Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics

Transcription

1 Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics