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Stabilization for Wireless Sensor/Actuator Networks with Reordering-Based Sample Rate Adaptation

A novel sample rate adaptation technique is proposed for the wireless sensor/actuator networks with packet reordering. Considering network-induced delay and packet loss, a new technique for adapting the sampling interval to network parameters is studied in terms of the Reorder Density, such that a closed-loop switched delay system is presented by constructing state feedback controller. Furthermore, a criterion for stochastic stability of WSANs is derived based on the stochastic theory due to the subsystems subject to Markov chain. An adaptive controller is designed by solving linear matrix inequalities (LMI). An example is given to show the effectiveness of the proposed method.


Introduction
Wireless sensor networks (WSNs), wireless communication tools equipped sensor nodes, have been applied in the industry field increasingly. Sensors are emplaced to detect the physical world in the spatially distributed control systems, and actuators, carrying out appropriate actuation according to sensors measurements, are also introduced into the WSNs, which forms up the wireless sensor/actuator networks (WSANs) [1,2] . In the WSANs, the controllers are introduced to make up the closed-loop control systems, they can be also called wireless networked control systems and have potential use in the practical industry control fields. The wireless transmissions are subject to obstacles, reflections, multi-path effects, and fading, all of which affects the quality of transmissions [3] , and conspires to long or unbounded delay, severe packet reordering and high packet loss rate, such that packets are received much more unreliable than in wired media. So, WSANs provide many new challenging control problems. Then, it is the most urgent problem to how to analyze and control the WSANs by the approaches that are sensitive to the wireless networks, such that the WSANs can be applied in the practical application preferably.
Sample rate adaptation is a form of improving the performance of networks and control systems. The sampling policy determines the choice of sampling interval, and the appropriate sampling interval should be deduced based on the appended Quality of Service of the networks (QoS), as well as knowledge about the control Quality-of-Performance (QoP). Up to date, the sampling interval is generally regulated based on the delay and loss rate. The early attempts have been made [4][5][6]. An LMI 'girding' approach was proposed in [7], wherein the controller was designed so that the system is robust to variations in sampling period.
Traffic trends over the Internet indicate a significant increase of packet reordering, out-order packet. In addition, ad hoc routing and heterogeneous hand-off technologies in wireless domain introduce additional parallelism. All these point to an increasing trend of packet reordering [8] . However, noted that packet reordering is not analyzed in the above literature. To the best of the authors' knowledge, packet reordering and controller design of the WSANs have not been fully investigated to date. Especially for sample rate adaptation of the WSANs in terms of packet reordering, no results have been available in the literature so far, which motivates the present study.
In this paper, reordering-based sample rate adaptation technique is investigated for the WSANs with packet reordering. Considering network-induced delay and packet loss, a sampling interval variation strategy based on Reorder Density (RD) in the WSANs is put forward, which is a contribution in this paper. A sufficient condition for the stochastic stability and an adaptive controller design method of the WSANs are derived by the approach combining Markov theory and LMI. The example and simulations demonstrate the effectiveness of the proposed method.

