Adaptive sampling rate control for networked systems based on statistical characteristics of packet disordering.

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Adaptive sampling rate control for networked systems based on statistical characteristics of packet disordering Jin-Na Li a,b,c,n, Meng-Joo Er d, Yen-Kheng Tan c, Hai-Bin Yu b, Peng Zeng b a

Lab of Operation and Control, Shenyang University of Chemical Technology, Liaoning 110142, PR China1 Key Laboratory of Networked Control Systems, Shenyang Institute of Automation, Chinese Academy of Sciences, Liaoning 110016, PR China1 c Energy Research Institute @ Nanyang Technological University, 637553, Singapore d Division of Control and Instrumentation, School of Electrical and Electronic Engineering, College of Engineering, Nanyang Technological University, 639798, Singapore b

art ic l e i nf o

a b s t r a c t

Article history: Received 24 July 2014 Received in revised form 15 November 2014 Accepted 18 April 2015 This paper was recommended for publication by Dr. Q.-G. Wang.

This paper investigates an adaptive sampling rate control scheme for networked control systems (NCSs) subject to packet disordering. The main objectives of the proposed scheme are (a) to avoid heavy packet disordering existing in communication networks and (b) to stabilize NCSs with packet disordering, transmission delay and packet loss. First, a novel sampling rate control algorithm based on statistical characteristics of disordering entropy is proposed; secondly, an augmented closed-loop NCS that consists of a plant, a sampler and a state-feedback controller is transformed into an uncertain and stochastic system, which facilitates the controller design. Then, a sufficient condition for stochastic stability in terms of Linear Matrix Inequalities (LMIs) is given. Moreover, an adaptive tracking controller is designed such that the sampling period tracks a desired sampling period, which represents a significant contribution. Finally, experimental results are given to illustrate the effectiveness and advantages of the proposed scheme. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Networked control systems (NCSs) Sampling rate Packet disordering Stochastic stability

1. Introduction 1.1. Motivation Packet disordering, meaning that a packet sent earlier may arrive at the destination node after some packets sent later or vice versa, is an increasingly common phenomenon in communication networks. Packet disordering is introduced into networked control systems (NCSs) since information is exchanged or transmitted via network media in NCSs [1,2]. More importantly, it can impact end-to-end application performances significantly, irrespective of its causes. In User Datagram Protocol (UDP)-based applications that are highly sensitive to delay, e.g., IP telephony, an out-of-order packet (a packet that arrives late at the destination node) that arrives after the elapse of playback time is treated as lost thereby decreasing the perceived quality of voice [3]. For delay sensitive NCS applications wherein UDP is chosen usually and information is exchanged or

n Corresponding author at: Present address: Energy Research Institute @ Nanyang Technological University, 637553, Singapore. Tel: þ 86 13664104881, þ 866582441826. E-mail addresses: [email protected], [email protected] (J.-N. Li), [email protected] (H.-B. Yu). 1 Permanent address of J.-N. Li.

transmitted via network media, out-of-order packets inevitably existing in communication are treated as lost thereby degrading the reliability of communication networks, which might have adverse impact on control performances of NCSs. However, we know that wireless-based industrial applications, such as packaging, manufacturing, wood machining or plastic extrusion, have a strict requirement of reliability. For instance, normal channel load, Radio Frequency (RF) conditions and recommended Quality of Service (QoS) of network, the expected application-level packet loss is very small (less than 1 in 100,000) [4]. Hence, we advocate that packet disordering should be explicitly taken into account, and an alternative solution should be explored for NCSs to reduce such effect and to improve system performances. In this context, we propose an adaptive sampling rate control scheme for NCSs with packet disordering. So far, the majority of NCSs research efforts have focused on controller design to provide sufficient stability conditions in the presence of packet disordering [5–10]. Significant works have also been reported on how to describe packet disordering, such as comparing the sampling instants of received signals [5], identifying packet disordering by displacements of packets [6,7], and comparing transmission delay [8]. Based on those, a so-called active compensation scheme, wherein late packets are discarded, has been used in [6–10]. In the sense of stabilizing the NCSs, this type of methodology is effective.

http://dx.doi.org/10.1016/j.isatra.2015.04.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Li J-N, et al. Adaptive sampling rate control for networked systems based on statistical characteristics of packet disordering. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.005i

J.-N. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

However, in the above-mentioned literature, there still exist two underlining shortcomings, which can be briefly summarized in the following. One is that active control techniques for avoiding heavy packet disordering have not been investigated. The measurement studies done by [3,11–14] showed the impact of packet disordering on end-to-end application performances and predicted the increasing trend of out-of-order deliveries. It should be pointed out that heavy packet disordering would bring challenges for the methods in [6–10] wherein the degree of packet disordering is ignored. The other is that statistical characteristics shown by packet disordering have not been analyzed in the aforementioned literature so far, such that the existing methods are either highly inefficient or not optimal in real networked control system applications. These motivate the present study.

E (k ) RD [i] i

D , D 1, ,D

d (h i

0,1,

1

i) ,h

Augmented system

T* Plant (Actuator)

uk

xk

Tk Sensor

uk

k

PDC k ca

Network k sc

K 1.2. Control strategy To overcome these shortcomings, we fully analyze the statistical characteristics of packet disordering and investigate an adaptive sampling rate control method. Moreover, we also show that adjusting sampling rate based on statistical characteristics of packet disordering is an alternative solution to reduce packet disordering. To illustrate the possible existence of a correlation between packet disordering and sampling rate, [2] a measurement setup and UDP flows across an IP backbone network were generated. And the experiments in [2] have shown that delivering data at the higher rates is more prone to packet disordering due to the small sending time interval though there are many other reasons which cause packet disordering. It seems likely that parallelism at the link layer will permit some packets to overtake those sent earlier but queued on a parallel link, causing this behavior [2]. Hence the sampling rate should be appropriately reduced if heavy packet disordering occurred within network communications, otherwise much more packets may arrive out of order leading to bad network performances. Thus, there is an obvious conflict between the two demands of high sampling rate and light packet disordering, while ensuring stability and desired control performances of NCSs. Our strategy is to design an adaptive controller in order to enable the sampling period to tend to a desired sampling period under the control actions. The so-called desired sampling period means a sampling interval under which networks can be under a good and steady condition at operation (light packet disordering) and the desired control performance can be guaranteed as well. The suggested scheme tries to improve QoS of networks, such that control performances of NCSs are improved. Hence, the proposed sampling rate control method is a tradeoff between QoS of networks and control performances of systems. Moreover, the statistical characteristics of packet disordering are used for stability analysis and control of NCSs. The established technology route for NCSs is depicted in Fig. 1. Different from the conventional NCSs, a Packet Disordering Calculator (PDC) is added to the closed-loop systems. PDC is responsible for computing displacement values of packets, disordering density and disordering entropy. As the sensor takes measurements of the plant state at each tk time instant (tk denotes the kth ðk ¼ 0; 1; …Þ sampling time instant), the information about the sampling period is additionally appended. Once the packet arrives at the controller node, the data are used to calculate control action (state feedback). After that, the achieved control action is sent to the PDC via the network. The actuator selects the newest control signal and then uses it to control the plant, as well as act on the sampling rate together with disordering entropy.

