State Estimation for Wireless Network Control System with Stochastic Uncertainty and Time Delay Based on Sliding Mode Observer

Abstract

State estimation problem is considered for a kind of wireless network control system with stochastic uncertainty and time delay. A sliding mode observer is designed for the system under the situation that no missing measurement occurs and system uncertainty happens in a stochastic way. The observer designed for the system can guarantee the system states will be driven onto the sliding surface under control law, and the sliding motion of system states on sliding surface will be stable. By constructing proper Lyapunov-Krasovskii functional, sufficient conditions are acquired via linear matrix inequality. Finally, simulation result is employed to show the effectiveness of the proposed method.

1. Introduction

During the past decades, state estimation problem is a hot issue in academical field, and researchers made fruitful research on both theoretical research and practical applications [1–4]. At the same time, research and applications in wireless network control system (WiNCS) attracted much interest from scholars [5–9]. Compared with wired network control system, WiNCS is more convenient in integrated wiring and could be arranged in hazardous area people cannot easily reach. Sensor nodes are distributed in advance to collect information and transmit signal back via signal channel through multihop technology, until they reach users’ terminal applications. Because of its low cost, mobility character, and convenience for using, WiNCS is widely used in hospital monitoring, military and urban affairs, and some other important cases [10–13]. However, as the whole system is becoming complicated and integrated, people want to keep track of useful measured values, and a rich body of literature has appeared on the topic of state estimation or observer-based design; see [14–16] and the references therein.

As a special class of complex networks, WiNCS has its own characteristics due mainly to large numbers of nodes distributed over the region of interest. In this case, sensor nodes may be in mobile motion, which is attributed to the uncertainty in system state, and much effort has been made on this issue [17–22]. For distributed state estimation problem, each sensor node in WiNCS estimates system state not only from its own measurement but also from its neighboring nodes, so some unpredictable factors which happen in a stochastic way will enhance the complexity in state estimation [23–25]. Reference [26] studied synchronization problem for Markovian jump neural networks with time-varying delay and variable samplings. Stochastic stability of the error system was guaranteed by two criteria, and mode-independent controller was designed based on the maximum sampling interval. Reference [27] constructed proper Lyapunov-Krasovskii functional and studied the adaptive fault estimation problems for stochastic Markovian jump systems (MJSs), combined with time delays. Adaptive fault detection observer was designed and the sufficient condition was proposed. Reference [28] concentrated on the proportional-integral control problems of stochastic Markovian jump systems (MJSs) with uncertain parameters. It transformed the controller design problem to an output feedback control problem, and a sufficient condition was proposed via LMI.

Nodes receive measurement information from other nodes in WiNCS which is time consuming, so time delay cannot be avoided in the research of WiNCS, and many scholars devoted to this problem [29–33]. Reference [34] considered the problem that both had discrete and distributed delays, and the delay dependent passivity condition was acquired. Reference [35] investigated the problem of sampled-data extended dissipative control for uncertain Markov jump systems. By proposing a new integral inequality, a novel exponential stability criterion and an extended dissipativity condition were established. In addition, a sufficient condition for desired mode-independent sampled-data controller was also obtained. Reference [36] dealt with the average consensus problem in directed networks of agents with switching topology and time delay. It proved that all the agents could reach average consensus under a proper time delay if the topology of agents was weakly connected and balanced.

Motivated by previous research stated above, our target is to deal with state estimation problem in wireless network control system, which contains stochastic uncertainty and time delay. A sliding mode observer is designed in two steps; by the use of Lyapunov stability theory, sufficient conditions are proposed to make sure that system states can be driven onto the sliding surface within finite time and make stable sliding motion on sliding surface.

The rest of the paper is organized as follows. In Section 2, the state estimation problem of WiNCS is formulated and some useful lemmas are introduced. In Section 3, we make our designing process in two steps. Besides, the gain of observer is acquired by LMI. An illustrated example is given in Section 4 to demonstrate the effectiveness of the proposed method. Finally, we give our conclusions in Section 5.

