Mathematical Problems in Engineering

Volume 2012, Article ID 852161, 16 pages

http://dx.doi.org/10.1155/2012/852161

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

^{1}Computer and Information Engineering College, Guangxi Teachers Education University, Nanning 530023, China^{2}School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, China

Received 25 April 2012; Accepted 26 August 2012

Academic Editor: Yiu-ming Cheung

Copyright © 2012 Ruliang Wang and Jie Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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: where is state, is control and is output vectors, respectively. is a time-varying disturbance. , , , are unknown continuous functions.

*Assumption 2.1. *The unknown function satisfies , where is a known constant.

*Assumption 2.2. *The time-varying disturbance satisfies , , where is a known constant.

Lemma 2.3. *, where , is an unknown constant.*

#### 3. RBF NN Approximation

In this paper, for a given and any continuous function defined on , there is a perfect RBF neural network, which satisfies where is the weight vector of the neural networks, is the number of the NN nodes, is the input vector, is defined by 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 be the estimate of , and define .

#### 4. Main Result

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

(I) when ,

Let us define error variables assistant functions and the virtual control , respectively, as follows: Define the following sets: where is a small constant. Define assistant functions as Define the virtual control as where

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 as
*Step 1*. For the first differential equation of the the first subsystem, by (4.1), (4.3), we can get
By
differentiating (4.10) and using (4.12), the inequality below can be obtained easily.
(1) If , then . Thus, substituting (4.4) and (4.7) into (4.14) results in
where , . .

If there is no item in (4.15), then
where . Thus is bounded.

(2) If , then , is bounded. By the integral median theorem, we can obtain
By Assumption 2.1 and (4.9), it can be concluded that is bounded.

Differentiating ,
where
is the number of neurons of the neural networks. Choose the parameter so that . Therefore
where . Thus is bounded. Because are all bounded, is bounded when .*Step **.* For the th subsystem, by utilizing (4.1)(4.3), we have
Differentiating (4.10) along track (4.21), we have

(1) If , then . Thus, substituting (4.4) and (4.7) into (4.22) results in
where , . .

If there is no item in (4.23), then
where . Thus is bounded.

(2) If , similar to step 1, we have is bounded.*Step *. This is the last step for the th subsystem, similarly to the th subsystem, if , then . Thus we have
where , . .

By (4.25), it is easy to have
where . Thus is bounded.

(1) If , similar to step 1, we have is bounded.

The is bounded when . In :
where , .

Then
where . Thus is bounded.

(II) When .

Let us define error variables assistant functions and the virtual control , respectively, as follows: Define the following sets: where is a small constant. Define assistant functions as Define the virtual control as where

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:
**
then the closed-loop system is semi-globally uniformly ultimately bounded.*

*Proof. *Define the Lyapunov-Kresovskii functional as
*Step 1*. For the first differential equation of the first subsystem, by (4.29), (4.31), We can get
By
differentiating (4.39) and using (4.41), the inequality below can be obtained easily.

(1) If , then . Thus, substituting (4.32) and (4.36) into (4.43) results in
where , . .

If there is no item in (4.44), then
where . Thus is bounded.

(2) If , then , is bounded. By the integral median theorem, we can obtain
By Assumption 2.1 and (4.38), it can be concluded that is bounded.

Differentiating ,
where
is the number of neurons of the neural networks. Choose the parameter so that . Therefore
where . Thus is bounded. Because are all bounded, is bounded when .*Step *. For the th subsystem, by utilizing (4.29), (4.31), we have
Differentiating (4.39) along track (4.50),we have

(1) If , then . Thus, substituting (4.32) and (4.36) into (4.51) results in
where , . .

If there is no item in (4.52), then
where . Thus is bounded.

(2) If , similar to step 1, we have is bounded.*Step *. This is the last step for the th subsystem, similarly to the th subsystem, If , then . Thus we have
where . .

By (4.54), it is easy to have
where . Thus is bounded.

(II) If , similar to step 1, we have is bounded

The is bounded. When . In : where .

Then where . Thus is bounded.

*Simulation Example*

Consider the nonlinear system with input and state delays as follows:
Define virtual control as
where , , , , , , , , .
The result of control scheme is in Figures 1 and 2.

