1 Introduction

A networked control system (NCS) is a control system in which the control loop is closed over communication networks. In contrast to traditional control systems whose components are usually connected via point-to-point wiring,an NCS is a flexible architecture with the advantages of low cost,and simple installation and maintenance. However,the generally existing transmission delays and transmission intervals often degrade the performance of the NCS and even cause instability^{[1, 2, 3, 4, 5, 6, 7, 8]}. Note that an NCS with time-varying transmission delays and intervals is also deemed as a sampled-data system under variable sampling with additional transmission delays^[3].

In existing literatures,three main approaches are proposed for the analysis and the synthesis of NCSs with time-varying transmission delays and intervals：

The first one is a discrete-time modeling approach,which discretizes the continuous-time NCSs and transforms the effects of transmission delays and intervals into its model parameters. However,this approach may be inappropriate for NCSs with parametric uncertainties or external disturbances^{[9, 10, 11]}.

The second one is an input-delay approach,which represents the NCSs as a continuous-time system with a delayed control input^{[12, 13, 14, 15]}. The Lyapunov-Krasovskii methods have been extensively applied to the NCSs along with various techniques estimating cross-product integral terms,such as free-weighting matrix technique^{[12, 13, 14, 16, 17, 18]},Jensen′s inequality based technique^[15],and so on. The conservatism of the derived results depends on the selected Lyapunov-Krasovskii functionals and the estimation techniques for cross-product integral terms. It is known that the input-delay approach is very conservative for the NCSs with time-varying transmission delays and intervals when the input delays are taken as an uncertain bounded process ignoring the time derivatives of the input delays^{[12, 13, 14, 15]}. Recently,input-delay approach has been revised by scaled small gain theorem^[19] and input-output approach^{[20, 21]} which exploit distributed-delay operators to derive less conservative results for NCSs with only variable sampling or transmission intervals. But considerable room is left for further improvement in the results.

The third one is an impulsive-model approach which models the NCSs as an impulsive system^{[4, 6, 22]}. For the NCSs with only variable sampling or transmission intervals,a discontinuous Lyapunov functional method combined with the free-weighting matrix technique is proposed to improve the existing results by additionally exploiting the time derivatives of input delays and the impulsive effects of feedback signals and input delays^[22],which motivates a novel time-dependent Lyapunov functional method^[23] and looped functional methods^{[24, 25]} to improve the existing results greatly in the framework of the input-delay approach. However,for NCSs with time-varying transmission delays and intervals,the existing results are still very conservative due to the lack of an appropriate Lyapunov functional to efficiently characterize the system behavior of the NCSs based on the time derivatives of input delays and the impulsive effects of input delays and feedback signals^{[3, 26, 27, 28]}.

In this paper,we consider to derive new stability criteria for an NCS with time-varying transmission delays and intervals based on a new discontinuous Lyapunov functional method in the framework of input-delay approach. Note that the NCSs with transmission delays and intervals have been addressed based on discontinuous Lyapunov functional methods in ^{[3, 26, 27, 28]}. For a control signal performed on the zero-order holder (ZOH),the round-trip transmission delay is taken as a component of the input delays in ^{[3, 26, 27, 28]}. In this paper,the evolution of transmission delays is additionally taken into account by describing the accumulating transmission delays as the potential input delays before the corresponding control signal is imposed on the ZOH. Meanwhile,the actual input delays are related to the time when the corresponding control signal is performed on the ZOH,and take the round-trip transmission delay as a component as in ^{[3, 26, 27, 28]}. The time derivatives of the potential and actual input delays are both equal to 1. Based on these thoughts,the impulsive effects of input delays and feedback signals are revealed more fully than the ones in ^{[3, 26, 27, 28]},and are rationally treated along with the time derivatives of input delays by the new discontinuous Lyapunov functional method. This leads to the reduced conservatism in the derived stability criteria. A numerical example is presented to substantiate the advantages of our proposed method.

Notations The expression P >0 means that P is real,symmetric,and positive definite. I and 0 denote an identity matrix and a zero matrix with appropriate dimensions,respectively. The space of functions φ ： [- ,0]→Rⁿ ,which are absolutely continuous functions on [- ,0] and have a finite lim θ→0^- φ (θ) and square integrable first-order derivatives,is denoted by W with its norm as

Denote is denoted as a block matrix or vector with the component matrices or vectors in ｛ ｝ arranged as a column.

2 Problem formulation

Considering a linear system of the form：

where x (t)∈R^n_x is the state vector, u (t)∈R^n_u is the control input,and A and B are the system matrix and the input matrix with appropriate dimensions respectively.

Assuming that： 1) The sensor,the controller and the ZOH are connected through communication networks. 2) Feed-back signals and control signals are transmitted in single packet,and there exist time-varying transmission delays in the channels from the sensor to the controller and from the controller to the ZOH. 3) Only new control signals are applied to updating the ZOH,and u (t)= 0 before the first updating instant of the ZOH.

In the presence of transmission delays and intervals,the networked control based on state feedback and the ZOH is given as

where k∈R is the number of the new control signal arriving at the ZOH,s_k the sampling instant of the sensor,t_k the updating instant of the ZOH, x (s_k) the state feedback,and K the controller gain. For the state feedback x (s_k),sc_k denotes the transmission delay from the sensor to the controller, ca_k the transmission delay from the controller to the ZOH,and _k= sc_k+ ca_k=t_k-s_k the round-trip trans-mission delay from the sensor to the ZOH. Note here that the computation delay of the controller is taken as a component of ca_k. With the control law (2),the NCS is formulated as an input-delay system of the form：

where t∈［t_k,t_k+1),and η_k(t)=t-s_k denotes the input de-lays related to the state feedback x (s_k) for t∈［t_k,t_k+1). Note that η_k(t) consists of the transmission delay _k and the holding time t-t_k of the ZOH for t∈［t_k,t_k+1). Let _k=t_k+1-s_k denote the maximum input delay related to x (s_k). It is assumed that there exist the scalars τ ≥0,>0 and >0 such that

Before continuing,the following lemma is given：

Lemma 1 ^［23] Let there exist positive scalars β₁,β₂ and a function V∶ R×W×L₂[- ,0]→R such that

Let the function (t)=V(t,_t,· _t) is continuous from the right for _t satisfying (3),absolutely continuous for t≠s_k,t_k,and satisfies

Given λ≥0,if along the NCS in (3)：

then (3) is exponentially stable with the decay rate λ. 3 Main results 3. 1 Impulsive effects in the NCS

For a continuous-time process,impulsive effects mean that there are discrete-time jumps arising in the evolution of the continuous-time process. To fully reveal the impulsive effects of input delays and feedback signals in the NCS,the accumulating transmission delays are taken as the potential input delays for the associated feedback signal in this paper. Namely,for a state feedback x (s_k),τ_k(t)=t-s_k are the potential input delays for t∈[s_k,t_k) while η_k(t) are the actual input delays for t∈［t_k,t_k+1). The potential input delays τ_k(t) vary with a time derivative being 1 for t∈［s_k,t_k),which is same as the one of η_k(t) for t∈［t_k,t_k+1). At t_k,the potential input delay is incorporated into the actual input delay η_k(t_k) with a value of the round-trip transmission delay _k . For t∈［t_k,t_k+1),the actual input delays η_k(t) coexist with at least one potential input delays with and in the NCS (3). For the convenience of theoretical analysis,it is assumed in this paper that

This means that only the potential input delays τ_k+1(t) co-exist with the actual input delays η_k(t) for t∈［s_k+1,t_k+1). For this context,an example of the time sequence ｛(s_k,t_k),k∈R｝ is plotted in Fig. 1.

