Robust Adaptive Control for a Class of Uncertain Nonlinear Systems with Time-Varying Delay

Abstract

We present adaptive neural control design for a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form. Based on a neural controller, it is obtained by constructing a quadratic-type Lyapunov-Krasovskii functional, which efficiently avoids the controller singularity. The proposed control guarantees that all closed-loop signals remain bounded, while the output tracking error dynamics converge to a neighborhood of the desired trajectories. The simulation results demonstrate the effectiveness of the proposed control scheme.

1. Introduction

In the practical control process, control system is usually required to meet the stability and the corresponding performance index, which affects the system stability factors mainly including the uncertainties and time delays. On the study of the uncertain time-delay many scholars have achieved valuable fruits [1, 2]. Paper [3] has analyzed and designed the optimal 𝐻_∞ feedback controller by the LMI method. In recent decades, the delay nonlinear systems with neural network research have received extensive attention [4–20]. Paper [4] has solved the problem of chaotic synchronization phenomenon by the neural network method. In [5–11], the study of the nonlinear continuous system and discrete nonlinear system is based on adaptive neural network control. The tracking and stabilization problem of nonlinear systems has been studied by neural network backstepping method [12, 13]. In [14], neural network control has been applied to a piece of triangle structure of multiple-input multiple-output nonlinear time-delay system, in which a dynamic system neural network is used mainly for unknown function approximation and separation. In multiple input and multiple output nonlinear system, [15] presents a new adaptive neural network controller design method but does not consider with external disturbance and time-varying delay. In [18], the problem of the adaptive neural networks control for a class of nonlinear state-delay systems with unknown virtual control coefficients is considered. In [19], a control scheme combined with backstepping, radial basis function (RBF) neural networks and adaptive control are proposed for the stabilization of nonlinear system with input and state delay.

This paper mainly aims at studying the simultaneous presence of uncertainties and time-varying delay MIMO nonlinear system. By defining the new quadratic Lyapunov-Krasovskii functionals, it has analyzed and designed the adaptive neural network controller by neural network approximation method in [15, 16].

2. Description of the Problem

Let us consider the following block-triangular structure with the disturbance of nonlinear MIMO systems with time-varying delays: 𝑥̇_{𝑗,𝑖𝑗}=𝑓_{𝑗,𝑖𝑗}𝑥_{𝑗,𝑖𝑗}+𝑔_{𝑗,𝑖𝑗}𝑥_{𝑗,𝑖𝑗}𝑥_{𝑗,𝑖𝑗+1}+ℎ_{𝑗,𝑖𝑗}𝑥_{𝜏𝑗𝑗,𝑖}+𝜔_{𝑗,𝑖𝑗}(𝑡),𝑥̇_{𝑗,𝑚𝑗}=𝑓_{𝑗,𝑚𝑗}𝑋,𝑢_𝑗−1+𝑔_{𝑗,𝑚𝑗}𝑋,𝑢_𝑗−1𝑢_𝑗+ℎ_{𝑗,𝑚𝑗}𝑋_𝜏+𝜔_{𝑗,𝑚𝑗}(𝑡),𝑦_𝑗=𝑥_𝑗,1,𝑗=1,…,𝑛,𝑖_𝑗=1,…,𝑚_𝑗−1, (1) where 𝑥_{𝑗,𝑖𝑗}=[𝑥_𝑗,1,…,𝑥_{𝑗,𝑖𝑗}]^𝑇∈𝑅^𝑖𝑗 are the state variable for the 𝑖_𝑗 differential equations of the 𝑗th subsystem; 𝑋=[𝑥^𝑇₁,…,𝑥^𝑇_𝑛]^𝑇, where 𝑥_𝑗=[𝑥_𝑗,1,…,𝑥_{𝑗,𝑚𝑗}]^𝑇∈𝑅^𝑚𝑗 are the state vector of the 𝑗th subsystem; 𝑥_{𝜏𝑗𝑗,𝑖}=𝑥_{𝑗,𝑖𝑗}(𝑡−𝜏_{𝑗,𝑖𝑗(𝑡)}), where 𝜏_{𝑗,𝑖𝑗(𝑡)} are unknown time-varying delay of the states, and |𝜏_{𝑗,𝑖𝑗}(𝑡)|≤𝜏_{𝑗,𝑖𝑗},|𝜏̇_{𝑗,𝑖𝑗}(𝑡)|≤𝜏₁<1,𝜏₀=max{𝜏_{𝑗,𝑖𝑗}∣1≤𝑗≤𝑛,1≤𝑖_𝑗≤𝑚_𝑗}, 𝑥_{𝜏𝑗𝑗,𝑖}=[𝑥_𝜏𝑗,1,…,𝑥_{𝜏𝑗𝑗,𝑖}]^𝑇, 𝑋_𝜏=[𝑥_𝜏1,1,…,𝑥_𝜏11,𝑛,…,𝑥_𝜏𝑛,1,…,𝑥_{𝜏𝑛𝑛,𝑚}]^𝑇, the output 𝑦=[𝑦₁,…,𝑦_𝑛]^𝑇∈𝑅^𝑛; 𝑢_𝑗=[𝑢₁,…,𝑢_𝑗]^𝑇 are input vector of the 𝑗th subsystem; 𝑓_{𝑗,𝑖𝑗}(⋅), 𝑔_{𝑗,𝑖𝑗}(⋅), and ℎ_{𝑗,𝑖𝑗}(⋅) are unknown smooth nonlinear function. 𝜔_{𝑗,𝑖𝑗}(𝑡) is the disturbance input and |𝜔_{𝑗,𝑖𝑗}(𝑡)|≤𝑑_{𝑗,𝑖𝑗}<1. Let 𝑥_{𝑗,𝑖𝑗}(𝑡)=𝛽_{𝑗,𝑖𝑗}(𝑡), with 𝑡∈[−𝜏₀,0]; assume 𝛽_{𝑗,𝑖𝑗}(𝑡) is smooth and bounded.

