𝑘_𝑖𝑙1,𝑘_𝑖𝑟0, and 𝑘_𝑖𝑟1, such that 0<𝑘_𝑖𝑙0≤𝑔^′_𝑖𝑙𝑣_𝑖≤𝑘_𝑖𝑙1,∀𝑣_𝑖∈−∞,𝑏_𝑖𝑙,0<𝑘_𝑖𝑟0≤𝑔^′_𝑖𝑟𝑣_𝑖≤𝑘_𝑖𝑟1,∀𝑣_𝑖∈𝑏_𝑖𝑟,+∞, (3) and 𝛽_𝑖0≤min{𝑘_𝑖𝑙0,𝑘_𝑖𝑟0} is an unknown positive constant, where 𝑔^′_𝑖𝑙(𝑣_𝑖)=(𝑑𝑔_𝑖𝑙(𝑧)/𝑑𝑧)|_{𝑧=𝑣𝑖} and 𝑔^′_𝑖𝑟(𝑣_𝑖)=(𝑑𝑔_𝑖𝑟(𝑧)/𝑑𝑧)|_{𝑧=𝑣𝑖}.

Remark 2. Assumption 1 is similar to the assumption in [40] where 𝛽_𝑖0 is a known constant. However, Assumption 1 does not require it to be known. So, Assumption 1 relaxes the limitation in [40].

Based on Assumption 1, the dead zone (2) can be rewritten as [40] 𝑢_𝑖=𝐷𝑣_𝑖=𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+𝑑_𝑖𝑣_𝑖, (4) where Φ_𝑖(𝑡)=𝜑_𝑖𝑟(𝑡),𝜑_𝑖𝑙(𝑡)^𝑇,𝜑_𝑖𝑟(𝑡)=1,𝑣_𝑖(𝑡)>𝑏_𝑖𝑙,0,𝑣_𝑖(𝑡)≤𝑏_𝑖𝑙,𝜑_𝑖𝑙(𝑡)=1,𝑣_𝑖(𝑡)<𝑏_𝑖𝑟,0,𝑣_𝑖(𝑡)≥𝑏_𝑖𝑟,𝐾_𝑖(𝑡)=𝐾_𝑖𝑟𝑣_𝑖(𝑡),𝐾_𝑖𝑙𝑣_𝑖(𝑡)^𝑇,𝐾_𝑖𝑟𝑣_𝑖(𝑡)=0,𝑣_𝑖(𝑡)≤𝑏_𝑖𝑙,𝑔^′_𝑖𝑟𝜉_𝑖𝑟𝑣_𝑖(𝑡),𝑏_𝑖𝑙<𝑣_𝑖(𝑡)𝐾<+∞,_𝑖𝑙𝑣_𝑖(𝑡)=𝑔^′_𝑖𝑙𝜉_𝑖𝑙𝑣_𝑖(𝑡),−∞<𝑣_𝑖(𝑡)<𝑏_𝑖𝑟,0,𝑣_𝑖(𝑡)≥𝑏_𝑖𝑟,𝑑_𝑖𝑣_𝑖=−𝑔^′_𝑖𝑟𝜉_𝑖𝑟𝑣_𝑖𝑏_𝑖𝑟,𝑣_𝑖≥𝑏_𝑖𝑟,−𝑔^′_𝑖𝑙𝜉_𝑖𝑙𝑣_𝑖+𝑔^′_𝑖𝑟𝜉_𝑖𝑟𝑣_𝑖𝑣_𝑖,𝑏_𝑖𝑙<𝑣_𝑖<𝑏_𝑖𝑟,−𝑔^′_𝑖𝑙𝜉_𝑖𝑙𝑣_𝑖𝑏_𝑖𝑙,𝑣_𝑖≤𝑏_𝑖𝑙, (5) where 𝜉_𝑖𝑙(𝑣_𝑖)∈(𝑣_𝑖,𝑏_𝑖𝑙), if 𝑣_𝑖<𝑏_𝑖𝑙;  𝜉_𝑖𝑙(𝑣_𝑖)∈(𝑏_𝑖𝑙,𝑣_𝑖), if 𝑏_𝑖𝑙≤𝑣_𝑖<𝑏_𝑖𝑟;  𝜉_𝑖𝑟(𝑣_𝑖)∈(𝑏_𝑖𝑟,𝑣_𝑖), if 𝑏_𝑖𝑟<𝑣_𝑖; 𝜉_𝑖𝑟(𝑣_𝑖)∈(𝑣_𝑖,𝑏_𝑖𝑟), and if 𝑏_𝑖𝑙<𝑣_𝑖≤𝑏_𝑖𝑟, and |𝑑_𝑖(𝑣_𝑖)|≤𝑝^∗_𝑖, 𝑝^∗_𝑖 is an unknown positive constant with 𝑝^∗=(𝑘_𝑖𝑟1+𝑘_𝑖𝑙1)max{𝑏_𝑖𝑟,−𝑏_𝑖𝑙}.

Remark 3. As shown in [40], there exist some other expressions for the case of linear dead zone outside the deadband, but (4) includes the most actual cases and 𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)∈[𝛽_𝑖0,𝑘_𝑖𝑙1+𝑘_𝑖𝑟1]⊂(0,+∞).

