Switching control analysis and design in queue networks

Switching Control Analysis and Design in Queue Networks

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Switching Control Analysis and Design in Queue Networks } ´ Lorinc Marton PII: DOI: Reference:

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Switching Control Analysis and Design in Queue Networks L˝ orinc M´ artona a Department of Electrical Engineering, Sapientia Hungarian University of Transylvania, Tirgu Mures, Romania

Abstract This paper deals with the switching traffic control of such communication networks that can be modeled using interconnected queue systems. A general method is proposed to analyze the stability and to compute the expected upper bounds of the queue backlogs and recovery times in controlled queue networks which apply switching control algorithms. It is also discussed, how the state switching control algorithms can be designed for network traffic control applications by applying the principle of lexicographic optimization. This optimization method ensures a unified approach to design different types of switching traffic controllers. The delay control problem in networked control systems or known data traffic control methods, as the additive increase/multiplicative decrease algorithm, can be treated in the proposed analysis and design approach. Experimental measurements are also presented to show the effectiveness of the switching data traffic controllers in delay-critical wireless telerobotic applications. Keywords: Network traffic control, Networked control systems, Stability analysis, Queue systems

1

1. Introduction

2

Networked control systems apply communication networks to implement the

3

information exchange among some of their components. Typical examples are

4

Multi-Agent Systems [1] or Wireless Sensor Network-based control applications Preprint submitted to Journal of The Franklin Institute

November 12, 2019

5

[2]. The performances of such control systems are largely influenced by the

6

characteristics of the used communication channels [3]. In the applied commu-

7

nication links it is essential to implement such traffic control algorithms which

8

are able to assure proper communication conditions in almost all circumstances.

9

The congestion control mechanisms are meant to ensure that the commu-

10

nication network is not overloaded, by regulating the output flow rates at the

11

source side. This is done based on feedback information (e.g. round trip time or

12

packet loss) obtained from a congestion detection mechanism [4]. In many cases,

13

different transfer rate computation formulas are applied in different operating

14

conditions; this yields to switching control schemes such as the increase/decrease

15

algorithms [5].

16

In the case of the networked control systems, besides congestion avoidance,

17

other, control-specific criteria have to be fulfilled by the traffic control algo-

18

rithm. Since the communication delay directly influences the control perfor-

19

mances in networked feedback loops, its expected value should not exceed pre-

20

defined, application-specific limits. In the study [6] the effects of delay and

21

delay variation on the stability, performance of wireless networked control sys-

22

tems were treated, and a set of basic requirements were presented that support

23

successful deployment of such networked control systems.

24

For traffic control design it is beneficial to have a model which can suitably

25

describe the behavior of the communication traffic in the network. The inter-

26

connected queue systems are accepted models to describe the behavior of many

27

types of communication networks [7]. More recently the theory of stochastic

28

queue networks was proposed for control analysis and design in communication

29

networks [8].

30

The deterministic, continuous queue system models allow the application of

31

conventional control methods to design network traffic controllers. The paper [9]

32

introduces a discrete-time H∞ controller for congestion control in data networks

33

with communication lags. A robust L2 stable networked controller for packet

34

data queue level control between two nodes with an Internet connection was

35

introduced in the study [10]. Integral control approaches were also developed 2

36

for network traffic control to suppress the effect of unmodeled disturbances,

37

see e.g. [11]. The cooperative control concept was applied in [12] for queue

38

backlog control in a class of communication systems. Stability analysis methods

39

for network congestion control systems were introduced based on continuous

40

bottleneck link models of the communication networks, see e.g. [13].

41

The interconnected stochastic queues can capture adequately the behavior

42

of the communication systems [7], but the control analysis and design based

43

on such models are more challenging. A promising approach is based on the

44

Lyapunov theory extended to stochastic queue systems [8]. This was effectively

45

applied to analyze and design backpressure routing algorithms [8] or to design a

46

communication framework in radio sensor networks for smart grid applications

47

[14]. Lyapunov techniques were also applied to determine the average delays

48

and queue backlogs in input-queued cell-based switches [15]. The enumerated

49

methods in these previous works impose restrictions on the drift of the Lyapunov

50

functions to conclude on stability and performances of controlled queue systems.

51

The approach presented in this paper does not necessitate the Lyapunov drift

52

condition or other restrictions on the Lyapunov function candidate. It only

53

imposes that the control input should be bounded to conclude on the system

54

stability and performances. The assumptions related to the boundedness of the

55

control are more suitable for practical applications.

56

The significant contributions of this paper are:

57

• An analysis method is introduced for a general class of switching control

58

algorithms that are applied in queue networks. By using the proposed

59

method, the expected value of the cumulative maximum queue backlog

60

and the expected recovery time can be predicted based on the controller

61

and queue parameters.

62

63

• A unified approach for switching control algorithm design in queue systems is proposed, which is based on lexicographic optimization.

64

• A switching delay control method is presented for wireless networked con-

65

trol systems. Wireless networked robot control experiments were per3

66

formed, which apply the proposed switching delay control in the commu-

67

nication channels.

68

The rest of this paper is organized as follows: Section 2 introduces the

69

basic modeling assumptions. Section 3 presents the stability analysis method

70

for queue networks that are controlled using switching control algorithms. The

71

switching control design problem in queue networks is treated in section 4. The

72

additive increase/multiplicative decrease control and the delay control problems

73

are discussed here using the proposed analysis and design approaches. Case

74

studies are presented in section 5 to show the applicability of the proposed

75

analysis and design methods. Finally, section 6 concludes this study.