E R (k )

yk

Controller system

Sampling rate Plant state

Fig. 1. Architecture of NCS with the sampling rate adaptive control scheme.

1.3. Main contributions The main contributions of this work are threefold. First, a sampling rate algorithm based on computed disordering entropy is proposed, which extends the existing results [5–10] wherein the stability analysis and synthesis of NCSs with packet disordering are investigated under the condition of the constant sampling rate. Second, an adaptive tracking controller used to cooperatively control the sampling rate and the plant is designed, such that the sampling period tracks the desired sampling period and the system can be stabilized. To our knowledge, designing a controller to drive the sampling rate to a desired value has not been investigated up to now. In addition, Bernoulli statistical characteristic shown by disordering entropy is utilized during constructing the augmented model of systems and investigating the stability of NCSs, which is the third contribution of this paper. Experimental results are given to demonstrate the effectiveness of the proposed approach. This paper is organized as follows. The problem statement is presented in Section 2. Section 3 is dedicated to stability analysis and controller design of NCSs. Section 4 presents the adaptive tracking controller design. Experimental verification is performed in Section 5. Conclusions are stated in Section 6. Notation: We use “n” to represent a term that is induced by symmetry, diagð⋯Þ stands for a block-diagonal matrix, and LnðÞ stands for log e ðÞ. ‖A‖2 for a matrix A denotes the 2-norm, which is the maximum singular value of A. ProbðÞ stands for the probability operator, and EðÞ stands for the mathematical expectation operator.

2. Problem statement A state equation of system under zero input is first considered _ ¼ AxðtÞ xðtÞ

ð1Þ

where xðtÞ is the state vector, A is some constant matrix of appropriate dimensions. Since a digital controller and zero-order hold transformation are usually applied in typical control systems communicating over networks, it is quite natural to analyze NCSs from the discrete-time point of view [15,16]. Then, system (1) is discretized as xk þ 1 ¼ eAT k xk

ð2Þ



where xk ¼ xðt k Þ, xk þ 1 ¼ xðt k þ 1 Þ, T k ¼ t k þ 1 t k and Tk is the sampling period. We assume T min r T k r T max , where Tmax and Tmin are the maximum and minimum of sampling period, respectively. Since there exists the intrinsic and internal relation between packet disordering and sampling rate, a sampling period control algorithm is given below T k þ 1 ¼ T k þ ck ER ðkÞT k

ð3Þ

where ck is a step-size. ER(k) is the disordering entropy at the sampling time instant tk, and it is defined as [14] ER ðkÞ ¼ ð 1Þ

iX ¼ DT

ðRDk ½i LnðRDk ½iÞÞ

ð4Þ

i ¼ DT

where RDk ½i is a discrete probability distribution of packet displacement i (i is a positive integer) at tk time instant, and DT is the upper bound of packet displacement values [14]. Proposition 1. Disordering entropy ER(k) satisfies 0 rER ðkÞ r lnð2DT þ 1Þ. Proof. Since RDk ½i denotes the discrete probability distribution of packet displacement i at tk time instant, then 0 r RDk ½i r 1. Thus we have lnðRDk ½iÞ r 0. By (4), ER ðkÞ Z 0 is easily obtained. When all of focused packets arrive at the PDC in order, i.e., RDk ½0 ¼ 1, the disordering entropy ER ðkÞ ¼ 0. On the other hand, when all of RDk ½i ði ¼ DT ; DT þ 1; …; DT Þ uniformly equal, the ER(k) will reach to the maximum value based on the definition of entropy. More specifically, when packets are displaced uniformly with equal probabilities 1=ð2DT þ1Þ, the smallest upper bound for entropy is obtained as follows: ER ðkÞ r

iX ¼ DT i ¼ DT

1 lnð2DT þ 1Þ 2DT þ 1

¼ lnð2DT þ 1Þ

ð5Þ

Remark 1. Disordering entropy is a metric of packet disordering, and it shows the degree of packet disordering [14]. From the proof of Proposition 1, one can see that the bigger the disordering entropy is, the heavier the packet disordering becomes. Specially, ER ðkÞ ¼ 0 when all of packets arrive in order. As shown in Fig. 1, the sampling rate and the plant are integrated as an augmented system, followed that a controller is designed to control this augmented system. Let ξk ¼ ½xTk T k T , the combination of (2) and (3) gives the following augmented system:

ξk þ 1 ¼ A k ξk yk ¼ C ξk where " Ak ¼

ð6Þ

eAT k

0

0

1 þ ck ER ðkÞ

# ;

C ¼ ½0; 1:

A control sequence uk is introduced into the system (6), which can be rewritten as

ξk þ 1 ¼ A k ξk þ Buk yk ¼ C ξk

ð7Þ

where B is some constant matrix of appropriate dimensions. Note that " # B1 Buk ¼ u B2 k Thus, the equivalent form of system (7) can be obtained xk þ 1 ¼ eAT k xk þ B1 uk T k þ 1 ¼ T k þ ck ER ðkÞT k þ B2 uk

yk ¼ T k

3

ð8Þ

It is worth pointing out that the function of control input uk includes both stabilizing the controlled plant and enabling the sampling period to track a desired sampling period. Remark 2. Up to the present, the research on sampling rate in NCSs may be generally divided into three categories, considering the uncertain or time-varying feature of sampling period [17,18], actively adjusting sampling rate based on the state transition probability or transmission delay [19–21] and recent eventdriven sampling policy [22,23]. In [22,23], the sampler is activated when the specific performance index cannot be satisfied. Compared with the above-mentioned literature, the most significant difference of the proposed method in this paper is that it utilizes the control actions to control sampling rate to reach the desired sampling period for NCSs. In addition, note that the uncertain step-size ck in the sampling scheme (3) might produce negative effect on control performances of NCSs, since the blindness and spontaneity of the increment size of sampling variation in (3) potentially exist. The similar problem can be also seen in [19–21]. Thus, the proposed strategy that the sampling rate is driven to the desired sampling rate under control actions is an alternative solution. In the following subsections, the statistical analysis of packet disordering, the uncertain system model and the stochastic system model are presented. 2.1. Statistical analysis of packet disordering In this subsection, the packet disordering is described first and then the statistical characteristic is analyzed. In this paper, the sensor and the actuator are driven synchronously at each tk time instant, and the controller is event-driven. The controlled plant state xk and sampling period Tk encapsulated in a packet ξk are sent to the controller via network media. In the considered NCSs, network-induced delays and packet losses exist in both channels from the sensor to the controller and from the controller to the actuator, simultaneously. Two natural assumptions are made as follows: 1. The network transmission delay is bounded, that is 0 r τk ¼ τksc þ τkca r m1 T min , where τksc and τkca denote the transmission delays from the sensor to the controller and from the controller to the actuator, respectively. m1 is some positive integer and T min ðT min 4 0Þ is the minimum of sampling period. 2. The number of packet losses from the sensor to the actuator is bounded. m2 (m2 is some positive integer) denotes the maximum number of consecutive packet losses. Here, if the control input arrives at the actuator late, it is discarded, such that the newest control input will be executed by the actuator. Based on the nodes’ (sensor, controller and actuator) driven modes, the control input uðtÞ is constant during a sampling interval ½t k ; t k þ 1 Þ. We know ξk m1 must arrive at the PDC if it is not lost before t k time instant (including t k time instant) due to 0 r τk ¼ τksc þ τkca rm1 T min and T min r T k r T max . Otherwise, τk m1 4T k 1 þ T k 2 þ ⋯ þT k m1 Z m1 T min . Thus a conflict is caused because of τk m1 r m1 T min . While, the packets ξk m1 þ 1 ; ξk m1 þ 2 ; …; ξk sent after packet ξk m1 may be absent due to transmission delay or packet loss, and they also may be present. Because at the most m2 packets might be lost, if the consecutive m2 packets including ξk m1 is lost during transmission, the packet ξk m1 m2 (namely ξk h ) will be used to calculate the control input uk . So, there are h þ 1 ðh ¼ m1 þ m2 Þ cases. The control input associated with ξk acts on the controlled process



