Adaptive Neural Control for a Class of Outputs Time-Delay Nonlinear Systems

Abstract

This paper considers an adaptive neural control for a class of outputs time-delay nonlinear systems with perturbed or no. Based on RBF neural networks, the radius basis function (RBF) neural networks is employed to estimate the unknown continuous functions. The proposed control guarantees that all closed-loop signals remain bounded. The simulation results demonstrate the effectiveness of the proposed control scheme.

1. Introduction

The study of the time-delay systems has been one of the most active research topics in recent years [1–15]. The time-delay systems can be divided into four types: systems with input delay [1–5], systems with state delay [6–9, 16–18], systems with both input and state delays, and systems with both input and output delays [19]. The effect of time delay on stability and asymptotic performance has been investigated in [20]. In [21], Lyapunov-Krasovskii functionals were used with backstepping to obtain a robust controller for a class of single-input single-output (SISO) nonlinear time-delay systems with known bounds on the functions of delayed states, but it was commented that results could not be constructively obtained in [22]. In [23], the problem of the adaptive neural-networks control for a class of nonlinear state-delay systems with unknown virtual control coefficients is considered. In [24], An adaptive control scheme combined with radius basis function (RBF) neural networks, backstepping, and adaptive control is proposed for the output tracking control problem of a class of MIMO nonlinear system with input delay and disturbances. Neural networks are employed to estimate the unknown continuous functions; the control scheme ensures that the closed-loop system is semiglobally uniformly ultimately bounded (SGUUB). In [11] A control scheme combined with backstepping, radius basis function (RBF) neural networks, and adaptive control is proposed for the stabilization of nonlinear system with input and state delay.

In this paper, we present an adaptive neural controller design procedure for a class of output time-delay nonlinear systems with perturbed, based on backstepping, adaptive control, and neural networks. RBF neural network is employed to the unknown continuous function. A numerical example is provided to show the effectiveness of the control scheme.

2. Problem Formulation and Preliminaries

Consider the nonlinear time-delay system is described as follows: 𝑥̇_𝑖=𝑥_𝑖+1+𝑔_𝑖𝑦+𝑓_𝑖𝑦(𝑡−𝜏)+𝜈_𝑖(𝑡)𝑥̇,1≤𝑖≤𝑛−1,_𝑛=𝑢(𝑡)+𝑔_𝑛𝑦+𝑓_𝑛𝑦(𝑡−𝜏)+𝜈_𝑛(𝑡),𝑦=𝑥₁, (2.1) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 is state, 𝑢∈𝑅 is control and 𝑦∈𝑅 is output vectors, respectively. 𝜈_𝑖(𝑡)(𝑖=1,2,…,𝑛) is a time-varying disturbance. 𝑓_𝑖(𝑦(𝑡−𝜏)), (𝑖=1,2,…,𝑛), 𝑔_𝑖(𝑦), (𝑖=1,2,…,𝑛) are unknown continuous functions.

Assumption 2.1. The unknown function 𝑓_𝑖(𝑦) satisfies 𝑓²_𝑖(𝑦)≤𝑘_𝑖, where 𝑘_𝑖(𝑖=1,2,…,𝑛) is a known constant.

Assumption 2.2. The time-varying disturbance 𝜈_𝑖(𝑡) satisfies |𝜈_𝑖(𝑡)|≤𝑑_𝑖<1, 1≤𝑖≤𝑛, where 𝑑_𝑖(𝑖=1,2,…,𝑛) is a known constant.

Lemma 2.3. 𝑥(𝑡)∈Ω_𝑀,𝑡∈[−𝜏_max,𝜏], where Ω_𝑀={𝑥∣‖𝑥‖≤𝑀}, 𝑀 is an unknown constant.

3. RBF NN Approximation

In this paper, for a given 𝛿>0 and any continuous function 𝐻_𝑖(𝜂_𝑖) defined on Ω_𝑖, there is a perfect RBF neural network, which satisfies 𝐹_𝑖𝜁_𝑖=𝑊^𝑇_𝑖𝑆_𝑖𝜁_𝑖+𝛿_𝑖𝜁_𝑖, (3.1) where |𝛿_𝑖(𝜁_𝑖)|≤𝛿𝑊_𝑖∈𝑅^𝑚𝑖 is the weight vector of the neural networks, 𝑚_𝑖 is the number of the NN nodes, 𝜁_𝑖∈Ω_𝑖 is the input vector, 𝑆_𝑖(𝜁_𝑖)=[𝑠_𝑖1,…,𝑠_𝑖𝑚𝑖]^𝑇 is defined by 𝑠_𝑖𝑗𝜁_𝑖−𝜁=exp^𝑖−𝜇_𝑖𝑗^𝑇𝜁_𝑖−𝜇_𝑖𝑗𝜉². (3.2) According to the discussion in [21, 22], denote the best weight vector as follows: 𝑊^∗_𝑖=argmin_{𝑤𝑖∈𝑅𝑚𝑖}sup_{𝜁𝑖∈Ω𝑖}𝑊^𝑇_𝑖𝑆_𝑖𝜁_𝑖−𝐻_𝑖𝜁_𝑖 (3.3) which is unknown and needs to be estimated in control design. Let 𝑊_𝑖 be the estimate of 𝑊^∗_𝑖, and define 𝑊_𝑖=𝑊_𝑖−𝑊^∗_𝑖.

