2. Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China
With the rapid advancements of network technology, in a number of control engineering applications, sensors and actuators connect with a remote controller over a multipurpose networks, allowing the networks to exchange information among distributed spatial system components. Such systems are well known as networked control system (NCSs) and have gained special interests due to their attractive features such as simple installation and maintenance, increased system flexibility, higher system testability and resource utilization, lower cost, and reduced weight and power [1][8]. Typical examples can be seen in a wide range of areas including industrial automation, satellite clusters, traffic control, and mobile robots [9][11]. Consequently, in the past decades, research on NCSs has attracted considerable attentions in control and network community, e.g., [12][20]. Despite the advantages of NCSs, the use of a shared network result in communication constrains due to the inherent limitation of network bandwidth. It is well recognized that network induced delays and packet dropouts (NIDs & PDs) are two important constrains which are the potential causes of performance deterioration or instability of the NCSs [21]. How NIDs & PDs affect the stability and performance of NCSs has thus become an important issue. Several methodologies have been developed for modeling, analyzing and synthezing for NCSs in the presence of NIDs & PDs. Some important categories are mentioned for examples as follows:
1) Time delay system methodology models packet dropouts as a kind of delays, which are lumped with network induced delays in the system. Then, the NCS is modeled as a time delay system with mixed delays [22], [23].
2) Stochastic system method holds the property of an NCS that intermittent packet dropouts and network induced delays are random, and is viewed as an important strategy to handle random phenomenon in NCS. One of the stochastic methods uses a linear function of stochastic variable to model random delays, e.g., [1], [24], or random packet dropouts, e.g., [21], [25]; the other one is to use the Markovian chains to model random delays, e.g., [26], [27], or random packet dropouts, e.g., [28], [29].
3) Switched system method provides a powerful tool for modeling of an NCS with time delays and packet dropouts due to the fact that, whether the transmission of a data packet is successful or not, the switching and delay characteristics of the NCS are obvious, and thus, a switched system model can naturally capture the properties of the NCS with time delays and packet dropouts, e.g., [22], [30][32].
In the aforementioned results concerning NCSs with NIDs & PDs, it should be pointed out that a time delay system method mixes the packet dropouts with the delay term, the effects of dropouts are not clearly formulated. On the other hand, a stochastic system method mainly focuses on the discrete system model of NCSs, and more importantly, it is usually assumed that the NIDs & PDs meet certain probability distributions. A switched system method, which has made great advances in the study of the theories and methods [33][36], has found a good application for NCSs [22], [30], [31]. However, to the best knowledge of authors, existing results concerning NCSs suffering from NIDs & PDs have mainly focused on either delay or switching behaviors [30], [37]. The coexistence and interaction of switchings and delays in an NCS due to NIDs & PDs, which leads to a more complicated stability analysis, have not been fully studied [38]. The importance of the study of this issue, arises from the natural property of the NCS, and the requirement of performance improvement of the system. This motivates the present study.
As far as a switched delay system is concerned, it usually means a switched system with time delays, where the delay may be contained in the system state, control input or switching signal [39]. Distinctive features of switched delay systems are the so called mixedmode [40], i.e., when switching occurs, owing to the influence of time delays, both the present subsystem and the previous subsystems determine the state of the switched delay system. Consequently, it is natural to understand that the stability analysis problem for switched delay systems is not easy to handle based on the LyapunovKrasovskii functional theory, and there are still more space to improve existing results, e.g., [38], [40], especially for a switched timevarying delay system with unstable subsystems, which constitutes a major difficulty in dealing with the mixedmodes in timevarying delay case.
As is well known, switched delay systems can model a class of important systems that include switchings and delays, and an NCS can be seen as a typical example. Due to limited bandwidth, NIDs & PDs are viewed as important phenomena, and if we see packet dropouts occurring on an NCS as the switching behavior between the system with, and without packet dropouts, a switched delay system is regarded as a suitable model for real processes involving the NIDs & PDs, see [22], [30], [32], [41]. Therefore, stability analysis considering the effect of mixedmodes, from the viewpoint of the switched delay system, should be taken into account for an NCS. Unfortunately, the mixedmodes have not been considered in existing results for the NCSs with switched delay system models, see e.g. [22], [30], [41].
In this paper, stabilization of NCSs using a mixedmode based switched delay system method is studied. The main contributions of this paper are as follows. Firstly, significantly capturing the properties of NCSs which lead to the controllers switching, an NCS with NIDs & PDs will be modeled as a continuoustime switched linear timevarying delay system. Secondly, when mixedmodes are considered in the stability analysis of NCSs based on the LyapunovKrasovskii theory, it is necessary for the corresponding subsystems to decompose the integral interval of the constructed LyapunovKrasovskii functional, and the estimation of the integral terms resulted by the processing of LyapunovKrasovskii functional is implemented by the interpolatory quadrature formula. Meanwhile, the average dwell time approach is used and the threshold of the delay upper bound which leads to the switching is selected by an optimization technique rather than a prespecified constant, e.g. [41], [42]. Utilizing the Finsler's Lemma, new exponential stabilizability conditions for NCS with certain tolerated large delay and packet dropout rate are established. Distinctly different from exiting results that concerned with switched strategy for NCSs, e.g. [22], [30], the switched delay system model, and the stabilizability conditions allow the NCS to be unstabilizable during the period of the packet dropouts. Finally, the proposed method is demonstrated by an illustrative example.
Notations:
Consider the following linear continuoustime system described by
$ \begin{align}\label{10s1} \dot{x}(t)&=Ax(t)+Bu(t) \end{align} $  (1) 
where
Noted that introducing the network usually leads to the NIDs & PDs [1], [23]. This paper assumes that a state feedback controller (realized by a zeroorderhold (ZOH)) with piecewise continuous function is applied to synthesize the NCS with NIDs & PDs. The following assumption is given to derive stabilizability conditions.