Based on mean value theorem [51], 𝑓_𝑖,𝑗(⋅) in (1) can be described as 𝑓_𝑖,𝑗𝑥_𝑖,𝑗,𝑥_𝑖,𝑗+1=𝑓_𝑖,𝑗𝑥_𝑖,𝑗,𝑥⁰_𝑖,𝑗+1+ℎ_{𝜇𝑖,𝑗}𝑥_𝑖,𝑗+1−𝑥⁰_𝑖,𝑗+1,𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},𝑢_𝑖=𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},𝑢⁰_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑢_𝑖−𝑢⁰_𝑖, (6) where smooth function 𝑓_𝑖,𝑗(⋅) is explicitly analyzed between 𝑓_𝑖,𝑗(𝑥_𝑖,𝑗,𝑥_𝑖,𝑗+1) and 𝑓_𝑖,𝑗(𝑥_𝑖,𝑗,𝑥⁰_𝑖,𝑗+1), ℎ_{𝜇𝑖,𝑗}∶=ℎ_𝑖,𝑗(𝑥_𝑖,𝑗,𝑥_{𝜇𝑖,𝑗})=(𝜕𝑓_𝑖,𝑗(𝑥_𝑖,𝑗,𝑥_𝑖,𝑗+1)/𝜕𝑥_𝑖,𝑗+1)|_{𝑥𝑖,𝑗+1=𝑥𝜇𝑖,𝑗},𝑥_{𝑖,𝑛𝑖+1}=𝑢_𝑖,  𝑥_{𝜇𝑖,𝑗}=𝜇_𝑖,𝑗𝑥_𝑖,𝑗+1+(1−𝜇_𝑖,𝑗)𝑥⁰_𝑖,𝑗+1,  0<𝜇_𝑖,𝑗<1,𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑛_𝑖.

Next, by substituting (6) and (4) into (1), and choosing 𝑥⁰_𝑖,𝑗+1=0,  𝑢⁰_𝑖=0, we obtain 𝑥̇_𝑖,𝑗=ℎ_{𝜇𝑖,𝑗}𝑥_𝑖,𝑗+1+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0+𝜑_𝑖,𝑗𝑦,1≤𝑗≤𝑛_𝑖𝑥̇−1,_{𝑖,𝑛𝑖}=ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖+𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0+𝜑_{𝑖,𝑛𝑖}𝑦,𝑦_𝑖=𝑥_𝑖,1. (7)

Assumption 4 (see [40]). The signs of ℎ_{𝜇𝑖,𝑗}, 1≤𝑖≤𝑁,1≤𝑗≤𝑛_𝑖, do not change, and there exist unknown constants 𝑏_𝑚 and 𝑐 such that 0<𝑏_𝑚≤ℎ_{𝜇𝑖,𝑗}≤𝑐<∞. (8)

Remark 5. Assumption 4 means that ℎ_{𝜇𝑖,𝑗} are strictly either positive or negative. Without loss of generality, it is assumed that 0<𝑏_𝑚≤ℎ_{𝜇𝑖,𝑗}. In addition, by means of Assumptions 1 and 4, it can be further supposed that 0<𝑏≤ℎ_{𝜇𝑖,𝑗},1≤𝑖≤𝑁,1≤𝑗≤𝑛_𝑖−1,0<𝑏≤ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡), (9) where 𝑏=min{𝑏_𝑚,𝑏_𝑚𝛽_𝑖0} is an unknown constant.

Assumption 6 (see [31]). For uncertain nonlinear functions 𝜑_𝑖,𝑗(𝑦) in (1), there exist unknown smooth functions 𝜑_{𝑖,𝑗,𝑙}(𝑦_𝑙) such that for 1≤𝑖≤𝑁,  1≤𝑗≤𝑛_𝑖, 𝜑_𝑖,𝑗𝑦²≤^𝑁_𝑙=1𝜑²_{𝑖,𝑗,𝑙}𝑦_𝑙, (10) where 𝜑_{𝑖,𝑗,𝑙}(0)=0,𝑙=1,2,…,𝑁.

Noting that 𝜑_{𝑖,𝑗,𝑙}(𝑦_𝑙) in (10) is smooth function with 𝜑_{𝑖,𝑗,𝑙}(0)=0, so there exist unknown smooth functions 𝜑_{𝑖,𝑗,𝑙}(𝑦_𝑙) such that 𝜑_𝑖,𝑗𝑦²≤^𝑁_𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑗,𝑙}𝑦_𝑙. (11) In what follows, RBF neural networks are applied to approximate any continuous function 𝑓(𝑋)∶𝑅^𝑛→𝑅, 𝑓_𝑛𝑛(𝑋)=𝑊^𝑇𝜙(𝑋), (12) where 𝑋∈Ω_𝑍⊂𝑅^𝑞 is the input vector with 𝑞 being the neural networks input dimension, weight vector 𝑊=[𝑤₁,𝑤₂,…,𝑤_𝑙]^𝑇∈𝑅^𝑙, 𝑙>1 is the neural networks node number, and 𝜙(𝑋)=[𝜙₁(𝑋),𝜙₂(𝑋),…,𝜙_𝑙(𝑋)]^𝑇 means the basis function vector with 𝜙_𝑖(𝑋) being chosen as the commonly used Gaussian function of the form 𝜙_𝑖(𝑋)=exp−𝑋−𝜇_𝑖^𝑇𝑋−𝜇_𝑖𝜂²_𝑖,𝑖=1,2,…,𝑙, (13) where 𝜇_𝑖=[𝜇_𝑖1,𝜇_𝑖2,...,𝜇_𝑖𝑞]^𝑇 is the center of the receptive field and 𝜂_𝑖 is the width of the Gaussian function. In [52], it has been indicated that with sufficiently large node number 𝑙, the RBF neural networks (12) can approximate any continuous function 𝑓(𝑋) over a compact set Ω_𝑍⊂𝑅^𝑞 to arbitrary any accuracy 𝜀>0 as 𝑓(𝑋)=𝑊^∗𝑇𝜙(𝑋)+𝛿(𝑋),∀𝑧∈Ω_𝑧∈𝑅^𝑞, (14) where 𝑊^∗ is the ideal constant weight vector and defined as 𝑊^∗∶=argmin_{𝑊∈𝑅𝑙}sup_{𝑍∈Ω𝑍}𝑓(𝑋)−𝑊^𝑇𝜙(𝑋), (15) and 𝛿(𝑋) denotes the approximation error and satisfies |𝛿(𝑋)|≤𝜀.

