Sliding mode approach to congestion control in connection-oriented communication networks

Transcription

1 JOURNAL OF APPLIED COMPUTER SCIENCE Vol. xx. No xx (200x), pp. xx-xx Sliing moe approach to congestion control in connection-oriente communication networks Anrzej Bartoszewicz, Justyna Żuk Technical University of Łóź Institute of Automatic Control 18/22 Stefanowskiego St., Łóź, Polan Abstract. In this paper, a novel sliing moe flow controller esign for the connection-oriente communication networks is propose. The networks are moele as iscrete time systems with the available banwith acting as isturbance. The propose controller is esigne in such a way that the close-loop system stability an fast, finite time error convergence are ensure. In orer to avoi the problem of excessive control signal magnitue, a sliing moe controller with saturation is propose. When this controller is applie no bottleneck link buffer overflow an full utilization of its available banwith are guarantee. Furthermore, transmission rates generate by the controller are always upper boune an nonnegative. 1. Introuction High-spee connection-oriente communication networks may allow various kins of applications to run uner a uniform infrastructure. In these networks the sequence of application ata units are transmitte by a source an reach their estination via a path of intermeiate switches. On each switch a server scheules an forwars ata units along the path from their source to their estination in the network. The ifficulty of the flow control is mainly cause by long propagation elays in the network. If congestion occurs at a specific switch, information about these circumstances must be conveye to all the sources transmitting ata units through the switch. This information is use to ajust source rates an may affect the congeste switch after the roun trip propagation elay. Flow control in connection-oriente communication networks has recently become an exciting research fiel an valuable results have been reporte in many papers [3, 6, 10-15, 17]. Their authors propose on-off [3, 6], classical

2 48 A. Bartoszewicz, J. Żuk proportional-erivative (PD) [14], fuzzy proportional-integral-erivative (fuzzy PID) [17], stochastic [11], aaptive [13] an neural network base [12] controllers. Due to the significant propagation elays several researchers also applie the Smith preictors [3, 10, 15] for the flow control in such networks. On the other han, it is well known that sliing moe control is an attractive an efficient strategy which offers robustness an goo ynamic performance of the controlle systems [2, 5, 7, 16, 18]. Therefore, in this paper we attempt to apply iscrete time sliing moe approach [1, 4, 8, 9] to the flow control in a connection-oriente communication network. We consier a moel of the networks which provie feeback mechanism. An example of such networks is Available Bit Rate (ABR) service in Asynchronous Transfer Moe (ATM) stanar. The propose control algorithms employ an appropriately efine sliing hyperplane, which ensures the close-loop system stability an finite time error convergence to zero. In orer to avoi the problem of excessive control signal magnitue, we propose a sliing moe controller with saturation. When this controller is applie no ata loss an full link banwith utilization are ensure. These esirable properties are explicitly prove. Moreover, the relation between the ata flow rate an the consume banwith is erive. The remainer of this paper is organize as follows. In Section 2, etaile escription of the network moel is given. Afterwars, in Section 3, the propose sliing moe flow controller esign an the system performance when the controller is applie are presente. In this section the important properties of the controlle network are also state (in a lemma an three theorems) an explicitly prove. A simulation example, illustrating the iscusse properties, is presente in Section 4. Finally, Section 5 conclues the paper. 2. Network moel In this paper a single virtual circuit in a connection-oriente communication network is consiere. Furthermore, it is assume that there is only one bottleneck noe in the network. The source sens ata (as etermine by the controller at the bottleneck noe) an special control units. The control units carry information about the network state. After reaching their estination, they are immeiately sent back to the source, along the same path they arrive. The information carrie by the control units is use to ajust the amount of ata transmitte by the source at each control perio. The control units are processe by the intermeiate noes on a priority basis, i.e. they are not queue but sent to the next noe without elay. Consequently, the roun trip time of the control units in the virtual circuit is constant. Moreover, this time can be expresse as the sum of forwar an backwar propagation elays enote as T F an T B, respectively = T + T (1) F B