Assumption 1: The sensor is clockdriven, the controller and actuator are eventdriven, and the data is transmitted with a single packet.
In the following, we will model the NCS with NIDs & PDs as a continuoustime switched delay system.
With the state feedback controller, the NCS (1) with NIDs & PDs can be depicted in Fig. 1.
Download:


Fig. 1 Schematic of NCS with NIDs & PDs 
The sampling period is assumed to be a positive constant
Download:


Fig. 2 Diagram of signal transmitting in NCS with delays and dropouts 
Now, impose the piecewise linear continuous state feedback control via a network channel.
First, assume that at the sampling instant
$ \begin{align*} u(t_{k})=Kx(i_{k}h), \;k=1, 2, \ldots \end{align*} $ 
where
$ \begin{align}\label{controller_1} &u(t)=Kx(td(t))\\ & t\in [i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1}), \, k\in\{1, 2, \ldots\}. \end{align} $  (2) 
The network induced delays
$ \begin{align}\label{delaybound_1} 0\leq\eta_{k}\leq\eta_{m}, \quad k=1, 2, \ldots \end{align} $  (3) 
where
Second, suppose that the sampling instant
$ \begin{align*} u(t_{k})=Kx(i_{k}hl(k)h), \;k=1, 2, \ldots \end{align*} $ 
Furthermore, considering the behavior of the ZOH yields
$ \begin{align}\label{controller_2} u(t)&=Kx(td(t)l(k)h), t\in[i_{k}h+\eta_{m}, i_{k+j}h+\eta_{k+j}) \end{align} $  (4) 
where
When
$ \begin{align}\label{eq_tau} \eta_{k}\leq ti_{k}h<h+\eta_{k+1}. \end{align} $  (5) 
Note that the admissible upper bound of network delays satisfy
$ \begin{align} d(t)<h+\eta_{m}=\tau_{m}.\end{align} $  (6) 
Remark 1: For an instant
By virtue of the technique of controller switching, imposing the controllers (2) and (4) to system (1) gives rise to the following system
$ \begin{align}\label{10s2} \dot{x}(t)=\left\{\begin{array}{ll} Ax(t)+&\!\!\!\!\!\!BKx(td(t)), \\ &t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1})\\ &\mathit{\rm{or}}\, t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{m})\\ Ax(t)+&\!\!\!\!\!\!BKx(td(t)l(k)h), \\ &t\in[i_{k+1}h+\eta_{m}, i_{k+j}h+\eta_{k+j}).\end{array} \right. \end{align} $  (7) 
Introducing the following notions
$ \begin{align}\label{10s33} \dot{x}(t)=Ax(t)+BKx(td_{\sigma(t)}(t)) \end{align} $  (8) 
where the right continuous function
For the development of the main results, the upper bound of
$ \begin{align}d_{2}(t)=d(t)+l(k)h\leq\tau_{M}\end{align} $  (9) 
and obviously,
Remark 2: The switched delay system model (8) is capturing the switching property of NCS when packet dropouts or long delays occur. It should be mentioned that in the recent references [41], [42], a switched delay system model in the continuoustime case for NCS has been studied based on an idea of control missing. It is obvious that there are distinct structure differences between the two switched delay system models, and importantly, the switching signal used for the NCS in this paper are generated by the real signal from the NIDs & PDs.
The following definitions are needed.
Definition 1 [43]: For any
Definition 2: For any
Definition 3: System (1) is said to be exponentially
stabilizable under the control law
$ \x(t)\\leq \kappa \x_{t_{0}}\_{cl}e^{\lambda (tt_{0})}, \;\forall t\geq t_{0} $ 
for some constants
To end this section, some useful lemmas are needed.
Lemma 1 (Finsler's Lemma) :Let
ⅰ)
ⅱ)
Lemma 2: For any constant matrix
$ \begin{align*} &\int_{td(t)}^{t}\dot{\omega}^{T}(\xi)S\dot{\omega}(\xi)d\xi\leq\upsilon^{T}(t)\tau^{1}S\upsilon(t) \end{align*} $ 
where
Proof: Following the same line used in the proof of Lemma 1 [44] gives directly this lemma.
Ⅲ. MAIN RESULTSIn this section, a mixedmode based switched delay system method is used for stabilizing controller design of the NCS with NIDs & PDs. To begin, how the mixed modes affect the stability of the switched delay system will be studied in the LyapunovKrasovskii sense.
In order to study the stability of NCS (8), consider the following form of LyapunovKrasovskii functional
$ \begin{align}\label{10LF0} V_{\sigma(t)}(x_{t})&=x^{T}(t)P_{\sigma(t)}x(t)\\ &~~~~\!+\!\int_{\tau_{\sigma(t)}}^{0}\int_{t+\theta}^{t}\dot{x}^{T}(s)e^{\chi_{\sigma(t)}(ts)}S\dot{x}(s)dsd\theta\!\!\!\! \end{align} $  (10) 
where
Proposition 1: When system (1) suffers packet dropout or
large delay, the manifold
$ \begin{align}\label{10e10} \mathcal {U}(\zeta)\leq q\mathcal {U}(t) \end{align} $  (11) 
for
$ \begin{equation}\label{CONpro0} A^{T}S+SA\beta S\!\geq\! q_{0}I \end{equation} $  (12) 
where
Proof: See Appendix A.
According to the average dwell time strategy, the switched system guarantees the stability (or the stabilizability) if the proportion rate is relatively small between the total activation time of the unstable (or unstabilizable) subsystems and the one of the stable (or stabilizable) subsystems. The following condition is needed.
