Modeling and Optimization of Traffic Flow in Urban Areas - Czech ...

2.4 Control of The Simple Intersection Model 13w(k)q(k)Simple intersection modelx M(k)t swq q 0 t swZOHC∙Fig. 2.4: LQR control loopAssuming a control law in the formq ′ (k) = (q ′ 1(k), q ′ 2(k)) T = κ x M (k) (2.15)and solving the LQR Riccati equation gives the optimal feedback gain[ ]−0.001 −0.020 0.000 −0.012κ =0.001 0.016 −0.001 −0.062This control law produces a potentially unbounded result q ′ (k), whichcannot be directly applied to the intersection traffic control. Instead, fromthis result we compute the switching time t sw ast sw (k) = Tq ′ 1 (k)q ′ 1 (k) + q′ 2 (k) (2.16)The final control law q(k) is obtained by combining this expression and (2.12).The control law is computed at the start of the cycle time and is held for thewhole cycle time. Control loop schema is shown on Fig. 2.4.The LQR was applied to the simple intersection model control (2.9). Thesimulated response to real input data during one day (i.e. 86400 seconds) isshown in Fig. 2.5(a). The resulting average waiting times in the two queuesare significantly different, the error caused by the linearization of the model.2.4.2 Non-Linear Model Predictive ControllerNext, we consider controlling the waiting times in the simple intersectionmodel using a non-linear model predictive controller (NMPC) [Find 02,