3 Sliing moe approach to congestion control 49 The block iagram of the flow control system consiere in this paper is shown in Figure. y Source Forwar elay T F + Queue integrator y Controller u Backwar elay T B Fig. 1. Network moel Further in the paper, T represents the iscretisation perio, y(kt) enotes the bottleneck queue length at time instants kt, k = 0, 1, 2,, an y > 0 is the eman value of y(kt). It is assume that before setting up the connection, the bottleneck buffer is empty, i.e. y(kt < 0) = 0. Moreover, in this paper we assume that the roun trip time is a multiple of the iscretisation perio, i.e. = m T, where m is a positive integer. The amount of ata to be sent is generate by the controller place at the bottleneck noe. The controller output at time kt is enote as u(kt). This amount of ata will be sent by the source after backwar elay T B an will arrive at the bottleneck noe T F later. Consequently, the bottleneck buffer for any time kt remains empty. It is assume that before setting up the connection u( kt < 0) = 0 (2) The amount of ata which may leave the bottleneck buffer at time kt is moelle as an a priori unknown boune function of time (kt), for k = 0, 1, 2,. The maximum value of (kt) is enote by max an h(kt) represents the amount of ata actually leaving the bottleneck noe at time kt. Consequently k 0 0 h( kt ) ( kt ) max (3) Initially, the bottleneck buffer is empty y( kt = 0) = 0 (4) Then, for any k 1, the queue length can be expresse as follows

4 50 A. Bartoszewicz, J. Żuk k 1 k 1 k m 1 k 1 (5) ( ) = ( ) ( ) = ( ) ( ) y kt u jt h jt u jt h jt j= 0 j= 0 j= 0 j= 0 3. Sliing moe controllers In this section, the flow control problem for the escribe network is consiere. First, a chattering free iscrete time sliing moe controller is esigne so that fast an finite time error convergence to zero is achieve. Then, we propose a moifie sliing moe control strategy which takes into account physical constraints an ensures that the maximum amissible flow rate is never exceee Propose control strategy This subsection focuses on the esign of a iscrete time sliing moe flow controller for the communication network, whose moel was introuce in Section 2. For this purpose, first a iscrete time state space moel of the controlle system is formulate. Then, an appropriate sliing plane is introuce an its parameters are etermine in such a way that the close-loop system is stable an the error converges to zero in finite time. Let us consier the following iscrete time moel of the network x[ ( k + 1) T] = Ax ( kt ) + bu( kt ) + ph( kt ) ( ) T y kt = q x( kt ) (6) where x(kt) = [x 1 (kt) x 2 (kt) x n (kt)] T is the state vector with x 1 (kt) = y(kt), A is n n state matrix, b, p an q are n 1 vectors A = b = p = q = (7) an n = m + 1. Alternatively, the state space equation can be written as follows

5 Sliing moe approach to congestion control 51 x [( ) ] ( ) ( ) ( ) 1 k + 1 T = x1 kt + x2 kt h kt x [( ) ] ( ) 2 k + 1 T = x3 kt x [( ) ] ( ) 3 k + 1 T = x4 kt x [( ) ] ( ) n 1 k + 1 T = xn kt x [( 1) ] ( ) n k + T = u kt (8) In this moel the available banwith h(kt) is represente as unmatche isturbance. The esire state of the system is enote by x = [x 1 x 2 x n ] T. It can be notice from (8) that all components x i of vector x for i = 2,, n are equal to zero when h(kt) = 0. Let us enote the first state variable x 1 representing the eman queue length by y. We introuce a sliing hyperplane escribe by the following equation ( ) T s kt = c e ( kt ) = 0 (9) where c T = [c 1 c 2 c n ] is such a vector that c T b 0. Similarly as it is usually one when esigning the control systems, now we neglect the effect of isturbance h(kt) in the controller esign process. However, this oes not imply that the isturbance is isregare in the paper, it will be given full consieration when analysing the system performance. The close-loop system error is enote as e(kt) = x x(kt). Hence, substituting (6) into equation c T e[(k + 1)T ] = 0 the following feeback control law can be erive 1 ( ) ( T ) T ( ) u kt = c b c x Ax kt (10) When this control signal is applie, the close-loop system state matrix has the following form A ( T ) 1 T c = I n b c b c A. Then the characteristic polynomial of A c can be foun as follows et c c c c c c I A (11) n n n n 1 n n n 2 ( ) z n c = z + z + z + + z cn cn cn which leas to the conition c n 0. Asymptotic stability of the iscrete time system is ensure if an only if all its eigenvalues are locate insie the unit circle. Moreover, in orer to ensure the close-loop system error convergence to zero in finite time, the characteristic polynomial shoul have all roots equal to zero. Therefore, (11) has to satisfy et ( z ) n c n I A = z (12)