We make the following assumptions for the system (1).

Assumption 1. The desired trajectories 𝑦_𝑑𝑗,𝑗=1,2,…,𝑛, have the 𝑛th derivation and the derivation is continuous and bounded.

Assumption 2. We use 𝑔_{𝑗,𝑖𝑗}(⋅) to represent some given function. There exist constant 𝑔_𝑗0 and unknown smooth functions 𝑔_{𝑗,𝑖𝑗}(⋅), such that 0<𝑔_𝑗0≤|𝑔_{𝑗,𝑖𝑗}(⋅)|≤𝑔_{𝑗,𝑖𝑗}(⋅)<∞. Without loss of generality, we further assume that 𝑔_{𝑗,𝑖𝑗}(⋅)>𝑔_𝑗0>0.

Lemma 3 (see [16]). There exists smooth positive function 𝜓_𝑗(𝜂_𝑗)∶𝑅^𝑚𝑗→𝑅(𝑗=1,2,…,𝑛) with 𝜓_𝑗(0)=0 for all continuous functions ℎ(𝜂₁,…,𝜂_𝑛)∶𝑅^𝑚1×⋯×𝑅^𝑚𝑛→𝑅 with ℎ(0,…,0)=0, where 𝜂_𝑗∈𝑅^𝑚𝑗(𝑗−1,2,…,𝑛,𝑚_𝑗>0), such that |ℎ(𝜂₁,…,𝜂_𝑛)|≤∑^𝑛_𝑗=1𝜓_𝑗(𝜂_𝑗).

Lemma 4 (see [14]). On any normal number 𝜉>0 and random variable 𝑙∈𝑅 have lim_𝑙→0tanℎ²(𝑙/𝜉)/𝑙=0.

In this paper, the following radial basis function neural network is used to approximate unknown continuous function (in [13] once had been put forward): 𝑓(𝑍)=𝑊^𝑇𝑆(𝑍)+𝜃(𝑍),|𝜃(𝑍)|≤𝜀,(𝜀>0), (2) where the input vector 𝑍∈Ω_𝑍⊂𝑅^𝑛; 𝑊=[𝑤₁,𝑤₂,…,𝑤_𝑙]^𝑇 is the weight vector; the number of neural network node 𝑙>1 and 𝑆(𝑍)=[𝑠₁(𝑍),𝑠₂(𝑍),…,𝑠_𝑙(𝑍)]^𝑇, where 𝑠_𝑖=exp[−(𝑍−𝜇_𝑖)^𝑇(𝑍−𝜇_𝑖)/𝜙²_𝑖],𝑖=1,2,…,𝑙,𝜇_𝑖=[𝜇_𝑖1,𝜇_𝑖2,…,𝜇_𝑖𝑛]^𝑇 is the center of the receptive field, and 𝜙_𝑖 is the width of the Gaussian function.

3. Adaptive Neural Network Controller Design

In this section, we will introduce a novel adaptive NN control design procedure. There are 𝑚_𝑗 design steps in the design procedure for the 𝑗th subsystem. In each step, the unknown nonlinear function 𝑓_{𝑗,𝑖𝑗}(𝑍_{𝑗,𝑖𝑗}) will be approximated by a radial neural network approximation function. Define an unknown constant as 𝛼_𝑗=1𝑔_𝑗0max𝑊_{𝑗,𝑖𝑗}²∶1≤𝑖_𝑗≤𝑚_𝑗, (3) where the constant 𝑔_𝑗0 is defined as in Assumption 2; function 𝑓_{𝑗,𝑖𝑗} and vector 𝑍_{𝑗,𝑖𝑗} will be specified in each step. Furthermore, for 𝑗=1,2,…,𝑛 and 𝑖_𝑗=1,2,…,𝑚_𝑗−1, choose the virtual control laws as follows: 𝜆_{𝑗,𝑖𝑗}=−𝑘_{𝑗,𝑖𝑗}+1𝑧_{𝑗,𝑖𝑗}−12𝑎²_{𝑗,𝑖𝑗}𝛼_𝑗𝑧_{𝑗,𝑖𝑗}𝑆^𝑇𝑍_{𝑗,𝑖𝑗}𝑆𝑍_{𝑗,𝑖𝑗}, (4) where 𝑘_{𝑗,𝑖𝑗}>0 and 𝑎_{𝑗,𝑖𝑗}>0 are design parameters, 𝛼_𝑗 represent the estimation of the unknown constant 𝛼_𝑗, and 𝑆(⋅) is the basis function vector, and define the variables 𝑧_{𝑗,𝑖𝑗} as follows: 𝑧_{𝑗,𝑖𝑗}=𝑥_{𝑗,𝑖𝑗}−𝜆_{𝑗,𝑖𝑗−1},𝑧_𝑗,1=𝑥_𝑗,1−𝑦_𝑑𝑗, (5) for 𝑗=1,…,𝑛,𝑖_𝑗=2,…,𝑚_𝑗. Choose the adaptive laws 𝛼̇_𝑗 as follows: 𝛼̇_𝑗=^𝑚𝑗_𝑖𝑗=1𝑟_𝑗2𝑎²_{𝑗,𝑖𝑗}𝑧²_{𝑗,𝑖𝑗}𝑆^𝑇𝑍_{𝑗,𝑖𝑗}𝑆𝑍_{𝑗,𝑖𝑗}−𝑏_𝑗𝛼_𝑗, (6) where 𝑟_𝑗>0 and 𝑏_𝑗>0 are design parameters.