2. Problem Formulation

In this paper, we consider the following discrete stochastic and time delay model: 𝑥(𝑘+1)=𝐴+𝛼_𝑘Δ𝐴𝑥(𝑘)+𝐴_𝑑^𝑁_𝑖=1𝑥()(𝑘),𝑦(𝑘)(𝑘),𝑘−𝑖+𝐵𝑢=𝐶𝑥 (1) where 𝑥(𝑘)∈ℝ^𝑛 is the system state vector, 𝑦(𝑘)∈ℝ^𝑛 is the measured output, Δ𝐴 is internal perturbation arising from uncertain factors, 𝑢(𝑘) is system input, 𝐴, 𝐴_𝑑, 𝐵, and 𝐶 are constant matrices with appropriate dimensions, and 𝛼_𝑘 is Bernoulli distributed white sequences governed by Prob𝛼_𝑘=1=𝔼𝛼_𝑘𝛼=𝛼,Prob_𝑘=0=1−𝔼𝛼_𝑘=1−𝛼, (2) where 𝛼∈[0,1] is a known constant.

Remark 1. Most nodes in WiNCS may be in dynamic motion; they collect information from areas of interest. However, they may be affected by external environment, such as temperature, humidity, and topography, or linkage status inside the system, so system uncertainty may happen in a stochastic way, which increases the complexity of the system.

We make the following assumptions for system model (1).

Assumption 2. The perturbation parameter of the system satisfies Δ𝐴=𝐺𝐷(𝑘)𝐻. (3)
Respectively, 𝐺 and 𝐻 are known constant matrix, 𝐷(𝑘) is time-delay uncertain matrix, yet Lebesgue-measurable, and 𝐷^𝑘(𝑡)𝐷(𝑘)≤𝐼.

Assumption 3. 𝐶^𝑇𝐶 is full rank.

Assumption 4. All the states of the system can be measured and no missing measurement occurs.

We construct the following sliding mode observer: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑^𝑁_𝑖=1𝑥()(𝑘)𝑦(𝑘)−𝑦(𝑘(𝑘),𝑦(𝑘)𝑥(𝑘),𝑘−𝑖+𝐵𝑢+𝐿)+𝑤=𝐶 (4) where 𝐿 is the gain of observer to be designed and 𝑤(𝑘) is nonlinear item in observer.

So the state error 𝑒_𝑥(𝑘) and output error 𝑒_𝑦(𝑘) of the system are as follows: 𝑒_𝑥(𝑘)(𝑘)−𝑥(𝑘),𝑒=𝑥_𝑦(𝑘)=𝑦(𝑘)−𝑦(𝑘). (5)

In this case, the system error model is 𝑒_𝑥(𝑘+1)=(𝐴−𝐾𝐶)𝑒_𝑥(𝑘)+𝛼_𝑘(𝑘)(𝑘)−𝛼Δ𝐴𝑥+𝛼Δ𝐴𝑥+𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥()(𝑘),𝑒𝑘−𝑖−𝑤_𝑦(𝑘+1)=𝐶𝑒_𝑥(𝑘). (6)

We define the sliding surface 𝑠(𝑘) as 𝑠(𝑘)=𝑒_𝑦(𝑘). (7)

Our target in this paper is to find the gain of observer 𝐿 and the nonlinear input 𝑤(𝑘) that can drive system state onto the sliding surface within finite time and, in addition, make sure that sliding motion of system states on sliding surface is stable. Two conditions should be satisfied when designing sliding mode observer [37]:(1)system error model is asymptotically stable when 𝑠(𝑘+1)=𝑠(𝑘)=0;(2)sliding mode manifold satisfies ‖𝑠(𝑘+1)‖<‖𝑠(𝑘)‖.

Remark 5. We set system output error as sliding surface; the benefit is, condition (1) guarantees that sliding motion on sliding surface is stable, and condition (2) ensures that system state can be driven onto sliding surface within finite time. In this case, we convert state estimation problem into sliding mode observer design problem.

Besides, some useful and important lemmas that will be used in deriving out results are introduced below.

Lemma 6. Let 𝑌=𝑌^𝑇, 𝐷, 𝐸, and 𝐹(𝑡) be real matrix of proper dimensions, and 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼; then inequality 𝑌+𝐷𝐹𝐸+(𝐷𝐹𝐸)^𝑇<0 holds if there exists a constant 𝜀, which makes the following equation hold: 𝑌+𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸<0. (8)

Lemma 7 (Schur complement). Given a symmetric matrix 𝑆=^𝑆11𝑆12_𝑆21𝑆22, where 𝑆₁₁ is 𝑟×𝑟 dimensional, the following three conditions are equivalent:(1)𝑆<0;(2)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(3)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Lemma 8. For any 𝑥,𝑦∈𝑅^𝑛, 𝜇>0, the following equation holds: 2𝑥^𝑇𝑦≤𝜇𝑥^𝑇1𝑥+𝜇𝑦^𝑇𝑦. (9)

3. Main Results

In this section, two theorems will be given. The first one ensures that system states can reach the sliding surface within finite time.

Theorem 9. For system error model (6) which meets Assumptions 2, 3, and 4, and 𝑤(𝑘)=𝑤₁(𝑘)+𝑤₂(𝑘), where 𝑤₁(𝑘)=𝐴_𝑑∑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖), 𝑤₂(𝑘)=𝐺𝐻𝑥(𝑘), system states will be driven onto the sliding surface within finite time if there exists a general matrix 𝐿∈𝑅^{𝑛𝑥×𝑛𝑦} making (10) hold: −𝐶^𝑇𝐶√3+𝛼(𝐴−𝐿𝐶)^𝑇𝐶^𝑇∗−𝐼<0. (10)

Proof. Considering 𝑠(𝑘+1)=𝑒_𝑦(𝑘+1)=𝐶𝑒_𝑥(𝑘+1) and ‖𝑠(𝑘+1)‖<‖𝑠(𝑘)‖, we have 𝔼𝑠^𝑇(𝑘+1)𝑠(𝑘+1)−𝑠^𝑇(𝑘)𝑠(𝑘𝑒)=𝔼^𝑇_𝑦(𝑘+1)𝑒_𝑦(𝑘+1)−𝑒^𝑇_𝑦(𝑘)𝑒_𝑦(𝑘)=𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)−𝐶^𝑇𝐶𝑒_𝑥(𝑘)+2𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶Δ𝐴𝑥(𝑘)+2𝑒^𝑇_𝑥(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶𝑤(𝑘)+2^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇𝐶𝑤(𝑘)+2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶𝑤(𝑘)+𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇(𝑘)+𝐶Δ𝐴𝑥_𝑁𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇𝐶^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+𝑤^𝑇(𝑘)𝐶^𝑇𝐶𝑤(𝑘). (11) According to Lemma 8, we have 2𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶Δ𝐴𝑥(𝑘)≤𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶Δ𝐴𝑥(𝑘),2𝑒_𝑥(𝑘)^𝑇(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘)(𝑘−𝑖)≤𝑒_𝑥(𝑘)^𝑇(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+^𝑁_𝑖=1𝑒_𝑥(𝑘)^𝑇(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘)(𝑘−𝑖),2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)≤𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶Δ𝐴𝑥(𝑘)+𝛼^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖),2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶𝑤(𝑘)≤𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+𝑤^𝑇(𝑘)𝐶^𝑇𝐶𝑤(𝑘),2^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇(𝑘)≤𝐶𝑤_𝑁𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+𝑤^𝑇(𝑘)𝐶^𝑇𝐶𝑤(𝑘),2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶𝑤(𝑘)≤𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝐶^𝑇𝐶Δ𝐴𝑥(𝑘)+𝛼𝑤^𝑇(𝑘)𝐶^𝑇𝐶𝑤(𝑘). (12)(13)(14)(15)(16)(17)
Substituting (12) to (17) into (11), we have 𝔼𝑠^𝑇(𝑘+1)𝑠(𝑘+1)−𝑠^𝑇(𝑘)𝑠(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)−𝐶^𝑇𝐶𝑒_𝑥(𝑘)+𝑥^𝑇(𝑘)(3+𝛼)Δ𝐴^𝑇𝐶^𝑇𝐶Δ𝐴_𝑁𝑖=1𝑥(𝑘)𝑒^𝑇_𝑥(𝑘−𝑖)(3+𝛼)𝐴^𝑇_𝑑𝐶^𝑇𝐶𝐴^𝑇_𝑑×^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+(3+𝛼)𝑤^𝑇(𝑘)𝐶^𝑇𝐶𝑤(𝑘)=𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)−𝐶^𝑇𝐶𝑒_𝑥(𝑘)+Δ𝐴𝑥(𝑘)−𝑤₂(𝑘)^𝑇(3+𝛼)𝐶^𝑇𝐶Δ𝐴𝑥(𝑘)−𝑤₂(𝑘+𝐴_{𝑑𝑁𝑖=1})𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)^𝑇×(3+𝛼)𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇×𝐶^𝑇𝐶(𝐴−𝐿𝐶)−𝐶^𝑇𝐶𝑒_𝑥(𝑘)+𝐺𝐻𝑥(𝑘)−𝑤₂(𝑘)^𝑇(3+𝛼)𝐶^𝑇𝐶𝐺𝐻𝑥(𝑘)−𝑤₂(𝑘+𝐴_{𝑑𝑁𝑖=1})𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)^𝑇×(3+𝛼)𝐶^𝑇𝐶𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘). (18)
We make 𝑤₁(𝑘)=𝐴_𝑑∑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖), 𝑤₂(𝑘)=𝐺𝐻𝑥(𝑘), so we have 𝔼𝑠^𝑇(𝑘+1)𝑠(𝑘+1)−𝑠^𝑇(𝑘)𝑠(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝐶^𝑇𝐶(𝐴−𝐿𝐶)−𝐶^𝑇𝐶𝑒_𝑥(𝑘)≤0. (19)
By Schur complement, inequality (19) is equivalent to −𝐶^𝑇𝐶√3+𝛼(𝐴−𝐿𝐶)^𝑇𝐶^𝑇∗−𝐼<0. (20) The proof of Theorem 9 is complete.