The switching signals
$ \begin{align}\label{dwt} \triangle_{d}=\frac{T_{d}(T_{1}, T_{2})}{T_{2}T_{1}}<\frac{\alpha\alpha^{\ast}}{\alpha+\beta} \end{align} $  (13) 
where
For the convenience of disposal of the mixedmodes for the NCS using
a switched delay system mothod, it is helpful to exchange the
integral order of the term
$ \begin{align}\label{10LF} V&_{\sigma(t)}(x_{t})=x^{T}(t)P_{\sigma(t)}x(t)\\ &~~~~~~~~~~~\!~\!\!+\!\!\int_{t\!\!\tau_{\sigma(t)}}^{t}\!(s\!\!t\!+\!\tau_{\sigma(t)})\dot{x}^{T}(s)e^{\chi_{\sigma(t)}(t\!\!s)}S\dot{x}(s)ds.\!\! \end{align} $  (14) 
Before developing the stability conditions for the mixedmodes based switched delay systems, a preliminary result is presented. The following lemma is based on the interpolatory quadrature formula (see, e.g. [45]).
Lemma 3: Let
$ W_{1}(t)=\int_{t\tau_{m}}^{t}(st+\tau_{m})\dot{x}^{T}(s)e^{\alpha(ts)}S\dot{x}(s)ds $ 
and
$ W_{2}(t)=\int_{t\tau_{M}}^{t}(st+\tau_{M})\dot{x}^{T}(s)e^{\beta(ts)}S\dot{x}(s)ds $ 
when
$ \begin{array}{l} \int_{t  {\tau _m}}^{{t_k}} {(s  t + {\tau _m}){\cal U}(s)} ds\\ \le q\frac{{{t_k}  t + {\tau _m}}}{{{\tau _m}}} \times \int_{t  {\tau _m}}^t {(s  t + {\tau _m}){\cal U}(s)} ds \end{array} $  (15) 
$ {W_2}(t) \le {\left( {\frac{{{\tau _M}}}{{{\tau _m}}}} \right)^2} \times {e^{(\alpha + \beta ){\tau _M}}} \times {W_1}(t). $  (16) 
Proof: See Appendix B.
With the help of Lemma 3, the following lemma facilitates us to give the exponential decay and increase estimation of LyapunovKrasovskii functionals corresponding to the NCS without packet dropouts and, the NCS with packet dropouts, respectively.
Lemma 4: For given positive constants
$ \begin{equation}\label{eqLF1} V_{1}(x_{t})\leq e^{\alpha(tt_{0})}V_{1}(x_{t_{0}}) \end{equation} $  (17) 
when
$ \begin{equation}\label{eqLF2} V_{2}(x_{t})\leq e^{\beta(tt_{0})}V_{2}(x_{t_{0}}) \end{equation} $  (18) 
when
$ \begin{align}\label{CONpro} &\max \quad \tau_{m}, \;\tau_{M} \\ &\begin{array}{ll} \mathit{\rm{ s.t.}}&\!\!\Theta_{i}:=\left[\begin{array}{ccc} \varphi_{11}^{i}\;&\;\varphi_{12}^{i}\;&\;\varphi_{13}^{i}\\ *&\varphi_{22}^{i}&\varphi_{23}^{i}\\ *&*&\varphi_{33}^{i} \end{array}\right]<0, \quad i=1, 2 \end{array} \end{align}$  (19) 
is solvable, where
$ \begin{align*} \varphi_{11}^{i}&=A^{T}P_{i}\!+\!P_{i}A\!+\!K^{T}B^{T}P_{i}\!+\!P_{i}BK\!+\!\chi_{i} P_{i}\!+\!X_{i}\!+\!X_{i}^{T}\\ \varphi_{12}^{i}&=P_{1}BKX_{i}+Y_{i}^{T}\\ \varphi_{13}^{i}&=(A+BK)^{T}SX_{i}+Z_{i}^{T}\\ \varphi_{22}^{1}&=\left[\tau_{m}^{1}e^{\alpha\tau_{m}}q(\alpha\alpha^{\ast})e^{\beta\tau_{m}}\right]S Y_{1}Y_{1}^{T}\\ \varphi_{22}^{2}&=\tau_{M}^{1}e^{\alpha\tau_{M}}SY_{1}Y_{1}^{T}\\ \varphi_{23}^{i}&=K^{T}B^{T}SY_{i}Z_{i}^{T}\\ \varphi_{33}^{i}&=\tau_{i}^{1}SZ_{i}Z_{i}^{T}\quad(i=1, 2). \end{align*} $ 
Proof: See Appendix C.
Remark 3: In order to deal with the mixedmodes based stability analysis for NCS with NIDs & PDs, the interpolatory quadrature formula is key to deriving the exponential decay or increase estimation conditions for the LyapunovKrasovskii functionals. It should be noted that simple bounding techniques may not give the available solution of the problem addressed.
Remark 4: By the introduction of the Finsler's lemma, the solvability problem of
We are now in a position to give the stabilization conditions for the NCS with NIDs & PDs.
Theorem 1: For given positive constants
ⅰ) the average dwell time is satisfied with
$ \begin{align}\label{eq26} \frac{\ln\mu\varepsilon}{\alpha}<\frac{\ln\mu\varepsilon}{\alpha^{*}}=T_{\alpha}^{*}<T_{\alpha} \end{align} $  (20) 
where
$ {\rm{and}}\;\mu = \mathop {\inf }\limits_\nu \left\{ {\nu \left {\begin{array}{*{20}{c}} {{P_i} \le \nu {P_j}, \;\forall i, j \in {\cal P}} \end{array}} \right.} \right\} $  (21) 
in
which
ⅱ) the packet dropouts rate meets (3).
Proof: The proof is similar to the one in [38], and thus is omitted.
Remark 5: By using the switched delay system method, an NCS with NIDs & PDs can be modeled as a switched delay system with both stable and unstable subsystems, which extremely extends the delay upper bound for an NCS. Note that the large delay case has been viewed as packet dropout, so the packet dropouts rate can be depicted as the occurrence frequency of the unstable subsystem, i.e., we can obtain the packet dropouts rate by (13) and the optimal solutions of (12) and (19).