Step  𝑗⋅1(1≤𝑗≤𝑛). For the first differential equation of the 𝑗th subsystem, choose the Lyapunov function candidate 𝑉_𝑧𝑗,1=12𝑧²_𝑗,1+𝑔_𝑗02𝑟_𝑗𝛼², (7) where 𝑧_𝑗,1=𝑥_𝑗,1−𝑦_𝑑𝑗,𝛼=𝛼_𝑗−𝛼_𝑗. Taking the time derivative of 𝑉_𝑧𝑗,1, we obtain 𝑉̇_𝑧𝑗,1=𝑧_𝑗,1𝑓_𝑗,1+𝑔_𝑗,1𝜆_𝑗,1−𝑦̇_𝑑𝑗+ℎ_𝑗,1𝑥_𝜏𝑗,1+𝜔_𝑗,1(𝑡)+𝑧_𝑗,1𝑔_𝑗,1𝑧_𝑗,2−𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝛼̇_𝑗. (8) With Lemma 3, existence of positive function 𝑄^{𝑗,𝑖𝑗}_𝑗,𝑙(𝑥_{𝜏𝑗,𝑙})𝑙=1,2,…,𝑖_𝑗, such that ℎ_{𝑗,𝑖𝑗}𝑥_{𝜏𝑗𝑗,𝑖}≤^𝑖𝑗_𝑙=1𝑄^{𝑗,𝑖𝑗}_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}. (9) Then, we have 𝑧_𝑗,1ℎ_𝑗,1𝑥_𝜏𝑗,1≤𝑧_𝑗,1𝑄^𝑗,1_𝑗,1≤12𝑧²_𝑗,1+12𝑄^𝑗,1_𝑗,1𝑥_𝜏𝑗,1². (10) Substituting (10) into (8) yields 𝑉̇_𝑧𝑗,1≤𝑧_𝑗,1𝑓_𝑗,1+𝑔_𝑗,1𝜆_𝑗,1−𝑦̇_𝑑𝑗+12𝑧_𝑗,1+𝜔_𝑗,1(𝑡)+𝑧_𝑗,1𝑔_𝑗,1𝑧_𝑗,2+12𝑄^𝑗,1_𝑗,1𝑥_𝜏𝑗,1²−𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝛼̇_𝑗. (11) To overcome the time-varying delay terms of (11), consider the following Lyapunov-Krasovskii functional: 𝑉_𝑗,1=𝑉_𝑧𝑗,1+𝑉_𝑢𝑗,1, (12) where 𝑉_𝑢𝑗,1=^𝑡_{𝑡−𝜏𝑗,1(𝑡)}121−𝜏₁𝑄^𝑗,1_𝑗,1𝑥_𝑗,1(𝑠)²𝑑𝑠. (13) Take the time derivative of 𝑉_𝑢𝑗,1: 𝑉̇_𝑢𝑗,1≤121−𝜏₁𝑄^𝑗,1_𝑗,1𝑥_𝑗,1(𝑡)²−12𝑄^𝑗,1_𝑗,1𝑥_𝑗,1𝑡−𝜏_𝑗,1(𝑡)²; (14) from (11) and (14), one has 𝑉̇_𝑗,1≤𝑧_𝑗,1𝑓_𝑗,1𝑍_𝑗,1+𝑔_𝑗,1𝜆_𝑗,1+𝜔_𝑗,1(𝑡)−𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝛼̇_𝑗+𝑧_𝑗,1𝑔_𝑗,1𝑧_𝑗,2+1−2tanℎ²𝑧_𝑗,1𝜂_𝑗,1𝑈_𝑗,1, (15) where 𝑍_𝑗,1=𝑥_𝑗,1,𝑦_𝑑𝑗,𝑦̇_𝑑𝑗,𝛼_𝑗^𝑇,𝑈_𝑗,1=121−𝜏₁𝑄^𝑗,1_𝑗,1𝑥_𝑗,1²,𝑓_𝑗,1𝑍_𝑗,1=𝑓_𝑗,1−𝑦̇_𝑑𝑗+12𝑧_𝑗,1+2𝑧_𝑗,1tanℎ²𝑧_𝑗,1𝜂_𝑗,1𝑈_𝑗,1, (16) and 𝜂_𝑗,1 is a positive constant.