Next, we will prove that sliding motion on sliding surface is stable.

Theorem 10. For system error model (6) which meets Assumptions 2, 3, and 4, and 𝑤(𝑘)=𝑤₁(𝑘)+𝑤₂(𝑘), where 𝑤₁(𝑘)=𝐴_𝑑∑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖), 𝑤₂(𝑘)=𝐺𝐻𝑥(𝑘), sliding motion of system states on sliding surface will be stable if there exists a general matrix 𝐿∈𝑅^{𝑛𝑥×𝑛𝑦} and 𝑃∈𝑅^{𝑛𝑥×𝑛𝑥} making (21) hold: −𝑃√3+𝛼(𝐴−𝐿𝐶)^𝑇𝑃∗−𝑃<0. (21)

Proof. We construct the following Lyapunov-Krasovskii functional: 𝑉(𝑘)=𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘). (22)
Making difference of 𝑉(𝑘), we have 𝔼𝑒^𝑇_𝑥(𝑘+1)𝑃𝑒_𝑥(𝑘+1)−𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)=𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+2𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃Δ𝐴𝑥(𝑘)+2𝑒^𝑇_𝑥(𝐴−𝐿𝐶)^𝑇𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃𝑤(𝑘)+2^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝑃𝑤(𝑘)+2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃𝑤(𝑘)+𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇(𝑘)+𝑃Δ𝐴𝑥_𝑁𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝑃^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+𝑤^𝑇(𝑘)𝑃𝑤(𝑘). (23)
According to Lemma 8, we have 2𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃Δ𝐴𝑥(𝑘)≤𝛼𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃Δ𝐴𝑥(𝑘),2𝑒_𝑥(𝑘)^𝑇(𝐴−𝐿𝐶)^𝑇𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘)(𝑘−𝑖)≤𝑒_𝑥(𝑘)^𝑇(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖),2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)≤𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃Δ𝐴𝑥(𝑘)+𝛼^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖),2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃𝑤(𝑘)≤𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+𝑤^𝑇(𝑘)𝑃𝑤(𝑘),2^𝑁_𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑(𝑘)≤𝑃𝑤_𝑁𝑖=1𝑒^𝑇_𝑥(𝑘−𝑖)𝐴^𝑇_𝑑𝑃𝐴_𝑑×^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+𝑤^𝑇(𝑘)𝑃𝑤(𝑘),2𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃𝑤(𝑘)≤𝛼𝑥^𝑇(𝑘)Δ𝐴^𝑇𝑃Δ𝐴𝑥(𝑘)+𝛼𝑤^𝑇(𝑘)𝑃𝑤(𝑘). (24)(25)(26)(27)(28)(29)
Substituting (24) to (29) into (23), we have 𝔼𝑒^𝑇_𝑥(𝑘+1)𝑃𝑒_𝑥(𝑘+1)−𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+𝑥^𝑇(𝑘)(3+𝛼)Δ𝐴^𝑇𝑃Δ𝐴_𝑁𝑖=1𝑥(𝑘)𝑒^𝑇_𝑥(𝑘−𝑖)(3+𝛼)𝐴^𝑇_𝑑𝑃𝐴^𝑇_𝑑×^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)+(3+𝛼)𝑤^𝑇(𝑘)𝑃𝑤(𝑘)=𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+Δ𝐴𝑥(𝑘)−𝑤₂(𝑘)^𝑇(3+𝛼)𝑃Δ𝐴𝑥(𝑘)−𝑤₂(𝑘+𝐴_{𝑑𝑁𝑖=1})𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)^𝑇×(3+𝛼)𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+𝐺𝐻𝑥(𝑘)−𝑤₂(𝑘)^𝑇(3+𝛼)𝑃𝐺𝐻𝑥(𝑘)−𝑤₂(𝑘+𝐴_{𝑑𝑁𝑖=1})𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘)^𝑇×(3+𝛼)𝑃𝐴_𝑑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖)−𝑤₁(𝑘). (30)
We make 𝑤₁(𝑘)=𝐴_𝑑∑^𝑁_𝑖=1𝑒_𝑥(𝑘−𝑖),𝑤₂(𝑘)=𝐺𝐻𝑥(𝑘), so we have 𝔼𝑒^𝑇_𝑥(𝑘+1)𝑃𝑒_𝑥(𝑘+1)−𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)≤𝑒^𝑇_𝑥(𝑘)(3+𝛼)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)≤0. (31)
By Schur complement again, inequality (31) is equivalent to −𝑃√3+𝛼(𝐴−𝐿𝐶)^𝑇𝑃∗−𝑃<0. (32) The proof of Theorem 10 is complete.