4. Main Result

In this section, we will consider system (2.1).

(I) when 𝜈_𝑖(𝑡)=0,𝑖=1,2,…,𝑛,

Let us define error variables 𝑧_𝑖 assistant functions and the virtual control 𝛼_𝑖, respectively, as follows: 𝑧₁=𝑥₁,𝑧_𝑖=𝑥_𝑖−𝛼_𝑖−1,2≤𝑖≤𝑛. (4.1) Define the following sets: Ω_𝑧𝑖=𝑧_𝑖∈𝑅∣𝑧_𝑖<𝜆_𝑖,Ω^𝑜_𝑧𝑖=𝑧_𝑖∈𝑅∣𝑧_𝑖≥𝜆_𝑖, (4.2) where 𝜆_𝑖 is a small constant. Define assistant functions as 𝐹₁=𝑔₁(𝑥)+12𝜆²₁𝑈₁𝑥₁𝑧₁,𝐹_𝑖=𝑔_𝑖(𝑥)+12𝜆²_𝑖𝑧_𝑖^𝑖−1_𝑗=1𝑈_𝑗𝑥₁−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑊_𝑗𝑊̇_𝑗−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑥_𝑗+1+𝑔_𝑗𝑥₁−12𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑧_𝑗,2≤𝑖≤𝑛. (4.3) Define the virtual control as 𝛼_𝑖=−𝑘_𝑖0𝑧_𝑖−𝑊^𝑇_𝑖𝑆_𝑖𝜉_𝑖,1≤𝑖≤𝑛, (4.4) where 𝑘₁₀=𝑐₁+𝑏₁2+32+12𝜆²₁𝜆²₂+𝜏_max𝑘₁,𝑘_𝑖0=𝑐_𝑖+𝑏_𝑖21+2+2𝜆²_𝑖𝜆²_𝑖+1+𝜏_max^𝑖−1_𝑗=1𝑘_𝑗𝑘,2≤𝑖≤𝑛−1,_𝑛0=𝑐_𝑛+𝑏_𝑛21+1+2𝜆²_𝑛𝜆²_𝑖+1+𝜏_max^𝑛−1_𝑗=1𝑘_𝑗,𝜉₁=𝑥₁,𝑥₁^𝑇,𝜉_𝑖=𝑥^𝑇_𝑖,𝑥^𝑇_𝑖,𝛼_𝑖−1,𝜕_𝑖−1𝜕𝑥_𝑖−1^𝑇𝜕_𝑖−1𝜕𝑥_𝑖−1^𝑇,𝜓_𝑖−1^𝑇𝜓,2≤𝑖≤𝑛,_𝑖−1=^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑊_𝑗𝜕𝑊̇_𝑗,2≤𝑖≤𝑛. (4.5)

Theorem 4.1. System (2.1) with both input delay and state delay satisfies Assumptions 2.1 and 2.2. The virtual control can be selected as (4.4). If the control law and the adaptive law are selected as follows: 𝑢(𝑡)=𝛼_𝑛,𝑊̇_𝑖=Γ_𝑖𝑧_𝑖𝑆_𝑖𝜉_𝑖−𝜎_𝑖𝑊_𝑖,1≤𝑖≤𝑛, (4.6)(4.7) then the closed-loop system is semi-globally uniformly ultimately bounded.