#### 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).

#### References

- H. Fang and Z. Lin, “A further result on global stabilization of oscillators with bounded delayed input,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 1, pp. 121–128, 2006. View at Publisher · View at Google Scholar - Z. Lin and H. Fang, “On asymptotic stabilizability of linear systems with delayed input,”
*IEEE Transactions on Automatic Control*, vol. 52, no. 6, pp. 998–1013, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Gu, V. L. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Birkhauser, Boston, Mass, USA, 2003. - V. B. Kolmanovskii and J.-P. Richard, “Stability of some linear systems with delays,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 5, pp. 984–989, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-J. Sun, J.-G. Hsieh, and H.-C. Yang, “On the stability of uncertain systems with multiple time-varying delays,”
*IEEE Transactions on Automatic Control*, vol. 42, no. 1, pp. 101–105, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Niu, J. Lam et al., “Adaptive control using backing and neural networks,”
*Journal of Dynamic Systems, Measurement, and Control*, vol. 127, pp. 313–526, 20052005. View at Google Scholar - T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,”
*Automatica*, vol. 43, no. 6, pp. 1021–1033, 2007. View at Publisher · View at Google Scholar - Z. Wang, Y. Liu, and X. Liu, “On global asymptotic stability of neural networks with discrete and distributed
delays,”
*Physics Letters A*, vol. 345, pp. 299–308, 2005. View at Google Scholar - S. Xu, J. Lam, D. W. C. Ho, and Y. Zou, “.Improved global robust asymptotic stability criteria for delayed cellualar neural
networks,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 35, pp. 1317–1321, 2005. View at Google Scholar - M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 7, pp. 1048–1060, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Zhu, S. m. Fei, T. P. Zhang, and T. Li, “Control for a class of time-delay nonlinear systems,”
*Neurocomputing*. In press. View at Publisher · View at Google Scholar - B. G. Xu and Y. Q. Liu, “An improved Razumikhin-type theorem and its applications,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 4, pp. 839–841, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Xu, J. Lam, and Y. Zou, “New results on delay-dependent robust ${H}_{\infty}$ control for systems with time-varying delays,”
*Automatica*, vol. 42, no. 2, pp. 343–348, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,”
*IEEE Transactions on Networks Neural, Networks Neural*, vol. 3, no. 6, pp. 837–863, 1992. View at Google Scholar - F. Hong, S. Zhi, S. Ge, and T. H. Lee, “Practical adaptive Neural Control of Nonlinear systems with unknown time delays,”
*IEEE Transactions on Systems*, vol. 35, no. 4, pp. 849–854, 2005. View at Google Scholar - H. Wu, “Adaptive stabilizing state feedback controllers of uncertain dynamical systems with multiple time delays,”
*IEEE Transactions on Automatic Control*, vol. 45, no. 9, pp. 1697–1701, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Yue and Q.-L. Han, “Delayed feedback control of uncertain systems with time-varying input delay,”
*Automatica*, vol. 41, no. 2, pp. 233–240, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Dugard and E. I. Verriest,
*Stability and Control of Time-Delay Systems*, vol. 228, Springer, London, UK, 1998. View at Publisher · View at Google Scholar - Q. Zhu, S. m. Fei, T. P. Zhang, and T. Li, “Control for a class of time-delay nonlinear systems,”
*Neuro Computing*. In press. View at Publisher · View at Google Scholar - V. B. Kolmanovskii, “Delay effects on stability,” in
*Asurvey Proceedings of 38th The CDC*, pp. 1993–1998, 1999. - S. K. Nguang, “Robust stabilization of a class of time-delay nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 45, no. 4, pp. 756–762, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Zhou, G. Feng, and S. K. Nguang, “Robust stabilization of a class of time-delay nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 9, p. 1586, 2002. View at Publisher · View at Google Scholar - S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,”
*IEEE Transactions on Systems, Man, and Cybernetics*, vol. 34, pp. 499–516, 2004. View at Google Scholar - Q. Zhu et al., “Adaptive tracking control for input delayed MIMO nonlinear systems,”
*Neurocomputing*, vol. 74, no. 1–3, pp. 472–480, 2010. View at Publisher · View at Google Scholar