6 52 A. Bartoszewicz, J. Żuk Comparing coefficients on the right-han sies of (11) an (12), the following form of vector c is obtaine [ ] c T = c (13) n Substituting (7) an (13) into (10) the following state feeback control can be erive n u( kt ) = y x ( kt ) (14) i= 1 Alternatively, from (8), one can get the state variables x i (i = 2, 3,, n) expresse in terms of the control signal generate by the controller at the previous n 1 samples i ( ) [( 1) ] i x kt = u k n+ i T for i = 2, 3,, n (15) Substituting these expressions into (14) an putting x 1 (kt) = y(kt), we obtain { [ ]} u ( kt ) = y ( ) ( ( 1) ) ( ( 2) ) ( 1 ) y kt + u k n T + u k n T + + u k T = n 1 ( ) ( ) ( ) ( ) = y y kt u k j T = y y kt u k j T = ( ) y y( kt ) u( jt ) u kt j= 1 j= 1 k 1 j= k m m = (16) which actually represents a ynamic sliing moe flow controller. This completes the esign of the flow control algorithm which guarantees the closeloop system stability an fast, finite time error convergence to zero in the consiere network. Unfortunately, the strategy propose in this section may generate initial flow rate of unacceptable magnitue. Therefore, in orer to avoi this unesirable effect, further in the paper a moifie sliing moe control strategy will be propose.

7 Sliing moe approach to congestion control Moifie control strategy In this section we introuce a new flow control strategy. The amount of ata to be sent by the source at time kt is now etermine by the controller accoring to the following formula ( ) u kt = min y y( kt ) u( jt ), u k 1 max (17) where u max > max is the maximum amissible value of the flow rate. One can easily notice that this strategy etermines ata transmission rates which never excee the preetermine value u max. Let us enote the first argument of the min{, } function in (17) as w(kt), i.e. k 1 ( ) ( ) ( ) w kt = y y kt u jt (18) It irectly follows from (17) that at any time instant kt 0 inequality u ( kt ) w( kt ) (19) is satisfie. Moreover, we will prove that the flow rate generate accoring to strategy (17) is always nonnegative. This is shown in the following lemma. Lemma If the propose control algorithm is applie, then ata transmission rate u(kt) is always nonnegative, i.e. k 0 u( kt ) 0 (20) Proof: At the initial time function w(kt = 0) = y. Therefore, the flow rate u(0) either equals y or u max. Consequently, inequality (20) is satisfie for k = 0. Furthermore, at any time instant kt > 0 if the amount of ata to be sent by the source is u max, then the flow rate u(kt) is also strictly positive. Hence, in orer to complete the proof it is only necessary to show that (20) is satisfie for any k > 0 when u(kt) = w(kt). For that purpose, let us notice that the following relation can be erive from (5) [ ] ( ) [( ) ] y ( kt ) = y ( k 1) T + u k m 1 T h k 1 T (21) which hols for any positive integer k. Since in the analyse case u(kt) = w(kt) an the bottleneck queue length satisfies (21), we obtain