4

during a sampling interval ½t k ; t k þ 1 Þ, or the control input associated with ξk 1 acts on the controlled process during a sampling interval ½t k ; t k þ 1 Þ, the rest may be deduced by analogy, or the control input associated with ξk h acts on the controlled process during a sampling interval ½t k ; t k þ 1 Þ. 2.1.1. Description of packet disordering To fully describe the phenomenon of packet disordering, a sequence of packets ξk h , ξk h þ 1 ; …; ξk transmitted over the network from the sensor is taken into account. For ξk h ; ξk h þ 1 ; …; ξk , it is well known that the corresponding expected arrival sequence numbers are 1; 2; …; h þ 1. Then, it is easily obtained that the expected arrival sequence number of packet ξk i is h þ 1 i ði ¼ 0; 1; …; hÞ. A receive_index lðl ¼ 1; 2; …; hþ 1Þ is assigned to each non-duplicate packet as it arrives at the point of measurement, which we refer to as PDC since the controller is event-driven. Rk ðiÞ and dk ðh þ 1 iÞ denote the receive_index and displacement value of packet ξk i , respectively. And dk ðh þ1 iÞ ¼ Rk ðiÞ ðh þ 1 iÞ. Without loss of generality, we assume that the packets not appeared or lost before the t k time instant (including the t k time instant) arrive at the PDC in order after the t k time instant. Moreover, if p (p r m2 and it is some positive integer) packets sent before the ξk i are dropped, then the receive_index value of ξk i is p more than its real value. For the packet ξk i arriving at the actuator before the t k time instant (including t k time instant), if dk ðh þ 1 iÞ a 0, then a ‘disordering event’ has occurred in communication. Packet ξk i is late if dk ðhþ 1 iÞ 4 0, early if dk ðh þ 1 iÞ o 0, and in order if dk ðhþ 1 iÞ ¼ 0 (see [5,6,9]). To guarantee the newest signals to be executed by the plant, the packets that arrive at the actuator late are discarded. Define ( 1 dk ðh þ1 iÞ r 0 δðdk ðh þ 1 iÞÞ ¼ ð9Þ 0 dk ðh þ1 iÞ 4 0 A state feedback controller is chosen, and based on the aforementioned analysis, we have uk ¼

h X

θk ðlÞK ξk l

ð10Þ

l¼0

where K is some constant matrix of appropriate dimensions. θk ðlÞ ¼ ∏lj ¼10 ð1 δðdk ðh þ 1 jÞÞÞδðdk ðh þ1 lÞÞ, ∏j¼10 ð1 δðdk ðh þ 1 jÞÞÞ ¼ 1, which can guarantee that the newest signals are executed. Then, the system (7) is transformed into the following:

ξk þ 1 ¼ A k ξk þ B yk ¼ C ξk

h X

θk ðlÞK ξk l

l¼0

ð11Þ

3. It is worth noting that θk ðlÞ ¼ 1 or 0, and θk ðlÞ ¼ 1. θk ðlÞ ¼ 1 or θk ðlÞ ¼ 0, which is determined in terms of the displacement values of packets, represents the signal ξk l is Remark Ph l¼0

executed or not executed by the plant, respectively (more details can be seen in [7]). To further understand packet disordering, an example shown in Fig. 2 is studied. Corresponding to Fig. 2, the receive-index values, displacement values, δ operator and disordering density for the transmitted sequence are listed in Table 1. In this example, DT ¼ 2 and packet ξk 2 is lost, thus p¼1. h is set to 4, which means that a group of 5ðh þ 1 ¼ 5Þ packets is used as the studying object. With respect to the packets ξk 4 ; ξk 3 and ξk 1 arriving at the actuator before the t k time instant, based on the displacement values computed in Table 1, packet ξk 4 is displaced by one unit from its position, packet ξk 3 is early by one position and packet ξk 1 is in order. Furthermore, θk ð0Þ ¼ 0; θk ð1Þ ¼ 1; θk ð2Þ ¼ 0; θk ð3Þ ¼ 0 and θk ð4Þ ¼ 0 can be easily calculated according to the formulation of

1 Packets

Expected arrival sequence numbers 2 3 4 k 3

k 4

k 2

5 k

k 1

k 1

Sensor

Actuator k 4

tk k

k 3

tk

1

k 1

Fig. 2. Illustration of packet disordering in NCS.

Table 1 An example in Fig. 2. Packets ξk i

ξk 4

ξk 3

Receive_index values Displacement values dk ð5 iÞ δðdk ð5 iÞÞ ði ¼ 4; 3; …; 0Þ Disordering density