Lemma 7 (see [53]). Consider the Gaussian RBF networks (12) and (13). Let 𝜌∶=(1/2)min_𝑖≠𝑗‖𝜇_𝑖−𝜇_𝑗‖; then an upper bound of ‖𝜙(𝑋)‖ is taken as 𝜙(𝑋)≤^∞_𝑘=03𝑞(𝑘+2)^𝑞−1𝑒^{−2𝜌2𝑘2/𝜂2}∶=𝑠. (16)

It has been shown in [38] that the constant 𝑠 in Lemma 7 has a limited value and is independent of the variable 𝑋 and the dimension of neural weights 𝑙.

3. Adaptive Neural Control Design

In this section, an adaptive neural backstepping control scheme will be developed. During the controller design, for the 𝑖th subsystem, RBF neural network 𝑊^𝑇_𝑖,𝑗𝜙(𝑋_𝑖,𝑗) will be utilized to approximate the packaged unknown function 𝐹_𝑖,𝑗(𝑋_𝑖,𝑗) at step 𝑗. Both the virtual control signals 𝛼_𝑖,𝑗(𝑋_𝑖,𝑗) and adaption laws ̇𝜃_𝑖 will be constructed in the following forms: 𝛼_𝑖,𝑗𝑋_𝑖,𝑗=−𝑘_𝑖,𝑗𝑧_𝑖,𝑗−12𝑎²_𝑖,𝑗𝑧_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝑋_𝑖,𝑗𝜙_𝑖,𝑗𝑋_𝑖,𝑗,̇𝜃_𝑖=^𝑛𝑖_𝑗=1𝜆_𝑖2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗𝜙^𝑇_𝑖,𝑗𝑋_𝑖,𝑗𝜙_𝑖,𝑗𝑋_𝑖,𝑗−𝛾_𝑖𝜃_𝑖, (17)(18) where 𝑖=1,2,…,𝑁,  𝑗=1,2,…,𝑛_𝑖, 𝑘_𝑖,𝑗, 𝑎_𝑖,𝑗, 𝜆_𝑖, and 𝛾_𝑖 are positive design parameters, 𝑋_𝑖,1=𝑥_𝑖,1,𝑋_𝑖,𝑗=[𝑥^𝑇_𝑖,𝑗,𝜃_𝑖]^𝑇,  (𝑗=2,…,𝑛_𝑖) with 𝑥_𝑖,𝑗=[𝑥_𝑖,1,𝑥_𝑖,2,…,𝑥_𝑖,𝑗]^𝑇, and 𝑧_𝑖,𝑗 satisfy the following coordinate transformation: 𝑧_𝑖,𝑗=𝑥_𝑖,𝑗−𝛼_{𝑖,𝑗−1} (19) with 𝛼_𝑖,0=0. 𝜃_𝑖 is used to estimate an unknown constant 𝜃_𝑖 which will be defined as 𝜃_𝑖1=max𝑏𝑊_𝑖,𝑗²;𝑗=1,2,…,𝑛_𝑖, (20) where 𝑏 is specified in Remark 5, and ‖𝑊_𝑖,𝑗‖ will be defined at the 𝑗th step. Specially, 𝛼_{𝑖,𝑛𝑖} is the actual control input signal 𝑢_𝑖.

For simplicity, in the following, the time variable 𝑡 and the state vector 𝑥_𝑖,𝑗 will be omitted from the corresponding functions and let 𝜙_𝑖,𝑗(𝑋_𝑖,𝑗)=𝜙_𝑖,𝑗.