Based on the assumption in (6),the time interval ［t_k,t_k+1) can be divided into two subintervals ［t_k,s_k+1) and ［s_k+1,t_k+1). The impulsive effects of feedback signals and input delays in the NCS are given as follows：

1) With t varying from t^-_k to t_k and the updating of the ZOH,the actual feedback signals jump from x (s_k-1) to x (s_k),the actual input delays jump from η_k-1(t^-_k) to η_k(t_k),and the potential input delays τ_k(t) which stop varying at t^-_k are incorporated into η_k(t_k). Then only the actual input delays η_k(t) exist for t∈［t_k,s_k+1).

2) At s_k+1,a new state feedback x (s_k+1) is sampled and the potential input delays τ_k+1(t) occur. Then the actual input delays η_k(t) coexist with the potential input delays τ_k+1(t) for t∈［s_k+1,t_k+1).

3) From t_k+1^- to t_k+1,the actual feedback signals jumps from x (s_k) to x (s_k+1) along with the updating of the ZOH,and the two aforementioned coexisting input delays stop increasing with the actual input delays jumping from η_k(t_k+1^-) to η_k+1(t_k+1) and the potential input delays τ_k+1(t_k+1^-) being incorporated into η_k+1(t_k+1). Then only the actual input delays η_k+1(t) exist for t∈［t_k+1,s_k+2).

Based on the analysis above,it is ready to propose the discontinuous Lyapunov functional. 3. 2 Construction of discontinuous Lyapunov functional

In this paper,a new discontinuous Lyapunov functional is proposed based on the impulsive effects of feedback signals and input delays and the time derivatives of input delays as follows：

For t∈［t_k,s_k+1),

and for t∈［s_k+1,t_k+1),

where

and ζ _k(t)=col｛ x (t),x (s_k)｝,ζ _k+1(t)=col｛ x (t),x (s_k+1)｝, λ≥0, P >0, Q ₁= Q T₁, Q ₂= Q T₂, Q ₃, Q ₄, R ₁>0, R ₂>0. Since _k-η_k(t)= _k+1-τ_k+1(t) for t∈［s_k+1,t_k+1),it follows that lim t→s^-_k+1 (t)= (s_k+1) and lim t→t^-_k+1 (t)= (t_k+1),that is,the whole Lyapunov functional is continuous for all t. The positivity of the discontinuous Lyapunov functional in (7) and (8) is guaranteed by the following conditions：

Lemma 2 For a scalar λ≥0,if there exist matrices P >0, Q ₁= Q T₁, Q ₂= Q T₂,Q ₃, Q ₄, R ₁>0, R ₂>0 with appropriate dimensions such that

then the discontinuous Lyapunov functional in (7) and (8) is positive for all t,where

we have

Note that is convex with respect to η_k(t)∈ andWith is equivalent to

With is equivalent to

and is equivalent to

With and is equivalent to

is equivalent to

and is equivalent to

Then it is concluded that > holds if and only if (9) is feasible,which guarantees the positivity of for t∈［t_k,s_k+1).

2) For t∈［s_k+1,t_k+1),the following inequality holds：

if

Note that is convex with respect toanWith is equivalent to

With is equivalent to

and is equivalent to

With and is equivalent to

is equivalent to

is equivalent to

and is equivalent to

Then it is concluded that holds if and only if (10) is feasible,which guarantees the positivity of for t∈［s_k+1,t_k+1). The proof of Lemma 2 is completed.

Remark 1 The potential input delays are important to guarantee the continuity of the discontinuous Lyapunov functional in (7) and (8). As seen in (7) and (8),by exploiting the potential input delays τ_k(t) for t∈［s_k,t_k) and the actual input delays η_k(t) for t∈［t_k,t_k+1),the discontinuous term V_k¹ is actually defined in the interval ［s_k,t_k+1) with V_k¹(s_k)=0 for nonzero x (s_k) and V_k¹(t_k+1^-)=0 for nonzero x (t_k+1^-). If the potential input delays τ_k(t) are ignored for t∈［s_k,t_k),V_k¹ can only be defined in the interval ［ t_k,t_k+1). When _k≠0 and x (s_k)≠ x (t_k),it follows that V_k¹(t_k)≠0 for nonzero x (s_k) and x (t_k),and V_k¹(t_k+1^-)=0 for nonzero x (t_k+1^-). Then the continuity of the whole discontinuous functional will be lost. Meanwhile,by further exploiting the potential input delays τ_k(t) for t∈［s_k,t_k),the discontinuous term V_k² defined in ［s_k,t_k) provides more freedom for the discontinuous Lyapunov functional in (7) and (8) to characterize the behavior of the NCS,which helps to reduce the conservatism in the derived results.