Proof. Define the Lyapunov-Kresovskii functional 𝑉(𝑡) as 𝑉_𝑧𝑖(𝑡)=12𝑧²_𝑖(𝑡),𝑉_𝑊𝑖(𝑡)=12𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊_𝑖,𝑉_𝑈𝑖(𝑡)=12^𝑖_𝑗=1^𝑡_{𝑡−𝜏ℎ}𝑈_𝑗𝑦(𝜎)𝑉𝑑𝜎,_𝑖(𝑡)=𝑉_𝑧𝑖(𝑡)+𝑉_𝑈𝑖(𝑡)+𝑉_𝑊𝑖(𝑡)𝑉(𝑡)=,1≤𝑖≤𝑛,_𝑛𝑖=1𝑉_𝑖(𝑡). (4.8)(4.9)(4.10)(4.11)Step 1. For the first differential equation of the the first subsystem, by (4.1), (4.3), we can get 𝑧̇₁=𝑥̇₁=𝑥₂+𝑔₁𝑥₁+𝑓₁𝑥₁𝑡−𝜏_ℎ=𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+𝑔₁𝑥₁𝑡−𝜏_ℎ. (4.12) By 𝑧₁𝑓₁𝑥₁𝑡−𝜏_ℎ≤12𝑧²₁+12𝑓²₁𝑥₁𝑡−𝜏_ℎ (4.13) differentiating (4.10) and using (4.12), the inequality below can be obtained easily. 𝑉̇₁=𝑧₁𝑧̇₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ=𝑧₁𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+𝑓₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁. (4.14) (1) If 𝑧₁∈Ω^𝑜_𝑧1, then |𝑧₁|≥𝜆₁. Thus, substituting (4.4) and (4.7) into (4.14) results in 𝑉̇₁≤𝑧₁𝑧₂+𝑧₁−𝑘₁₀𝑧₁−𝑊^𝑇₁𝑆₁𝜁₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁≤12𝑧₁²+12𝑧₂²−𝑐₁+𝑏₁2+32𝑧²₁+𝑧₁𝐹₁𝜁₁−𝑧²₁2𝜆₁𝜏_max𝑘₁+𝜆²₂−𝑧₁𝑊^𝑇₁𝑆₁𝜁₁+12𝑧²₁+12𝑧₁²+𝑊𝑧₁𝑆₁𝜁₁−𝜎₁𝑊₁≤12𝑧²₂−𝑐₁+𝑏₁2𝑧²₁−𝑧²₁2𝜆²₁𝜏_max𝑘₁+𝜆²₂+𝑧₁𝜀₁𝜉₁−𝜎₁𝑊^𝑇₁𝑊₁≤12𝑧²₂−𝑐₁+𝑏₁2𝑧²₁+𝑏₁2𝑧²₁+12𝑏₁𝜀^∗₁²−12𝜏_max𝑘₁+𝜆²₂−𝜎₁𝑊^𝑇₁𝑊₁≤−𝑐₁𝑧²₁−𝑉_𝑈1(𝑇)−𝜎₁2𝑊₁²+𝜎₁2𝑊^∗₁²+12𝑏₁𝜀^∗₁²+12𝑧²₂−𝜆²₂≤−𝑘₁𝑉₁+𝑏_𝑣1+Θ₁, (4.15) where 𝑘₁=min{2𝑐₁,𝜎₁/𝜆_max(Γ₁⁻¹),1}, 𝑏_𝑣1=(𝜎₁/2)‖𝑊^∗₁‖²+(1/2𝑏₁)(𝜀^∗₁)². Θ₁=(1/2)(𝑧²₂−𝜆²₂).
If there is no item Θ₁ in (4.15), then 0≤𝑉₁(𝑡)≤𝑉₁(0)−𝛿_𝑣𝑒^−𝑘1𝑡+𝛿_𝑣, (4.16) where 𝛿_𝑣=𝑏_𝑣1/𝑘₁. Thus 𝑉₁ is bounded.
(2) If 𝑧₁∈Ω_𝑧1, then |𝑧₁|<𝜆₁, 𝑧₁ is bounded. By the integral median theorem, we can obtain 12^𝑡_{𝑡−𝜏ℎ}𝑈₁𝑦(𝜎)1𝑑𝜎=2𝜏_ℎ𝑈₁𝑦(𝜃),𝜃∈𝑡−𝜏_ℎ,𝑡. (4.17) By Assumption 2.1 and (4.9), it can be concluded that 𝑉_𝑈1 is bounded.
Differentiating 𝑉_𝑤1, 𝑉_𝑤1=𝑊^𝑇₁𝑧₁𝑆₁𝜉₁−𝜎₁𝑊₁1≤−2𝜎₁−𝑘_𝑤1𝑊₁²+12𝜎₁𝑊^∗₁²+1𝑘_𝑤1𝑧²₁𝑆₁𝜁₁²≤−𝑘₁𝑉_𝑤1+𝑏₁, (4.18) where 𝑘₁=𝜎₁−𝑘_𝑤1𝜆_maxΓ₁⁻¹,𝑏₁=12𝜎₁𝑊^∗₁²+𝑚₁𝑘_𝑤1𝜆²₁. (4.19)𝑚₁ is the number of neurons of the neural networks. Choose the parameter so that 𝑘_𝑤1<𝜎₁,𝑘₁>0. Therefore 0≤𝑉_𝑤1(𝑡)≤𝑉_𝑤1(0)−𝛿_𝑤1𝑒^−𝑘1𝑡+𝛿_𝑤1, (4.20) where 𝛿_𝑤1=𝑏₁/𝑘₁. Thus 𝑉_𝑤1(𝑡) is bounded. Because 𝑉_𝑧1,𝑉_𝑈1,𝑉_𝑤1 are all bounded, 𝑉₁ is bounded when 𝑧₁∈Ω_𝑧1.
Step 𝑖. For the 𝑖th (2≤𝑖≤𝑛−1) subsystem, by utilizing (4.1)(4.3), we have 𝑧̇_𝑖=𝑥̇_𝑖−𝛼̇_𝑖−1=𝑥_𝑖+1+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ−𝛼̇_𝑖−1=𝑧_𝑖+1+𝛼_𝑖+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑥_𝑗+1+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑤_𝑗𝑊̇_𝑗. (4.21) Differentiating (4.10) along track (4.21), we have 𝑉̇_𝑖≤𝑧_𝑖𝑧_𝑖+1+𝛼_𝑖+𝐹_𝑖𝜁_𝑖+12𝑧²_𝑖+12^𝑖_𝑗=1𝑓²_𝑗𝑥₁𝑡−𝜏_ℎ−𝑧²_𝑖2𝜆²_𝑖^𝑖−1_𝑗=1𝑈_𝑗𝑥₁+12^𝑖−1_𝑗=1𝑈_𝑗𝑥₁−𝑈_𝑗𝑥₁𝑡−𝜏_ℎ≤𝑧₁𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+𝑓₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖+𝑧_𝑖𝑣_𝑖(𝑡)≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖+𝑧_𝑖𝑣_𝑖(𝑡)≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑧²_𝑖+^𝑖−1_𝑗=1𝑈_𝑗𝑥₁𝑧1−^2𝑖𝜆²_𝑖+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖. (4.22)
(1) If 𝑧_𝑖∈Ω^𝑜_𝑧𝑖, then |𝑧_𝑖|≥𝜆_𝑖. Thus, substituting (4.4) and (4.7) into (4.22) results in 𝑉̇_𝑖≤𝑧_𝑖𝑧_𝑖+1+𝑧_𝑖𝐹_𝑖𝜁_𝑖12𝑧²_𝑖−𝑐_𝑖+𝑏_𝑖2+2𝑧²_𝑖−𝑧²_𝑖2𝜆²_𝑖𝜏_max^𝑖−1_𝑗=1𝑘_𝑗+𝜆²_𝑖+1−𝑧_𝑖𝑊^𝑇_𝑖𝑆_𝑖𝜁_𝑖+𝑊^𝑇_𝑖𝑧_𝑖𝑆_𝑖𝜁_𝑖−𝜎_𝑖𝑊_𝑖≤𝑧_𝑖𝑧_𝑖+1−𝑐_𝑖+𝑏_𝑖2+1𝑧²_𝑖−12𝜏_max^𝑖−1_𝑗=1𝑘_𝑗+𝜆²_𝑖+1+𝑧_𝑖𝜀_𝑖𝜉_𝑖−𝜎_𝑖𝑊^𝑇_𝑖𝑊_𝑖+12𝑑²_𝑖≤−𝑐_𝑖𝑧²_𝑖−𝜏_max2^𝑖−1_𝑗=1𝑘_𝑗−𝜎2𝑊_𝑖²+𝜎2𝑊^∗_𝑖²+12𝑏_𝑖𝜀^∗_𝑖²+12−𝑧²_𝑖+𝑧²_𝑖+1−𝜆²_𝑖+1≤−𝑘_𝑖𝑉_𝑖+𝑏_𝑣𝑖+Θ_𝑖, (4.23) where 𝑘_𝑖=min{2𝑐_𝑖,𝜎_𝑖/(𝜆_max(Γ_𝑖⁻¹)),1}, 𝑏_𝑣𝑖=(𝜎_𝑖/2)‖𝑊^∗_𝑖‖²+(1/2𝑏_𝑖)(𝜀^∗_𝑖)². Θ_𝑖=(1/2)(−𝑧²_𝑖+𝑧²_𝑖+1−𝜆²_𝑖+1).
If there is no item Θ_𝑖 in (4.23), then 0≤𝑉_𝑖(𝑡)≤𝑉_𝑖(0)−𝛿_𝑣𝑒^{−𝑘𝑖𝑡}+𝛿_𝑣, (4.24) where 𝛿_𝑣=𝑏_𝑣𝑖/𝑘_𝑖. Thus 𝑉_𝑖 is bounded.
(2) If 𝑧_𝑖∈Ω_𝑧𝑖, similar to step 1, we have 𝑉_𝑖 is bounded.
Step 𝑛. This is the last step for the 𝑛th subsystem, similarly to the 𝑖th subsystem, if 𝑧_𝑛∈Ω^𝑜_𝑧𝑛, then |𝑧_𝑛|≥𝜆_𝑛. Thus we have 𝑉̇_𝑛≤−𝑐_𝑛𝑧²_𝑛−𝜏_max2^𝑛−1_𝑗=1𝑘_𝑗−𝜎2𝑊_𝑛²+𝜎2𝑊^∗_𝑛²+12𝑏_𝑛𝜀^∗_𝑛²−12𝑧²_𝑛≤−𝑘_𝑛𝑉_𝑛+𝑏_𝑣𝑛+Θ_𝑛, (4.25) where 𝑘_𝑛=min{2𝑐_𝑛,𝜎_𝑛/𝜆_max(Γ_𝑛⁻¹),1}, 𝑏_𝑣𝑛=(𝜎_𝑛/2)‖𝑊^∗_𝑛‖²+(1/2𝑏_𝑛)(𝜀^∗_𝑛)². Θ_𝑛=−(1/2)𝑧²_𝑛.
By (4.25), it is easy to have 0≤𝑉_𝑛(𝑡)≤𝑉_𝑛(0)−𝛿_𝑣𝑒^{−𝑘𝑛𝑡}+𝛿_𝑣, (4.26) where 𝛿_𝑣=𝑏_𝑣𝑛/𝑘_𝑛. Thus 𝑉_𝑛 is bounded.
(1) If 𝑧_𝑛∈Ω_𝑧𝑛, similar to step 1, we have 𝑉_𝑛 is bounded.
The 𝑉_𝑖,(1≤𝑖≤𝑛) is bounded when 𝑧_𝑖∈Ω_𝑧𝑖,(1≤𝑖≤𝑛). In 𝑧_𝑖∈Ω^𝑜_𝑧𝑖,(1≤𝑖≤𝑛): 𝑉̇(𝑡)≤^𝑛_𝑖=1𝑉̇_𝑖≤−^𝑛_𝑖=1𝑘_𝑖𝑉_𝑖+^𝑛_𝑖=1𝑏_𝑣𝑖+^𝑛_𝑖=1Θ_𝑖≤−𝑘_𝑣𝑉(𝑡)+𝑏_𝑣, (4.27) where 𝑘_𝑣=min{𝑘₁,𝑘₂,…,𝑘_𝑛}, 𝑏_𝑣=∑^𝑛_𝑖=1𝑏_𝑣𝑖.
Then 0≤𝑉(𝑡)≤𝑉(0)−𝛿_𝑣𝑒^{−𝑘𝑣𝑡}+𝛿_𝑣, (4.28) where 𝛿_𝑣=𝑏_𝑣/𝑘_𝑣. Thus 𝑉(𝑡) is bounded.