Now, we present the procedure to get the maximum packet dropouts
rate
Step 1: Preset the values of positive constants
Step 2: Bidirectionally adjust the constants
Step 3: Select an appropriate parameter
Remark 6: The delay upper bounds
Remark 7: Literally, the main results obtained in this paper are similar to those of [19] due to that the switchings, networked delays, the ``mixed" properties and the average switching techniques are studied in both papers. However, compared with [19], stability analysis of NCSs considering of the coexistence and interaction of switchings and delays, i.e., the effect of mixedmode, is the obvious the main task of this paper.
Ⅳ. AN ILLUSTRATIVE EXAMPLEIn this section, a numerical example is introduced to demonstrate the mixedmode based switched delay system method for NCS with NIDs & PDs.
Consider the same NCS (1) as in [38], where
$ \begin{equation*} A=\left[\begin{array}{cc}0.8&0.01\\1&0.1\end{array}\right], \quad B=\left[\begin{array}{c}0.4\\0.1\end{array}\right]. \end{equation*} $ 
It is obvious that the openloop system is not stable.
With the networked control scheme, we can model the following switched delay system
$ \begin{align*}\dot{x}(t)=Ax(t)+BKx(td_{\sigma(t)}(t)). \end{align*} $ 
By the mixedmode based switching delay system method proposed in
this paper, we can obtain the critical delay bound
$ K=\left[\begin{array}{cc}7.2893&5.6621\end{array}\right] $ 
and the LyapunovKrasovskii functional (14) with
$ P_{1}=\left[ \begin{array}{cc} 11.1070& 6.2535\\ 6.2535 & 8.4610 \end{array} \right], \quad P_{2}=\left[ \begin{array}{cc} 10.2466& 3.9710\\ 3.9710 & 7.2438 \end{array} \right] $ 
and
$ S=\left[ \begin{array}{cc} 153.6669& 13.6497\\ 13.6497& 132.9459 \end{array} \right] $ 
are calculated with the MATLAB LMI control Toolbox based on conditions (19).
Note that, though the effect of the mixedmode of the delays and
dropouts is considered, by introducing the Finsler's lemma to
deal with the network induced delays and dropouts, we can still
establish the conditions of less conservativeness, which enables us
to get larger critical delay bound
By selecting the parameters
We simulate the network environment such that the NIDs & PDs
satisfy the conditions of Theorem 1. By selecting the initial state
as
Download:


Fig. 3 The switching signal resulted by the packet dropouts 
Download:


Fig. 4 The control input 
Download:


Fig. 5 The state response of NCS with NIDs & PDs 
This paper addressed the modeling of NCS with network induced delays and packet dropouts as a continuoustime switched linear delay system. The effect of mixedmode to the stability analysis, from the viewpoint of a switched delay system, was investigated for NCSs based on a novel constructed LyapunovKrasovskii functional, and NewtonCotes quadrature formula was taken into account for the decay rate estimation of the switched Lyapunov Krasovskii functional. With the help of the Finsler's lemma, new exponential stabilizability conditions with less conservativeness were given for NCSs. Finally, an example was given to illustrate the effectiveness of the developed results. Issues of the stabilization problem for NCSs using the mixedmode based switching delay system method with more general Lyapunov functionals, or for NCSs with multiple packets transmission and other communication constraints, are interesting and challenging, and thus deserve further investigation in the future.
APPENDIX A PROOF OF PROPOSITION 1Consider the derivative of
For
$ \begin{align}\!\dot{z}(s)\!=\!Az(s)\!+\!(1\!\!\dot{d}_{2}(s))BKz(s\!\!d_{2}(s)).\end{align} $  (A1) 
Note that the sawtooth structure of the artificial delays, i.e.,
$ \begin{align}\label{unsys1}\dot{z}(s)=Az(s).\end{align} $  (A2) 
Fixing
$ \begin{equation} \dot{\mathcal {U}}(s)=z^{T}(s)e^{\beta(ts)}\left[A^{T}S+SA\beta S\right]z(s).\end{equation} $  (A3) 
The conditions of (12) help us to get
$ \begin{align}\dot{\mathcal {U}}(s)\geq\frac{q_{0}}{\lambda_{\min}(S)}\mathcal {U}(s).\end{align} $  (A4) 
Integrating (A4) from
$ \begin{equation} \mathcal {U}(\zeta)\leq q \mathcal {U}(t). \end{equation} $  (A5) 
Consider the NewtonCotes quadrature formula [45].