2 1 1 1 0 1 RDk ½1 ¼ 2=5 RDk ½ 1 ¼ 1=5 RDk ½ 2 ¼ 0

ξk 2

ξk 1

ξk

5 2 0

4 6 0 1 1 0 RDk ½0 ¼ 1=5 RDk ½2 ¼ 1=5

θk ðlÞ. Thus, the selected newest signal ξk 1 is used to compute the control action. This means uk ¼ K ξk 1 . Moreover, disordering entropy ER ðkÞ ¼ 1:3322 can also be calculated in light of (4). Especially for the case of RDk ½0 ¼ 1, namely all packets arrive at the actuator in order, RDk ½i ¼ 0 ði ¼ 2; 1; 1; 2Þ are naturally obtained. Thus, ER ðkÞ ¼ 0, which indicates no packet disordering event has occurred in the network. 2.1.2. Statistics of disordering entropy In an actual network communication, disordering entropy is not constant usually, but displays some irregular behavior. To illustrate the statistical characteristics of actual packet disordering, an IP-based local network is simulated in OPNET software, wherein the packet loss is absent, then m2 ¼ 0. Fig. 3(a) demonstrates the measured time-varying transmission delay. Choosing m1 ¼ 2, we have h ¼ m1 þ m2 ¼ 2. Then, 3 ðh þ 1 ¼ 3Þ packets are grouped together, packet displacement values can be obtained and shown in Fig. 3(b). From Fig. 3(b), we know DT ¼ 2. Based on it, RDk ½i ði ¼ 2; 1; 0; 1; 2Þ is computed and the frequency histogram of disordering entropy is also given in Fig. 3(c). Further, the cumulative distribution function shown in Fig. 3(d) can be obtained. From Fig. 3(c), we know that disordering entropy ER ðkÞ mainly focuses on the range from 0.5 to 0.8, which indicates the non-uniform distribution of the disordering entropy. From Fig. 3(d), we know Probð0 r ER ðkÞ r 0:62Þ ¼ 0:65 and Probð0:62 rER ðkÞ r 1:6Þ ¼ 0:35 with 0:62 0 o 1:6 0:62, as well as Probð0 r ER ðkÞ r 0:75Þ ¼ 0:885 and Probð0:75 r ER ðkÞ r 1:6Þ ¼ 0:115 with 0:75 0 r 1:6 0:75, which shows that disordering entropy falls into lower interval with higher probability. Without loss of generality, we assume disordering entropy belongs to the interval ½0; Bd with higher probability than the interval ½Bd ; Lnð2DT þ 1ÞÞ, where Bd is a positive number. In general, Lnð2DT þ 1Þ Bd is far larger than Bd 0. To our knowledge, from the statistical characteristics of out-of-order packets point of view, no results have been available for analysis and control of NCSs. According to the aforementioned analysis, we have ProbðER ðkÞ A ½0; Bd ÞÞ ¼ β , ProbðER ðkÞ A ½Bd ; Lnð2DT þ 1ÞÞ ¼ 1 β , where β A ½0; 1; Bd is a bound


Displacement of packet

Transmission delay


0.2 0.1 0 0

10

20

30

40

50

60

70

2 1 0 −1

0

Time (second)

Cumulative probability

Frequency

30 20 10

0

0.5

0.8

50

100

150

200

packets

40

0

5

1

1.5

1 0.885 0.65 0.5

0

0

Disordering entropy

0.62 0.75

1.4

Disordering entropy Fig. 3. The communication network.

which means the probability of ER ðkÞ A ½0; Bd Þ equals to β. Set Ω1 ¼ fkjER ðkÞ A ½0; Bd Þg, Ω2 ¼ fkjER ðkÞ A ½Bd ; Lnð2DT þ 1Þg, and define a stochastic variable βk as ( 1; k A Ω1 βk ¼ ð12Þ 0; k A Ω2 It is clear that Ω1 [ Ω2 ¼ Z, Ω1 \ Ω2 ¼ ϕ, where Z and ϕ denote set of non-negative integer and empty set, respectively. Based on the above definition and combined with stochastic theory, disordering entropy associated with the stochastic variable βk is subject to Bernoulli distribution [24] with Probðβ k ¼ 1Þ≔β , Probðβ k ¼ 0Þ≔1 β . Since the fast sampling will easily result in heavy packet disordering, the proposed sampling policy tries to slow down sampling frequency when the disordering entropy value is larger than Bd , otherwise, we will speed up sampling frequency. Thus, in (3), we set ( c; k A Ω1 ck ¼ ð13Þ c; k A Ω2 Remark 4. By (3) and (13), we know when k A Ω1 , the sampling period is reduced by cER ðkÞT k . This means that we can speed up the sampling such that control performances of NCSs will be improved under the case of light packet disordering. Moreover, the reduction scale of T k is small as ER ðkÞ becomes small, which means that the sampling frequency is increased slowly when the network becomes better. A special case is when ER ðkÞ ¼ 0 the sampling period remains constant, namely T k þ 1 ¼ T k . This means that the sampling period remains unchanged when the network is in good status (all packets are in order). When k A Ω2 , we increase the sampling period by cER ðkÞT k so as to avoid serious packet disordering. Remark 5. It should be pointed out when k A Ω1 and the entropy ER ðkÞ is not equal to zero, the sampling frequency is always increased, which probably results in increased packet disordering. When k∈Ω2 ; ck jumps from c to c. Thus, the sampling frequency becomes slow, which attempts to reduce packet disordering. In summary, the proposed sampling policy in this paper attempts to

slow down the sampling frequency when the disordering entropy value is larger than β d . Otherwise, we will increase the sampling frequency except for ER ðkÞ ¼ 0. Frequent switching of ck might destabilize the NCSs, and when ER ðkÞ ¼ 0, the sampling period will probably be large resulting in the degradation of control performances of NCSs, so we add the control input into the sampling rate control algorithm (3) in order to drive the sampling rate to the desired sampling rate (see the second equation in (8)). The sampling rate control strategy proposed in this paper not only utilizes the statistical characteristics of packet disordering, but also eliminates the negative effect brought about by ck switching on the control performance of NCSs. Actually, this solution is inspired by [25] where data rate was modeled as a state-space model and it was updated based on congestion degree and control signal uk . Remark 6. Compared with the existing results [5–10], two apparent differences are given as follows. One is that the model of NCSs constructed in this paper embodies the statistical information of packet disordering. If the statistical characteristics of out-of-order packet are properly utilized, it is taken for granted that the less conservative results are expected to be achieved in the analysis and synthesis of NCSs. The other is that the sampling rate is controlled to avoid increasing packet disordering. Thus, the network performance can be improved on some level, such that the control performance associated with the network performance will be improved naturally. Remark 7. It is worth noting that our scheme is executed for NCSs with transmission delay and packet loss, in this sense, the suggested method also provides a general framework for analysis and synthesis of NCSs. Here, we assume that the disordering entropy defined in (4) is an Ergodic process, which means that its statistical properties can be deduced from a single and sufficiently long sample of the process. Thus, we will design the following algorithm to obtain β for a given Bd . Algorithm 1. Step 1. Choose positive integers kmax and k0 , and set k ¼ k0 ;



6

Step 2. Calculate the packet displacement value dk ðh þ 1 iÞ of packet ξk i ði ¼ 0; 1; …; hÞ, we obtain the disordering density RDk l ðl ¼ DT ; DT þ 1; …; DT Þ; Step 3. Calculate the disordering entropy ER ðkÞ in terms of (4); Step 4. Set k ¼ k þ1, if k r kmax , we return to Step 2, otherwise, continue to Step 5; Step 5. Estimate the cumulative distribution function of disordering entropy in terms of sequence ER ðk0 Þ; ER ðk0 þ 1Þ; …ER ðkmax Þ, we can thus calculate β for the given Bd . Exit. Remark 8. In general, k0 Z h, which is used to guarantee enough packets available for calculating disordering entropy. For example, for h ¼ 2, if we set k0 ¼ 1, the displacement values of the sequence ξ 1 ; ξ0 ; ξ1 will be needed to obtain ER ð1Þ. However, it is not possible to get ξ 1 . For the current sampling time instant t k , if t kmax ¼ t k , we perform an on-line measurement algorithm. Actually, the cumulative distribution function, shown in Fig. 3(d), is obtained using Algorithm 1 by an off-line approach. Here, the so-called off-line approach means that we measure packet disordering with disordering entropy over a long period of time before NCSs are closed. 2.2. Uncertain system modeling Note that the matrix A k varies with the time-varying sampling period T k , which brings out a great difficulty for stability analysis and controller design of NCSs. To address this, let GðT k Þ ¼ eAT k , we come up with the following solution. For ease of notation, eðÞ is expressed as expðÞ in this paper. Lemma 1. GðT k Þ can be expressed as follows: GðT k Þ ¼ GðT nom Þ þ Δðτk ÞAGðT nom Þ ð14Þ R τk where Δðτk Þ ¼ 0 expðArÞ dr; T nom is a constant to be chosen, T min r T nom rT max , and τk ¼ T k T nom . Proof. H and FðT k Þ are respectively defined as A 0 H¼ ; FðT k Þ ¼ expðHT k Þ 0 0 we have [26] expðAT k Þ FðT k Þ ¼ 0 further, GðT k Þ ¼ ½I 0FðT k Þ