(II) When 𝜈_𝑖(𝑡)≠0,𝑖=1,2,…,𝑛.

Let us define error variables 𝑧_𝑖 assistant functions and the virtual control 𝛼_𝑖, respectively, as follows: 𝑧₁=𝑥₁,𝑧_𝑖=𝑥_𝑖−𝛼_𝑖−1,2≤𝑖≤𝑛. (4.29) Define the following sets: Ω_𝑧𝑖=𝑧_𝑖∈𝑅∣𝑧_𝑖<𝜆_𝑖,Ω^𝑜_𝑧𝑖=𝑧_𝑖∈𝑅∣𝑧_𝑖≥𝜆_𝑖, (4.30) where 𝜆_𝑖 is a small constant. Define assistant functions as 𝐹₁=𝑔₁(𝑥)+12𝜆²₁𝑈₁𝑥₁𝑧₁,𝐹_𝑖=𝑔_𝑖(𝑥)+12𝜆²_𝑖𝑧_𝑖^𝑖−1_𝑗=1𝑈_𝑗𝑥₁−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑊_𝑗𝑊̇_𝑗−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑥_𝑗+1+𝑔_𝑗𝑥₁−12𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑧_𝑗,2≤𝑖≤𝑛. (4.31) Define the virtual control as 𝛼_𝑖=−𝑘_𝑖0𝑧_𝑖−𝑊^𝑇_𝑖𝑆_𝑖𝜉_𝑖,1≤𝑖≤𝑛, (4.32) where 𝑘₁₀=𝑐₁+𝑏₁2+32+12𝜆²₁𝜆²₂+𝜏_max𝑘₁+𝑑²₁,𝑘_𝑖0=𝑐_𝑖+𝑏_𝑖21+2+2𝜆²_𝑖𝜆²_𝑖+1+𝜏_max^𝑖−1_𝑗=1𝑘_𝑗+𝑑²_𝑖𝑘,2≤𝑖≤𝑛−1_𝑛0=𝑐_𝑛+𝑏_𝑛21+1+2𝜆²_𝑛𝜆²_𝑖+1+𝜏_max^𝑛−1_𝑗=1𝑘_𝑗+𝑑²_𝑛,𝜉₁=𝑥₁,𝑥₁^𝑇𝜉_𝑖=𝑥^𝑇_𝑖,𝑥^𝑇_𝑖,𝛼_𝑖−1,𝜕_𝑖−1𝜕𝑥_𝑖−1^𝑇𝜕_𝑖−1𝜕𝑥_𝑖−1^𝑇,𝜓_𝑖−1^𝑇𝜓,2≤𝑖≤𝑛,_𝑖−1=^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑊_𝑗𝜕𝑊̇_𝑗,2≤𝑖≤𝑛. (4.33)(4.34)

Theorem 4.2. System (2.1) with both input delay and state delay satisfies Assumptions 2.1 and 2.2. The virtual control can be selected as (4.32). If the control law and the adaptive law are selected as follow: 𝑢(𝑡)=𝛼_𝑛,𝑊̇_𝑖=Γ_𝑖𝑧_𝑖𝑆_𝑖𝜉_𝑖−𝜎_𝑖𝑊_𝑖,1≤𝑖≤𝑛, (4.35)(4.36) then the closed-loop system is semi-globally uniformly ultimately bounded.