8 54 A. Bartoszewicz, J. Żuk k 1 ( ) ( ) ( ) ( ) u kt = w kt = y y kt u jt = = y [( ) ] ( ) [( ) y k 1 T u k m 1 T + h k 1 T] u( jt ) = k 1 = y y[ ( k 1) T] u( jt ) + h[ ( k 1) T] = 1 k 2 k 1 = y y[ ( k 1) T] u( jt ) u[ ( k 1) T] + h[ ( k 1) T] = 1 = w[ ( k 1) T] u[ ( k 1) T] + h[ ( k 1) T] (22) Taking into account inequality (19) an the fact that the consume banwith is always nonnegative, we obtain ( ) [( ) ] u kt h k 1 T 0 (23) which shows that inequality (20) inee hols at any time instant kt > 0 when u(kt) = w(kt). This conclusion ens the proof of the lemma. Thus we conclue that controller (17) generates ata transmission rate which is always nonnegative an upper boune, i.e. k 0 0 u ( kt ) umax (24) Clearly, this property is of utmost importance for the practical implementation of the strategy in any real network. In the sequel, three theorems stating further important properties of the propose flow control scheme are presente. The first one gives the conition which must be satisfie in orer to eliminate the risk of ata loss as a consequence of exceeing the bottleneck noe buffer capacity. Afterwars, the secon theorem provies a sufficient conition for the full bottleneck link banwith utilization. Finally, a relation between the control signal u(kt) an the consume banwith is formulate in the thir theorem. Theorem 1. If the propose strategy is applie, then the queue length in the bottleneck buffer is always upper boune by its eman value, i.e. k 0 y( kt ) y (25)

9 Sliing moe approach to congestion control 55 Proof: As it has alreay been prove, ata transmission rate u(kt) is nonnegative at any time instant kt. On the other han, by efinition u(kt) is smaller than or equal to w(kt). Therefore, the following relation hols for any time kt 0 k 1 y ( ) ( ) ( ) ( ) y kt u jt = w kt u kt 0 (26) Hence, the queue length satisfies k 1 ( ) ( ) y kt y u jt (27) Again taking into account that u(kt) is always nonnegative one conclues that the queue length inee never excees its eman value. This ens the proof of Theorem 1. Another esirable property of the analyze system is full bottleneck link banwith utilisation. Since the bottleneck link banwith (kt) is fully use if the queue length y[(k + 1)T] is strictly greater than zero, then the next theorem specifies a conition which guarantees that the queue length in our scheme is always strictly positive. Theorem 2. If u max > max an the eman value of the queue length y satisfies the following inequality ( 1) max y > m + u (28) then for any k m + 1 the queue length in the bottleneck buffer is always strictly positive. Proof: Let us efine an auxiliary function ϕ k 1 ( kt ) y( kt ) u ( jt ) = + (29) This function represents the amount of ata currently waiting in the bottleneck buffer queue, an the amount of in flight ata, i.e. this ata which has alreay been sent by the source but not yet arrive at the bottleneck noe, an that ata which will be sent by the source because the controller has alreay sent out an appropriate comman signal to the source. Substituting formula (5) into (29), one can express function ϕ (kt) as

10 56 A. Bartoszewicz, J. Żuk ϕ k 1 k m 1 k 1 k 1 ( kt ) y( kt ) u ( jt ) u( jt ) h( jt ) u( jt ) = + = + = j= k m j= 0 j= 0 j= k m k 1 k 1 u( jt ) h( jt ) (30) = j= 0 j= 0 Hence, taking into account conition (28) for k = 0 ϕ ( kt ) ϕ ( 0) 0 y ( m 1) u max y u max = = < + < (31) Furthermore, if for some k the following inequality ϕ(kt) < y u max is satisfie, then k 1 ϕ max (32) ( ) ( ) ( ) ( ) w kt = y y kt u jt = y kt > u which implies that u(kt) = u max. Consequently, since u max > max, we conclue that if ϕ(kt) < y u max, then function ϕ increases at least at the rate u max max. Moreover, since for any time kt < the consume banwith h(kt) = 0, then if ϕ(kt) < y u max an conition (28) is satisfie, then ϕ(kt) increases at the rate u max, reaching m u max at the time m T. On the other han, the consume banwith for any time kt satisfies inequality h(kt) max. This implies that ϕ(kt) can ecrease at most at the rate max. Further, we will show that function ϕ(kt) after reaching m u max never ecreases below this value, i.e. we will emonstrate that the following inequality hols for any kt > ϕ ( kt ) > mumax (33) In orer to prove this we will apply the principle of mathematical inuction. Let us first check whether (33) hols for k = m + 1. If conition (28) is satisfie, then ϕ(m T) = m u max < y u max. This implies that u(m T) = u max. Consequently ϕ ( m 1) T ϕ ( m T ) u( m T ) h( m T ) + = + = max max ( ) = m u + u h m T m u + u > m u max max max max (34) which shows that (33) is inee true for k = m + 1. Now, let us assume that (33) hols for some k m + 1. We will show that this implies that (33) is also