0

ð15Þ

ð16Þ

I

I

ð17Þ

0

Using a saturation device in the actuator, the following relationship holds 0 r T min r T k rT max o1

8k

by the For a given T nom , T min T nom r τk r T max T nom . Note that FðT k Þ ¼ Fðτk þ T nom Þ ¼ Fðτk ÞFðT nom Þ ¼ FðT nom Þ þ ðFðτk Þ IÞFðT nom Þ Utilizing the following relationship [27]: Z τk Fðτk Þ I ¼ expðHrÞH dr 0 Z τk I ¼ expðArÞ dr 0 0 ½A 0

ð18Þ (18),

we

have

ð19Þ

Lemma 2 (Suh [28]). Let α1 be the maximum real part of the eigenvalues of A and α2 be the maximum real part of the eigenvalues of A. The Schur decomposition of A is given by U T AU ¼ D þ N where U is an orthogonal matrix, D is a diagonal matrix, and N is a strictly upper triangular matrix. The following is satisfied Z τ k expðArÞdr ð21Þ rβ 0

2

where

β¼

min |ffl{zffl}

maxfβðT min T k Þ; βðT max T k Þg:

T min r T k r T max

Moreover, T nom is the sample period corresponding to which β attains its minimum, i.e.,

β ¼ βðT nom Þ

ð22Þ

and

8 n 1 X > ð 1Þj expðα1 τk Þ > > > ‖N‖j2 j þ 1 þ > > α1 > α1 j¼0 > > > ! > > j i j > X ð 1Þi τk > > > if τk Z 0; α1 a 0 > > > αi ðj iÞ! > i¼0 1 > > > n1 > α2 > >j¼0 > ! > > j > X > ð 1Þi ∣τk ∣j i > > if τk r 0; α2 a 0 > > > αi2 ðj iÞ! > i¼0 > > > n1 > X ‖N‖j > > 2 > > ∣τk ∣j þ i otherwise > > : j ¼ 0ðj þ 1Þ!

ð23Þ

Then, by Lemmas 1 and 2, we have A k ¼ G~ 1 þ I~ Δðτk ÞG~ 2 þ C~ ER ðkÞ, where GðT nom Þ 0 I G~ 1 ¼ ; I~ ¼ ; ‖Δðτk Þ‖2 o β ; 0 1 0 " # 0 0 ~ ~ : G 2 ¼ ½AGðT nom Þ 0; C k ¼ 0 ck h i T T T Let ηTk ¼ ξk ξk 1 … ξk h , the state equation of system (11) is expressed as

ηk þ 1 ¼ ðG^ 1 þ I^ Δðτk ÞG^ 2 þ C^ k ER ðkÞ þ B^ Λðθk ð0Þ; …; θk ðhÞÞK Þηk

ð24Þ h i ~ ^ ~ ^ G2 ¼ G2 0 … 0 , where C k ¼ diagðC k ; 0; …; 0Þ, Λðθk ð0Þ; …; θk ðhÞÞ ¼ θk ð0ÞI θk ð1ÞI … θk ðhÞI , K ¼ diagðK; K; …; KÞ, and 2 3 2 3 2 3 B I~ G~ 1 0 ⋯ 0 0 6 7 6 7 607 607 6 I 7 0 ⋯ 0 0 6 7 ^ ^ 6 7 7 G^ 1 ¼ 6 7: 6 ⋮ ⋮ ⋱ ⋮ ⋮ 7; I ¼ 6 ⋮ 7; B ¼ 6 4⋮5 4 5 4 5 0

0

⋯

I

0

0

0

2.3. Stochastic system with uncertainty modeling

ð20Þ

By (17), (19) and (20), we can obtain (14). The proof is completed.□ Note that Δðτk Þ in (14) is time-varying, the following lemma is given to treat it as a norm bounded uncertainty.

Note that the uncertain parameters θk ðiÞ ði ¼ 1; 2; …; hÞ and ck also pose great challenges to stability analysis and controller design of NCSs. So, we utilize stochastic theory to construct NCS as a stochastic T system model with uncertainty. Define dk ¼ θk ð0Þ; θk ð1Þ; …; θk ðhÞ , P since θk ðlÞ ¼ 1 or 0ðl ¼ 0; 1; …; hÞ and hl¼ 0 θk ðlÞ ¼ 1, then, there are hþ 1 possible values for dk , obviously. Similar to our prior effort [7], for ease of notation, we define a vector-valued function f : dk -σ ðkÞ to



map the vector dk into a scalar number σ ðkÞ A I ¼ f1; 2; …rg, where r ¼ h þ 1. σ ðkÞ ¼ i denotes dk ¼ ½0; …; 1; ⋯; 0T , namely, θk ði 1Þ ¼ 1, which means that the control input uk ¼ K ξk ði 1Þ , and also denotes the No. i ði A IÞ subsystem of NCS (24). Here, we shall assume that the subsystems switched is subject to Markovian chain [9,29]. Moreover, transition probability associated with the switched subsystem is P defined as π ij ¼ Probðσ ðk þ 1Þ ¼ jjσ ðkÞ ¼ iÞ. Obviously, j A I π ij ¼ 1. To design an adaptive controller, the controller gain K i adapts to the No. i subsystem. Actually, the adaptive controller means the gain scheduling controller. Set M ¼ G^ 1 þ I^ Δðτk ÞG^ 2 , then the system (24) is rewritten as

7

and V 3k ¼ ER2 ðkÞηTk W i ER2 ðkÞηk . Due ER2 ðkÞ r Lnð2DT þ 1Þ, we have X EV k þ 1 V k r π ij zTk EððΥ i þ ðβk βÞΓ ÞT

to

ER1 ðkÞ r Bd

and

jAI

ðP j þ B2d Q j þ ðLnð2DT þ 1ÞÞ2 W j Þ ðΥ i þ ðβk βÞΓ ÞÞzk diagðP i ; Q i ; W i Þ o 0

ð32Þ

Combining Eðβk Þ ¼ β and Eððβk βÞ2 Þ ¼ βð1 β Þ, EV k þ 1 V k o 0. Theorem 1 is completed.□

we

yield

ð25Þ

Theorem 2. For given scalars DT ; β and Bd , the closed-loop system (27) is stochastically stable and Ki (determined by

where K i ¼ diagðK i ; K i ; …; K i Þ. Indeed, (25) is a Markovian jumping |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

