Wireless Ad Hoc and Sensor Networks

90 Wireless Ad Hoc and Sensor NetworksFACT 3.3.1The activation functions are bounded by known positive values so that≤ ϕ 1max , |ˆ ϕ2( k)| ≤ ϕ2max,| ϕ̃1( k)| ≤ ϕ̃max1 , and | ϕ̃2( k)| ≤ ϕ̃2maxThe traffic rate using the feedback, uk ( ), is obtained as|ˆ ϕ 1 ( k)|1ukT x k W T( ) = ( d( + 1) − ˆ ( k )ˆ ϕ2( k ) + k v e ( k ))(3.19)and the closed-loop dynamics becomeTek ( + ) = kek ( ) + e( k) + W ( k) ̃ϕ ( k) + ε( k) + dk ( )1 v i2(3.20)where the traffic-flow modeling error is defined byTei( k) = W ̃ ( k) ̃ϕ 2 ( k).(3.21)3.3.1.1 Weight Initialization and Online LearningFor the NN control scheme that is presented in this chapter, there is nopreliminary offline analysis required for tuning the weights. Theweights are simply initialized at zero. Figure 3.3 shows that the controlleris just a simple and conventional rate-based proportional andderivative controller because the NN outputs become zero. As a result,the unknown traffic accumulation at the switch is not approximatedby the NNs initially or during the transient phase when the transmissionstarts. Standard results (Jagannathan and Talluri 2000) indicatethat for controlling traffic congestion, a conventional rate controllermay result in bounded buffer-occupancy errors if the gains are suitablyselected. Then, the closed-loop system remains stable until the NNsystem begins to learn. Subsequently, the buffer occupancy errorsbecome smaller and the cell losses eventually become small as well.Providing offline training will minimize the cell losses in the transientcondition, and the weights from offline can serve as initial weights foronline training. Therefore, the NN weights are tuned online, and theQoS improves as the NN learns the unknown traffic accumulation,f( x( k)).A comparison is shown in terms of QoS between multilayerNN with, and without, an initial learning phase. Here, thebackpropagation algorithm is used to train the NN, and the training isdescribed in subsection 3.3.3.