76

2. Controlled Queue Systems

77

78

Let a system of queues in which the evolution of each queue is described by the Lindley’s recursion (see e.g. [16]): Qj [k + 1] = max(0, Qj [k] + Uj [k]), 0 ≤ E[Qj [0]] < ∞, j = 1 . . . n < ∞.

(1)

79

Here E[·] denotes the expected value, Qj [k] is the jth queue’s backlog in the

80

integer time slot k ≥ 0, Uj [k] is the control input. The Lindley’s recursion is an

81

accepted model to describe links with storing capacity in real communication

82

networks, see e.g. [17] and the references therein.

83

It is considered that the expected value of the control input is upper bounded,

84

as it is formulated in the following assumptions.

85

Assumption 1. E[Qj [k]Uj [k]] < E[Qj [k]]UM j , ∀j, k, UM j > 0.

86

Assumption 2. E[ 21 (Uj [k])2 ] < UM j , ∀j, k, UM j > 0.

87

88

89

90

(2)

(2)

(2)

Denote UM = maxj (UM j ) = max(UM 1 , . . . , UM j , . . . , UM n ), and UM = (2)

maxj (UM j ). The stability of the queue systems is commonly related to the boundedness of the expected value of queue backlogs.

4

91

Definition 1. [18] The queue network is strongly stable if: n k−1

XX 1 limk→∞ sup E[Qj [i]] < ∞. k j=1 i=0 92

93

(2)

A method to study the stability of queue networks is based on following Lyapunov function candidate: n

1X 2 Q [k] 2 j=1 j

L(Q[k]) =

94

(3)

where Q[k] = (Q1 [k] Q2 [k] . . . QM [k])T .

95

The control input Uj [k] can be defined as the difference between number

96

of arrived items (Aj [k]), and the number of served/departed items (Dj [k]). A

97

common approach to assure stability under mild conditions is to design such

98

control policies which assure that the expected values of the arrivals are strictly

99

smaller then the expected value of the departures [19], i.e. E[Uj [k]] < −Umj ∀j, k,

(4)

100

where Umj > 0 is the minimum value of the difference between E[Aj [k]] and

101

E[Dj [k]]. Let Um = minj (Umj ) = min(Um1 , . . . , Umj , . . . , Umn ).

102

Theorem 1. [18] Consider a system of queues modeled by (1). If there exists

103

UM > 0 and Um > 0 such that

(2)

(2)

E[L(Q[k + 1]) − L(Q[k])|Q[k]] ≤ UM − Um 104

n X

106

(5)

then the queue system is strongly stable and n k−1 (2) U E[L[0]] 1 XX E[Qj [k]] ≤ M + . nk j=1 i=0 Um Um nk

105

E[Qj [k]],

j=1

(6)

The conditional expectation E[L(Q[k+1])−L(Q[k])|Q[k]] is called Lyapunov drift.

5

107

3. Analysis of Queue Systems with Switching Control

108

The stability condition (4) for the expected value of the control is restrictive

109

as it imposes that the expected value of the control input should always be

110

strictly negative.

111

The control strategies that are based on the condition (4) do not explore the

112

storing capability of the queues. A better strategy is to enforce the condition

113

(4) only if the queue backlog overpasses a predefined limit.

114

115

116

117

Moreover, many control strategies cannot instantly satisfy the condition (4). It is the case of control algorithms with integral terms. For a more efficient and flexible utilization of the queues, consider the following switching control algorithm for each queue:   U [k], if E[Q [k]] > Q , j− j ε Uj [k] =  U [k], otherwise.

(7)

j+

118

where Qε is a predefined queue backlog threshold value.

119

No restriction is imposed on Uj+ . It may be set by the applications, which

120

use the queue system, to satisfy the demands of the tasks implemented over the

121

queue system.

122

123

124

125

To ensure stability and to compute an upper bound for the average queue backlog, here more relaxed conditions for the control input are proposed. The condition below is similar to (4) but it imposes a restriction on the control only if the expected queue backlog overpasses Qε . Assumption 3. E[Qj [k]Uj− [k]] < −E[Qj [k]]Umj , if E[Qj [k]] > Qε ∀j, k.

126

The next condition is less restrictive as it does not enforce instantaneous

127

sign restriction when the condition E[Qj [k]] ≤ Qε is not satisfied:

128

Assumption 4. If in the l-th time slot Qj [l] overpasses Qε , Uj− assures that

129

E[Qj [k]] returns under Qε within 0 ≤ Pl < ∞ successive time slots. 6

130

Let the recovery time be P = maxl (Pl ).

131

Assumption 4 involves that there exists a number of time slots in which

132

Assumption 3 is satisfied but it also allows a finite number of such time slots

133

when Assumption 3 is violated.

134

The Assumption 4 is suitable to analyze such switching control algorithms

135

which have terms with integral character, e.g. terms in the form Uj− [k + 1] =

136

Uj− [k] − ∆j (Qj [k]), where ∆j (·) is the control decrement function. Such

137

control actions cannot instantly ensure that Assumption 3 is satisfied. In the

138

next section it will be shown that common switching control algorithms, such

139

as the additive increase/multiplicative decrease or the switching delay control

140

algorithm obey to Assumption 4.

141

142

(U )

(U )

Bounded control input change is assumed. (U )

(U )

Assumption 5. E[|Uj [k + 1] − Uj [k]|] < δM j , ∀j, k, δM j > 0. (U )

(U )

143

Let δM = maxj (δM j ).

144

The following theorem summarizes the stability results for different types of

145

switching control algorithms. It also gives the average queue backlog bound in

146

the function of control input bounds.

147

Theorem 2. Let a system of queues that are described by (1) and apply the

148

control (7). Then,

149

(I) if the Assumptions 1, 2 and 3 hold, the queue system is strongly stable

150

and the average of the expected cumulative queue backlog in each time slot

151

k > 0 satisfies   n k−1 (2) UM 1 XX UM E[L[0]] E[Qj [i]] ≤ + Qε +1 + . nk j=1 i=0 Um Um Um nk

(8)

152

(II) if the Assumptions 1, 2, 4 and 5 hold, the queue system is strongly

153

stable and the average of the expected cumulative queue backlog in the kth

154

time slot satisfies   n k−1 (2) U UM E[L[0]] 1 XX (U ) +1 + . E[Qj [i]] ≤ M + (2Qε + δM P ) nk j=1 i=0 Um Um Um nk 7

(9)

Proof: From the model (1) it results that

155

Qj [k + 1]2 = (max(0, Qj [k] + Uj [k]))2 ≤ (Qj [k] + Uj [k])2 .

Consider the Lyapunov function candidate (3). By the inequality (10) it

156

157

(10)

results that n n X 1X 2 L[k + 1] − L[k] ≤ Uj [k] + Qj [k]Uj [k]. 2 j=1 j=1

(11)

By taking the expectations, the Lyapunov drift reads as  n   X 1 E[L[k + 1] − L[k]|Q[k]] ≤ E Uj [k]2 + Qj [k]Uj [k])|Q[k] . 2 j=1

158

(12)

It yields

159

k−1 X i=0

(E[L[i + 1] − L[i]|Q[i]]) ≤

n  k−1 XX i=0 j=1

 1 E[ Uj [i]2 |Q[i]] + E[Qj [i]Uj [i]|Q[i]]] . (13) 2

160

Denote N− [i] the set of queues that in the ith iteration satisfy E[Qj [i]] > Qε .

161

The set of the other queues in the system is denoted by N+ [i]. The cardinality

162

of the set N+ [i] is n− [i] = card(N− [i]), and let n+ [i] = card(N+ [i]). Note that

163

n+ [i] + n− [i] = n ∀i.

164

The sum in the relation (13) is separated corresponding to the sets N− [i]

165

and N+ [i]. Using the law of telescoping sums and by Assumption 2 it results

166

that: (2)

E[L[k + 1]] − E[L[0]] ≤ nkUM + 167

168

169

E[Qj [i]Uj [i]] +

k−1 − [i] X nX

E[Qj [i]Uj [i]]. (14)

i=0 j=1

i=0 j=1

It was explored that E[E[X|Y ]] = E[X]. As E[L[k + 1]] ≥ 0 and by using the Assumption 1, the relation above simplifies as: −E[L[0]] ≤

170

k−1 + [i] X nX

(2) nkUM

+ Qε UM

k−1 X

n+ [i] +

i=0

k−1 − [i] X nX

E[Qj [i]Uj [i]].

(15)

i=0 j=1

(I) First, consider that Assumption 3 is satisfied. Then (2)

−E[L[0]] ≤ nkUM + Qε UM

k−1 X i=0

8

n+ [i] −

k−1 − [i] X nX i=0 j=1

Um E[Qj [i]],

(16)

171

k−1 − [i] X nX

E[Qj [i]] ≤

i=0 j=1

172

As

Pk−1 Pn+ [i] j=1

i=0

k−1 n XX i=0 j=1

E[Qj [i]] ≤ Qε (2)

E[Qj [i]] ≤

Pk−1

k−1 (2) knUM UM X E[L[0]] + Qε . n+ [i] + Um Um i=0 Um

Pk−1 i=0

knUM + Qε Um



(17)

n+ [i] it results that

 k−1 X UM E[L[0]] +1 . n+ [i] + Um Um i=0

(18)

n+ [i] ≤ kn, the relation (8) directly results.

173

Since

174

The strong stability yields by taking the limit k → ∞ of the inequality (8):

i=0

limk→∞

  k−1 n (2) nUM UM 1 XX E[Qj [i]] ≤ + nQε +1 . k i=0 j=1 Um Um

(19)

175

(II) Second, consider that Assumption 4 is satisfied.

176

Let N± [i] the set of queues that in the ith iteration satisfy both E[Qj [i]] >

177

Qε and the Assumption 3. The set of the queues in the system that satisfy

178

E[Qj [i]] > Qε but does not meet Assumption 3 is denoted by N= [i]. Let n± [i] =

179

card(N± [i]) and n= [i] = card(N= [i]). Note that n± [i] + n= [i] = n− ∀i.

180

With these notations the inequality (15) takes the form: −E[L[0]] ≤

181

182

i=0

n+ [i] − Um

k−1 = [i] X nX

E[Qj [i]] + UM

i=0 j=1

k−1 ± [i] X nX

E[Qj [i]]. (20)

i=0 j=1

(U )

yields: E[Qj [i]] ≤ δM Pl + Qε , ∀j, l < i ≤ l + Pl . It results that

i=0 j=1

184

+ Qε UM

k−1 X

If in the lth iteration Qj [i] overpasses Qε , from the Assumptions 4 and 5

k−1 = [i] X nX 183

(2) knUM

E[Qj [i]] ≤

k−1 k−1 (2) X knUM UM X UM (U ) + Qε n+ [i] + (δM P + Qε ) n± [i] + E[L[0]]. (21) Um Um i=0 Um i=0

Pk−1 Pn+ [i] Pk−1 Pk−1 Pn± [i] (U ) E[Qj [i]] ≤ Qε i=0 n+ [i] and i=0 j=1 E[Qj [i]] ≤ (δM P + As i=0 j=1 Pk−1 Qε ) i=0 n± [i] it results that k−1 n XX

(2)

knUM E[Qj [i]] ≤ + Qε Um i=0 j=1 +





 k−1 X UM +1 n+ [i] Um i=0

 k−1 X UM (U ) + 1 (δM P + Qε ) n± [i] + E[L[0]]. Um i=0 9

(22)

Pk−1

(n+ [i] + n± [i]) ≤ kn, the relation (9) directly results.

185

Since

186

The strong stability also yields by taking the limit k → ∞ of the inequality

187

i=0

(9): limk→∞

  k n (2) nUM 1 XX UM (U ) + n(2Qε + δM P ) +1 . E[Qj [i]] ≤ k i=1 j=1 Um Um

(23)

188



189

If Qε = 0 and P = 0, the queue backlog bounds in the Theorem 2 simplify

190

to the bound obtained in the Theorem 1. The bound obtained in (8) can be

191

viewed as a generalization of (6) for the switching control case. The relation (9)

192

further generalizes the computed backlog bound for the case of such switching

193

controllers that contain terms with integral character.

194

The Theorem 2 treats the stability of queue systems and gives an estimate

195

of the average queue backlog under restrictions on the control action when a

196

critical queue backlog is overpassed. The condition formulated in Assumption 4

197

requires a known upper bound for the recovery time (P ). In the next section, it

198

will be presented, how P can be computed for different switching traffic control

199

algorithms.

200

4. Switching Control Design

201

4.1. Lexicographic Optimization Approach for Switching Control Design in Queue

202

Systems

203

Consider a communication network in which the transmission links are mod-

204

eled by the equation (1). The network is used by applications which initiate data

205

flows among the different nodes of the network.

206

In realistic communication scenarios, the queues in the communication links

207

have an upper bound of the backlog (QM ). A queue is called here overloaded if

208

E[Q[k]] ≥ QM .

209

A bottleneck link is a communication link that is shared by a high number

210

of data flows which could generate together the overload of the link. Assume a

10

211

number of N data flows that share the bottleneck link, i.e. Q[k + 1] = max(0, Q[k] +

N X

U` [k])

(24)

`=1

PN

where

213

arrived items from the individual flows and the served items. The applications

214

are considered to be able to gather information about the state of the used

215

bottleneck link, and are also able the regulate their own transmission rates.

216

Here A` [k] is considered to be controlled by the application which generates the

217

flow (source based control).

218

219

220

221

222

223

224

225

`=1

U` [k] =

PN

212

`=1

A` [k]−D[k], i.e. the difference between the sum of the

During the transmission rate computation, several objectives should be fulfilled. First, the overload of the links in the communication network should be avoided. Second, the communication medium should assure the data transmission rate requested by the applications. When high transmission rates are expected, the two objectives above could be contradictory.

226

It can also be affirmed that the first objective generally has higher priority

227

since a possible overload of the bottleneck link compromises all the flows which

228

pass through it. The traffic control design can handle this by using the principle

229

of the lexicographic optimization [20].

230

Lexicographic optimization method with two objective functions: Consider

231

the objective functions F (1) (A) and F (2) (A) depending on a decision variable A.

232

The optimization problem can be formally written as: minimize (F (1) (A) F (2) (A))

233

subject to gi (A) ≤ 0 ∀i, where gi (A) ≤ 0 represents the ith inequality constraint.

234

According to the principle of the lexicographic ordering [21], the objective

235

functions are aligned in function of their priorities. Let F (1) the objective func-

236

tion with the higher priority. In the first step find such decision variables for

237

which F (1) (A) ≤ Fε , where Fε

238

(1)

jective F

(1)

(1)

is an acceptably low limit value for the ob-

. The second step can be formalized as: minimize F (2) (A) subject

11

239

(1)

to F (1) (A) ≤ Fε

and gi (A) ≤ 0 ∀i.

240

Data traffic control design in the view of lexicographic optimization:

241

Define two objective functions.

242

The first objective function (F (1) ) is formulated such to assure that the

243

overload is avoided. It can be formulated as a strictly increasing function of the

244

queue backlog.

245

The second objective function (F (2) ) can be defined in function of A` such to

246

assure the desired transfer rate (sent data packets over a sending period) for the

247

data flow. The minimum of F (2) corresponds to the case when the real transfer

248

rate is equal to the desired transfer rate.