From Lemma 4, the function (1/𝑧)tanℎ²(𝑧/𝜂) is defined at 𝑧=0 and can be approximated by a neural network. Therefor the function 𝑓_𝑗,1 will be approximated by the NN 𝑊^𝑇_𝑗,1𝑆(𝑍_𝑗,1), such that, for given 𝜀_𝑗,1>0, 𝑓_𝑗,1𝑍_𝑗,1=𝑊^𝑇_𝑗,1𝑆𝑍_𝑗,1+𝜃_𝑗,1𝑍_𝑗,1,𝜃_𝑗,1𝑍_𝑗,1≤𝜀_𝑗,1, (17) where 𝜃_𝑗,1(𝑍_𝑗,1) is the approximation error. Furthermore, a straightforward calculation shows that 𝑧_𝑗,1𝑓_𝑗,1𝑍_𝑗,1≤12𝑎²_𝑗,1𝑔_𝑗0𝑧²_𝑗,1𝛼_𝑗𝑆^𝑇𝑍_𝑗,1𝑆𝑍_𝑗,1+12𝑎²_𝑗,1+12𝑔_𝑗0𝑧²_𝑗,1+12𝜀²_𝑗,1𝑔⁻¹_𝑗0. (18) In additions, from (6), we obtain that for any initial conditions 𝛼_𝑗(𝑡₀)≥0, 𝛼_𝑗(𝑡)>0 for all 𝑡>𝑡₀. Therefor 𝑧_𝑗,1𝑔_𝑗,1𝜆_𝑗,1𝑔≤−^𝑗02𝑎²_𝑗,1𝛼_𝑗𝑧²_𝑗,1𝑆^𝑇𝑍_𝑗,1𝑆𝑍_𝑗,1−𝑘_𝑗,1+1𝑔_𝑗0𝑧²_𝑗,1,𝑧_𝑗,1𝜔_𝑗,1(𝑡)≤12𝑔_𝑗0𝑧²_𝑗,1+12𝑑²_𝑗,1𝑔⁻¹_𝑗0. (19)(20) Substituting (18)–(20) into (15) yields that 𝑉̇_𝑗,1≤𝑘_𝑗,1𝑔_𝑗0𝑧²_𝑗,1+12𝑎²_𝑗,1+𝜀²_𝑗,1𝑔⁻¹_𝑗0+𝑑²_𝑗,1𝑔⁻¹_𝑗0+𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝑟_𝑗2𝑎²_𝑗,1𝑧²_𝑗,1𝑆^𝑇𝑍_𝑗,1𝑆𝑍_𝑗,1−𝛼̇_𝑗+𝑧_𝑗,1𝑔_𝑗,1𝑧_𝑗,2+1−2tanℎ²𝑧_𝑗,1𝜂_𝑗,1𝑈_𝑗,1. (21)

