The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method.

Delay (or memory) systems represent a class of
infinite-dimensional systems largely used to describe propagation and transport
phenomena or population dynamics [

Neutral delay systems constitute a more general class than
those of the retarded type. Stability of these systems proves to be a more
complex issue because the system involves the derivative of the delayed state. Especially in the past few decades, increased attention has been devoted to the
problem of robust delay-independent stability or delay-dependent stability and
stabilization via different approaches (e.g., model transformation
techniques [

Among the existing results on neutral delay
systems, the linear matrix inequality (LMI) approach is an efficient method to
solve many problems such as stability analysis, stabilization [

In the recent
literature on neutral systems, He et al. in [

In this paper, we develop new stability criteria for the stability analysis of the neutral systems with nonlinear parameter perturbations based on a descriptor model transformation. The dynamical system under consideration consists of time-varying neutral, discrete, and distributed delays without any restriction on upper bounds of derivatives of time-varying delays. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of LMIs and can be easily solved by existing convex optimization techniques. Two numerical examples are given to demonstrate the less conservatism of the proposed results over some existence results in the literature.

Consider a class of linear neutral systems with different time-varying
neutral, discrete, and distributed delays and nonlinear parameter perturbations represented by

In this case,

Throughout
the paper, the following assumptions are needed to enable the
application of Lyapunov’s method for the stability of neutral systems [

let the difference
operator

all the eigenvalues of the matrix

Before ending this section, we recall the following lemmas, which will be used in the proof of our main results.

For any arbitrary column
vectors

Given matrices

In this section, new delay-range-dependent sufficient conditions for the
asymptotic stability of the neutral system (

Under (A1), for given scalars

Firstly, we represent (

On the other
hand, noting that

Using Lemma

Following [

The
results given in Theorem

The reduced conservatism of Theorem

In this section, we will discuss the uncertainty
characterization for the linear neutral system (

The first
class of uncertainty frequently encountered in practice is the polytopic
uncertainty [

Under
(A1), for given scalars

It follows
directly from the proof of Theorem

There are also other uncertainties that cannot
be reasonably modeled by a polytopic uncertainty set with a number of
vertices. In such a case, it is assumed that the deviation of the system
parameters of an uncertain system from their nominal values is norm bounded [

let the difference
operator

all the eigenvalues of the matrix

Under

If the matrices

It
is noted that
our approach is
different from that in the reference [

In this section, two examples are provided to illustrate the effectiveness of the results obtained in the previous sections.

Consider the neutral system (

Assume that

Comparative results for

Results of [ | 0.6811 | 0.5467 | 0.6129 | 0.4950 |

Results of [ | 1.3279 | 0.6743 | 1.2503 | 0.5716 |

Results of [ | 2.742 | 1.142 | 1.875 | 1.009 |

Results of [ | 3.744 | 1.471 | 2.443 | 1.299 |

Results of this paper | 3.8205 | 1.6350 | 2.7105 | 1.3580 |

Assume that

Comparative results for

Results of [ | 0.7437 | 0.5131 | 0.3112 | 0.1398 |

Results of [ | 0.7749 | 0.5658 | 0.3859 | 0.2357 |

Results of this paper | 0.8429 | 0.6903 | 0.4504 | 0.3015 |

Consider the system (

Upper bounds of delays

1.6025 | 1.5875 | 1.5215 | 1.4050 | 1.1510 |

Upper bounds of delays

1.2560 | 1.2358 | 1.2125 | 1.1950 | 1.1045 |

The problem of stability analysis has been presented in this paper for a class of neutral systems with different time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations using an appropriate Lyapunov-Krasovskii functional. By combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new feasibility conditions, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent, have been developed to ensure that the considered system is asymptotically stable. The conditions were presented in terms of linear matrix inequalities (LMIs) and solved by existing convex optimization techniques. Two numerical examples were given to demonstrate the less conservatism of the proposed results over some existence results in the literature.

This work has been partially funded by the European Union (European Regional Development Fund) and the Ministry of Science and Innovation (Spain) through the coordinated research project DPI2008-06699-C02-01. H. R. Karimi is grateful to the grant of Juan de la Cierva program of the Ministry of Science and Innovation (Spain), and M. Zapateiro is grateful to the FI grant of the Department for Innovation, University and Enterprise of the Government of Catalonia (Spain). The authors would also like to thank the associate editor and anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.