Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 89bounding constant known. For suitable approximation properties, it isnecessary to select a large-enough number of hidden-layer neurons. It isnot known how to compute this number for general multilayer NN. Typically,the number of hidden-layer neurons is selected after careful analysis.ε Nˆ( ( )) ˆ ( ˆ ( )),3.3.1 Controller StructureDefining the NN functional estimate in the controller byT=2 1Tf x k W ϕ V ϕ k(3.16)with Wk ˆ ( ) and Vk ˆ ( ) being the current NN weights, the next step is todetermine the weight updates so that the performance of the closed-loopbuffer error dynamics of the network is guaranteed. The structure of thecontroller is shown in Figure 3.3 and it is envisioned that a controllerexists at every ATM switch.Let W and V be the unknown ideal weights required for the approximationto hold in Equation 3.1 and assume they are bounded by knownvalues so that| | , | |W ≤W V ≤Vmaxmax(3.17)Then, the error in the weights during estimation is given by:̃W ( k) = W − Wk ˆ ( ), and Ṽk ( ) = V−Vkˆ ( ), ̃Z ( k) = Z−Zkˆ( ), (3.18)W 0where Z = ⎡ ⎤ ⎣ and0 V ⎦, ˆ WZ = ⎡ 0⎤ˆ.⎣ 0 V ⎦ˆj(⋅)NeuralnetworkZ −1Z −1d(k)x de(k)??(k) Networkk v+ +Z −1+ sources/switch x(k)FIGURE 3.3Neural network (NN) controller structure.