Step  𝑗⋅𝑖_𝑗(𝑖_𝑗=2,…,𝑚_𝑗−1). Define the Lyapunov-Krasovskii functional as 𝑉_{𝑧𝑗𝑗,𝑖}=12𝑧²_{𝑗,𝑖𝑗}; (22) differentiating 𝑉_{𝑧𝑗𝑗,𝑖} yields 𝑉̇_{𝑧𝑗𝑗,𝑖}=𝑧_{𝑗,𝑖𝑗}𝑓_{𝑗,𝑖𝑗}+𝑔_{𝑗,𝑖𝑗}𝑥_{𝑗,𝑖𝑗+1}−𝜆̇_{𝑗,𝑖𝑗−1}+ℎ_{𝑗,𝑖𝑗}𝑥_{𝜏𝑗𝑗,𝑖}+𝜔_{𝑗,𝑖𝑗}(𝑡). (23) From (10), we have 𝑧_{𝑗,𝑖𝑗}ℎ_{𝑗,𝑖𝑗}≤^𝑖𝑗_𝑘=112𝑧²_{𝑗,𝑖𝑗}+12𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_{𝜏𝑗,𝑘}²; (24)𝜆̇_{𝑗,𝑖𝑗−1}(𝑍_{𝑗,𝑖𝑗−1}) can be expressed as 𝜆̇_{𝑗,𝑖𝑗−1}=^𝑖𝑗−1_𝑘=1𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘𝑓_𝑗,𝑘+𝑔_𝑗,𝑘𝑥_𝑗,𝑘+1+𝜔_𝑗,𝑘+^𝑖𝑗−1_𝑘=0𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑦^(𝑘)_𝑑𝑗𝑦^(𝑘+1)_𝑑𝑗+𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+^𝑖𝑗−1_𝑘=1𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘ℎ_𝑗,𝑘𝑥_{𝜏𝑗,𝑘}. (25) Similar to (24), we can get −𝑧_{𝑗,𝑖𝑗}^𝑖𝑗−1_𝑘=1𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘ℎ_𝑗,𝑘𝑥_{𝜏𝑗,𝑘}≤^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑧²_{𝑗,𝑖𝑗}𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘²+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑄^𝑗,𝑘_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}². (26) Substituting (24)–(26) into (23) yields that 𝑉̇_{𝑧𝑗𝑗,𝑖}≤𝑧_{𝑗,𝑖𝑗}𝑓_{𝑗,𝑖𝑗}+𝑔_{𝑗,𝑖𝑗}𝑥_{𝑗,𝑖𝑗+1}+𝜔_{𝑗,𝑖𝑗}−^𝑖𝑗−1_𝑘=1𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘𝑓_𝑗,𝑘+𝑔_𝑗,𝑘𝑥_𝑗,𝑘+1+𝜔_𝑗,𝑘+^𝑖𝑗_𝑘=112𝑧_{𝑗,𝑖𝑗}−^𝑖𝑗−1_𝑘=0𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑦^(𝑘)_𝑑𝑗𝑦^(𝑘+1)_𝑑𝑗+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑧_{𝑗,𝑖𝑗}𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘²−𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑄^𝑗,𝑘_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}²+^𝑖𝑗_𝑘=112𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_{𝜏𝑗,𝑘}². (27) To overcome the delay terms in (27), let us consider the following Lyapunov-Krasovskii functional: 𝑉_{𝑗,𝑖𝑗}=𝑉_{𝑧𝑗𝑗,𝑖}+𝑉_{𝑢𝑗𝑗,𝑖}, (28) where 𝑉_{𝑢𝑗𝑗,𝑖}=^𝑖𝑗_𝑘=1^𝑡_{𝑡−𝜏𝑗,𝑘}121−𝜏₁𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_𝑗,𝑘(𝑠)²+𝑑𝑠_{𝑖𝑗𝑘𝑘=1𝑙=1𝑡𝑡−𝜏𝑗,𝑙}121−𝜏₁𝑄^𝑗,𝑘_𝑗,𝑙𝑥_𝑗,𝑙(𝑠)²𝑑𝑠. (29) Differentiating 𝑉_{𝑢𝑗𝑗,𝑖} yields 𝑉̇_{𝑢𝑗𝑗,𝑖}=^𝑖𝑗_𝑘=1121−𝜏₁𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_𝑗,𝑘(𝑡)²+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=1121−𝜏₁𝑄^𝑗,𝑘_𝑗,𝑙𝑥_𝑗,𝑙(𝑡)²−^𝑖𝑗_𝑘=1121−𝜏₁𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_{𝜏𝑗𝑗,𝑖}²1−𝜏̇_𝑗,𝑘−^{𝑖𝑗−1𝑘}_𝑘=1𝑙=1121−𝜏₁𝑄^𝑗,𝑘_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}²1−𝜏̇_𝑗,𝑙≤𝑈_{𝑗,𝑖𝑗}−^𝑖𝑗_𝑘=112𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_{𝜏𝑗𝑗,𝑖}²−^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑄^𝑗,𝑘_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}²≤𝑧_{𝑗,𝑖𝑗}2𝑧_{𝑗,𝑖𝑗}tanℎ²𝑧_{𝑗,𝑖𝑗}𝜂_{𝑗,𝑖𝑗}𝑈_{𝑗,𝑖𝑗}+1−2tanℎ²𝑧_{𝑗,𝑖𝑗}𝜂_{𝑗,𝑖𝑗}𝑈_{𝑗,𝑖𝑗}−^𝑖𝑗_𝑘=112𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_{𝜏𝑗𝑗,𝑖}²−^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑄^𝑗,𝑘_𝑗,𝑙𝑥_{𝜏𝑗,𝑙}², (30) where 𝑈_{𝑗,𝑖𝑗}=^𝑖𝑗_𝑘=1121−𝜏₁𝑄^{𝑗,𝑖𝑗}_𝑗,𝑘𝑥_𝑗,𝑘(𝑡)²+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=1121−𝜏₁𝑄^𝑗,𝑘_𝑗,𝑙𝑥_𝑗,𝑙(𝑡)². (31) Then, combining (27) and (30) results in 𝑉̇_{𝑗,𝑖𝑗}≤𝑧_{𝑗,𝑖𝑗}𝜑_{𝑗,𝑖𝑗}−𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+𝑔_{𝑗,𝑖𝑗}𝑧_{𝑗,𝑖𝑗}𝑧_{𝑗,𝑖𝑗+1}+1−2tanℎ²𝑧_{𝑗,𝑖𝑗}𝜂_{𝑗,𝑖𝑗}𝑈_{𝑗,𝑖𝑗}+𝑧_{𝑗,𝑖𝑗}𝑓_{𝑗,𝑖𝑗}+𝑔_{𝑗,𝑖𝑗}𝜆_{𝑗,𝑖𝑗+1}+𝜔_{𝑗,𝑖𝑗}, (32) where 𝑓_{𝑗,𝑖𝑗}=𝑓_{𝑗,𝑖𝑗}−^𝑖𝑗−1_𝑘=1𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘𝑓_𝑗,𝑘+𝑔_𝑗,𝑘𝑥_𝑗,𝑘+1+𝜔_𝑗,𝑘−^𝑖𝑗−1_𝑘=0𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑦^(𝑘)_𝑑𝑗𝑦^(𝑘+1)_𝑑𝑗+^𝑖𝑗_𝑘=112𝑧_{𝑗,𝑖𝑗}+^{𝑖𝑗−1𝑘}_𝑘=1𝑙=112𝑧_{𝑗,𝑖𝑗}𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝑥_𝑗,𝑘²+2𝑧_{𝑗,𝑖𝑗}tanℎ²𝑧_{𝑗,𝑖𝑗}𝜂_{𝑗,𝑖𝑗}𝑈_{𝑗,𝑖𝑗}−𝜑_{𝑗,𝑖𝑗}. (33) The NN 𝑊^𝑇_{𝑗,𝑖𝑗}𝑆(𝑍_{𝑗,𝑖𝑗}) is used to approximate 𝑓_{𝑗,𝑖𝑗} such that for given 𝜀_{𝑗,𝑖𝑗}>0 we have 𝑓_{𝑗,𝑖𝑗}=𝑊^𝑇_{𝑗,𝑖𝑗}𝑆𝑍_{𝑗,𝑖𝑗}+𝜃_{𝑗,𝑖𝑗}𝑍_{𝑗,𝑖𝑗},𝜃_{𝑗,𝑖𝑗}𝑍_{𝑗,𝑖𝑗}≤𝜀_{𝑗,𝑖𝑗}, (34) where 𝜃_{𝑗,𝑖𝑗}(𝑍_{𝑗,𝑖𝑗}) represent the approximation error. Similar to (18) and (20), we have 𝑉̇_{𝑗,𝑖𝑗}≤−𝑘_{𝑗,𝑖𝑗}𝑔_𝑗0𝑧²_{𝑗,𝑖𝑗}+12𝑎²_{𝑗,𝑖𝑗}+𝜀²_{𝑗,𝑖𝑗}𝑔⁻¹_𝑗0+𝑑²_{𝑗,𝑖𝑗}𝑔⁻¹_𝑗0+𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝑟_𝑗2𝑎²_{𝑗,𝑖𝑗}𝑧²_{𝑗,𝑖𝑗}𝑆^𝑇𝑍_{𝑗,𝑖𝑗}𝑆𝑍_{𝑗,𝑖𝑗}+𝑧_{𝑗,𝑖𝑗}𝜑_{𝑗,𝑖𝑗}−𝜕𝜆_{𝑗,𝑖𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+𝑔_{𝑗,𝑖𝑗}𝑧_{𝑗,𝑖𝑗}𝑧_{𝑗,𝑖𝑗+1}+1−2tanℎ²𝑧_{𝑗,𝑖𝑗}𝜂_{𝑗,𝑖𝑗}𝑈_{𝑗,𝑖𝑗}. (35)