11 Sliing moe approach to congestion control 57 satisfie for k + 1. For this purpose we will consier the following two cases: the first one when u(kt) = w(kt) an the secon one when u(kt) = u max. In the first situation, from (28), (32) an inequality u max > max we obtain ( k 1) T ( kt ) u( kt ) h( kt ) ( kt ) w( kt ) h( kt ) ϕ + = ϕ + = ϕ + = ( kt ) y ϕ ( kt ) h( kt ) y h( kt ) = ϕ + = y > y u > m u max max max (35) Then, in the secon situation, i.e. when u(kt) = u max, we can write ( k 1) T ( kt ) u( kt ) h( kt ) ( kt ) u h( kt ) ϕ + = ϕ + = ϕ + max ( ) ϕ ( ) ϕ kt + u > kt > m u max max max (36) Therefore, we conclue that relation (33) actually hols for any time kt >. Finally, taking into account relations (29), (33) an the fact that the flow rate generate by our controller is always upper boune by u max, for any time kt >, we get ( ) ϕ ( ) ( ) k 1 max max 0 (37) y kt = kt u jt > m u m u = which ens the proof of Theorem 2. The theorem shows that using strategy (17) with conition (28) we ensure full bottleneck link banwith utilisation for any time kt >. Further, in the next theorem, a relation between the flow rate an the consume banwith is state an prove. Theorem 3. If the esigne sliing moe flow controller is applie, the eman queue length y > u max an the maximum flow rate u max > max, then there exists such a nonnegative integer k 0 satisfying k y u max 0 < + umax max 1 (38) that for any k > k 0 the following relation hols ( ) [( 1) ] u kt = h k T (39) Furthermore, when y u max, relation (39) is satisfie for any k 1. Proof: First, let us consier the situation when inequality y u max hols. It will be shown that then the following relation is always satisfie

12 58 A. Bartoszewicz, J. Żuk k 0 w( kt ) umax (40) which irectly implies u(kt) = w(kt). In orer to prove that relation (40) is inee satisfie for any time kt 0, we apply the principle of mathematical inuction. At the initial time w(0) = y u max. Therefore, inequality (40) hols for k = 0. Now let us assume that (40) is true for some k 0 an we will show that it is also satisfie for k + 1. Using equations (18) an (21), an taking into account that u(kt) = w(kt) we get ( 1) ( 1) ( ) w k + T = y y k + T u jt = j= k m + 1 ( ) ( ) ( ) ( ) k 1 ( ) ( ) ( ) ( ) = w( kt ) u ( kt ) + h( kt ) = h( kt ) < u k = y y kt u k m T + h kt u jt = max k j= k m + 1 = y y kt u jt u kt + h kt = max (41) This ens the proof of inequality (40). Since it follows from (40) that at any time instant kt 0 the flow rate u(kt) = w(kt), then using expression (22), for any k 1, we obtain ( ) [( 1) ] [( 1) ] [( 1) ] [( 1) ] u kt = w k T u k T + h k T = h k T (42) Equation (42) shows that, if y u max, then (39) inee hols for any positive integer k. Now let us consier the situation when y > u max. If for some k inequality ϕ(kt) < y u max is satisfie, then it follows from equation (30) an assumption u max > max, that function ϕ increases at least at the rate u max max. Thus, there exists such a finite time instant k 0 T, when the following conition ϕ ( kt ) y umax (43) becomes satisfie for the first time. We will etermine the latest time instant when inequality (43) can become satisfie for the first time. Since function ϕ(kt) is smaller than the ifference y u max until k < k 0, then