Remark 2 In ［3,26-28],the potential input delays are completely ignored,and the time-varying^{[26, 27]} or constant^{[3, 28]} round-trip transmission delays are taken as components of the actual input delays. There are no discontinuous terms V_k¹ and V_k² for t∈[s_k,t_k) in the Lyapunov functionals of ［3,26-28]. In ^[3],the discontinuous terms of the Lyapunov functional are mainly based on the extended Wirtinger′s inequality. In ^[26],the whole Lyapunov functional is discontinuous in value as the one in ^[22],where the discontinuous terms are actually given as

where Z _i>0,i∈｛1,2,3,4｝,and φ (t) is a nonnegative bounded function with φ (t^-_k)=0 and for all t≠t_k. However,the discontinuous terms V_d1 and V_d2 do not directly depend on the state feedback x (s_k),and the discontinuous term V_d4 is only related to the delays in [0,t-t_k) which excludes the actual input delays η_k(t). These terms often result in much conservatism in the derived results. Although the term V_d3 helps to relieve this situation,it actually takes the actual input delays η_k(t) as an uncertain bounded process and ignores the time derivatives of η_k(t) as some continuous integral terms do in the Lyapunov functional of ［26]. Then much room is still left for the reduction of the conservatism. In ^[27],the time-varying transmission delays of an event-triggered the NCS have been ignored by the discontinuous terms in the Lyapunov functional of the form：

where Z _i>0,i∈｛1,2｝. These discontinuous terms may lead to ineffectivity in the derived results for the concerned NCS. To guarantee the effectivity of the results,some continuous integral terms have been introduced into the Lyapunov functional in ^[27],which brings more conservatism in the derived results. In ^[28],to guarantee the continuity of the whole Lyapunov functional for the event-triggered NCS with constant round-trip transmission delays,the discontinuous term V_k¹ for t∈[t_k,t_k+1) of this paper is modified as follows：

As pointed out above for the discontinuous terms in ［26],the modified term V_k¹ in ^[28] does not directly depend on the state feedback x (s_k) and the sampling instant s_k of the sensor,and leads to more conservatism in the derived results along with some continuous integral terms. 3. 3 Stability analysis

Theorem 1 Given (4) and (6),a scalar λ≥0 and the controller gain matrix K in (2),the NCS in (3) is exponentially stable with a decay rate λ,if there exist matrices P >0,Q _i= Q _i^T(i∈｛1,2｝),Q _i(i∈｛3,4｝),R _i>0(i∈｛1,2｝),L _1j,M _1j(j∈｛1,2,3｝),L _2j,M _2j,N _ij(i∈｛1,2｝,j∈｛1,2,3,4｝) of appropriate dimensions such that (9) and (10),and

where are defined as in Lemma 2,and

Proof

1) Along the trajectories of the closed-loop NCS in (3) for t∈[t_k,s_k+1),it follows that

By Newton-Leibniz formula,we introduce the following identities with free weighting matrices：

where . Note that

Then we have

for t∈［t_k,s_k+1). By Schur complement,it follows that

if

Note that is convex with respect to η_k(t)∈ and for t∈［t_k,s_k+1). Similar to is sufficiently guaranteed by (11) for t∈［t_k,s_k+1).

2) Along the trajectories of the closed-loop system (3) for t∈［s_k+1,t_k+1),it follows that

By Newton-Leibniz formula and system equation,we introduce the following identities with free weighting matrices：

where . Note that

Then we have

for t∈［s_k+1,t_k+1). By Schur complement,it follows that

if

Note that is convex with respect to and . Similar to is sufficiently guaranteed by (12) for t∈［s_k+1,t_k+1).