Step  𝑗⋅𝑚_𝑗(1≤𝑗≤𝑛). In the final step of the 𝑗th subsystem to construct the actual control law 𝑢_𝑗, let us consider the following Lyapunov-Krasovskii function: 𝑉_{𝑗,𝑖𝑗}=12𝑧²_{𝑗,𝑚𝑗}+𝑉_{𝑢𝑗𝑗,𝑚}, (36) where 𝑉_{𝑢𝑗𝑗,𝑚}=^𝑛𝑚_𝑗=1^𝑗_𝑘=1^𝑡_{𝑡−𝜏𝑗,𝑘}121−𝜏₁𝑄^{𝑗,𝑚𝑗}_𝑗,𝑘𝑥_𝑗,𝑘(𝑠)²+𝑑𝑠_{𝑚𝑗−1𝑘𝑘=1𝑙=1𝑡𝑡−𝜏𝑗,𝑙}121−𝜏₁𝑄^𝑗,𝑘_𝑗,𝑙𝑥_𝑗,𝑙(𝑠)²𝑑𝑠 (37) and 𝑧_{𝑗,𝑚𝑗}=𝑥_{𝑗,𝑚𝑗}−𝜆_{𝑗,𝑚𝑗−1}. Similar to (32) we get𝑉̇_{𝑗,𝑚𝑗}≤𝑧_{𝑗,𝑚𝑗}𝜑_{𝑗,𝑚𝑗}−𝜕𝜆_{𝑗,𝑚𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+1−2tanℎ²𝑧_{𝑗,𝑚𝑗}𝜂_{𝑗,𝑚𝑗}𝑈_{𝑗,𝑚𝑗}+𝑧_{𝑗,𝑚𝑗}𝑓_{𝑗,𝑚𝑗}+𝑔_{𝑗,𝑚𝑗}𝑢_𝑗+𝜔_{𝑗,𝑚𝑗}, (38) where 𝑓_{𝑗,𝑚𝑗}(𝑧_{𝑗,𝑚𝑗}) can be defined by (33) with 𝑖_𝑗=𝑚_𝑗.

We use the NN 𝑊^𝑇_{𝑗,𝑚𝑗}𝑆(𝑍_{𝑗,𝑚𝑗}) to approximate 𝑓_{𝑗,𝑚𝑗} such that, for given 𝜀_{𝑗,𝑚𝑗}>0, we have 𝑓_{𝑗,𝑚𝑗}=𝑊^𝑇_{𝑗,𝑚𝑗}𝑆𝑍_{𝑗,𝑚𝑗}+𝜃_{𝑗,𝑚𝑗}𝑍_{𝑗,𝑚𝑗},𝜃_{𝑗,𝑚𝑗}𝑍_{𝑗,𝑚𝑗}≤𝜀_{𝑗,𝑚𝑗}, (39) where 𝜃_{𝑗,𝑚𝑗}(𝑍_{𝑗,𝑚𝑗}) express the approximation error.