13 Sliing moe approach to congestion control 59 k 2 k ( 1) ( ) ( ) ϕ k T = u jt h jt < y u (44) 0 max j= 0 j= 0 Moreover, since the flow rate for any k < k 0 is equal to u max, then inequality (44) can be rewritten as 0 2 ( 1) ( ) k k u h jt < y u (45) 0 max max j= 0 Number k 0 in this equation is the biggest, when for any time from 0 up to (k 0 2)T, the consume banwith has its greatest possible value max. Consequently, from relation (45) we get the following inequality ( )( ) k u < y u (46) 0 1 max max max which gives the estimate of k 0 specifie by relation (38). We will now emonstrate that for any time kt > k 0 T conition (43) is inee satisfie. For that purpose we take into account some k > k 0 an we consier the two cases: the first one when w(kt) u max, an the secon one when w(kt) > u max. In the first case from relations (18) an (29) we obtain w( kt ) = y ( ) ϕ kt umax (47) From this inequality it can be easily notice that conition (43) actually hols for any k > k 0. Now let us consier the secon case, i.e. the situation when w(kt) > u max. In this situation, in orer to show that conition (43) hols for any kt > k 0 T, one can apply the principle of mathematical inuction. We have alreay emonstrate that there exists such a moment k 0 T, when inequality (43) is satisfie. Now, let us assume that for some instant kt > k 0 T the consiere conition hols, an we will show that this implies that the conition is also satisfie at the time instant kt + T. Since in the analyse case w(kt) > u max, then u(kt) = u max. Taking into account equation (30) an inequality u max > max, we get ( k 1) T ( kt ) u( kt ) h( kt ) ( kt ) u h( kt ) ϕ + = ϕ + = ϕ + max ( ) ϕ ( ) ϕ kt + u > kt y u max max max (48) Consequently, we conclue that for any k > k 0 inequality (43) is always satisfie. Conition (43) implies that for any kt > k 0 T, w(kt) u max an u(kt) = w(kt). Therefore, it immeiately follows from equation (22) that relation (39) is inee satisfie for any k > k 0. This ens the proof of Theorem 3.

14 60 A. Bartoszewicz, J. Żuk 4. Simulation example In orer to verify the properties of the sliing moe flow control strategy propose in this paper computer simulations of the network escribe by equations (6) (8) have been performe in Matlab-Simulink environment. First, the moel of the network was constructe accoring to the escription given in Section 2. Then the system parameters were chosen as follows: the iscretisation perio T was selecte as 1 ms an the roun trip time in the virtual circuit was assume to be = m T = 10 ms (T F = 3 ms, T B = 7 ms). Consequently, the system orer n = m + 1 = 11. The maximum available banwith of the bottleneck link was set as max = 4.8 Mb per secon, an the maximum amissible flow rate as u max = 6.1 Mb per secon. The banwith actually available for the ata transfer is shown in Fig.2. Suen changes of function, visible in the figure, reflect the most rigorous networking conitions. Accoring to Theorem 2, when strategy (17) is applie, the eman value of the queue length require to assure full bottleneck link banwith utilization in the analyze network must be greater than Mb. Consequently, y = Mb, which is equivalent to 170 ATM cells, is chosen. The transmission rate generate by the controller an the queue length evolution are shown in Figs.3 an 4, respectively. It can be clearly seen from the figures that the transmission rate is always nonnegative an never excees the maximum value u max. Furthermore, the queue length actually never grows beyon its eman value y, which ensures no ata loss in the network an no nee for ata retransmission. Moreover, after initial perio of 11 ms, i.e. for any time greater than (m + 1)T = 11 ms, the queue length is strictly positive, which implies full utilization of the bottleneck link available banwith.