249

250

251

252

(1)

In the view of the lexicographic optimization a limit value Fε (1)

chosen such that F (1) < Fε

has to be

is equivalent to E[Q[k]] < Qε ∀k, where Qε < QM .

Then the lexicographic optimization-based queue control can be implemented as the following switching control strategy:

253

• If E[Q[k]] ≥ Qε , compute A` such to ensure the decrease of F (1) .

254

• Otherwise, compute A` such to ensure the decrease of F (2) .

255

The constraints in the optimization problem can be the bounds of the control

256

inputs in (24), see e.g. the Assumptions 1, 2.

257

4.2. Additive Increase/Multiplicative Decrease (AIMD) Algorithm

258

The AIMD algorithm is a wide-spread data traffic regulation method in wide-

259

area computer networks as it can simultaneously assure congestion avoidance

260

and fairness [5, 22].

261

The control algorithms, meant to avoid congestion, are generally based on a

262

congestion detection mechanism that gives an estimate of the cumulative back-

263

logs of the queues on the route used by the data flow. The well-known conges-

264

tion detection methods apply acknowledgment package loss, Round-Trip Time

265

measurements or Explicit Congestion Notification [4]. The route is considered

266

congested if the estimated queue backlog overpasses a threshold value (Qε ).

12

267

This value has to be chosen such to avoid the overload of the queues on the

268

route.

269

Let a data flow which uses a communication channel that includes a bot-

270

tleneck link. To apply the principle lexicographic optimization for congestion

271

avoidance, two objectives are formulated. The first objective with higher prior-

272

ity can be formulated as: assure that the bottleneck queue backlog is smaller

273

than the prescribed threshold (Qε ). The second, lower priority objective is to

274

assure as high data rate for the data flow as possible. The scalable AIMD policy

276

[4] solves this problem by applying the following algorithm:   µA [k − 1], if E[Q[k]] ≥ Q , ` ε A` [k] =  A [k − 1] + α, otherwise.

277

in [23] in the case of real communication networks. The first term in (25) assures

278

the decrease of the sent packets to avoid the overload in critical situations. The

279

second term assures the increase of the sent packets whether it is possible.

275

(25)

`

The controller parameters α > 0 and 0 < µ < 1 can be chosen as it is proposed

280

As follows it will be shown that the switching control algorithm (25) assures a

281

finite recovery time, i.e. the control (25) is in concordance with the Assumption

282

4.

283

284

Proposition 1. Consider the bottleneck link model (24) in which each data PN (N ) flow applies the control (25). If E[Q[0]] = Q0 > Qε , `=1 A` [0] = A0 and

285

E[D[k]] ≥ Dm > 0, ∀k, then the expected value of the queue backlog becomes

286

smaller the Qε within 1 P = Dm

287

288

289

(N )

A Q0 − Qε + 0 1−µ

!

(26)

time slots. Proof: Consider the successive time slots k = 0 . . . P when E[Q[k]] ≥ Qε . In this case the expected queue backlog satisfies: E[Q[k + 1]] = E[Q[k]] +

N X `=1

13

A` [k] − E[D[k]].

(27)

290

If E[Q[k]] ≥ Qε , A` [k] = µA` [k − 1] = µk A` [0]. It yields: E[Q[P + 1]] = Q0 +

P X N X

k=1 `=1 291

µk A` [0] −

(N ) 1

E[Q[P + 1]] ≤ Q0 + A0 292

P X

E[D[k]].

(28)

k=1

− µP +1 − P Dm . 1−µ

(29)

Since 0 < µ < 1 (N )

E[Q[P + 1]] ≤ Q0 + 293

294

A0 − P Dm . 1−µ

(30) 

By equating this upper bound with Qε , yields the equation (26). 4.3. Delay Control

295

Due to the delay dependency of the control performances in networked con-

296

trol systems, it is necessary to design such data rate control algorithms which

297

assure that the expected end-to-end delay (E[T [k]]) in the communication links

298

of the networked control system remains under a predefined bound Tε > 0, i.e.

299

E[T [k]] ≤ Tε .

300

It is considered that the data flows under consideration pass through a bot-

301

tleneck link, the queuing delay of which is the dominant component of the

302

end-to-end delay.

303

The queuing delay is closely related to the expected queue backlog. This

304

relation is formulated in the Little’s law for queue systems [24], or more specific

305

formulas are given in data communication networks which also take into consid-

306

eration in the propagation delay (Tm ≥ 0) [25]. Here a general relation between

307

the queuing delay and the queue backlog in a bottleneck link is considered: E[Q[k]] = a[k](E[T [k]] − Tm )

308

309

310

(31)

where E[T [k]] ≥ Tm and 0 < am ≤ a[k] ≤ aM is finite. During the control design, the delay measurement errors should also be taken into consideration. Here it is assumed that |Tb[k] − E[T [k]]| ≤ ∆ 14

(32)

311

312

313

314

315

316

where Tb[k] denotes the measured delay and ∆ > 0 is the finite upper bound of the measurement error.

It is assumed that the prescribed delay bound Tε is chosen such that Tε > max(∆, Tm ). From the relations (31) and (32) yield that, if Tb[k] < Tε , then E[Q[k]] ≤ Qε

where

Qε = aM (Tε + ∆ − Tm ).