Step 1. Based on 𝑧_𝑖,1=𝑥_𝑖,1,  𝑧_𝑖,2=𝑥_𝑖,2−𝛼_𝑖,1, the error dynamic 𝑧_𝑖,1 satisfies 𝑧̇_𝑖,1=ℎ_𝜇𝑖,1𝑧_𝑖,2+ℎ_𝜇𝑖,1𝛼_𝑖,1+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0+𝜑_𝑖,1𝑦. (21)

Consider a Lyapunov function candidate as 𝑉_𝑖,1=(1/2)𝑧²_𝑖,1=(1/2)𝑦²_𝑖. Then, the time derivative of 𝑉₁ along (21) satisfies 𝑉̇_𝑖,1=𝑦_𝑖ℎ_𝜇𝑖,1𝑧_𝑖,2+ℎ_𝜇𝑖,1𝛼_𝑖,1+𝑓_𝑖,1𝑥_𝑖,1,0+𝜑_𝑖,1𝑦. (22) By using (11) and the completion of squares, we get 𝑦_𝑖𝜑_𝑖,1𝑦≤12𝑦²_𝑖+12^𝑁_𝑙=1𝑦²_𝑙𝜑²_𝑖,1,𝑙𝑦_𝑙. (23) Further, (22) can be rewritten as 𝑉̇_𝑖,1=𝑦_𝑖ℎ_𝜇𝑖,1𝑧_𝑖,2+ℎ_𝜇𝑖,1𝛼_𝑖,1+𝑓_𝑖,1𝑥_𝑖,1,0+12𝑦_𝑖+12^𝑁_𝑙=1𝑦²_𝑙𝜑²_𝑖,1,𝑙𝑦_𝑙. (24)

Step 𝑗(2≤𝑗≤𝑛_𝑖−1). According to (19), one has 𝑧̇_𝑖,𝑗=ℎ_{𝜇𝑖,𝑗}𝑧_𝑖,𝑗+1+ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0+𝜑_𝑖,𝑗𝑦−𝛼̇_{𝑖,𝑗−1}, (25) where 𝛼̇_{𝑖,𝑗−1}=^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+𝜑_𝑖,𝑘𝑦+𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖̇𝜃_𝑖. (26) Take Lyapunov function 𝑉_𝑖,𝑗=(1/2)𝑧²_𝑖,𝑗; then we have 𝑉̇_𝑖,𝑗=𝑧_𝑖,𝑗𝑔_{𝜇𝑖,𝑗}𝑧_𝑖,𝑗+1+𝑔_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0+𝜑_𝑖,𝑗𝑦−^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+𝜑_𝑖,𝑘𝑦−𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖̇𝜃_𝑖. (27) Following the same line as the procedures used in (23), one has −𝑧_𝑖,𝑗^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝜑_𝑖,𝑘𝑦≤12𝑧²_𝑖,𝑗^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘²+12^𝑗−1𝑁_𝑘=1𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑘,𝑙}𝑦_𝑙,𝑧_𝑖,𝑗𝜑_𝑖,𝑗𝑦≤12𝑧²_𝑖,𝑗+12^𝑁_𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑗,𝑙}𝑦_𝑙. (28)(29) It can be easily verified from (27) to (29) that 𝑉̇_𝑖,𝑗≤𝑧_𝑖,𝑗ℎ_{𝜇𝑖,𝑗}𝑧_𝑖,𝑗+1+ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+𝑓_𝑖,𝑗𝑥_𝑖,𝑗−,0_{𝑗−1𝑘=1}𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+12𝑧_𝑖,𝑗+12𝑧_𝑖,𝑗^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘²−𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖̇𝜃_𝑖+12^𝑗𝑁_𝑘=1𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑘,𝑙}𝑦_𝑙. (30)

Step 𝑛_𝑖. Similar to (25), the following equation can be obtained: 𝑧̇_{𝑖,𝑛𝑖}=ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖+𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0+𝜑_{𝑖,𝑛𝑖}𝑦−𝛼̇_{𝑖,𝑛𝑖−1}, (31) where 𝛼̇_{𝑖,𝑛𝑖−1} is shown in (26) with 𝑗=𝑛_𝑖. Choose a Lyapunov function as 𝑉_{𝑖,𝑛𝑖}=12𝑧²_{𝑖,𝑛𝑖}+𝑏2𝜆_𝑖𝜃²_𝑖, (32) where 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖 is the parameter error and 𝜆_𝑖>0 is a design parameter.