Based on the discussion in 1) and 2),it follows that along the trajectories of the NCS in (3) for all t≠s_k,t_k. Then the NCS in (3) is exponentially stable according to Lemma 1. The proof of Theorem 1 is completed.

Remark 3 The conservatism of Theorem 1 mainly comes from 4 aspects： 1) The proposed discontinuous Lyapunov functional in (7) and (8) cannot perfectly characterize the system behavior of the NCS. A more advanced discontinuous Lyapunov functional is still needed to more rationally exploit the impulsive effects of input delays and feedback signals and the time derivatives of input delays. 2) The applied free weighting matrix technique is still conservative to some extent in estimating the cross-product integral terms. 3) The quantity e_k^2λη(t) in (14) is estimated as e^2λ by which leads to some conservatism in the derived results. If the quantity e2λη_k(t) is directly taken by ,a larger number of LMIs will be introduced into Theorem 1 by the convexity technique. 4) There is some conservatism in (11) with respect to for the presence of ρ₁ in . Meanwhile,(12) is relatively conservative with respect to due to the existence of .

Remark 4 The computational complexities of the stability conditions of ［3,26] and Theorem 1 are given in Tab. 1. The number of the decision variables in Theorem 1 is larger than the one in ^[3],and is smaller than the one in ^[26]. The number of the linear matrix inequalities (LMIs) in Theorem 1 is larger than the ones in ^{[3, 26]}. The total dimensions of the LMIs in Theorem 1 are larger than the ones in ^[3],and are smaller than the ones in ^[26]. By the applied convexity technique,the four inequalities results in the 42 LMIs in Theorem 1. The reduced conservatism of Theorem 1 is related to the increased number of the LMIs.

3. 4 Controller design

Based on Theorem 1,we have the following controller design method：

Theorem 2 Given (4),(6),and scalars λ≥0,σ₁₁=1,σ_1j,j∈｛2,3｝,σ_2j,j∈｛1,2,3,4｝,the NCS in (3) with the controller gain K = YX ^-T is exponentially stable with a decay rate λ,if there exist matrices ,and Y of appropriate dimensions,such that

where is defined as in Lemma 2,and are defined as in Theorem 1,respectively,with replaced by respectively.

Proof Define L _1j=σ_1j L ₁₁(j∈｛2,3｝), L _2j=σ_2j L ₁₁(j∈ ｛1,2,3,4｝). Define X = L ^-1₁₁, Y = KX ^T= XPX ^T,(i∈｛1,2｝,j∈｛1,2,3,4｝). We have (15) and (16) from (9) and (10),and (11) and (12) by congruence transformation with diag ｛ X ,X ｝ and diag ｛ X ,X ,X ｝,and diag ｛ X ,X ,X ｝ and diag ｛ X ,X ,X ,X ｝,respectively. For more details of the congruence transformation,please refer to ［12-13]. The proof of Theorem 2 is completed. 4 A numerical example

Example 1 Considering the following system^[22]：

Tab. 2 gives the maximal values of computed by Theorem 1 and the existing methods under different and ,where = means that the transmission delays are constant. It is seen from Tab. 2 that： a) For a same ,a larger leads to a larger maximal value of ； b) The maximal values of decrease with the increase of ； c) Our numerical results are all better than those of ^[22],^[26],^[3] and ^[28]. So our proposed approach is advantageous over the existing ones. 5 Conclusions

In this paper,a new stability criteria are derived for an NCS with time-varying transmission delays and intervals. The evolution of transmission delays is additionally taken into account by describing the accumulating transmission delays as the potential input delays before the corresponding control signal is imposed on the ZOH. A new discontinuous Lyapunov functional method is proposed to rationally consider the impulsive effects of input delays and feedback signals and the time derivatives of input delays,which leads to less conservatism in the derived results. A numerical example is presented to substantiate the advantage of the proposed approach.