(33)

317

If Tε is chosen such that Qε is below QM , then the delay control implicitly

318

assures the avoidance of the overload.

319

320

For the lexicographic approach of the delay control design, let the first objective function be the measured delay: F (1) [k] = Tb[k].

(34)

321

Many applications, such as video streaming, need to transfer a predefined

322

number of data packets within a given number of time slots. If it is not possible,

323

the service offered by the application can also be functional but of lower quality.

324

Consider that in the bottleneck link each data flow is required to send a (d)

325

desired number of A`

326

objective function as:

data packets in each time slot. Formulate the second

N

F (2) [k] =

1 X (d) (A` − A` [k])2 . 2

(35)

`=1

327

The following switching control algorithm solves the lexicographic optimiza-

329

tion problem with the objective functions (34) and (35), see [26]:   A [k − 1] − γ Tb[k], if Tb[k] > T , ` ε A` [k] =  A [k − 1] + α(A(d) − A [k − 1]), otherwise. ` ` `

330

The first term of (36) decreases the number of the sent data packets. It implicitly

331

yields to the decrease of the queue backlog and queuing delay, respectively. The

332

second term assures the decrease of the objective function (35).

328

(36)

where the decrement and increment gains are γ > 0, and 0 < α < 1 respectively.

15

333

334

335

336

The control (36) assures a finite recovery time as it is presented in the proposition below. Proposition 2. Consider the bottleneck link model (24) in which each data PN (N ) flow applies the control (36). If E[Q[0]] = Q0 > Qε , `=1 A` [0] = A0 and

337

E[D[k]] ≥ Dm > 0 ∀k, then the expected value of the queue backlog becomes

338

smaller then Qε , defined in (33), within P time slots, where P is the positive

339

340

341

solution of the equation:   γN (Tε − ∆) 2 γN (Tε − ∆) (N ) P − A0 − Dm + P + Qε − Q0 = 0. 2 2

Proof: Consider the successive time slots k = 0 . . . P when Tb[k] ≥ Tε . The

control input satisfies:

A` [k] = A` [0] − γ

k X i=1

Tb[i],

(38)

A` [k] ≥ A` [0] − γk(Tε − ∆). 342

P −1 X k=1

(N )

E[Q[P ]] ≤ Q0 + P (A0 E[Q[P ]] ≤ Q0 + P

344

345

346

347

348

(39)

If Tb[k] ≥ Tε , the expected queue backlog satisfies: E[Q[P ]] = Q0 +

343

(37)



(N )

A0

N X `=1

!

A` [k] − E[D[k]] ,

− Dm ) − γN (Tε − ∆)

− Dm − γN (Tε − ∆)

P −1 X k=1

P −1 2

k,

(41)



(42)

Equate the computed upper bound with Qε , given by (33):   P −1 (N ) Q0 + P A0 − Dm − γN (Tε − ∆) = Qε 2 The equation above corresponds to (37). Remark: Since

γN (Tε −∆) 2

(40)

.

(43) 

> 0 and Qε − Q0 < 0 the quadratic equation (37)

always admits a real positive solution. The benefits of the presented delay control algorithm in time-critical networked control applications is shown in section 5.

16

349

4.4. Implementation Issues

350

The control algorithms (25) or (36) can be implemented using timer event

351

functions, that are invoked repeatedly with a predefined interval (period). The

352

implementation is exemplified through the algorithm (36). Consider that an

353

application intends to use a bottleneck link for data transmission such that the

354

communication delay stays around a given threshold Tε . If possible, the source

355

application intends to send a number of A`

356

Ts .

(d)

packages within the time interval

357

A possible implementation of the control algorithm (36) for the `th applica-

358

tion, which computes the number of data packets (A` ) to be sent in the current

359

sending period, reads as follows:

360

Delay Control : (d)

get A` , γ, α, Tε

361 362

A` = 0

363

repeat {with interval Ts } get Tb

364

if (Tb > Tε )

365 366

else

367

A` = max(0, A` − γ Tb);

(d)

A` = max(0, A` + α(A`

368

− A` ));

end

369

end

370 371

end

372

4.5. Simulation of Delay Control Algorithm

373

Simulation experiments were performed to analyze the transient perfor-

374

mances of the delay control algorithm (36). Consider the bottleneck link model

375

given in the equation (24). Communication packets are sent through this link.

376

The link can serve D = 10000 packets in a discrete-time slot. It was consid-

377

ered that the delay in the bottleneck link is given by the relation (31) with the

378

parameters Tm = 1ms and a = 5E5.

17

379

Let an application (` = 1) which uses the bottleneck link and applies the

380

delay control algorithm (36). The control parameters were chosen as γ = 500,

381

α = 0.01 and A1 = D.

(d)

382

It is assumed that, besides this communication channel, other channels also

383

use the same bottleneck link, and they send together 5000 packets in each dis-

384

crete time slot.

385

Figure 1 shows the behavior of the queue state Q, the delay value T and the

386

computed control signal A1 during the transient state of the control for different

387

Tε values. It can be seen that, if Tε decreases, the attenuation of the transient

388

oscillations also decreases. It is because in the switching control algorithm (36)

389

the term that ensures the delay decrease is proportional to T . This effect can

390

be compensated by increasing the controller parameter γ for larger Tε values.

