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The stability for a class of uncertain linear systems with interval time-varying delays is studied. Based on the delay-dividing approach, the delay interval is partitioned into two subintervals. By constructing an appropriate Lyapunov-Krasovskii functional and using the convex combination method and the improved integral inequality, the delay-dependent stability criteria with less conservation are derived. Finally, some numerical examples are given to show the effectiveness and superiority of the proposed method.

Time delay arises in many systems like manufacturing, telecommunications, chemical industry, power, transportation, and so on. It is generally regarded as a main source of instability and poor performance, which has a negative impact on the performance of the system [

In recent years, some scholars have put forward many effective methods in order to reduce the conservation of the existing results, solve the time-delay problem of system, and make the system more stable. For example, in the process of analyzing the time-delay system, the free weighting matrix method is used in [

Motivated by the above research, this paper considers the problem of delay-dependent stability for uncertain systems with interval time-varying delay. By constructing an appropriate Lyapunov-Krasovskii functional and using the convex combination method and the improved integral inequality, a new less conservative delay-dependent stability criterion is proposed. The proposed method is verified by the classical numerical examples and applied to the WSCC 3-machine 9-bus system. The results suggest that the proposed method is less conservative than some known results.

Consider the following uncertain time-delay system:

Time-delay interval is divided into two sections, namely,

Firstly, some related lemmas are given in this section.

For any positive-definite matrix

Suppose

For given real matrices

Lemma

Given matrices

Lemma

Suppose

Substituting

For any given constant

The Lyapunov-Krasovskii functional is constructed as follows:

The derivative of

By Lemma

Function

The first case, when

Using Lemma

According to (

By Lemma

In view of Lemma

Besides, based on the Lemma

For (

The second case, when

By Lemma

Similar to the first case, we can obtain

Similarly,

When the time-delay function

For any given constant

The definitions of other symbols are the same as (

The delay-dividing technique requires

In this section, the validity of Theorem

Consider the uncertain linear system (

When the accuracy of

Maximum allowable delay bound for

Method | | ||
---|---|---|---|

0.3 | 0.5 | 0.9 | |

[ | 1.0280 | 0.9322 | 0.7590 |

[ | 1.0281 | 0.9561 | 0.8919 |

[ | 1.0943 | 1.0043 | 0.9131 |

Theorem | 1.1415 | 1.1387 | 1.1388 |

Consider the time-delay system (

When the accuracy of

Maximum allowable delay bound for

Method | | ||
---|---|---|---|

0.5 | 0.9 | Unknown | |

[ | 2.07 | 1.74 | 1.74 |

[ | 2.15 | 2.12 | 2.12 |

[ | ( | ( | ( |

Theorem | ( | ( | ( |

Consider WSCC 3-machine 9-bus system, the wiring diagram of the system is shown in Figure

The generator equation of system can be expressed as

By calculating, the state variables at the equilibrium point are obtained:

Wiring diagram of WSCC 3-machine 9-bus system.

The coefficient matrices in the system model are as follows:

When the accuracy of

Maximum allowable delay bound for

Method | | ||
---|---|---|---|

0.3 | 0.5 | 0.9 | |

[ | 0.2692 | 0.2499 | 0.1779 |

[ | ( | ( | ( |

Theorem | ( | ( | ( |

The paper investigates the stability of uncertain linear systems with interval time-varying delay. According to the delay-dividing approach, the delay interval is partitioned into two subintervals and a new Lyapunov-Krasovskii functional is constructed, which makes use of the information on the some delayed sufficiently. The delay-dependent stability criteria are presented by using convex combination technique and improved integral inequality. In the example of WSCC 3-machine 9-bus system, the calculation results show that the upper bound of time delay is larger than that of the previous references. Therefore, the delay-dependent stability criterion presented in this paper is less conservative. Based on the work of this paper, then the analytical method of the uncertain time-delay systems can be extended to the nonlinear uncertain time-delay systems, and the stability of nonlinear uncertain time-delay systems will be analyzed.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Program of National Natural Science Foundation, China (Grant no. 61473325).