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The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stable criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

As one of important dynamical systems, neutral system has been received considerable attention in past years. Large numbers of monographs and papers on the stability of neutral type system with or without time delays have been published. A wide variety of methods disposing the stability problems of neutral system have been proposed [

Motivated by the afore-mentioned analysis, in this paper, based on the equivalent equation of the zero which is similar to [

Consider uncertain neutral system with mixed delays [

The nominal

For further discussion, we first introduce the following lemmas.

Given constant symmetric matrices

For given matrices

For any real vector

In this section, we will analyse the stability problem of uncertain neutral systems with mixed delays described by (

For the asymptotic stability of system (

For any given matrix

Constructing a new Lyapunov functional candidate for system (

Hence,

Motivated by the results obtained in [

The transformation from system (

When

For given positive scalars

Based on Theorem

For any given matrix

Replacing

For given positive scalars

In this section, some numerical examples will be presented to show the validity and improvement of the main results derived earlier.

Consider the following neutral system presented in Park and Kwon [

Stability bounds of time delays (Example

[ | [ | [ | [ | [ | [ | Ours | |
---|---|---|---|---|---|---|---|

Consider the following neutral system studied in He et al. [

For this example, we calculate the allowable upper bound of

Some comparison for allowable upper bounds on

[ | [ | [ | [ | [ | [ | [ | [ | [ | Ours | |
---|---|---|---|---|---|---|---|---|---|---|

0.3 | 0.71 | 0.74 | 0.8844 | 1.3718 | 1.6525 | 1.6525 | 1.6525 | 2.2254 | 2.2254 |

Consider the neutral system studied in Liu et al. [

Calculated allowable size of distributed delay

Liu [ | cc | 0 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.4 |

0.67 | 0.62 | 0.56 | 0.51 | 0.46 | 0.41 | 0.36 | 0.32 | 0.28 | ||

0.79 | 0.78 | 0.73 | 0.66 | 0.58 | 0.50 | 0.41 | 0.37 | 0.21 | ||

Ours | ||||||||||

2.15 | 2.13 | 2.11 | 2.09 | 2.07 | 2.04 | 2.02 | 1.99 | 1.97 | ||

2.61 | 2.61 | 2.61 | 2.61 | 2.61 | 2.61 | 2.61 | 2.61 | 2.61 |

Calculated allowable size of distributed delay

Liu [ | cc | 0 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.4 |

0.67 | 0.62 | 0.56 | 0.51 | 0.46 | 0.41 | 0.36 | 0.32 | 0.28 | ||

0.79 | 0.78 | 0.73 | 0.66 | 0.58 | 0.50 | 0.41 | 0.37 | 0.21 | ||

Ours | 0.3 | |||||||||

1.82 | 1.79 | 1.77 | 1.75 | 1.72 | 1.70 | 1.67 | 1.63 | 1.60 | ||

1.94 | 1.94 | 1.94 | 1.94 | 1.94 | 1.94 | 1.94 | 1.94 | 1.94 |

State trajectories of system (12) with

State trajectories of system (12) with

State trajectories of system (12) with

State trajectories of system (12) with

Consider the following uncertain neutral system [

Comparison of

cc | 0 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 |

Han [ | 3.13 | 2.98 | 2.83 | 2.66 | 2.49 | 2.31 | 2.12 | 1.93 |

Han [ | 1.77 | 1.63 | 1.48 | 1.33 | 1.16 | 0.98 | 0.79 | 0.59 |

He et al. [ | 2.39 | 2.05 | 1.75 | 1.49 | 1.27 | 1.08 | 0.91 | 0.76 |

Theorem | 3.45 | 3.21 | 3.02 | 2.88 | 2.62 | 2.54 | 2.51 | 2.23 |

By constructing a new Lyapunov function, some new robust stable criteria for a class of neutral control systems with mixed delays are obtained. These criteria are formulated in the forms of linear matrix inequalities. Compared with some previous publications, our results are less conservative. Numerical examples and simulations show that our results are valid.

This work was supported by the program for New Century Excellent Talents in University (NCET-06-0811), the National Basic Research Program of China (2010CB732501), and the Research Fund for the Doctoral Program of Guizhou College of Finance and Economics (200702).