391

5. Telerobotic Application

392

5.1. Video-supported Teleoperation Systems

393

Networked telerobotic systems are a special class of networked control sys-

394

tems that are designed to assure the remote control of a distant robot by a

395

human operator [27]. Figure 2 presents end nodes and the communication chan-

396

nels in a common telerobotic system. The two end nodes are represented by

397

the master side computer, used by the human operator-controlled master robot,

398

and by the slave side computer, used by the distant robot. Through the PCh

399

channel, the reference position is sent continuously from the master robot to

400

the remote mobile slave robot. In the case of bilateral teleoperation [28] the

401

environmental forces are sent from the slave to the master through the FCh

402

channel to ensure haptic feedback for the human operator. The VCh channel is

403

used to transmit video data from the salve side to the master side.

404

The time-critical data flows are represented by PCh and FCh, as the de-

405

lays in these communication channels are responsible for control performance

406

degradation [27]. The cumulative transfer rate (sent data packets over a sending

407

period), necessary for the teleoperation system, is R = RV + RP + RF where

18

6000

A1 (packets)

5000 4000 3000 2000 1000 0

0

500

1000 1500 Discrete time (s/sending period)

2000

0

500

1000 1500 Discrete time (s/sending period)

2000

0

500

1000 1500 Discrete time (s/sending period)

2000

4

4

x 10

Q (packets)

3

2

1

0

0.1

T (s)

0.08 0.06 0.04 0.02 0

Figure 1: Transient behavior of the delay control for different Tε values (Tε = 24ms - red; Tε = 12ms - blue; Tε = 6ms - green)

408

RV , RP and RF denote the transfer rates in VCh, PCh and FCh respectively.

409

The video channel (VCh) needs a substantially greater transfer rate than the

410

PCh and FCh.

411

In wireless communication networks, the ensured service rates for the end

412

nodes are determined based on the observed radio signal strength among the

413

end nodes and the access point. It is controlled by the access points’ dynamic

414

rate scaling algorithm such to assure a reliable access point - end node com-

415

munication link [29]. The service rate regulation is application-independent.

19

WLAN

VCh PCh FCh

PCh FCh VCh

Figure 2: Video assisted teleoperation of a mobile manipulator

416

The rate can take predefined values in function of the measured wireless signal

417

strength. If the mobile robotic slave (the end node) moves away from the access

418

point, its service rate is decremented by this algorithm. The wireless channel

419

represents the bottleneck link in most of the mobile telerobotic systems.

420

The communication delays in the PCh and FCh channels influence the con-

421

trol performances of teleoperation, it can even compromise the stability of the

422

bilateral teleoperation systems [28]. The delay can be controlled by regulating

423

the transfer rates in the bottleneck link. The transport layer endpoint multi-

424

plexing algorithm ensures that the flow of each end node has the same chance of

425

transmission [23], independent of its transfer rate. In the case of teleoperation

426

systems, by decreasing the amount of sent video data, the packets in the PCh

427

and FCh can get faster through the wireless access point and lower communica-

428

tion delay in these channels can be obtained. Hence, by modifying the transfer

429

rate in VCh, the delay in the FCh and PCh can be adjusted, RV can be applied

430

as control input in the data traffic control algorithm of the video supported

431

telerobotic system. The transfer rate in the VCh can be regulated by several

432

approaches: by changing the sending rate of the video data, by modifying the

433

size of the video frames, or by varying the video frame quality (e.g. by applying

434

compression to the video frames at different levels).

20

435

5.2. Real-time Data Transfer Rate Control Experiment

436

Experimental measurements were performed to show the applicability of

437

the delay control approach presented in subsection 4.3 in wireless teleoperation

438

systems. On the master side, two Sensable Phantom Omni haptic devices were

439

used as master robots. A KUKA Youbot mobile manipulator was used as the

440

slave robot. The first haptic device was applied to control the mobile platform;

441

the second one assured the remote control of the manipulator. A video camera

442

was placed on the mobile platform to follow the motion and the actions of the

443

slave robot. The master and the slave sides were connected by using a wireless

444

TPLINK TL-WR941ND access point.

445

The sending periods in the time-critical channels (PCh and FCh) were chosen

446

Ts = 5ms. The frames from the slave camera were captured with 20Hz. Each

447

video frame has 640 × 480 pixels with 24 bit color resolution. JPEG compression

448

was applied to reduce the amount of video data that has to be sent. The JPEG

449

compression rate is between 0% and 100%. Before sending, the compressed video

450

data was decomposed into 8 kbyte packets. The received video frame packets

451

were reconstructed at the master side. Figure 5.2 presents the quality of a

452

displayed video frame at the master side for u = 10% and u = 90% respectively.

453

The data traffic controller which regulates the video transfer rate was im-

454

plemented at the slave side, the average delay was measured in the FCh. The

455

delay was calculated using the clock synchronization technique between the

456

master and slave side computers, see e.g. [30].

457

For the validation of the performed rate control measurements, the data

458

traffic in the WLAN was observed by using an Aircap NX wireless data packet

459

capturing device. It was used to monitor the data traffic with such measure-

460

ments that are independent of the rate control application. The wireless signal

461

strength at the slave robot and the data transfer rates from the slave were

462

measured using this equipment.