K i ¼ F i X i 1 ; K i ¼ diagðK i ; K i ; …; K i Þ) are the controller gains if there |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

system. Based on the defined βk that follows Bernoulli stochastic process, we have Eðβ k Þ ¼ β and Eððβ k β Þ2 Þ ¼ β ð1 βÞ. Define ( E R ðkÞ; k A Ω1 E R1 ðkÞ ¼ 0; k A Ω2 ( E R ðkÞ; k A Ω2 ð26Þ E R2 ðkÞ ¼ Bd ; k A Ω1

exist matrices X i 4 0, F i ði A IÞ and a scalar ε 4 0, such that 2 3 ^T Φ Φ X G 11 12 i 2 6 7 6 n Φ22 0 7 4 5o0 n n ε 1I

ηk þ 1 ¼ ðM þ B^ Λi K i Þηk þ C^ k ER ðkÞηk ðh þ 1Þ

ðh þ 1Þ

With the above as a background, we rewrite the system (25) as

ηk þ 1 ¼ ðM þ B^ Λi K i Þηk βC^ ER1 ðkÞηk þ ð1 β ÞC^ ER2 ðkÞηk ðβk βÞC^ ER1 ðkÞηk þ ðβ βk ÞC^ ER2 ðkÞηk

ð27Þ

Ξ i1 Ξ i1 0

⋯

Ψ i1

0

⋯

0

0

n

⋱

⋮

⋮

n

n

Ωir

0

n

n

n

ρ1 1 X r

0

3

7 Ξ ir 7 5; Ξ ir 3 7 7 7 7; 7 7 5

T

T

Φ11 ¼ diagð X i ; r i X i ; qi X i Þ, Δij ¼ αij X i C^ , Ξ ij ¼ βij X i C^ ,

~ ¼ X G^ T þ F T ΛT B^ T , Ψ ¼ ε 1 α α β 2 I^ I^ T ðs ¼ 1; 2; …; r X i ¼ P i 1 , Π i i 1 is is ir i i 1Þ,

2

T

Ωij ¼ ρj 1 X j þ ε 1 α2ij β I^ I^ ,

Fi ¼ K iXi,

ρj ¼ 1 þB2d rj þ

ðLnð2DT þ1ÞÞ2 qj ðj ¼ 1; 2; …; rÞ.

ηk þ 1 ¼ ðM þ B^ Λi K i Þηk þ C^ ER ðkÞηk

ð29Þ

ηk þ 1 ¼ Υ i zk þ ðβk βÞΓ zk

ð30Þ

T Further, set zk ¼ ηTk ER1 ðkÞηTk ER2 ðkÞηTk , we have

3. Stability analysis and controller design In this section, we will present the sufficient condition for stochastic stability and the method of network controller design. Lemma 3 (Petersen [30]). For any matrices W , M; N; FðkÞ with F T ðkÞFðkÞ o I and any scalar ε 4 0, the following inequality holds W þ MFðkÞN þ N T F T ðkÞM T r W þ εMM T þ ε 1 N T N Theorem 1. For given scalars DT ; β and Bd , the closed-loop system (27) is stochastically stable if there exist matrices P i 4 0, Q i 4 0 and W i 4 0ði A IÞ, such that X π ij ðΥ Ti ðP j þ B2d Q j þ ðLnð2DT þ1ÞÞ2 W j ÞΥ i jAI

þ β ð1 βÞΓ ðP j þ B2d Q j þ ðLnð2DT þ 1ÞÞ2 W j ÞΓ Þ T

diagðP i ; Q i ; W i Þ o0

αir Π~ i βΔir ⋯ ð1 βÞΔi1 ⋯ ⋯

0

ρ1 1 X 1

n

and

And if k A Ω2 , (27) is changed into

i h i where Υ i ¼ M þ B^ Λi K i βC^ ð1 β ÞC^ , Γ ¼ 0 C^ C^ .

Ωi1

6 n 6 6 n Φ22 ¼ 6 6 6 n 4

ð28Þ

where C^ ¼ diagðC~ ; 0; …; 0Þ, 0 0 C~ ¼ 0 c

h

αi1 Π~ i 6 Φ12 ¼ 6 4 βΔi1 ð1 β ÞΔi1 2

More specifically, if k A Ω1 , (27) is changed into

ηk þ 1 ¼ ðM þ B^ Λi K i Þηk C^ ER ðkÞηk

where 2

ð33Þ

ð31Þ

Proof. Choose a Lyapunov–Krasovskii functional candidate V k ¼ V 1k þ V 2k þV 3k , where V 1k ¼ ηTk P i ηk , V 2k ¼ ER1 ðkÞηTk Q i ER1 ðkÞηk ,

Proof. To facilitate the controller design, we choose Q j ¼ r j P j and W j ¼ qj P j , where r j 40; qj 4 0. By Lemma 3, the uncertainty Δðτk Þ is deleted due to ‖Δðτk Þ‖2 o β . Further, by Schur complement, (33) can be derived. Theorem 2 is completed.□ Remark 9. It should be pointed out that the inequalities in Theorem 2 do not meet linear matrix inequality (LMI) conditions because of the terms r j and ρj . Based on the search algorithm [1], we find r j and ρj satisfied (33) until the controller gains K i ði; j A IÞ are obtained. 4. Adaptive tracking controller design In this section, we design the following adaptive tracking controller in order to not only stabilize the NCS (27), but also enable the sampling period to track the desired sampling period. uk ¼ K i ξ k þ N i T n

ð34Þ

where K i is the controller gain that stabilizes the NCS (27), T n is the desired sampling period, and N i is a parameter to be designed. Actually, the desired sampling rate may be obtained based on network performances and control performances, and [31] has provided a solution. In this paper, our efforts are focused on providing a control scheme that guarantees the sampling period to track the desired value. Hence, for the given desired sampling period, we present a solution.



8

Note that the parameter K i has been obtained in Section 3, and the rest is to design the parameter N i such that the desired performance can be achieved, i.e., yk converges to T n T k →T n when the NCS (27) gets close to a steady state for the given desired sampling period (reference input). Inspired by [32], in which the method of zero steady-state tracking error was investigated. Here, we first assume that the NCS has been stabilized and the desired (steady-state) value ys ¼ T n has been achieved. In this context, the steady values of state ξk and control input uk are taken as ξs and us for the reference input T n , respectively. We have us ¼ K i ξs þ N i T n

ð35Þ

For simplification, let ξs ¼ N ξ T and us ¼ N u T , and by (34) and (35), we have uk ¼ K i ξk þ ðN u K i N ξ ÞT n . Then, N i ¼ N u K i N ξ . In this case, a problem to be addressed is how to determine the matrices N ξ and N u . Under the desired sampling rate, the network is at steady state, then A k converges to the matrix below " # 0 eA T n ð36Þ G¼ 0 1 cER n

n

where ER is a steady value of disordering entropy and is obtained easily by measuring network. It is obvious that the disordering entropy remains at steady state and belongs to the interval ½0; Bd Þ when the network remains steady. Thus ck ¼ c in (13), and (7) can be represented as