Proof. Define the Lyapunov-Kresovskii functional 𝑉(𝑡) as 𝑉_𝑧𝑖(𝑡)=12𝑧²_𝑖(𝑡),𝑉_𝑊𝑖(𝑡)=12𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊_𝑖,𝑉_𝑈𝑖(𝑡)=12^𝑖_𝑗=1^𝑡_{𝑡−𝜏ℎ}𝑈_𝑗𝑦(𝜎)𝑉𝑑𝜎,_𝑖(𝑡)=𝑉_𝑧𝑖(𝑡)+𝑉_𝑈𝑖(𝑡)+𝑉_𝑊𝑖(𝑡)𝑉(𝑡)=,1≤𝑖≤𝑛,_𝑛𝑖=1𝑉_𝑖(𝑡). (4.37)(4.38)(4.39)(4.40)Step 1. For the first differential equation of the first subsystem, by (4.29), (4.31), We can get 𝑧̇₁=𝑥̇₁=𝑥₂+𝑔₁𝑥₁+𝑓₁𝑥₁𝑡−𝜏_ℎ+𝜔₁(𝑡)=𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+ℎ₁𝑥₁𝑡−𝜏_ℎ+𝜔₁(𝑡). (4.41) By 𝑧₁𝑓₁𝑥₁𝑡−𝜏_ℎ≤12𝑧²₁+12𝑓²₁𝑥₁𝑡−𝜏_ℎ (4.42) differentiating (4.39) and using (4.41), the inequality below can be obtained easily. 𝑉̇₁=𝑧₁𝑧̇₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ=𝑧₁𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+𝜔₁(𝑡)+𝑓₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+𝑧₁𝜔₁(𝑡). (4.43)
(1) If 𝑧₁∈Ω^𝑜_𝑧1, then |𝑧₁|≥𝜆₁. Thus, substituting (4.32) and (4.36) into (4.43) results in 𝑉̇₁≤𝑧₁𝑧₂+𝑧₁−𝑘₁₀𝑧₁−𝑊^𝑇₁𝑆₁𝜁₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+𝑧₁𝜈₁(𝑡)≤12𝑧₁²+12𝑧₂²−𝑐₁+𝑏₁2+32𝑧²₁+𝑧₁𝐹₁𝜁₁−𝑧²₁2𝜆₁𝜏_max𝑘₁+𝜆²₂+𝑑²₁−𝑧₁𝑊^𝑇₁𝑆₁𝜁₁+12𝑧²₁+12𝑧₁²+𝑊𝑧₁𝑆₁𝜁₁−𝜎₁𝑊₁+12𝑑²₁≤12𝑧²₂−𝑐₁+𝑏₁2𝑧²₁−𝑧²₁2𝜆²₁𝜏_max𝑘₁+𝜆²₂+𝑧₁𝜀₁𝜉₁−𝜎₁𝑊^𝑇₁𝑊₁≤12𝑧²₂−𝑐₁+𝑏₁2𝑧²₁+𝑏₁2𝑧²₁+12𝑏₁𝜀^∗₁²+12𝑑²₁−12𝜏_max𝑘₁+𝜆²₂+12𝑑²₁−𝜎₁𝑊^𝑇₁𝑊₁≤−𝑐₁𝑧²₁−𝑉_𝑈1(𝑇)−𝜎₁2𝑊₁²+𝜎₁2𝑊^∗₁²+12𝑏₁𝜀^∗₁²+12𝑧²₂−𝜆²₂≤−𝑘₁𝑉₁+𝑏_𝑣1+Θ₁, (4.44) where 𝑘₁=min{2𝑐₁,𝜎₁/(𝜆_max(Γ₁⁻¹)),1}, 𝑏_𝑣1=(𝜎₁/2)‖𝑊^∗₁‖²+(1/2𝑏₁)(𝜀^∗₁)². Θ₁=(1/2)(𝑧²₂−𝜆²₂).
If there is no item Θ₁ in (4.44), then 0≤𝑉₁(𝑡)≤𝑉₁(0)−𝛿_𝑣𝑒^−𝑘1𝑡+𝛿_𝑣, (4.45) where 𝛿_𝑣=𝑏_𝑣1/𝑘₁. Thus 𝑉₁ is bounded.
(2) If 𝑧₁∈Ω_𝑧1, then |𝑧₁|<𝜆₁, 𝑧₁ is bounded. By the integral median theorem, we can obtain 12^𝑡_{𝑡−𝜏ℎ}𝑈₁𝑦(𝜎)1𝑑𝜎=2𝜏_ℎ𝑈₁𝑦(𝜃),𝜃∈𝑡−𝜏_ℎ,𝑡. (4.46) By Assumption 2.1 and (4.38), it can be concluded that 𝑉_𝑈1 is bounded.
Differentiating 𝑉_𝑤1, 𝑉_𝑤1=𝑊^𝑇₁𝑧₁𝑆₁𝜉₁−𝜎₁𝑊₁1≤−2𝜎₁−𝑘_𝑤1𝑊₁²+12𝜎₁𝑊^∗₁²+1𝑘_𝑤1𝑧²₁𝑆₁𝜁₁²≤−𝑘₁𝑉_𝑤1+𝑏₁, (4.47) where 𝑘₁=𝜎₁−𝑘_𝑤1𝜆_maxΓ₁⁻¹,𝑏₁=12𝜎₁𝑊^∗₁²+𝑚₁𝑘_𝑤1𝜆²₁. (4.48)𝑚₁ is the number of neurons of the neural networks. Choose the parameter so that 𝑘_𝑤1<𝜎₁,𝑘₁>0. Therefore 0≤𝑉_𝑤1(𝑡)≤𝑉_𝑤1(0)−𝛿_𝑤1𝑒^−𝑘1𝑡+𝛿_𝑤1, (4.49) where 𝛿_𝑤1=𝑏₁/𝑘₁. Thus 𝑉_𝑤1(𝑡) is bounded. Because 𝑉_𝑧1,𝑉_𝑈1,𝑉_𝑤1are all bounded, 𝑉₁ is bounded when 𝑧₁∈Ω_𝑧1.