Then, the following result holds. 𝑉̇_{𝑖,𝑛𝑖}=𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖+𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0+𝜑_{𝑖,𝑛𝑖}𝑦−𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝜃_𝑖̇𝜃_𝑖−^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+𝜑_𝑖,𝑘𝑦−𝑏𝜆_𝑖𝜃_𝑖̇𝜃_𝑖. (33) Repeating the same derivations as (28)–(30) produces 𝑉̇_{𝑖,𝑛𝑖}≤𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖+𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0−^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+12𝑧_{𝑖,𝑛𝑖}^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘²+12𝑧_{𝑖,𝑛𝑖}−𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝜃_𝑖̇𝜃_𝑖+12^𝑛𝑖𝑁_𝑘=1𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑘,𝑙}𝑦_𝑙−𝑏𝜆_𝑖𝜃_𝑖̇𝜃_𝑖. (34) Now, consider a Lyapunov function for the whole system as 𝑉=^𝑁𝑛_𝑖=1^𝑖_𝑗=1𝑉_𝑖,𝑗=^𝑁_𝑖=112𝑦²_𝑖+^𝑛𝑖_𝑗=212𝑧²_𝑖,𝑗+𝑏2𝜆_𝑖𝜃²_𝑖. (35) Then, combining (24) together with (30) and (34), one has 𝑉̇≤^𝑁_𝑖=1𝑦_𝑖ℎ_𝜇𝑖,1𝛼_𝑖,1+𝑓_𝑖,1𝑥_𝑖,1,0+12𝑦_𝑖+12𝑦_𝑖^𝑁𝑛_𝑙=1^𝑙𝑠_𝑠=1𝑘=1𝜑²_{𝑙,𝑘,𝑖}𝑦_𝑖+^𝑁𝑛_𝑖=1^𝑖−1_𝑗=2𝑧_𝑖,𝑗ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+ℎ_{𝜇𝑖,𝑗}𝑧_{𝑖,𝑗−1}+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0−^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+12𝑧_𝑖,𝑗+12𝑧_𝑖,𝑗^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘²+^𝑁_𝑖=1𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖+𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0−^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+12𝑧_{𝑖,𝑛𝑖}^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘²+12𝑧_{𝑖,𝑛𝑖}−^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖̇𝜃_𝑖−^𝑁_𝑖=1𝑏𝜆_𝑖𝜃_𝑖̇𝜃_𝑖, (36) where the fact of 12^𝑁𝑛_𝑖=1^𝑖𝑠_𝑠=1^𝑁_𝑘=1𝑙=1𝑦²_𝑙𝜑²_{𝑖,𝑘,𝑙}𝑦_𝑙=12^𝑁𝑁_𝑖=1^𝑛_𝑙=1^𝑙𝑠_𝑠=1𝑘=1𝑦²_𝑖𝜑²_{𝑙,𝑘,𝑖}𝑦_𝑖 (37) has been used in above inequality.

By using adaptive laws defined in (18) and rearranging sequence, we can obtain −^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖̇𝜃_𝑖=−^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖^𝑛𝑖_𝑘=1𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘−𝛾_𝑖𝜃_𝑖=^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖𝛾_𝑖𝜃_𝑖−^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖^𝑗−1_𝑘=1𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘−^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖^𝑛𝑖_𝑘=𝑗𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘≤^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖𝛾_𝑖𝜃_𝑖−^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝑧_𝑖,𝑗𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖^𝑗−1_𝑘=1𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘+^𝑁𝑛_𝑖=1^𝑖_𝑗=2𝜆_𝑖2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗^𝑗_𝑘=2𝑧_𝑖,𝑘𝜕𝛼_{𝑖,𝑘−1}𝜕𝜃_𝑖. (38) Taking |𝑑_𝑖(𝑣_𝑖)|≤𝑝^∗_𝑖 into account, one has 𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝑑_𝑖𝑣_𝑖≤12𝑧²_{𝑖,𝑛𝑖}ℎ²_{𝜇𝑖𝑖,𝑛}+12𝑝_𝑖^∗2. (39)

By substituting (38) and (39) into (36), it follows 𝑉̇≤^𝑁_𝑖=1𝑦_𝑖ℎ_𝜇𝑖,1𝛼_𝑖,1+𝐹_𝑖,1𝑋_𝑖,1+^𝑁𝑛_𝑖=1^𝑖−1_𝑗=2𝑧_𝑖,𝑗ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+𝐹_𝑖,𝑗𝑋_𝑖,𝑗+^𝑁_𝑖=1𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+𝐹_{𝑖,𝑛𝑖}𝑋_{𝑖,𝑛𝑖}+12^𝑁_𝑖=1𝑝_𝑖^∗2−12^𝑁𝑛_𝑖=1^𝑖_𝑗=1𝑧²_𝑖,𝑗−^𝑁_𝑖=1𝑏𝜆_𝑖𝜃_𝑖̇𝜃_𝑖, (40) where the packaged functions 𝐹_𝑖,𝑗(𝑋_𝑖,𝑗),  𝑖=1,2,…,𝑁, are specified as 𝐹_𝑖,1𝑋_𝑖,1=𝑓_𝑖,1𝑥_𝑖,1,0+𝑦_𝑖+12𝑦_𝑖^𝑁𝑛_𝑙=1^𝑙𝑠_𝑠=1𝑘=1𝜑²_{𝑙,𝑘,𝑖}𝑦_𝑖,𝐹_𝑖,𝑗𝑋_𝑖,𝑗=ℎ_{𝜇𝑖,𝑗}𝑧_{𝑖,𝑗−1}+𝑓_𝑖,𝑗𝑥_𝑖,𝑗,0−^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+𝑧_𝑖,𝑗+12𝑧_𝑖,𝑗^𝑗−1_𝑘=1𝜕𝛼_{𝑖,𝑗−1}𝜕𝑥_𝑖,𝑘²+𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖𝛾_𝑖𝜃_𝑖−𝜕𝛼_{𝑖,𝑗−1}𝜕𝜃_𝑖^𝑗−1_𝑘=1𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘+𝜆_𝑖2𝑎²_𝑖,𝑗𝑧_𝑖,𝑗^𝑗_𝑘=2𝑧_𝑖,𝑘𝜕𝛼_{𝑖,𝑘−1}𝜕𝜃_𝑖,𝑗=2,…,𝑛_𝑖𝐹−1,_{𝑖,𝑛𝑖}𝑋_{𝑖,𝑛𝑖}=𝑓_{𝑖,𝑛𝑖}𝑥_{𝑖,𝑛𝑖},0−^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘𝑓_𝑖,𝑘𝑥_𝑖,𝑘+1+12𝑧_{𝑖,𝑛𝑖}^𝑛𝑖−1_𝑘=1𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝑥_𝑖,𝑘²+12𝑧_{𝑖,𝑛𝑖}ℎ²_{𝜇𝑖𝑖,𝑛}+𝑧_{𝑖,𝑛𝑖}+𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝜃_𝑖𝛾_𝑖𝜃_𝑖−𝜕𝛼_{𝑖,𝑛𝑖−1}𝜕𝜃_𝑖^𝑛𝑖−1_𝑘=1𝜆_𝑖2𝑎²_𝑖,𝑘𝑧²_𝑖,𝑘𝜙^𝑇_𝑖,𝑘𝜙_𝑖,𝑘+𝜆_𝑖2𝑎²_{𝑖,𝑛𝑖}𝑧_{𝑖,𝑛𝑖}^𝑛𝑖_𝑘=2𝑧_𝑖,𝑘𝜕𝛼_{𝑖,𝑘−1}𝜕𝜃_𝑖. (41) Because 𝑓_𝑖,𝑗, ℎ_{𝜇𝑖,𝑗}, and 𝜑_{𝑙,𝑘,𝑖} are unknown smooth functions, 𝐹_𝑖,𝑗(𝑋_𝑖,𝑗),𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑛_𝑖 cannot be used to define the virtual control input 𝛼_𝑖,𝑗 and the real controller 𝑢_𝑖. Then, RBF neural network 𝑊^𝑇_𝑖,𝑗𝜙_𝑖,𝑗(𝑋_𝑖,𝑗) is utilized to model 𝐹_𝑖,𝑗(𝑋_𝑖,𝑗), such that for a given positive constant 𝜀_𝑖,𝑗, 𝐹_𝑖,𝑗𝑋_𝑖,𝑗=𝑊^𝑇_𝑖,𝑗𝜙_𝑖,𝑗𝑋_𝑖,𝑗+𝛿_𝑖,𝑗𝑋_𝑖,𝑗, (42) where 𝛿_𝑖,𝑗(𝑋_𝑖,𝑗) is approximation error and satisfies |𝛿_𝑖,𝑗(𝑋_𝑖,𝑗)|<𝜀_𝑖,𝑗. Subsequently, by Young's inequality, one has 𝑧_𝑖,𝑗𝐹_𝑖,𝑗𝑋_𝑖,𝑗=𝑧_𝑖,𝑗𝑊^𝑇_𝑖,𝑗𝑊_𝑖,𝑗𝜙_𝑖,𝑗𝑊_𝑖,𝑗+𝑧_𝑖,𝑗𝛿_𝑖,𝑗𝑋_𝑖,𝑗≤𝑏2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗+12𝑎²_𝑖,𝑗+12𝑧²_𝑖,𝑗+12𝜀²_𝑖,𝑗, (43) where 𝑖=1,2,…,𝑁,  𝑗=1,2,…,𝑛_𝑖 and the unknown constant 𝜃_𝑖=max{(1/𝑏)‖𝑊_𝑖,𝑗‖²;𝑗=1,2,…,𝑛_𝑖}.

Substituting (42) into (40) and using (43) produces 𝑉̇≤^𝑁_𝑖=1𝑧_𝑖,1ℎ_𝜇𝑖,1𝛼_𝑖,1+𝑏2𝑎²_𝑖,1𝑧_𝑖,1𝜃_𝑖𝜙^𝑇_𝑖,1𝜙_𝑖,1+^𝑁𝑛_𝑖=1^𝑖−1_𝑗=2𝑧_𝑖,𝑗ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗+𝑏2𝑎²_𝑖,𝑗𝑧_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗+^𝑁_𝑖=1𝑧_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖+𝑏2𝑎²_{𝑖,𝑛𝑖}𝑧_{𝑖,𝑛𝑖}𝜃_𝑖𝜙^𝑇_{𝑖,𝑛𝑖}𝜙_{𝑖,𝑛𝑖}+12^𝑁_𝑖=1𝑝_𝑖^∗2+^𝑁𝑛_𝑖=1^𝑖_𝑗=112𝑎²_𝑖,𝑗+12𝜀²_𝑖,𝑗−^𝑁_𝑖=1𝑏𝜆_𝑖𝜃_𝑖̇𝜃_𝑖. (44) Now, construct the virtual control signals 𝛼_𝑖,𝑗=−𝑘_𝑖,𝑗𝑧_𝑖,𝑗−(1/2𝑎²_𝑖,𝑗)𝑧_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗, where 𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑛_𝑖, and 𝛼_{𝑖,𝑛𝑖}=𝑢_𝑖. Then, by using (9), we can obtain 𝑧_𝑖,𝑗ℎ_{𝜇𝑖,𝑗}𝛼_𝑖,𝑗≤−𝑘_𝑖,𝑗𝑏𝑧²_𝑖,𝑗−𝑏2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗,𝑗=1,…,𝑛_𝑖𝑧−1,_{𝑖,𝑛𝑖}ℎ_{𝜇𝑖𝑖,𝑛}𝐾^𝑇_𝑖(𝑡)Φ_𝑖(𝑡)𝑣_𝑖≤−𝑘_{𝑖,𝑛𝑖}𝑏𝑧²_{𝑖,𝑛𝑖}−𝑏2𝑎²_{𝑖,𝑛𝑖}𝑧²_{𝑖,𝑛𝑖}𝜃_𝑖𝜙^𝑇_{𝑖,𝑛𝑖}𝜙_{𝑖,𝑛𝑖}, (45) where 𝑖=1,2,…,𝑁.