ξs ¼ Gξs þ Bus ys ¼ C ξs

ð37Þ

To compute N ξ and N u , we substitute ξs ¼ N ξ T and us ¼ N u T into (37) and let ys ¼ T n leading to n

n

Nξ T n ¼ GN ξ T n þ BN u T n T n ¼ CN ξ T n

Further, we have " # Nξ GI B 1 0 ¼ Nu I C 0

ð38Þ

ð39Þ

As the adaptive controller parameters K i ðiA IÞ are presented in Section 3, we have Ni ¼ Nu K iNξ

ð40Þ

In summary, the basic idea of tracking controller design proposed in this paper is that we first obtain stabilizing controller gains K i , then parameter N i is calculated by virtue of (39) and (40), thus we obtain the tracking controller (34). Apparently, if N ξ and N u in (40) take the values obtained from (39), the sampling period will track the desired value, since us acts the plant and sampling period when NCS is at steady-state (referring to (35)). Remark 10. It should be pointed out that the desired sampling rate maybe changes according to network allocation and control system parameters. In this case, the controller parameters K i and N i can be updated by the above method. 5. Experimental verifications In this section, we verify the effectiveness of the control strategy proposed for NCSs with packet disordering. First, we show that control performances of NCSs are highly sensitive to the degree of packet disordering. Therefore, one should carefully analyze statistical characteristics shown by packet disordering and design a kind of effective control method to improve control performances of NCSs. Second, for a specific network, we show that the presented control technique can not only ensure the design criterion of sampling rate, but also significantly improve control performances of NCSs with packet disordering. Finally,

comparative studies are carried out to clearly demonstrate the advantages of the suggested approach in this paper. As shown in Fig. 4, the necessary experimental setup is developed in our experimental platform. The experimental apparatus is comprised of a two-step DC motor, a PC with Windows and a local board. The PC is used to implement the networked controller, and the local board can send the motor state to the PC controller and convert the control signal into pulsewidth modulation for DC motor. The DC motor exchanges data with the controller over the simulated network based on IP rather than the actual network [32]. The main reason is that the experimental results, obtained by applying different methods, can be compared under the same network condition. The DC motor has the following system matrices: 0:2246 0:1230 A¼ 0:0487 0:6559 and the control input matrix is given 2 3 0:0008 0:5088 0:2102 6 7 B ¼ 4 0:6103 0:7825 0:8239 5 0:7664 0:2896 0:0410

5.1. Experimental results of NCS over different communication networks In this paper, Buk ¼ BK ξk , which means that the plant and sampling period can be controlled simultaneously under the state feedback control law. If we set B2 ¼ 0, K T ¼ 0 in the following equation and B1 is used as control input matrix, it is clear that only the plant is controlled. " # " # xk B1 ½K x K T BK ξk ¼ Tk B2 " # B 1 K x xk þ B 1 K T T k ¼ ð41Þ B 2 K x xk þ B 2 K T T k Similar to Section 2.1.2, we construct three IP-based local networks by OPNET software. Different degrees of packet disordering corresponding to these networks are shown in Fig. 5. Without loss of generality, these three networks are absent of packet loss. By Algorithm 1, it is not difficult to calculate β ¼0.9406, β ¼ 0.8812 and β ¼0.8614 for Bd ¼ 0:75 in Net 1, Net 2 and Net 3, which means that ProbðER ðkÞ r 0:75Þ ¼ 0:9406 for Net 1, ProbðER ðkÞ r 0:75Þ ¼ 0:8812 for Net 2, and ProbðER ðkÞ r 0:75Þ ¼ 0:8614 for Net 3. With a fixed sampling period T ¼ 0:05 s and a given controller gain (42), the state responses of NCS communicated over Net 1, Net 2 and Net 3 appear in Fig. 6(a), (b) and (c), respectively. Moreover, a comparison of convergence speed is presented in Table 2. We know if disordering entropy value occurs in a low interval ½0; Bd with small probability β, this means that the chance of disordering entropy value appearing in the high interval ðBd , Lnð2DT þ 1Þ is high. Since high disordering entropy value usually indicates heavy packet disordering, the small β for the same Bd means the heavier

Fig. 4. DC networked control system setup.



5.2. Experimental verification of NCS using the proposed theoretical method For Net 3, the observed transmission delay and calculated packet displacements are provided in Fig. 7. Similar to the analysis in Section 2.1.2, h ¼ 2 is derived. Based on the packet displacements, θk ðiÞ ði ¼ 0; 1; 2Þ can be calculated. Thus, the jumping process takes values in a finite set I ¼ f1; 2; 3g, standing for I ¼ f½1; 0; 0; ½0; 1; 0; ½0; 0; 1g, and governs the switching among the different system modes, then r ¼ 3 ½1; 0; 0-1, which means uðkÞ ¼ K ξk acts on the plant; ½0; 1; 0-2, which means

uðkÞ ¼ Kξk 1 ; ½0; 0; 1→3, which means uðkÞ ¼ K ξk 2 . Correspondingly, the following state transition matrix is calculated using the function “hmmestimate” in Matlab. 2 3 0:4615 0:5385 0 6 7 0 0:7143 5 ð43Þ 4 0:2857 0:9090 0 0:0910

Table 2 Comparisons of convergence time. Parameters

Net 1

Net 2

Net 3

β Convergence time (s)

0.9406 20

0.8812 24

0.8416 48

Transmission delay

degree of disordering owned by communication network. Thus, it is found that the packet disordering in Net 3 is the heaviest, the packet disordering in Net 2 is heavier and the packet disordering in Net 1 is slight in the constructed three networks. As expected, as the degree of packet disordering increases, the convergence speeds of systems become slow, which can be seen clearly in Fig. 6 and Table 2. It seems that the statistical characteristics of packet disordering are worth investigating for achieving the improved control performance of NCSs. 2 3 0:2934 0:2843 6 7 0:9154 5 K x ¼ 4 0:9359 ð42Þ 1:5672 1:4688

9

0.2 0.15 0.1 0.05 0 −0.05

0

10

20

30

40

50

60

70

80

Time (second)

1

Packet displacement value

Cumulative probability

0.9 0.9406 0.8812

0.6

0.8614 0.3

0

Net1 Net2 Net3 0

0.75 Disordering entropy

−1 50

100

150

200

Fig. 7. Transmission delay and displacement values of packets in Net 3.

20 x1

State variables

State variables

0

Packet

x2 0

0

20 Time (second)

x1 x2 0

−20

80

0

24

80

Time (second)

10

10 x1 x2

State variables

State variables

1

0

200

0

−10

2

1.4

Fig. 5. Cumulative density functions of disordering entropy.

−200

3

0

48 Time (second)

80

x1 x2 0

−10

0 7

80 Time (second)

Fig. 6. Responses of state under the different distributions of packet disordering.