Step 𝑖. For the 𝑖th (2≤𝑖≤𝑛−1) subsystem, by utilizing (4.29), (4.31), we have 𝑧̇_𝑖=𝑥̇_𝑖−𝛼̇_𝑖−1=𝑥_𝑖+1+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ−𝛼̇_𝑖−1+𝜈_𝑖(𝑡)=𝑧_𝑖+1+𝛼_𝑖+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ+𝜈_𝑖(𝑡)−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝑥_𝑗+1+𝑔_𝑖𝑥₁+𝑓_𝑖𝑥₁𝑡−𝜏_ℎ−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑤_𝑗𝑤̇_𝑗. (4.50) Differentiating (4.39) along track (4.50),we have 𝑉̇_𝑖≤𝑧_𝑖𝑧_𝑖+1+𝛼_𝑖+𝐹_𝑖𝜁_𝑖+12𝑧²_𝑖+12^𝑖_𝑗=1𝑓²_𝑗𝑥₁𝑡−𝜏_ℎ−𝑧²_𝑖2𝜆²_𝑖^𝑖−1_𝑗=1𝑈_𝑗𝑥₁+12^𝑖−1_𝑗=1𝑈_𝑗𝑥₁−𝑈_𝑗𝑥₁𝑡−𝜏_ℎ≤𝑧₁𝑧₂+𝛼₁+𝐹₁𝜁₁−12𝜆²₁𝑈₁𝑥₁𝑧₁+𝜈_𝑖(𝑡)+𝑓₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇₁Γ₁⁻¹𝑊̇₁+12𝑈₁𝑥₁−12𝑈₁𝑥₁𝑡−𝜏_ℎ+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖+𝑧_𝑖𝜈_𝑖(𝑡)≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑈₁𝑧1−²¹𝜆²₁+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖+𝑧_𝑖𝜈_𝑖(𝑡)≤𝑧₁𝑧₂+𝑧₁𝛼₁+𝑧₁𝐹₁𝜁₁+12𝑧²₁+12𝑧²_𝑖+12𝑑²_𝑖+^𝑖−1_𝑗=1𝑈_𝑗𝑥₁𝑧1−^2𝑖𝜆²_𝑖+𝑊^𝑇_𝑖Γ_𝑖⁻¹𝑊̇_𝑖. (4.51)
(1) If 𝑧_𝑖∈Ω^𝑜_𝑧𝑖, then |𝑧_𝑖|≥𝜆_𝑖. Thus, substituting (4.32) and (4.36) into (4.51) results in 𝑉̇_𝑖≤𝑧_𝑖𝑧_𝑖+1+𝑧_𝑖𝐹_𝑖𝜁_𝑖12𝑧²_𝑖−𝑐_𝑖+𝑏_𝑖2+2𝑧²_𝑖−𝑧²_𝑖2𝜆²_𝑖𝑑²_𝑖+𝜏_max^𝑖−1_𝑗=1𝑘_𝑗+𝜆²_𝑖+1−𝑧_𝑖𝑊^𝑇_𝑖𝑆_𝑖𝜁_𝑖+𝑊^𝑇_𝑖𝑧_𝑖𝑆_𝑖𝜁_𝑖−𝜎_𝑖𝑊_𝑖≤𝑧_𝑖𝑧_𝑖+1−𝑐_𝑖+𝑏_𝑖2+1𝑧²_𝑖−12𝜏_max^𝑖−1_𝑗=1𝑘_𝑗+𝑑²_𝑖+𝜆²_𝑖+1+12𝑑²_𝑖+𝑧_𝑖𝜀_𝑖𝜉_𝑖−𝜎_𝑖𝑊^𝑇_𝑖𝑊_𝑖+12𝑑²_𝑖≤−𝑐_𝑖𝑧²_𝑖−𝜏_max2^𝑖−1_𝑗=1𝑘_𝑗−𝜎2𝑊_𝑖²+𝜎2𝑊^∗_𝑖²+12𝑏_𝑖𝜀^∗_𝑖²+12−𝑧²_𝑖+𝑧²_𝑖+1−𝜆²_𝑖+1≤−𝑘_𝑖𝑉_𝑖+𝑏_𝑣𝑖+Θ_𝑖, (4.52) where 𝑘_𝑖=min{2𝑐_𝑖,𝜎_𝑖/𝜆_max(Γ_𝑖⁻¹),1}, 𝑏_𝑣𝑖=(𝜎_𝑖/2)‖𝑊^∗_𝑖‖²+(1/2𝑏_𝑖)(𝜀^∗_𝑖)². Θ_𝑖=(1/2)(−𝑧²_𝑖+𝑧²_𝑖+1−𝜆²_𝑖+1).
If there is no item Θ_𝑖 in (4.52), then 0≤𝑉_𝑖(𝑡)≤𝑉_𝑖(0)−𝛿_𝑣𝑒^{−𝑘𝑖𝑡}+𝛿_𝑣, (4.53) where 𝛿_𝑣=𝑏_𝑣𝑖/𝑘_𝑖. Thus 𝑉_𝑖 is bounded.
(2) If 𝑧_𝑖∈Ω_𝑧𝑖, similar to step 1, we have 𝑉_𝑖 is bounded.
Step 𝑛. This is the last step for the 𝑛th subsystem, similarly to the 𝑖th subsystem, If 𝑧_𝑛∈Ω^𝑜_𝑧𝑛, then |𝑧_𝑛|≥𝜆_𝑛. Thus we have 𝑉̇_𝑛≤−𝑐_𝑛𝑧²_𝑛−𝜏_max2^𝑛−1_𝑗=1𝑘_𝑗−𝜎2𝑊_𝑛²+𝜎2𝑊^∗_𝑛²+12𝑏_𝑛𝜀^∗_𝑛²−12𝑧²_𝑛≤−𝑘_𝑛𝑉_𝑛+𝑏_𝑣𝑛+Θ_𝑛, (4.54) where 𝑘_𝑛=min{2𝑐_𝑛,𝜎_𝑛/𝜆_max(Γ_𝑛⁻¹),1},𝑏_𝑣𝑛=(𝜎_𝑛/2)‖𝑊^∗_𝑛‖²+(1/2𝑏_𝑛)(𝜀^∗_𝑛)². Θ_𝑛=−(1/2)𝑧²_𝑛.
By (4.54), it is easy to have 0≤𝑉_𝑛(𝑡)≤𝑉_𝑛(0)−𝛿_𝑣𝑒^{−𝑘𝑛𝑡}+𝛿_𝑣, (4.55) where 𝛿_𝑣=𝑏_𝑣𝑛/𝑘_𝑛. Thus 𝑉_𝑛 is bounded.

