Wireless Ad Hoc and Sensor Networks
Wireless Ad Hoc and Sensor Networks Wireless Ad Hoc and Sensor Networks
Congestion Control in ATM Networks and the Internet 115Substituting the buffer error dynamics Equation 3.42 in Equation 3.47yields( v )2 2∆J ≤− 1−k max |( e k)|.(3.48)The closed-loop system is globally asymptotically stable.COROLLARY 3.4.1Let the desired buffer length x d be finite and the network disturbancebound, d M be equal to zero. Let the source rate for Equation 3.37 be provided byEquation 3.41, then the packet losses ek ( ) approach zero asymptotically.REMARK 1This theorem shows that in the absence of bounded disturbing traffic patterns,the error in buffer length ek ( ) and packet losses converges to zero asymptotically,for any initial condition. The rate of convergence and transmission delays andnetwork utilization depends upon the gain matrix k v .THEOREM 3.4.2 (GENERAL CASE)Let the desired buffer length x d be finite and the disturbance bound d M be a knownconstant. Take the control input for Equation 3.37 as Equation 3.41 with anestimate of network traffic such that the approximation error, ̃f (), ⋅ is boundedabove by f , then the error in buffer length ek ( ) is GUUB, providedM0 < k < Iv(3.49)PROOF Let us consider the following Lyapunov function candidate,Equation 3.46, and substitute Equation 3.42 in the first difference Equation3.47 to getTT∆J = ( k e+ f̃+ d) ( k e+ f̃+ d) −e ( k) e( k).vv(3.50)∆J ≤ 0 if and only if( k | e| + f̃+ d ) < | e|vmaxM(3.51)or| e|>f +M dM( 1− k )vmax(3.52)
116 Wireless Ad Hoc and Sensor NetworksHere, any rate-based scheme can be used, including the robust controlapproach as long as the bound on the approximate error in traffic accumulationis known.COROLLARY 3.4.2 (GENERAL CASE)Let the desired buffer length x d be finite and the disturbance bound, d M , be aknown constant. Take the control input for Equation 3.37 as Equation 3.43, thenthe packet losses are finite.REMARK 2This theorem shows that the error in queue length, e(k), is bounded, and thepacket losses are also bounded and finite for any initial state of the network. Therate of convergence depends upon the gain matrix.CASE II (UNKNOWN CONGESTION LEVEL/ARRIVAL AND SERVICE RATES)In this case, a neural network (NN)-based adaptive scheme is proposed but anytype of function approximator can be applicable. Assume, therefore, that thereexist some constant ideal weights W and V for a two-layer NN, so that thenonlinear traffic accumulation function in Equation 3.48 can be written as:T Tf( x) = W ϕ( V xk ( )) + ε( k)(3.53)where V is the input to hidden-layer weights kept constant throughoutTthe tuning process after selecting it randomly, ϕ( V xk ( )) is the hiddenlayersigmoidal activation function vector, and the approximation error|( ε k)|≤ εNwith the bounding constant ε N is known. Note, that by nottuning V and randomly selecting initially, the hidden-layer sigmoidalfunctions form a basis (Jagannathan 2006).3.4.2.2 Congestion Controller StructureDefining the NN traffic estimate in the controller byˆ( ( )) ˆ Tf x k = W ( k) ϕ( x( k))(3.54)with ˆ ( ) being the current value of the weights, let W be the unknownideal weights required for the approximation to hold in Equation 3.53 andassume they are bounded so thatWk| | maxW≤ W(3.55)
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116 <strong>Wireless</strong> <strong>Ad</strong> <strong>Hoc</strong> <strong>and</strong> <strong>Sensor</strong> <strong>Networks</strong>Here, any rate-based scheme can be used, including the robust controlapproach as long as the bound on the approximate error in traffic accumulationis known.COROLLARY 3.4.2 (GENERAL CASE)Let the desired buffer length x d be finite <strong>and</strong> the disturbance bound, d M , be aknown constant. Take the control input for Equation 3.37 as Equation 3.43, thenthe packet losses are finite.REMARK 2This theorem shows that the error in queue length, e(k), is bounded, <strong>and</strong> thepacket losses are also bounded <strong>and</strong> finite for any initial state of the network. Therate of convergence depends upon the gain matrix.CASE II (UNKNOWN CONGESTION LEVEL/ARRIVAL AND SERVICE RATES)In this case, a neural network (NN)-based adaptive scheme is proposed but anytype of function approximator can be applicable. Assume, therefore, that thereexist some constant ideal weights W <strong>and</strong> V for a two-layer NN, so that thenonlinear traffic accumulation function in Equation 3.48 can be written as:T Tf( x) = W ϕ( V xk ( )) + ε( k)(3.53)where V is the input to hidden-layer weights kept constant throughoutTthe tuning process after selecting it r<strong>and</strong>omly, ϕ( V xk ( )) is the hiddenlayersigmoidal activation function vector, <strong>and</strong> the approximation error|( ε k)|≤ εNwith the bounding constant ε N is known. Note, that by nottuning V <strong>and</strong> r<strong>and</strong>omly selecting initially, the hidden-layer sigmoidalfunctions form a basis (Jagannathan 2006).3.4.2.2 Congestion Controller StructureDefining the NN traffic estimate in the controller byˆ( ( )) ˆ Tf x k = W ( k) ϕ( x( k))(3.54)with ˆ ( ) being the current value of the weights, let W be the unknownideal weights required for the approximation to hold in Equation 3.53 <strong>and</strong>assume they are bounded so thatWk| | maxW≤ W(3.55)
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