Wireless Ad Hoc and Sensor Networks

Predictive Congestion Control for Wireless Sensor Networks 439REMARK 3The preceding theorem using the Lyapunov method shows that underthe ideal case of no errors in traffic estimation and with no disturbances,the control scheme will ensure that the actual queue level converges tothe target value asymptotically.THEOREM 9.3.2 (GENERAL CASE)Consider the desired buffer length, q id , to be finite, and the disturbancebound, d M , to be a known constant. Let the virtual-source rate for Equation 9.2be given by Equation 9.5 with the network traffic being estimated properly suchthat the approximation error ̃fi() ⋅ is bounded above by f M . Then, the bufferoccupancy feedback system is globally uniformly bounded provided 0< < 1.PROOF Let us consider the Lyapunov function candidate J = [ ebi( k)] 2 .Then, the first difference isk bv∆J = [ k e ( k) + f̃( u ( k)) + d( k)] −[ e ( k)]2 2bv bi i i+1bi(9.8)The stability condition ∆J ≤ 0 is satisfied if and only ife > ( f + d ) ( 1−k)M M bvmax(9.9)When this condition is satisfied, the first difference of the Lyapunovfunction candidate is negative for any time instance k. Hence, the closedloopsystem is globally uniformly bounded.REMARK 4The preceding theorem using the Lyapunov method shows that underthe general case of where errors in traffic estimation are upper-boundedand with bounded disturbances, the control scheme will ensure that theactual queue level converges close to the target value.Next the outgoing traffic function is estimated, using a vector of trafficparameters θ , by fi( ui+ 1( k)) = θ⋅ fi( k− 1) + ε( k), where fi( k− 1)is the pastvalue of the outgoing traffic and the approximation error ε( k)is assumedbounded by the known constant ε N . Now, define traffic estimate in thecontroller as f ˆi ( u i+ 1 ( k)) = ˆ( θ k) f i ( k −1) , where θiˆ ( k ) is the actual vector oftraffic parameters, f ˆi ( u i+1 ( k))is an estimate of the unknown outgoing trafficfi( ui+1( k)), and fi( k− 1)is the past value of the outgoing traffic.THEOREM 9.3.3 (NO TRAFFIC ESTIMATION ERROR)Given the aforementioned incoming rate selection scheme with variable θ i estimatedaccurately (no estimation error) and the backoff interval updated as in