ⅰ) Note that inequality (15) is equivalent to the following inequality
$ \begin{align} &\frac{q (t_{k}t)+(q1)\tau}{\tau}\int_{t\tau}^{t_{k}}(st+\tau)\mathcal {U}(s)ds\\ &\quad\quad+\frac{q (t_{k}t)+q\tau}{\tau} \int_{t_{k}}^{t}(st+\tau)\mathcal {U}(s)ds\geq0. \end{align} $  (A6) 
If (A6) can be guaranteed, the proof is finished. Now, consider the
left hand of (A6). According to the integral mean value theory, there exist
$ \begin{align} &\frac{q (t_{k}t)+(q1)\tau}{\tau}\int_{t\tau}^{t_{k}}(st+\tau)\mathcal {U}(s)ds\\ &\;\, =\frac{q (t_{k}t)\!+\!(q1)\tau}{\tau} (t_{k}\!t\!+\!\tau)[(\xi_{1}\!t\!+\tau)\mathcal {U}(\xi_{1})] \end{align} $  (A7) 
and
$ \begin{align} &\frac{q (t_{k}t)+q\tau}{\tau}\int_{t_{k}}^{t}(st+\tau)\mathcal {U}(s)ds\\ &\qquad=\frac{q (t_{k}t)+q\tau}{\tau}(tt_{k})[(\xi_{2}t+\tau)\mathcal {U}(\xi_{2})]. \end{align} $  (A8) 
Noticing that
$ \begin{align} &\frac{q (t_{k}t)+(q1)\tau}{\tau}\int_{t\tau}^{t_{k}}(st+\tau)\mathcal {U}(s)ds\\ &\qquad+\frac{q (t_{k}t)+q\tau}{\tau} \int_{t_{k}}^{t}(st+\tau)\mathcal {U}(s)ds\\ &\quad=\frac{q (t_{k}t)+(q1)\tau}{\tau} (t_{k}t+\tau)[(\xi_{1}t+\tau)\mathcal {U}(\xi_{1})]\\ &\qquad+\frac{q (t_{k}t)+q\tau}{\tau}(tt_{k})[(\xi_{2}t+\tau)\mathcal {U}(\xi_{2})]\\ &\quad\geq\frac{q (t_{k}t)+(q1)\tau}{\tau} (t_{k}t+\tau)[(\xi_{1}t+\tau)\mathcal {U}(\xi_{1})]\\ &\qquad+\frac{q (t_{k}t)+q\tau}{\tau}(tt_{k})[(\xi_{1}t+\tau)\mathcal {U}(\xi_{1})]\frac{1}{q} \end{align} $  (A9) 
noticing that
ⅱ) Considering the integral transformation for
$ \begin{align*}W_{2}(t)&=\int_{t\tau_{M}}^{t}(st+\tau_{M})U_{2}(s)ds\\ &=\left(\frac{\tau_{M}}{\tau_{m}}\right)^{2} \int_{t\tau_{m}}^{t}(xt+\tau_{m})\\ &~~~~\times U_{2}\left(\frac{\tau_{M}}{\tau_{m}}(xt+\frac{\tau_{m}}{2})+(t\frac{\tau_{M}}{2})\right)dx. \end{align*} $ 
Note that when
$ \begin{align*}&\frac{\tau_{M}}{\tau_{m}}\left(xt+\frac{\tau_{m}}{2}\right)+(t\frac{\tau_{M}}{2})x\\ =&\left(\frac{\tau_{M}}{\tau_{m}}1\right)(xt)\leq0. \end{align*} $ 
Note that
Firstly, we consider the following decomposition of the
LyapunovKrasovskii functional (14). Suppose that system (8) is in
the
$ \begin{align} V_{i}(x_{t})=&x^{T}(t)P_{i}x(t)\\ &+\int_{t\tau_{i}}^{t_{k}}(st+\tau_{i})\dot{x}^{T}(s)e^{\chi_{j}(ts)}S\dot{x}(s)ds\\ &+\int_{t_{k}}^{t}(st+\tau_{i})\dot{x}^{T}(s)e^{\chi_{i}(ts)}S\dot{x}(s)ds\dot{x}(s) \end{align} $  (A10) 
and when
$ \begin{equation} V_{i}(x_{t})=x^{T}(t)P_{i}x(t) +\int_{t\tau_{i}}^{t}(st+\tau_{i})\dot{x}^{T}(s)e^{\chi_{i}(ts)}S\dot{x}(s)ds. \end{equation} $  (A11) 
We will discuss the derivatives of the LyapunovKrasovskii
functionals in two cases, i.e.,
Case 1:
Note that in this case, the mixedmode may cover the whole switching
interval
$ \begin{align} \dot{V}_{i}(x_{t})=&\, 2x^{T}(t)P_{i}\dot{x}(t)+\tau_{i} \dot{x}^{T}(t)S\dot{x}(t)\nonumber\\ &\int_{t\tau_{i}}^{t_{k}}\dot{x}^{T}(\theta)e^{\chi_{j}(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &\int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{\chi_{i}(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\chi_{j}\int_{t\tau_{i}}^{t_{k}}(st+\tau_{i})\dot{x}^{T}(\theta)e^{\chi_{j}(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\chi_{i}\int_{t_{k}}^{t}(st+\tau_{i})\dot{x}^{T}(\theta)e^{\chi_{i}(t\theta)}S\dot{x}(\theta)d\theta. \end{align} $  (A12) 
If it is assumed that
