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

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.


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.

Problem Formulation and Preliminaries
Consider the nonlinear time-delay system is described as follows:

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 where According to the discussion in 21, 22 , denote the best weight vector as follows: 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 .

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:

4.1
Define the following sets: where λ i is a small constant.Define assistant functions as

4.3
Define the virtual control as where

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: then the closed-loop system is semi-globally uniformly ultimately bounded.
Proof.Define the Lyapunov-Kresovskii functional V t as U j y σ dσ, 4.9

4.11
Step 1.For the first differential equation of the the first subsystem, by 4.1 , 4.3 , we can get By

Mathematical Problems in Engineering
If there is no item Θ 1 in 4.15 , then where By the integral median theorem, we can obtain 4.17 By Assumption 2.1 and 4.9 , it can be concluded that

4.18
where m 1 is the number of neurons of the neural networks.Choose the parameter so that k w where Step i.For the ith 2 ≤ i ≤ n − 1 subsystem, by utilizing 4.1 4.3 , we have żi ẋi − αi−1

4.21
Differentiating 4.10 along track 4.21 , we have where

Mathematical Problems in Engineering
If there is no item Θ i in 4.23 , then where δ v b vi /k i .Thus V i is bounded.
2 If z i ∈ Ω z i , 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

4.25
where where
Let us define error variables z i assistant functions and the virtual control α i , respectively, as follows:

Mathematical Problems in Engineering 9
Define the following sets: where λ i is a small constant.Define assistant functions as

4.31
Define the virtual control as where then the closed-loop system is semi-globally uniformly ultimately bounded.

Mathematical Problems in Engineering
Proof.Define the Lyapunov-Kresovskii functional V t as 37 38

4.40
Step 1.For the first differential equation of the first subsystem, by 4.29 , 4.31 , We can get

4.41
By differentiating 4.39 and using 4.41 , the inequality below can be obtained easily.
Thus, substituting 4.32 and 4.36 into 4.43 results in where where By the integral median theorem, we can obtain

4.46
By Assumption 2.1 and 4.38 , it can be concluded that

4.47
where m 1 is the number of neurons of the neural networks.Choose the parameter so that k w where Step i.For the ith 2 ≤ i ≤ n − 1 subsystem, by utilizing 4.29 , 4.31 , we have żi ẋi − αi−1

4.50
Differentiating 4.39 along track 4.50 have where If there is no item Θ i in 4.52 , then where δ v b vi /k i .Thus V i is bounded. 2 If z i ∈ Ω z i , 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

4.56
where where δ v b v /k v .Thus V t is bounded.

4.58
Define virtual control as

4.60
The result of control scheme is in Figures 1 and 2.

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.

Figure 1 :Figure 2 :
Figure 1: The control input u t .
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 im i T is defined by