#### Abstract

The problem of stability analysis for a class of networked control systems (NCSs) with network-induced delay and packet dropout is investigated in this paper. Based on the working mechanism of zero-order holder, the closed-loop NCS is modeled as a continuous-time linear system with input delay. By introducing a novel Lyapunov-Krasovskii functional which splits both the lower and upper bounds of the delay into two subintervals, respectively, and utilizes reciprocally convex combination technique, a new stability criterion is derived in terms of linear matrix inequalities. Compared with previous results in the literature, the obtained stability criterion is less conservative. Numerical examples demonstrate the validity and feasibility of the proposed method.

#### 1. Introduction

Network control systems (NCSs) are the feedback control systems in which the control loops are closed via real-time networks [1]. NCSs have great advantages compared to the traditional point-to-point control systems, including high reliability, ease of installation and maintenance, and low cost, and they have been applied in many areas, such as computer integrated manufacturing systems, intelligent traffic systems, aircraft control, and teleoperation. However, due to the insertion of communication network in feedback control loops, time delay caused by data transmission and packet dropout in NCSs is always inevitable, which may degrade system performance or even lead to the potential system instability. Thus it is of significance to cope with the adverse influences of induced delay and packet dropout. Recently, the problem of robust stability analysis for NCSs has attained considerable attention, and a great number of research results have been reported [1–9]. In order to reduce the conservatism, Yu et al. [5] obtained the sufficient condition on the stabilization of NCSs; the admissible upper bounds of induced delay and packet dropout can be computed by using Lyapunov-Razumikhi function techniques and the quasiconvex optimization algorithm. The free weighting matrices method was proposed in [3] to solve the problem of network-based control. However, using too many free weighting matrices makes the system analysis complex; what is more, some terms were neglected directly, which brings the conservative. Peng et al. [9] indicated that if more information of induced delays in NCSs were utilized, conservatism in system analysis could be reduced. Moreover, it can build a bridge to connect quality of control (QoC). In [4], a stability criterion based on maximum allowable network-induced delay rate is proposed for choosing a reasonable sampling period.

Inspired by the above research results, in the present paper, the robust stabilization problem for a class of NCSs with network-induced delay and packet dropout is addressed. The NCSs are modeled as continuous-time linear systems on the basis of the input delay approach. By exploiting the information of lower and upper bounds of the delay, a new Lyapunov-Krasovskii functional is constructed. Moreover, reciprocally convex combination technique is introduced such that less conservative results are obtained. Two numerical examples are given to illustrate the validity and feasibility of the proposed results.

#### 2. System Description

In this paper, the sensor module is assumed to act in a clock-driven fashion with transmission period h; the controller and actuator modules are assumed to act in an event-driven fashion. A new packet will be used by the controller immediately after its arrival. Single packet transmission is considered, where all the data sent or received over the network are sampled at the same sampling instant and assembled together into one network packet.

The plant is described by the following linear plant model proposed in [7]: where , are the plant’s state and input vectors, respectively. and are known to be real constant matrices with proper dimensions. This control signal is based on the plant’s state at the instant . So, the control law can be described as where is the state feedback gain matrix, denotes the network-induced delay, and is the sampling period, ; due to the introduction of logical zero-order holder, the actuator will use the latest available control input, so . If , then no packet dropout occurred in the transmission. And the numbers of consecutive packet dropouts during the time interval can be described as follows:

Assume the existence of constants , . One has where and indicate the lower and the upper bounds of the total delay involving both transmission delays and packet dropouts, respectively.

From a straightforward combination of (1)–(4), the system can be rewritten as follows: where , the function which satisfies denotes the time-varying delay in the control signal, and is the state’s initial function.

The following technical lemmas are introduced, which are indispensable for the proof of the main result.

Lemma 1 (see [13]). *For any positive matrix , scalar , and a vector function such that the integration is well defined,
*

Lemma 2 (see [14]). *For any positive matrix , scalars and satisfying , and vector function , the following inequality holds:
*

Lemma 3 (see [15]). *Let have positive values for arbitrary value of independent variable in an open subset of . The reciprocally convex combination of in satisfies
**
subject to
*

#### 3. Main Results

Theorem 4. *For a given scalar , NCS (5) is asymptotically stable if there exist symmetric matrices , , , , , and proper dimensions matrice such that
**
where
**
with
*

*Proof. *Let us choose a Lyapunov-Krasovskii functional candidate as
where
Define an extended-state vector asand then NCS (5) is simplified as
Calculating the derivative of along the trajectory of NCS (5) yields
It is easy to get
The time derivative of can be represented as
By utilizing Lemma 1, we obtain
Define , ; by the reciprocally convex combination in Lemma 3, the following inequality holds:
Due to , according to Lemma 1 and inequalities (21), we have
where
Based on Lemma 2, an upper bound of can be estimated as

From (16)–(24), we have
If , then is negatively defined, based on the Lyapunov theory, we can conclude that NCS (5) is asymptotically stable.

*Remark 5. *The robust stability criteria presented in Theorem 4 are for the nominal system. However, it is easy to further extend Theorem 4 to uncertain systems, where the systems matrices and contain parameter uncertainties.

*Remark 6. *To reduce the conservatism of stability criteria, the Lyapunov-Krasovskii functional contains some triple-integral terms in [12]. However, the new Lyapunov-Krasovskii functional in our paper which not only contains some triple-integral terms but also divides the lower and upper bounds of the delay into two equal segments, respectively, is proposed. The results will be less conservative.

*Remark 7. *Unlike [2], is enlarged as which may lead to conservation. In this paper, is retained as well. Meanwhile, we deal with the integral terms of quadratic quantities by virtue of the reciprocally convex combination in Lemma 3; the obtained results are improved.

*Remark 8. *The relationship between the time-varying delay and its lower bound and upper bound is taken into account. As a result, fewer useful information is ignored in the derivative of the Lyapunov-Krasovskii functional.

*Remark 9. *The delay-departioning method in this paper reduces the conservativeness considerably, which arises from the LMI analysis of the interval . But when the interval is reduced, that is, when , the benefits of analysis method are shortened. The new strategy is to further make use of the information of the delay lower bound through departioning of the delay interval into equidistant subintervals.

#### 4. Numerical Example

*Example 1. *Consider the following NCS [10]:
Assume that the state feedback gain matrix ; when we do not consider the lower bound of the delay, that is, , by applying Theorem 4, the maximum upper bound of delay obtained is , while in [10], , and the maximum allowable value is in [8]. A more detailed comparison for different values of is provided in Table 1. As shown in the table, it can be seen that our results are less conservative than the ones in [8, 10].

*Example 2. *Consider the NCS (5) with the following parameters [11]:
Table 2 lists the maximum allowable upper bound of with respect to different conditions of along with some existing results from the literature. From Table 2, we can conclude that the criterion derived in this paper presents superior results.

#### 5. Conclusion

The problem of robust stability is addressed for a class of NCSs with network-induced delay and packet dropout. Improved and simplified stability criterion is established without involving any model transformation or free weighting matrices. The simulation results indicate that the criterion derived in this paper can exhibit better performance compared to that in the existing literature.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant no. 61203136 and Grant no. 61304064), the Natural Science Foundation Hunan Province of China (Grant no. 11JJ2038), and the Construct Program for the Key Discipline of electrical engineering in Hunan Province.