$ \begin{align} \dot{V}_{1}(x_{t})&+\alpha V_{1}(x_{t})\\=&\, 2x^{T}(t)P_{1}\dot{x}(t)+\tau_{m} \dot{x}^{T}(t)S\dot{x}(t)+x^{T}(t)\alpha P_{1}x(t)\nonumber\\ &\int_{t\tau_{m}}^{t_{k}}\dot{x}^{T}(\theta)e^{\beta(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &\int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{\alpha(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\beta\int_{t\tau_{m}}^{t_{k}}(st+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &\alpha\int_{t_{k}}^{t}(st+\tau_{m})\dot{x}^{T}(\theta)e^{\alpha(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\alpha\int_{t\tau_{m}}^{t_{k}}(st+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\alpha\int_{t_{k}}^{t}(st+\tau_{m})\dot{x}^{T}(\theta)e^{\alpha(t\theta)}S\dot{x}(\theta)d\theta.\ \end{align} $  (A13) 
Considering the effect of the mixedmode, during
$ \begin{align} &(\alpha+\beta)\int_{t\tau_{m}}^{t_{k}}(st+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t\theta)}S\dot{x}(\theta)d\theta\\ &\leq(\alpha+\beta)\cdot\frac{t_{k}t+\tau_{m}}{\tau_{m}}\cdot q\tau_{m} \int_{t\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta\\ &\leq(\alpha+\beta)\cdot\frac{\alpha\alpha^{\ast}}{\alpha+\beta}\cdot q\tau_{m} \int_{t\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta\\ &\leq q(\alpha\alpha^{\ast})\cdot\tau_{m} \int_{t\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta \end{align} $  (A14) 
and it is obvious that
$ \begin{align} &\int_{t\tau_{m}}^{t_{k}}\dot{x}^{T}(\theta)e^{\beta(t\theta)}S\dot{x}(\theta)d\theta \int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{\alpha(t\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &\leq\int_{t\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\alpha\tau_{m}}S\dot{x}(\theta)d\theta. \end{align} $  (A15) 
Substituting (A14) and (A15) into (A13) gives rise to
$ \begin{align} \dot{V}_{1}&(x_{t})+\alpha V_{1}(x_{t})\nonumber\\ \leq\, &2x^{T}(t)P_{1}[Ax(t)+BKx(td_{1}(t))]\nonumber\\ &\!+\!\tau_{m} [Ax(t)\!+\!BKx(t\!\!d_{1}(t))]^{T}S[Ax(t)\!+\!BKx(t\!\!d_{1}(t))]\\ &\!+\!x^{T}(t)\alpha P_{1}x(t)\int_{t\tau_{m}}^{t}\dot{x}^{T}(s)e^{\alpha\tau_{m}}S\dot{x}(s)ds\nonumber\\ &\!+\!q(\alpha\alpha^{\ast})\tau_{m} \int_{t\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta.\\ \leq\, &2x^{T}(t)P_{1}\dot{x}(t)+\tau_{m} \dot{x}^{T}(t)S\dot{x}(t)+x^{T}(t)\alpha P_{1}x(t)\nonumber\\ &\left[e^{\alpha\tau_{m}}q(\alpha\alpha^{\ast})\tau_{m} e^{\beta\tau_{m}}\right] \int_{td_{1}(t)}^{t}\dot{x}^{T}(\theta)S\dot{x}(\theta)d\theta. \end{align} $  (A16) 
Note that
$ \begin{align} &\int_{td_{1}(t)}^{t}\dot{x}^{T}(\theta)S\dot{x}(\theta)d\theta\leq\upsilon^{T}(t)\tau_{m}^{1}S\upsilon(t) \end{align} $  (A17) 
where
Let
$ \begin{align} \dot{V}_{1}(x_{t})+\alpha V_{1}(x_{t})\leq\xi^{T}(t)\Xi_{1}\xi(t) \end{align} $  (A18) 
where
$ \begin{equation*} \Xi_{1}:=\left[\begin{array}{ccc} \phi_{11}^{1}\;&\;\phi_{12}^{1}\;&\;\phi_{13}^{1}\\ *&\phi_{22}^{1}&\;\phi_{23}^{1}\\ *&*&\;\phi_{33}^{1} \end{array}\right]\end{equation*} $ 
with
$ \begin{align*} \phi_{11}^{1}&=A^{T}P_{1}+P_{1}A+K^{T}B^{T}P_{1}+P_{1}BK+\alpha P_{1}\\ \phi_{12}^{1}&=P_{1}BK\\ \phi_{13}^{1}&=(A+BK)^{T}S\\ \phi_{22}^{1}&=\left[\tau_{m}^{1}e^{\alpha\tau_{m}}q(\alpha\alpha^{\ast})e^{\beta\tau_{m}}\right]S\\ \phi_{23}^{1}&=K^{T}B^{T}S\\ \phi_{33}^{1}&=\tau_{m}^{1}S. \end{align*} $ 
Note that, on the other hand, when
$ \begin{align} \Xi_{1}+\mathcal {X}_{1}\mathcal {H}+\mathcal {H}^{T}\mathcal {X}_{1}^{T}<0 \end{align} $  (A19) 
where
$ \begin{align*} \mathcal {X}_{1}&=\left[ \begin{array}{cc} X_{1} \\ Y_{1} \\ Z_{1} \end{array} \right]\\ \mathcal {H}&=\left[ \begin{array}{ccc} I &I &I \\ \end{array} \right]. \end{align*} $ 
Obviously,
$ \begin{align}\tag{A20} V_{1}(x_{t})\leq e^{\alpha(tt_{0})}V_{1}(x_{t_{0}}) \end{align} $ 
holds apparently.
Case 2:
In this case, two subintervals:
Note that no time interval decomposition of the integral is needed.