Moreover, by (39) and (40), we obtain 2 3 2 3 2:2377 0:3973 6 7 6 7 N 1 ¼ 4 3:3792 5; N 2 ¼ 4 0:0803 5; 2:0756 0:2046

2

3

0:3690 6 7 N 3 ¼ 4 0:0823 5: 0:1954

2 Sampling period

To obtain the controller gains in terms of the proposed method in this paper, Table 3 summarizes some experimental parameters. According to T min and T max given in Table 3, we can obtain the β ¼ 0:1015; T nom ¼ 0:05 s by Lemmas 1 and 2. By Theorem 2, we have 2 3 0:2674 0:2662 2:4836 6 7 0:8704 2:9517 5; K 1 ¼ 4 0:8758 1:5755 1:5393 2:2050 2 3 0:0053 0:0013 0:0261 6 7 K 2 ¼ 4 0:0385 0:0323 0:0008 5; 0:0117 0:0207 0:0072 2 3 0:0021 0:0011 0:0023 6 7 0:0029 0:0009 5: ð44Þ K 3 ¼ 4 0:0031 0:0210 0:0231 0:0018

0.1

−2

0

10

20

For comparison purposes, the similar systems' responses in [7,20] are investigated. Note that the discretized systems with the constant sampling period are operated for NCSs in [7], we choose a constant sampling period T ¼ 0:05 s. Set B2 ¼ 0 and K T ¼ 0, then the DC motor system used in the experiment can be an object of study of [7]. In addition, [20] executed the sampling rate adaptation policy, where the sampling rate was adjusted based on the transmission delay. Similarly, we set B2 ¼ 0, K T ¼ 0 in [20]. According to [20], we determine the two sampling policies as follows: 8 0:02 s delay r 0:05 s > > > > < 0:1 s 0:05 s o delay r 0:1 s ð46Þ Tk ¼ 0:2 s 0:1 s o delay r 0:15 s > > > > : 0:3 s 0:15 s o delay Table 3 Experimental parameters. Parameters

Descriptions

Values

T min T max Tn ER c Δðτk Þ ξ0

Minimum sampling period Maximum sampling period Desired sampling period Desired disordering entropy Step size Uncertain parameter Initial state value

0s 0.15 s 0.1 s 0.6 0.5 sin(k) ½ 1 3 0T

50

60

70

80

50

60

70

80

0

0

10

20

ð45Þ

5.3. Experimental comparisons of NCS with the constant and adaptive sampling period [7,20]

40

2

−2

Under the presented adaptive tracking controller (34), the state response of NCS exchanging data via Net 3, the sampling period and tracking error are shown in Fig. 6(d), Fig. 8(a) and (b), respectively. Easily seen, both stability and the desired sampling period could be guaranteed using the proposed control method in this paper. Comparing with the corresponding result from Fig. 6(c), one can see that, as shown in Fig. 6(d), the short settling time and less oscillation times are obtained by the proposed method in this paper. Actually, the statistical analysis of disordering entropy (Bernoulli process) and sampling rate control play an importantly potential role in improving the control performance of NCS.

30

Time (second)

Tracking error

10

30

40

Time (second) Fig. 8. The curves of sampling period and tracking error.

8 > < 0:02 s T k ¼ 0:05 s > : 0:1 s

delay r0:05 s 0:05 s odelay r 0:1 s

ð47Þ

0:1 s o delay

Using the method of [7] and the aforementioned two policies (46) and (47), the state responses of an NCS are provided in Fig. 9(a), (b) and (c), respectively. Note that the passive impact brought about by the constant sampling period policy has taken place. In Fig. 9(a), we can see clearly the slower convergence speed and bigger amplitude than those in Fig. 6(d) obtained by the proposed sampling rate control technique in this paper. Moreover, note that two adaptive sampling schemes have entirely different effects on the control performance of an NCS in Fig. 9(b) and (c). Using policy (47), the system is stabilized in a short time, while the system fails to be stabilized when policy (46) is used to adjust the sampling period. One reason that can account for these results is the different increment size of sampling period in the two sampling policies. Since how to determine the scale of change is vague, then the blindness and spontaneity existing in the adaptive sampling scheme [20] may lead to invalidation of adaptive sampling policy for performance improvement of NCSs. To clearly demonstrate the efficacy of the presented sampling rate control scheme on packet disordering and further explain the simulation results shown in Figs. 6(d) and 9(a–c), the appearance probability of ER ðkÞ ¼ 0 under different sampling policies are given in Fig. 9(d). Note that the part of blue and green lines are not shown in Fig. 9(b), because they are beyond the showed limits. It is very easy to see that a great probability of ER ðkÞ ¼ 0 has happened under policy (46), which indicates the communication network is in a good status if policy (46) is operated. However, we find that the destabilization effect has taken place as shown in Fig. 9(b). The sampling period chosen in policy (46) is too long to update NCS timely, this leads to the noted destabilization effect. Compared with the constant sampling policy, policy (47) and the control policy proposed in this paper can provide us with improved communication quality (reduced packet disordering) presented in Fig. 9(d) and better stability results shown in Fig. 9(c) and Fig. 6(d). But, the different scales of sampling interval variation in [20] have the opposite effects on NCS. By contrast, the obvious advantages have been enjoyed by the approach suggested in this paper.



11

6

x1 x2

40

6 x1 x2

−1

20 0 −20

x 10

−2 −3 −4

0

20

40

60

80

100

Time (second)

−5

x1 x2

4 State variables

0

State variables

State variables

60

2 0 −2 −4

0

20

40

60

80

100

−6

0

20

40

60

80

100

Time (second)

Time (second)

0.8

Frequency

0.6 0.4 0.2 0

Constant sampling rate

Policy (46)

Policy (47)

Sampling rate control

Sampling methods Fig. 9. Responses of state and probability distribution of ER ðkÞ ¼ 0 under the different sampling methods. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

6. Conclusions

References

Avoiding more serious packet disordering is quite beneficial for the improvement of QoS of networks, and is also of significant importance to control performances of NCSs, since control performances of systems communicating over network media are highly sensitive to QoS of networks. In this paper, an adaptive sampling rate control strategy is investigated for NCSs for the purpose of reducing packet disordering and achieving a tradeoff between QoS of networks and control performance of systems. First, a novel sampling rate control algorithm is proposed based on the disordering entropy. Next, an augmented system model composed of the plant and sampling period is put forward. The time-varying properties of sampling rate and disordering entropy pose tough challenges to controller design. Robust control technique and stochastic theory have been successfully used to account for the difficulties. Thus, the stability condition and adaptive tracking controller design are presented. Different from the existing research on packet disordering in NCSs, the proposed control strategy enables the sampling period to track the desired sampling period, such that the desired control performance of NCSs can be obtained. Moreover, the statistical characteristic of packet disordering is also fully analyzed during executing the presented scheme. Finally, the effectiveness and advantages of the proposed control scheme have been demonstrated through experimental verifications.

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Acknowledgment The authors would like to acknowledge the National Natural Science Foundation of China under Grant 61104093, 61174119, 61034006, 60774070, 61104003, 61273011, the National High Technology Research, Development Program of China (863 Program) under Grant 2011AA040101, the Opening Project of Key Laboratory of Networked Control Systems, Chinese Academy of Sciences, the State Scholarship Fund of China as a Visiting Scholar under Grant 2011821213 and the Natural Science Foundation of Liaoning Province under Grant 2014020138.


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