Substituting (45) into (44) and using adaptive laws 𝜃̇_𝑖 in (18) result in 𝑉̇≤^𝑁𝑛_𝑖=1^𝑖_𝑗=1𝑧_𝑖,𝑗−𝑘_𝑖,𝑗𝑏𝑧_𝑖,𝑗+𝑏2𝑎²_𝑖,𝑗𝑧_𝑖,𝑗𝜃_𝑖𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗+^𝑁𝑛_𝑖=1^𝑖_𝑗=112𝑎²_𝑖,𝑗+12𝜀²_𝑖,𝑗+12^𝑁_𝑖=1𝑝_𝑖^∗2−^𝑁_𝑖=1𝑏𝜆_𝑖𝜃_𝑖^𝑛𝑖_𝑗=1𝜆_𝑖2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗−𝛾_𝑖𝜃_𝑖≤−^𝑁𝑛_𝑖=1^𝑖_𝑗=1𝑘_𝑖,𝑗𝑏𝑧²_𝑖,𝑗+^𝑁_𝑖=1𝛾_𝑖𝑏𝜆_𝑖𝜃_𝑖𝜃_𝑖+^𝑁𝑛_𝑖=1^𝑖_𝑗=112𝑎²_𝑖,𝑗+12𝜀²_𝑖,𝑗+12^𝑁_𝑖=1𝑝_𝑖^∗2≤−^𝑁_𝑖=1^𝑛𝑖_𝑗=1𝑘_𝑖,𝑗𝑏𝑧²_𝑖,𝑗+𝛾_𝑖𝑏2𝜆_𝑖𝜃²_𝑖+^𝑁𝑛_𝑖=1^𝑖_𝑗=112𝑎²_𝑖,𝑗+12𝜀²_𝑖,𝑗+𝛾_𝑖𝑏2𝜆_𝑖𝜃²_𝑖+12^𝑁_𝑖=1𝑝_𝑖^∗2, (46) where 𝜃_𝑖𝜃_𝑖≤−(1/2)𝜃_𝑖²+(1/2)𝜃²_𝑖 has been used in the above inequality.

Define 𝑎₀=min2𝑘_𝑖,𝑗𝑏,𝛾_𝑖,𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑛_𝑖;𝑏₀=^𝑁𝑛_𝑖=1^𝑖_𝑗=112𝑎²_𝑖,𝑗+12𝜀²_𝑖,𝑗+𝛾_𝑖𝑏2𝜆_𝑖𝜃²_𝑖+12^𝑁_𝑖=1𝑝_𝑖^∗2, (47) we have 𝑉̇≤−𝑎₀𝑉+𝑏₀,𝑡≥0. (48)

Furthermore, multiplying (48) by 𝑒^𝑎0𝑡 yields 𝑑𝑑𝑡𝑉𝑒^𝑎0𝑡≤𝑏₀𝑒^𝑎0𝑡. (49) Integrating (49) over [0,𝑡] gives (𝑡)≤𝑏0≤𝑉⁰𝑎₀+𝑉(0)−𝑏₀𝑎₀𝑒^−𝑎0𝑡, (50) which implies that 𝑉(𝑡)(0)+𝑏≤𝑉⁰𝑎₀,∀𝑡>0, (51) which means that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded.

Furthermore, it is easily obtained that 𝑉(𝑡)≤𝑏₀𝑎₀,𝑡⟶+∞. (52) Therefore, based on the definition of 𝑉 in (50), the error signals 𝑧_𝑖,𝑗 and 𝜃_𝑖 eventually converge to the compact set Ω_𝑠, which is specified as Ω_𝑠=𝑧_𝑖,𝑗,𝜃_𝑖∣𝑧_𝑖,𝑗≤2𝑏₀𝑎₀,𝜃_𝑖≤2𝜆_𝑖𝑏𝑏₀𝑎₀,1≤𝑖≤𝑁,1≤𝑗≤𝑛_𝑖. (53)

At the present stage, adaptive decentralized control procedure has been completed via backstepping technique. The above design procedures and stable analysis are summarized in the following theorem.

Theorem 8. Under Assumptions 1–6, consider the closed-loop nonlinear system consisted of (1), unknown dead zone nonlinearities (2), controller (17), and adaptive law (18). Under the action of controller (17), for any initial conditions [𝑧^𝑇_𝑖(0),𝜃_𝑖(0)]^𝑇∈Ω₀ (where Ω₀ is an appropriately chosen compact set), all the signals in the closed-loop system are semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals 𝑧_𝑖,𝑗 and 𝜃_𝑖 eventually converge to the compact set Ω_𝑠 in (53).