When
Similarly, it is easy to obtain
[1]  W. Zhang, M. S. Branicky, and S. M. Phillips, "Stability of networked control systems, " IEEE Control Syst. Mag., vol. 21, no. 1, pp. 8499, Feb. 2001. http://dx.doi.org/10.1109/37.898794 
[2]  G. C. Walsh, H. Ye, and L. G. Bushnell, "Stability analysis of networked control systems, " IEEE Trans. Control Syst. Technol., vol. 10, no. 3, pp. 438446, May 2002. http://dx.doi.org/10.1109/87.998034 
[3]  U. Tiberi, C. Fischione, K. H. Johansson, and M. D. Di Benedetto, "Energyefficient sampling of networked control systems over IEEE 802. 15. 4 wireless networks, " Automatica, vol. 49, no. 3, pp. 712724, Mar. 2013. http://dx.doi.org/10.1016/j.automatica.2012.11.046 
[4]  R. Sakthivel, S. Santra, K. Mathiyalagan, and H. Y. Su, "Robust reliable control design for networked control system with sampling communication, " Int. J. Control, vol. 88, no. 12, pp. 25102522, Jun. 2015. http://www.ingentaconnect.com/content/tandf/tcon/2015/00000088/00000012/art00010 
[5]  R. Sakthivel, P. Selvaraj, Y. Lim, and H. R. Karimi, "Adaptive reliable output tracking of networked control systems against actuator faults, " J. Franklin Inst., vol. 354, no. 9, pp. 38133837, Jun. 2017. http://dx.doi.org/10.1016/j.jfranklin.2016.06.022 
[6]  D. J. Du, B. Qi, M. R. Fei, and Z. X. Wang, "Quantized control of distributed eventtriggered networked control systems with hybrid wiredwireless networks communication constraints, " Inf. Sci. , vol. 380, pp. 7491, Feb. 2017. http://dx.doi.org/10.1016/j.ins.2016.03.033 
[7]  L. L. Su and G. Chesi, "Robust stability analysis and synthesis for uncertain discretetime networked control systems over fading channels, " IEEE Trans. Autom. Control, vol. 62, no. 4, pp. 19661971, Apr. 2017. http://paperity.org/p/54501117/onstabilityanalysisofdiscretetimeuncertainswitchednonlineartimedelaysystems 
[8]  Y. Sadi and S. C. Ergen, "Joint optimization of wireless network energy consumption and control system performance in wireless networked control systems, " IEEE Trans. Wireless Commun., vol. 16, no. 4, pp. 22352248, Apr. 2017. http://ieeexplore.ieee.org/abstract/document/7876829/ 
[9]  Y. Tipsuwan and M. Y. Chow, "Control methodologies in networked control systems, " Control Eng. Pract., vol. 11, no. 10, pp. 10991111, Oct. 2003. http://dx.doi.org/10.1016/S09670661(03)000364 
[10]  J. Finke, K. M. Passino, and A. G. Sparks, "Stable task load balancing strategies for cooperative control of networked autonomous air vehicles, " IEEE Trans. Control Syst. Technol., vol. 14, no. 5, pp. 789803, Sep. 2006. https://core.ac.uk/display/20979388 
[11]  S. J. Yoo, "Distributed adaptive containment control of networked flexiblejoint robots using neural networks, " Expert Syst. Appl., vol. 41, no. 2, pp. 470477, Feb. 2014. http://dx.doi.org/10.1016/j.eswa.2013.07.072 
[12]  J. Nilsson, B. Bernhardsson, and B. Wittenmark, "Stochastic analysis and control of realtime systems with random time delays, " Automatica, vol. 34, no. 1, pp. 5764, Jan. 1998. 
[13]  G. C. Goodwin, H. Haimovich, D. E. Quevedo, and J. S. Welsh, "A moving horizon approach to networked control system design, " IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 14271445, Sep. 2004. http://dx.doi.org/10.1109/TAC.2004.834132 
[14]  W. P. M. H. Heemels, A. R. Teel, N. van de Wouw, and D. Nesic, "Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance, " IEEE Trans. Autom. Control, vol. 55, no. 8, pp. 17811796, Aug. 2010. http://dx.doi.org/10.1109/TAC.2010.2042352 
[15]  C. Peng and T. C. Yang, "Eventtriggered communication and H_{∞} control codesign for networked control systems, " Automatica, vol. 49, no. 5, pp. 13261332, May 2013. http://dx.doi.org/10.1016/j.automatica.2013.01.038 
[16]  L. X. Zhang, H. J. Gao, and O. Kaynak, "Networkinduced constraints in networked control systemsa survey, " IEEE Trans. Ind. Inform. , vol. 9, no. 1, pp. 403416, Feb. 2013. http://dx.doi.org/10.1109%2FTII.2012.2219540 
[17]  R. N. Yang, G. P. Liu, P. Shi, C. Thomas, and M. V. Basin, "Predictive output feedback control for networked control systems, " IEEE Trans. Ind. Electron., vol. 61, no. 1, pp. 512520, Jan. 2014. https://www.mendeley.com/researchpapers/predictiveoutputfeedbackcontrolnetworkedcontrolsystems/ 
[18]  A. Sahoo and S. Jagannathan, "Stochastic optimal regulation of nonlinear networked control systems by using eventdriven adaptive dynamic programming, " IEEE Trans. Cybern., vol. 47, no. 2, pp. 425438, Feb. 2017. http://dx.doi.org/10.1109/TCYB.2016.2519445 
[19]  Y. Tang, H. J. Gao, W. B. Zhang, and J. Kurths, "Leaderfollowing consensus of a class of stochastic delayed multiagent systems with partial mixed impulses, " Automatica, vol. 53, pp. 346354, Mar. 2015. http://dialnet.unirioja.es/servlet/articulo?codigo=5046238 
[20]  Y. Tang, D. D. Zhang, D. W. C. Ho, and F. Qian, "Tracking control of a class of cyberphysical systems via a flexray communication network, " IEEE Trans. Cybern., vol. PP, no. 99, pp. 114, 2018, to be published. http://europepmc.org/abstract/MED/29993875 
[21]  Z. D. Wang, F. W. Yang, D. W. C. Ho, and X. H. Liu, "Robust H_{∞} control for networked systems with random packet losses, " IEEE Trans. Syst. Man Cybern. B: Cybern., vol. 37, no. 4, pp. 916924, Aug. 2007. Robust H∞ control for networked systems with random packet losses 
[22]  M. Yu, L. Wang, T. G. Chu, and G. M. Xie, "Stabilization of networked control systems with data packet dropout and network delays via switching system approach, " in Proc. 43rd IEEE Conf. Decision and Control, Nassau, Bahamas, 2004, pp. 35393544. http://dx.doi.org/10.1109/CDC.2004.1429261 
[23]  D. Yue, Q. L. Han, and J. Lam, "Networkbased robust H_{∞} control of systems with uncertainty, " Automatica, vol. 41, no. 6, pp. 9991007, Jun. 2005. 
