Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 83A general function,NN as()f( x) ∈C s , can be approximated using the two-layerT Tf( x( k)) = W ϕ ( V ϕ ( x( k))) + ε( k),2 1(3.1)where W and V are constant weights and ϕ2( V T ( k) ϕ1( x( k)), ϕ1( xk ( )) denotethe vectors of activation functions at the instant k, with ε( k)an NN functionalreconstruction error vector. The net output is defined asˆ( ( )) ˆ T( ˆ Tf x k = W ϕ V ϕ ( x( k))).2 1(3.2)( k)From now on, ϕ ( xk ( )) is denoted as ϕ and is denoted as11 ϕ (2 Vˆ ϕ ( xk ( )))1ˆ( k)ϕ 2 . The main challenge now is to develop suitable weight update equationssuch that Equation 3.1 and Equation 3.2 are satisfied.T3.2.2 Stability of SystemsTo formulate the discrete-time controller, the following stability notion isneeded (Jagannathan 2006). Consider the nonlinear system given by:xk ( + 1) = f( xk ( ), uk ( ))yk ( ) = hxk ( ( )),(3.3)where x(k) is a state vector, u(k) is the input vector, and yk ( ) is the outputvector. The solution is said to be uniformly ultimately bounded (UUB) if,for all xk ( 0) = x0,there exists an µ ≥ 0 and a number N( µ , x 0 ), such that|()| xk ≤ µ for all k≥ k + N0 .3.2.3 Network ModelingFigure 3.1a shows the popular parking lot network configuration forevaluating the proposed congestion control schemes. This configurationand its name are derived from theatre parking lots, which consist ofseveral parking areas connected via a single exit path as shown in thefigure. For computer networks, an n-stage parking lot configuration consistsof n switches connected in a series as n virtual circuits (VCs). Congestionhappens whenever the input rate is more than the available linkcapacity. In other words,∑Input rate>available link capacity(3.4)