4. Simulation Example

In this section, in order to illustrate the effectiveness of the proposed control scheme, consider the nonaffine interconnected nonlinear system with dead zones as 𝑥̇_1,1=1+𝑥²_1,1𝑥_1,2+𝑥³_1,2+𝑦₂ln1+𝑦²₁,𝑥̇_1,2=5+sin𝑥_1,1𝑥_1,2𝑢₁+0.2𝑢⁵₁+sin𝑦²₁𝑦₂,𝑦₁=𝑥_1,1,𝑥̇_2,1=3+sin𝑥_2,1𝑥_2,2+0.3𝑥⁵_2,2+𝑦₁cos𝑦²₂,𝑥̇_2,2=2+𝑥²_2,1𝑢₂+15sin𝑢₂+𝑦³₂𝑦₁,𝑦₂=𝑥_2,1, (54) where 𝑢_𝑖=𝐷(𝑣_𝑖) is defined as 𝑢₁=𝐷𝑣₁=1.5𝑣₁−2.5,𝑣₁≥2.5,0,−1<𝑣₁𝑣<2.5,0.5₁+1,𝑣₁𝑢≤−1,₂=𝐷𝑣₂=0.8𝑣₂−2,𝑣₂≥2,0,−1.8<𝑣₂𝑣<2,1.2₂+1.8,𝑣₂≤−1.8, (55) where 𝑥_1,1, 𝑥_1,2, 𝑥_2,1, and 𝑥_2,2 are the state variables and 𝑦_𝑖 is the system output. 𝑢_𝑖 and 𝑣_𝑖 are the output and input of the dead zone nonlinearity, respectively. It is obvious that the system is of nonaffine structure and satisfies Assumptions 1, 4, and 6. By using Theorem 8, the virtual control law, the actual controller, and the adaptive laws are constructed as 𝛼_𝑖,1=−𝑘_𝑖,1𝑧_𝑖,1−12𝑎²_𝑖,1𝑧_𝑖,1𝜃_𝑖𝜙^𝑇_𝑖,1𝜙_𝑖,1,𝑢_𝑖=−𝑘_𝑖,2𝑧_𝑖,2−12𝑎²_𝑖,2𝑧_𝑖,2𝜃_𝑖𝜙^𝑇_𝑖,2𝜙_𝑖,2,̇𝜃_𝑖=²_𝑗=1𝜆_𝑖2𝑎²_𝑖,𝑗𝑧²_𝑖,𝑗𝜙^𝑇_𝑖,𝑗𝜙_𝑖,𝑗−𝛾_𝑖𝜃_𝑖, (56) where 𝑧_𝑖,1=𝑥_𝑖,1,  𝑧_𝑖,2=𝑥_𝑖,2−𝛼_𝑖,1, and  𝑖=1,2. The simulation is run under the initial conditions [𝑥_1,1(0),𝑥_1,2(0),𝑥_2,1(0),𝑥_2,2(0)]^𝑇=[0.4,−0.1,0.2,0.3]^𝑇, and [𝜃₁(0),𝜃₂(0)]^𝑇=[0.1,0.2]^𝑇. In the simulation, design parameters are taken as follows: 𝑘_1,1=𝑘_1,2=𝑘_2,1=𝑘_2,2=11,𝑎_1,1=𝑎_1,2=𝑎_2,1=𝑎_2,2=3,  𝛾₁=𝛾₂=1, and 𝜆₁=𝜆₂=2.

The simulation results are indicated by Figures 1–4. Figure 1 shows the state variables 𝑥_1,1 and 𝑥_1,2 of the first subsystems. Figure 2 shows the second subsystems state variables 𝑥_2,1 and 𝑥_2,2. Figure 3 displays the control signals 𝑣₁ and 𝑣₂, and Figure 4 shows the response curve of the adaptive parameters 𝜃₁ and 𝜃₂. Apparently, simulation results show that good convergence performances are achieved and all the signals in the closed-loop system are bounded.

Figure 1 

State variables 𝑥1,1 and 𝑥1,2.

Figure 2 

State variables 𝑥2,1 and 𝑥2,2.

Figure 3 

The control signals 𝑣1 and 𝑣2.

Figure 4 

Adaptive laws 𝜃1 and 𝜃2.

5. Conclusion

In this paper, a robust decentralized adaptive neural control approach has been developed for a class of large-scale nonaffine nonlinear systems with dead zones. The proposed decentralized controller can guarantee that all the signals in the closed-loop systems are semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals eventually converge to small neighborhood around the origin. The main advantage of the proposed controller is that the prior knowledge of bounds of dead zone slopes is not required. In addition, only one adaptive parameter needs to be updated online for each 𝑛-order subsystem. In this way, the computational burden is significantly alleviated. Simulation results have been provided to further illustrate the effectiveness of our results.

There are some problems remaining to be considered, for example, how to generalize the result in this paper to stochastic pure-feedback nonlinear systems with dead zones and how to design an output-feedback adaptive neural controller for original system (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the Natural Science Foundation of China (61304002 and 61304003), the Program for New Century Excellent Talents in University (NECT-13-0696), the Program for Liaoning Innovative Research Team in University under Grant LT2013023, the Program for Liaoning Excellent Talents in University under Grant LR2013053, the Polish-Norwegian Research Programme operated by the National Centre for Research and 24 Development under the Norwegian Financial Mechanism 2009–2014 in the frame of Project Contract no. Pol-Nor/200957/47/2013, and the Education Department of Liaoning Province under the General Project Research Grant no. L2013424.