15 Sliing moe approach to congestion control 61 6 x max 4 (kt), Mb time, ms Fig. 2. Available banwith at the output link of the bottleneck noe 8 x u max u(kt), Mb time, ms Fig. 3. Transmission rate generate by the controller

16 62 A. Bartoszewicz, J. Żuk 0,08 y 0,06 y(kt), Mb 0,04 0, time, ms Fig. 4. Queue length 5. Conclusions A new iscrete time sliing moe flow control strategy for a single virtual circuit of a connection-oriente communication network has been presente. The strategy is esigne so that the close-loop system stability an fast, finite time error convergence are ensure. In orer to avoi the problem of excessive control signal magnitue, a sliing moe controller with saturation is propose. When this controller is applie, full bottleneck noe link utilization an no ata loss in the controlle network are guarantee. The conitions ensuring these favorable properties are formulate an explicitly prove. Consequently, the nee for ata retransmission is eliminate an the maximum throughput is achieve. Moreover, as the flow rate generate by the controller is always nonnegative an boune, the propose mechanism can be feasibly incorporate in real communication networks. Our further research focuses on aapting the control strategy propose in this paper for multi-source networks. Acknowlegement This work has been finance by the Polish State buget in the years as a research project N N Design of the switching surfaces for the sliing moe control".

17 Sliing moe approach to congestion control 63 References [1] Banyopahyay B., Janarhanan S.: Discrete-time sliing moe control. A multirate output feeback approach. Series: Lecture Notes in Control an Information Sciences, Vol. 323, Springer-Verlag Berlin Heielberg, [2] Bartolini G., Friman L., Pisano A., Usai E.: Moern Sliing Moe Control Theory. New Perspectives an Applications. Series: Lecture Notes in Control an Information Sciences, Vol. 375, Springer-Verlag Berlin Heielberg, [3] Bartoszewicz A.: Nonlinear flow control strategies for connection-oriente communication networks. IEE Proceeings on Control Theory an Applications, 2006, Vol. 153, No.1, pp [4] Bartoszewicz A.: Discrete time quasi-sliing moe control strategies. IEEE Transactions on Inustrial Electronics, 1998, Vol. 45, No.4, pp [5] Bartoszewicz A., Kaynak O., Utkin V.I. (eitors): Sliing moe control in inustrial applications. Special section: IEEE Transactions on Inustrial Electronics, 2008, Vol. 55, No. 11, pp [6] Chong S., Nagarajan R., Wang Y.: First-orer rate-base flow control with ynamic queue threshol for high-spee wie-area ATM networks. Computer Networks an ISDN Systems, 1998, Vol. 29, pp [7] DeCarlo R.S., Żak S., Mathews G.: Variable structure control of nonlinear multivariable systems: a tutorial. Proceeings of IEEE, 1988, Vol. 76, No.3, pp [8] Furuta K.: Sliing moe control of a iscrete system. Systems & Control Letters, 1990, Vol. 14, pp [9] Gao W., Wang Y., Homaifa A.: Discrete-time variable structure control systems. IEEE Transactions on Inustrial Electronics, 1995, Vol. 42, pp [10] Gómez-Stern F., Fornés J., Rubio F.: Dea-time compensation for ABR traffic control over ATM networks. Control Engineering Practice, 2002, Vol. 10, pp [11] Imer O., Compans S., Basar T., Srikant R.: Available bit rate congestion control in ATM networks. IEEE Control Systems Magazine, 2001, pp [12] Jagannathan S., Talluri J.: Preictive congestion control of ATM networks: multiple sources/single buffer scenario. Automatica, 2002, Vol. 38, pp [13] Laberteaux K., Rohrs Ch., Antsaklis P.: A practical controller for explicit rate congestion control. IEEE Transactions on Automatic Control, 2002, Vol. 47, pp [14] Lengliz I., Kamoun F.: A rate-base flow control metho for ABR service in ATM networks. Computer Networks, 2000, Vol. 34, pp [15] Mascolo S.: Congestion control in high-spee communication networks using the Smith principle. Automatica, 1999, Vol. 35, pp [16] Slotine J.J., Li W.: Applie Nonlinear Control, Prentice-Hall International Eitions, [17] Sun D.H., Zhang Q.H., Mu Z.C.: Single parametric fuzzy aaptive PID control an robustness analysis base on the queue size of network noe. Proceeings of 2004 International Conference on Machine Learning an Cybernetics, 2004, Vol. 1, pp [18] Utkin V.: Variable structure systems with sliing moes. IEEE Transactions on Automatic Control, 1977,Vol. 22, pp