(II) If 𝑧_𝑛∈Ω_𝑧𝑛, similar to step 1, we have 𝑉_𝑛 is bounded

The 𝑉_𝑖,(1≤𝑖≤𝑛) is bounded. When 𝑧_𝑖∈Ω_𝑧𝑖,(1≤𝑖≤𝑛). In 𝑧_𝑖∈Ω^𝑜_𝑧𝑖,(1≤𝑖≤𝑛): 𝑉̇(𝑡)≤^𝑛_𝑖=1𝑉̇_𝑖≤−^𝑛_𝑖=1𝑘_𝑖𝑉_𝑖+^𝑛_𝑖=1𝑏_𝑣𝑖+^𝑛_𝑖=1Θ_𝑖≤−𝑘_𝑣𝑉(𝑡)+𝑏_𝑣, (4.56) where 𝑘_𝑣=min{𝑘₁,𝑘₂,…,𝑘_𝑛},𝑏_𝑣=∑^𝑛_𝑖=1𝑏_𝑣𝑖.

Then 0≤𝑉(𝑡)≤𝑉(0)−𝛿_𝑣𝑒^{−𝑘𝑣𝑡}+𝛿_𝑣, (4.57) where 𝛿_𝑣=𝑏_𝑣/𝑘_𝑣. Thus 𝑉(𝑡) is bounded.

Simulation Example
Consider the nonlinear system with input and state delays as follows: 𝑥̇₁=𝑥₂+0.3𝑦²+sin2𝑦(𝑡−0.2)+𝜔₁,𝑥̇₂=𝑢(𝑡−0.20)+0.2𝑦²−sin2𝑦(𝑡−0.2)+𝜔₂,𝑦=𝑥₁. (4.58) Define virtual control as 𝛼₁=−𝑐₁+𝑏₁2+32+12𝜆²₁𝜏_max𝑘₁+𝜆²₂+𝑑²₁𝑧₁−𝑊^𝑇₁𝑆₁𝜉₁,𝛼₂=−𝑐₂+𝑏₂21+1+2𝜆²₂𝜏_max𝑘₁+𝑑²₂𝑧₂−𝑊^𝑇₂𝑆₂𝜉₂, (4.59) where 𝑐₁=𝑐₂=30, 𝑏₁=𝑏₂=2, 𝜆₁=𝜆₂=2, 𝜏_max=0.6, 𝑘₁=2.𝑑₁=𝑑₂=0.6, Γ₁=Γ₂=600, 𝜎₁=𝜎₂=0.006, 𝜔₁=0.05sin(2𝜋𝑡), 𝜔₂=0.05cos(2𝜋𝑡). 𝑊̇₁=Γ₁𝑧₁𝑆₁𝜉₁−𝜎₁𝑊₁,𝑊̇₂=Γ₂𝑧₂𝑆₂𝜉₂−𝜎₂𝑊₂,𝑧₁=𝑥₁,𝑧₂=𝑥₂−𝛼₁. (4.60) The result of control scheme is in Figures 1 and 2.

Figure 1 

The control input 𝑢(𝑡).

Figure 2 

System state 𝑥1(𝑡)(“−”) and 𝑥2(𝑡)(“−.−”).

5. Conclusion

For a class of outputs time-delay nonlinear systems with perturbed or not, a control scheme combined with adaptive control, backstepping, and neural network is proposed. The radius basis function (RBF) neural networks is employed to estimate the unknown continuous functions. It is shown that the proposed method guarantees the semi-globally uniformly ultimately boundedness of all signals in the adaptive closed-loop systems. Simulation results are provided to illustrate the performance of the proposed approach.

Acknowledgments

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