4. Numerical Simulations

In this section, a numerical simulation is given for testing the theorems developed in this paper. Consider the system model (1), where ,𝐴𝐴=6.180−35.641.0053.152.9703.3_𝑑=,1,,,𝐷(𝑘)=()()0(),𝑢(𝑘)()−3.380.150−0.2850.930.48−0.5551.172.235𝐶=2.88001.10480.01501.30950.76800.01950.09300.03151.0800,𝐵=1.20.7𝐺=0.10.2−0.100.23−0.760.870.5−0.32𝐻=0.1840.1120.230.097−0.16−0.156−0.277−0.069−0.1520.8sin0.7𝑘0000.8sin0.7𝑘000.8sin0.7𝑘=sin0.3𝑘,𝛼=0.15. (33)

The initial states are 𝑥(𝑘)=[1.30.427−0.92]^𝑇; according to Theorems 9 and 10, we have 𝐿=,6.3100−8.9663−2.70353.8252−4.31482.94142.8593−4.24003.0924𝑃=10⁵×.1.39870001.39870001.3987 (34)

Simulation results are shown in Figure 1; red line denotes real states value and blue line denotes observed value. From the figure, we can see that the estimated value tracks the real value very well, which can fully demonstrate the effectiveness of the proposed method. States error is shown in Figure 2, which is much smaller than observer design method proposed in [38], shown in Figure 3; this is because slide mode control is insensitive to uncertainty and external disturbance. However, we can see from Figure 2 that when system states are at turning point, states error becomes larger; this is because chattering is inevitable in slide mode control, which is our future work.

Figure 1 

Trajectories of system states and observed value.

Figure 2 

Trajectories of system states error.

Figure 3 

Trajectories of system states error by applying method proposed in [38].

5. Conclusion

This paper considers state estimation problem in wireless network control system with stochastic uncertainty and time delay. System uncertainty is assumed to occur in a stochastic way, which increases system complexity, and time delay is also considered. A sliding mode observer is designed in two steps; by constructing proper Lyapunov-Krasovskii functional, sufficient condition is acquired via linear matrix inequality. Finally, simulation result is employed to show the effectiveness of the proposed method.

Conflict of Interests

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

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant no. 61104109), the Natural Science Foundation of Jiangsu Province of China (Grant no. BK2011703), the Support of Science and Technology and Independent Innovation Foundation of Jiangsu Province of China (Grant no. BE2012178), and the Doctoral Fund of Ministry of Education of China (Grant no. 20113219110027).