By D. Bayliss (auth.), R. Conti, A. Ruberti (eds.)

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For such systems 51 TABLE 2 6. 8 Undersaturated systems The problem of optimal control of undersaturated intersections is not a par- ticularly critical one since the aggregate delays are fairly small in magnitude. An undersaturated junction is one where no queues exist on an arm at the end of the green period. It is not valid therefore to use the coarse averaging period of one cycle as for oversaturated intersections since it is the time spent queuing during the cycle which causes the delays. Thus for real time control of undersaturated systems, it may be necessary to sample as often as once a second.

P(N-l) ) Then the maximisation of the dual function overall optimum. function The dual function L (x,u,p) ~ (p) .... with respect to (9) p yields the ~ (p__) can be written in terms of the Lagrangian as = rain [ L(x,u,p) : equations (7) and (8) ] .... (iO) x_,u_ The Lagrangian L(x,u,~_) is defined as N-I L (x,u,p_) = [ IIxCk)II2 ÷ II_

1 (k) = O for k In the Tamura algorithm, less than zero. This can be 49 achieved by choosing the control variable to be deviations around zero. 475. 475 + Au . 225 where Au FO. 475j -60 6 and the cost f~mction is modified to: m-1 j=~ k=O 5. i I4 2 Example 1 In an initial study, the inflow rates qll ' q21 and q22 were chosen to be 20, 30 and 30 vehicles per cycle and initial queues were assumed to be: Xll = lOO , x21 = i00 , Xl2 = 40 , x22 = i00 . such a situation could arise during the peak of the rush hour when inflow rates are still quite large and big queues have built up.