Reorder Density
Several metrics have been proposed for packet reordering, such as Percentage of Reordered Packets, Reorder Density (RD), Reorder Buffer-occupancy Density (RBD), the Reordering Extent and N-Reordering [9] . These metrics face multiple challenges that include spatial and time complexities, real-time evaluation, robustness, etc. [8] . RD can capture both the amount and extent of reordering of packets in an arrival sequence Since RD uses a discrete density that represents the fractions of the sequence size with respect to packet displacements [9] . As pointed by in [8], RD fares better than the other metrics in all categories. Here, RD is chosen to evaluate the out-order packets. Considering the following system given by R are the state vector and control input vector, respectively; A and B are some constant matrices of appropriate dimensions. The state feedback controller can be expressed as where K is some constant matrix of appropriate dimensions. Since system (1) is controlled over a wireless network, the signal transmitting delay, packet loss and packet reordering will exist inevitably.
In this paper, a sensor samples a plant and transmits the sampled information to a remote controller at a particular sampling frequency, that is, the sensor is time-driven. The actuator receives the control action, and waits to apply the output at the next sampling instant, that is, the actuator is also time-driven, and the controller is event-driven, that is, the controller compute control action whenever it receives the signal from the sensor. T denotes the sampling period (sampling interval) of the sensor, and we assume that the network-induce delay is bounded, that is 0 k k k sc ca h T , and the maximum value of consecutive packet loss is 2 h ( 1 h and 2 h are positive integers, and k sc τ and k ca τ denote sensor to controller delay and controller to actuator delay, respectively). Set 1 2 h h h = + . Remark 1. Bounded network-induced delay and the maximum value of consecutive packet loss in the wireless networks can be guaranteed by adapting sampling interval to the changes of RD.
At each sampling instant, the current states are known by, e.g., the time-stamping technique. By the time-driven actuator, the newest control input can be executed by actuator. There are 1 h + cases, control action ( ) k u acts on the controlled process at sampling instant k t , or ( 1) k − u acts on the controlled process at sampling instant k t , by analogy, or ( ) k h − u acts on the controlled process at sampling instant k t . Recent studies of packet reordering based on measurements over the Internet validate reordering is an increasingly common phenomenon [9] . It is important to address the packet reordering proactively, especially for the WSANs. Without loss of generality, we consider a sequence of packets k h − x ,  (1, 2, , 1) h + is assigned to each non-duplicate packet as it arrives at the point of measurement, which we refer to as the destination (actuator). For an arrival sequence in which no loss or duplication of packets occur over the wireless networks, in the absence of reordering, the expected sequence number of the packet and the receive_index value are the same for each packet. If the receive_index assigned to packet with expected arrival sequence number m ( 1, 2, , )  [10] . For the presence of packet loss, one can wait till the arrival process is completed to assign receive_index values, so that receive_index values can be assigned correctly [8] . RD is defined as the histogram of k m d values, normalized with respect to the total number of packets, which has been adjusted for loss and duplication of packets. Table I illustrates  the expected arrival sequence numbers, assigned receive_index values, displacements and RD for a  sequence   4  3  2 x x transmitted over the network. Noted that packet 3 k − x has dropout and k x has no arrived at k t instant in the wireless communication. Remark 2. A packet whose sequence number has existed in early-arrival buffer is classified as a duplicate one, no receive_index value is assigned to it when it arrives the actuator. At sampling instant k t , the receive_index of k x is not assigned if it is not received at the actuator, which will not effect the measures of the packet reordering.

Sample Rate Adaptation
Sample rate is varied according to either control performance, network performance or a combination of the two. In general, poor network conditions are usually implied by long delay, high dropout rate and high degree of reordering. Observed from the experiments that the frequency of higher displacement increases, indicating higher degree of reordering, with increase in mean delay and packet loss rate, which means that RD may be used for network diagnosis purpose as well [11] . In other words, if  decreases, the sampling interval is increased to reduce the number of packets in the transmitting, such that QOS in the wireless networks is improved.
In this policy, the WASNs can be modeled as the discrete-time switched systems.

Modelling the Markov Jumping Systems
Due to the sample rate adaptation applied in the WSANs, the bounded network-induced delay and the maximum value of consecutive packet loss can be guaranteed. We have . It is well known it is possible to the newest signals received by nodes (actuator) subject to some probability distribution [12] , therefore, we may just as well assume that the newest signals transmitted over the wireless channel subject to Markovian chain, and define transition probability , we obtain the following augmented system

Stability Analysis and Controller Design
In this section, we will present the sufficient condition for the stochastic stability of the systems and the method of controller design.
Lemma 1 [13] For matrices A , > 0 ) If (7) is satisfied, the stochastic stability is obtained. This completes the proof. Note that Theorem 1 cannot be directly used to obtain the control law of the system (6). Therefore, we should design feedback gain i K , which makes system (6) stochastically stable corresponding to the dynastic changes of the wireless network.
Theorem 2 For a given scalar h , if there exist matrices Proof. By Lemma 1, (9) is true if there exists a matrix 0 i Y such that the following matrix inequality

A Numerical Example
To illustrate the effectiveness of the main results, a numerical example is presented. Here, the single WASN contained the following unstable plant A comparison of the network-induced delay in Fig. 1(a), under constant sampling interval T = 50ms and adaptive sampling, shows the significant reduction in delay with sample rate adaptation.
Since the upper bound of the network-induced delay is 1   Under the adaptive sampling policy (10) and the network-induced delay shown in Fig. 1(a), the subsystems are switched in the approach in Fig. 1(b). We assume that the subsystems of the WSAN subject to the Markovian process, whose state transition matrix is given in the following 11 12 Curves of state response of the WSAN.