[24]  F. W. Yang, Z. D. Wang, Y. S. Hung, and M. Gani, "H_{∞} control for networked systems with random communication delays, " IEEE Trans. Autom. Control, vol. 51, no. 3, pp. 511518, Mar. 2006. http://dx.doi.org/10.1109/TAC.2005.864207 
[25]  S. S. Hu and Q. X. Zhu, "Stochastic optimal control and analysis of stability of networked control systems with long delay, " Automatica, vol. 39, no. 11, pp. 18771884, Nov. 2003. https://www.mendeley.com/researchpapers/stochasticoptimalcontrolanalysisstabilitynetworkedcontrolsystemslongdelay6/ 
[26]  L. Q. Zhang, Y. Shi, T. W. Chen, and B. Huang, "A new method for stabilization of networked control systems with random delays, " IEEE Trans. Autom. Control, vol. 50, no. 8, pp. 11771181, Aug. 2005. http://dx.doi.org/10.1155/2012/834643 
[27]  Y. Shi and B. Yu, "Output feedback stabilization of networked control systems with random delays modeled by Markov chains, " IEEE Trans. Autom. Control, vol. 54, no. 7, pp. 16681674, Jul. 2009. http://www.ifacpapersonline.net/Detailed/48549.html 
[28]  J. Wu and T. W. Chen, "Design of networked control systems with packet dropouts, " IEEE Trans. Autom. Control, vol. 52, no. 7, pp. 13141319, Jul. 2007. http://pdfs.semanticscholar.org/247a/9bbf7efe1c6e7a762e0e3d26500e8df0a393.pdf 
[29]  J. L. Xiong and J. Lam, "Stabilization of linear systems over networks with bounded packet loss, " Automatica, vol. 43, no. 1, pp. 8087, Jan. 2007. http://dx.doi.org/10.1016/j.automatica.2006.07.017 
[30]  H. Lin and P. J. Antsaklis, "Stability and persistent disturbance attenuation properties for a class of networked control systems: switched system approach, " Int. J. Control, vol. 78, no. 18, pp. 14471458, Dec. 2005. 
[31]  W. A. Zhang and L. Yu, "Modelling and control of networked control systems with both networkinduced delay and packetdropout, " Automatica, vol. 44, no. 12, pp. 32063210, Dec. 2008. http://dialnet.unirioja.es/servlet/articulo?codigo=2791384 
[32]  M. C. F. Donkers, W. P. H. Heemels, N. van de Wouw, and L. Hetal, "Stability analysis of networked control systems using a switched linear systems approach, " IEEE Trans. Autom. Control, vol. 56, no. 9, pp. 21012115, Sep. 2011. 
[33]  R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, "Perspectives and results on the stability and stabilizability of hybrid systems, " Proc. IEEE, vol. 88, no. 7, pp. 10691082, Jul. 2000. http://dx.doi.org/10.1109/5.871309 
[34]  Z. D. Sun, Switched Linear Systems:Control and Design. New York: SpringerVerlag, 2005. 
[35]  R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, "Stability criteria for switched and hybrid systems, " SIAM Rev. , vol. 49, no. 4, pp. 545592, Nov. 2007. http://adsabs.harvard.edu/abs/2007SIAMR..49..545S 
[36]  L. J. Long and J. Zhao, "A smallgain theorem for switched interconnected nonlinear systems and its applications, " IEEE Trans. Autom. Control, vol. 59, no. 4, pp. 10821088, Apr. 2014. http://dx.doi.org/10.1109/TAC.2013.2286898 
[37]  Y. F. Su and J. Huang, "Stability of a class of linear switching systems with applications to two consensus problems, " IEEE Trans. Autom. Control, vol. 57, no. 6, pp. 14201430, Jun. 2012. 
[38]  Q. K. Li and H. Lin, "Effects of mixedmodes on the stability analysis of switched timevarying delay systems, " IEEE Trans. Autom. Control, vol. 61, no. 10, pp. 30383044, Oct. 2016. https://www.mendeley.com/researchpapers/effectsmixedmodesstabilityanalysisswitchedtimevaryingdelaysystems/ 
[39]  X. M. Sun, G. P. Liu, D. Rees, and W. Wang, "Delaydependent stability for discrete systems with large delay sequence based on switching techniques, " Automatica, vol. 44, no. 11, pp. 29022908, Nov. 2008. http://dialnet.unirioja.es/servlet/articulo?codigo=2766465 
[40]  S. Kim, S. A. Campbell, and X. Z. Liu, "Stability of a class of linear switching systems with time delay, " IEEE Trans. Circuit Syst. I: Regul. Paper, vol. 53, no. 2, pp. 384393, Feb. 2006. http://dx.doi.org/10.1109/TCSI.2005.856666 
[41]  W. A. Zhang and L. Yu, "Stabilization of sampleddata control systems with control inputs missing, " IEEE Trans. Autom. Control, vol. 55, no. 2, pp. 447452, Feb. 2010. http://dx.doi.org/10.1109/TAC.2009.2036325 
[42]  X. M. Sun, G. P. Liu, D. Rees, and W. Wang, "Stability of systems with controller failure and timevarying delay, " IEEE Trans. Autom. Control, vol. 53, no. 10, pp. 23912396, Nov. 2008. http://dx.doi.org/10.1109/tac.2008.2007528 
[43]  J. P. Hespanha and A. S. Morse, "Stability of switched systems with average dwelltime, " in Proc. 38th IEEE Conf. Decision and Control, Phoenix, AZ, USA, 1999, pp. 26552660. http://dx.doi.org/10.1109/CDC.1999.831330 
[44]  K. Gu, "An integral inequality in the stability problem of timedelay systems, " in Proc. 39th IEEE Conf. Decision and Control, Sydney, NSW, Australia, 2002, pp. 28052810. http://dx.doi.org/10.1109/cdc.2000.914233 
[45]  R. Kress, Numerical Analysis. New York: SpringerVerlag, 1998. 
[46]  L. S. Hu, T. Bai, P. Shi, and Z. M. Wu, "Sampleddata control of networked linear control systems, " Automatica, vol. 43, no. 5, pp. 903911, May 2007. http://dialnet.unirioja.es/servlet/articulo?codigo=2274037 