Choose the control law 𝑢_𝑗 as 𝑢_𝑗=−𝑘_{𝑗,𝑚𝑗}+1𝑧_{𝑗,𝑚𝑗}−12𝑎²_{𝑗,𝑚𝑗}𝛼_𝑗𝑧_{𝑗,𝑚𝑗}𝑆^𝑇𝑍_{𝑗,𝑚𝑗}𝑆𝑍_{𝑗,𝑚𝑗}. (40) Similar to (21) we have 𝑉̇_{𝑗,𝑚𝑗}≤12𝑎²_{𝑗,𝑚𝑗}+𝜀²_{𝑗,𝑚𝑗}𝑔⁻¹_𝑗0+𝑑²_{𝑗,𝑚𝑗}𝑔⁻¹_𝑗0+𝑔_𝑗0𝑟_𝑗𝛼_𝑗𝑟_𝑗2𝑎²_{𝑗,𝑚𝑗}𝑧²_{𝑗,𝑚𝑗}𝑆^𝑇𝑍_{𝑗,𝑚𝑗}𝑆𝑍_{𝑗,𝑚𝑗}−𝑘_{𝑗,𝑚𝑗}𝑔_𝑗0𝑧²_{𝑗,𝑚𝑗}+𝑧_{𝑗,𝑚𝑗}𝜑_{𝑗,𝑚𝑗}−𝜕𝜆_{𝑗,𝑚𝑗−1}𝜕𝛼_𝑗𝛼̇_𝑗+1−2tanℎ²𝑧_{𝑗,𝑚𝑗}𝜂_{𝑗,𝑚𝑗}𝑈_{𝑗,𝑚𝑗}. (41) Let 𝑉_{𝑛,𝑚𝑛}=∑^𝑛_𝑗=1∑^𝑚𝑗_𝑘=1𝑉_𝑗,𝑘. Combining (21), (35), and (41) gives that 𝑉̇_{𝑛,𝑚𝑛}≤−^𝑛𝑚_𝑗=1^𝑗_𝑘=1𝑘_𝑗,𝑘𝑔_𝑗0𝑧²_𝑗,𝑘+^𝑛𝑚_𝑗=1^𝑗_𝑘=112𝑎²_𝑗,𝑘+𝜀²_𝑗,𝑘𝑔⁻¹_𝑗0+𝑑²_𝑗,𝑘𝑔⁻¹_𝑗0+^𝑛_𝑗=1𝑔_𝑗0𝑟_𝑗𝛼_𝑗^𝑚𝑗_𝑘=1𝑟_𝑗2𝑎²_𝑗,𝑘𝑧²_𝑗,𝑘𝑆^𝑇𝑍_𝑗,𝑘𝑆𝑍_𝑗,𝑘−𝛼̇_𝑗+^𝑛𝑚_𝑗=1^𝑗_𝑘=11−2tanℎ²𝑧_𝑗,𝑘𝜂_𝑗,𝑘𝑈_𝑗,𝑘+^𝑛𝑚_𝑗=1^𝑗_𝑘=2𝑧_𝑗,𝑘𝜑_𝑗,𝑘−𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗𝛼̇_𝑗. (42) The control law design is thus completed.

4. Stability Analysis

Now, the main result in this paper can be presented as follows.

Theorem 5. Consider the nonlinear time-delay system (1) with the NN adaption law (6) and the control law (40) satisfying Assumptions 1–2. All the closed-loop trajectories can guarantee boundedness if the unknown function 𝑓_{𝑗,𝑖𝑗} can be approximated by neural network and the approximating error 𝜃_{𝑗,𝑖𝑗}is boundedness.

Proof. Define functions 𝜑_𝑗,𝑘, such that −^𝑛𝑚_𝑗=1^𝑗_𝑘=2𝑧_𝑗,𝑘𝜑_𝑗,𝑘−𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗𝛼̇_𝑗≤0. (43) Let 0<𝑆^𝑇(⋅)𝑆(⋅)<𝐿, where 𝐿 is the number of neural network weights.
From (6), we can get −^𝑚𝑗_𝑘=2𝑧_𝑗,𝑘𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗𝛼̇_𝑗≤^𝑚𝑗_𝑘=2𝑧_𝑗,𝑘𝑏_𝑗𝛼_𝑗𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗−^𝑘−1_𝑙=1𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗𝑟_𝑗2𝑎²_𝑗,𝑙𝑧²_𝑗,𝑙𝑆^𝑇𝑍_𝑗,𝑙𝑆𝑍_𝑗,𝑙+^𝑚𝑗_𝑘=2𝑧_𝑗,𝑘𝑟_𝑗𝐿2𝑎²_𝑗,𝑘𝑧^2𝑘_{𝑗,𝑘𝑙=2}𝑧_𝑗,𝑙𝜕𝜆_{𝑗,𝑙−1}𝜕𝛼_𝑗. (44) By choosing function 𝜑_𝑗,𝑘 as 𝜑_𝑗,𝑘=−𝑏_𝑗𝛼_𝑗𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗−𝑟_𝑗𝐿2𝑎²_𝑗,𝑘𝑧^2𝑘_{𝑗,𝑘𝑙=2}𝑧_𝑗,𝑙𝜕𝜆_{𝑗,𝑙−1}𝜕𝛼_𝑗+^𝑘−1_𝑙=1𝜕𝜆_{𝑗,𝑘−1}𝜕𝛼_𝑗𝑟_𝑗2𝑎²_𝑗,𝑙𝑧²_𝑗,𝑙𝑆^𝑇𝑍_𝑗,𝑙𝑆𝑍_𝑗,𝑙, (45) (43) holds.
In a similar way, we can get ^𝑛_𝑗=1𝑔_𝑗0𝑟_𝑗𝛼_𝑗^𝑚𝑗_𝑘=1𝑟_𝑗2𝑎²_𝑗,𝑘𝑧²_𝑗,𝑘𝑆^𝑇𝑍_𝑗,𝑘𝑆𝑍_𝑗,𝑘−𝛼̇_𝑗≤^𝑛_𝑗=1𝑔_𝑗0𝑟_𝑗−𝛼²_𝑗+𝛼²_𝑗. (46) Now, choose the Lyapunov function as 𝑉=𝑉_{𝑛,𝑚𝑛}.
Combining (42)–(46) gives that 𝑉̇_{𝑛,𝑚𝑛}≤−^𝑛𝑚_𝑗=1^𝑗_𝑘=1𝑘_𝑗,𝑘𝑔_𝑗0𝑧²_𝑗,𝑘−^𝑛_𝑗=1𝑔_𝑗0𝑏_𝑗𝑟_𝑗𝛼²_𝑗+^𝑛𝑚_𝑗=1^𝑗_𝑘=11−2tanℎ²𝑧_𝑗,𝑘𝜂_𝑗,𝑘𝑈_𝑗,𝑘+𝐷, (47) where 𝐷=^𝑛𝑚_𝑗=1^𝑗_𝑘=112𝑎²_𝑗,𝑘+𝜀²_𝑗,𝑘𝑔⁻¹_𝑗0+𝑑²_𝑗,𝑘𝑔⁻¹_𝑗0+^𝑛_𝑗=1𝑔_𝑗0𝑟_𝑗𝛼²_𝑗 (48) is a constant. Thus, by (47) the boundedness follows immediately from the same line used in the proof in [9–11].