463

Two experiments were performed. In both cases, the mobile robotic agent

464

was teleoperated by the human operator on the same track. The robot’s velocity

465

was about 0.4m/s. The motion started from the vicinity of the wireless access 21

466

point and the robot departed around 45m from the access point in an indoor

467

environment. After about 50 seconds the robot left the room in which the access

468

point was. In the second part of the motion (after about 110 seconds) the robot

469

returned to the wireless access point on the same path.

470

In the first experiment, the video frames from the slave were sent with the

471

constant u = 90% compression rate to the master. Figure 5 shows that in

472

the first part of the robot’s motion the wireless signal strength decreases and

473

during its return the detected signal strength increases (see Signal Strength).

474

This figure also shows that the available service rate of the mobile slave was

475

adjusted automatically by the access point according to the detected signal

476

strength, independent of the teleoperation application (see Service Rate).

477

The effect of the decreasing wireless signal strength on the delay can clearly

478

be observed, see Figure 6. Here TP denotes the delay from the master to the

479

slave and TF denotes the delay from the slave to the master in the time-critical

480

data flows (PCh and FCh in Figure 2). As the experimental measurements

481

show, the mean of the delay increases up to 300 ms.

482

During the second experiment, the delay control algorithm (36) was imple-

483

mented. Under constant sending period assumption, the video transfer rate can

484

be considered to be proportional to the number of sent video packets. If the

485

JPEG compression rate decreases, the number of video packets, which have to

486

be sent, also decreases. Hence u was used here as the control action. In the

487

algorithm (36) the following parameters were considered: γ = 100, α = 0.1,

488

u(d) = 90% and the prescribed upper bound for the communication delay was

489

set to Tε = 12ms. The advanced bilateral control algorithms, such as the time

490

domain passivity-based controllers, assure reliable teleoperation if the commu-

491

nication delay values are in the order of tens of milliseconds [31].

492

The behavior of the data flows with the proposed traffic control algorithm

493

is shown in Figure 7. When the robot departs from the access point and the

494

signal strength decreases, the delay tends to increase. As this figure shows, the

495

traffic controller reacts to the delay change and it adjusts the compression rate

496

u such that the delay is kept near to the threshold Tε . 22

Figure 3: Displayed video frame quality - u=10%

Figure 4: Displayed video frame quality - u=90%

497

The effect of the data traffic regulation on the communication performances

498

in the presence of varying wireless signal strength is presented in Figure 8. The

499

control algorithm assures that the average communication delay values TP and

500

TF in the delay-critical data flows (PCh and FCh) remain around the threshold

501

value (Tε = 12ms). At the same time, the controller ensures the best video

502

quality corresponding to these delay values and the momentary wireless radio

503

signal strength.

23

504

6. Conclusions

505

Switching control algorithms are a reasonable choice for many data traffic

506

regulation problems. The switching allows the application of different control

507

strategies in critical and regular cases respectively. In this study, an analysis

508

method for a general class of switching traffic controllers was introduced that is

509

applicable to such communication networks which can be modeled by intercon-

510

nected queues. The method concludes on stability and average queue backlog

511

bounds in function of the controller parameters. The proposed analysis method

512

can handle such switching control algorithms that include terms with integral

513

character. In this case, the stability can be assured if the controller assures a

514

bounded recovery time. It is presented, how the recovery time can be computed

515

for different types of switching control algorithms.

516

As the control objectives are often contradictory in the case of data traffic

517

regulation problems, the principle of lexicographical optimization can be ap-

518

plied to switching control design. This optimization method can be applied to

519

approach different types of data traffic control design problems in communica-

520

tion networks. As an example, a delay control algorithm design was presented

521

for networked control systems. The experimental measurements performed in a

522

wireless telerobotic system that shows the applicability and effectiveness of the

523

proposed switching data traffic control method.

524

In future works, it will be investigated how further data flow regulation tech-

525

niques, such as the Active Queue Management, can be treated in the proposed

526

analysis and design approach.

527

Acknowledgments

528

The author acknowledges Zolt´ an Sz´ ant´ o, Piroska Haller and Tam´ as Vajda,

529

Sapientia - Hungarian University of Transylvania for their contribution to the

530

experimental measurements. This work was supported in part by a grant of

531

the Romanian National Authority for Scientific Research CNCS - UEFISCDI,

532

project number PN-II-RU-TE-2011-3-0005. 24

533

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28

Service Rate (Mbs)

WLAN Monitor 100

50

0

0

50

100

150

200

150

200

Signal Strenght (dB)

Time (s) 0 −10 −20 −30 −40 −50 0

50

100 Time (s)

Figure 5: Signal strength and service rate during motion - Traffic controller not active

T F (ms)

1500 1000 500 0

0

50

100 Time (s)

150

200

0

50

100 Time (s)

150

200

T P (ms)

1500 1000 500 0

Figure 6: Delay during motion - Traffic controller not active

29

u (%)

100

50

Service Rate (Mbs)

100

Signal Strenght (dB)

0 0

0

50

100 Time (s)

150

200

50

100 Time (s)

150

200

50

100 Time (s)

150

200

50

0 0

−20 −40 0

Figure 7: Traffic controller behavior during motion 300

T F (ms)

TF Tǫ

200

100

0

0

50

100 Time (s)

150

200

0

50

100 Time (s)

150

200

T P (ms)

300

200

100

0

Figure 8: Delay during motion - Traffic controller active

30