5. Simulation Examples

In this section, we will give one example to demonstrate the effectiveness of the proposed method in this paper. Let us consider the following example: 𝑥̇_1,1=−𝑥_1,1+1+cos²𝑥_1,1𝑥_1,2+𝑥²_𝜏1,1+𝜔_1,1(𝑡),𝑥̇_1,2=𝑥_1,1𝑥_1,2+𝑥_2,1+𝑥_2,2+1+0.5cos²𝑥_2,2𝑢₁+𝑥_𝜏1,2+𝜔_1,2(𝑡),𝑥̇_2,1=−𝑥_2,1+𝑥_2,2+𝑥_𝜏2,1+𝜔_2,1(𝑡),𝑥̇_2,2=𝑥_1,2+𝑥_2,1𝑥_2,2−𝑥_1,1𝑢₁+2+sin²𝑢₁𝑢₂+𝑥_𝜏1,1𝑥_𝜏2,2+𝜔_2,2(𝑡), (49) where 𝑥_{𝜏𝑗𝑗,𝑖}=𝑥_{𝑗,𝑖𝑗}(𝑡−𝜏_{𝑗,𝑖𝑗}),𝑗=1,2,𝑖_𝑗=1,2.

And the time delays are chosen as 𝜏_1,1=0.9+0.1sin(𝑡),𝜏_1,2(𝑡),𝜏=1+0.5sin_2,1=0.4+0.1cos(𝑡),𝜏_2,2=2+0.1cos(𝑡), (50) given the reference output signals as 𝑦_𝑑1=0.5(sin(𝑡)+sin(0.5𝑡)),𝑦_𝑑2=0.5sin(𝑡)+sin(0.5𝑡). The control law is given by (40). The NN adaptation law is given by (6). Choose the design parameters 𝑘_1,1=𝑘_1,2=𝑘_2,1=𝑘_2,2𝑎=20,_1,1=𝑎_1,2=2,𝑎_2,1=𝑎_2,2𝑟=1,₁=𝑟₂𝑏=400,₁=𝑏₂=0.025. (51) Take the external disturbance as 𝜔_1,1(𝑡)=𝜔_1,2(𝑡)(),𝜔=0.04sin2𝜋𝑡_2,1(𝑡)=𝜔_2,2(𝑡)=0.04cos(2𝜋𝑡). (52)

The simulation is run under the initial conditions 𝑥_{𝑗,𝑖𝑗}(𝜗)=0,−𝜏₀≤𝜗≤0,𝑗=1,2,𝑖_𝑗=1,2, and [𝛼₁(0),𝛼₂(0)]^𝑇=[0,0]^𝑇. The result of control scheme is displayed in Figures 1–5. Figures 1 and 2 demonstrate the outputs of system and the reference signals. The responses of state variables 𝑥_1,2 and 𝑥_2,2 are shown in Figure 3. The control input signals 𝑢₁ and 𝑢₂ are illustrated in Figures 4 and 5which depict the boundedness of adaptive parameters 𝛼₁ and 𝛼₂.

Figure 1 

System output 𝑦1(𝑡)(“−”) and the reference 𝑦𝑑1(𝑡)(“−·−”).

Figure 2 

System output 𝑦2(𝑡)(“−”) and the reference 𝑦𝑑2(𝑡) (“−·−”).

Figure 3 

Responses of state variables 𝑥1,2 and 𝑥2,2.

Figure 4 

Control input signals 𝑢1 and 𝑢2.

Figure 5 

Adaptive parameters 𝛼1 and 𝛼2.

6. Conclusion

For a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form, an adaptive neural control design is presented. Although there are some fluctuations of the systems and control output under the influence of interference, the required performance can be achieved in a short period of time by using the controller designed in this paper and guarantees the boundedness of all the signals in the closed-loop system. It is further extended on the bases in [14, 15], which makes it suitable for wider range of applications. The effectiveness of the proposed approach is provided by a simulation example.

Acknowledgments

This work was jointly supported by the Natural Science Foundation of China (60864001) and Guangxi Natural Science Foundation (2011GXNSFA018161).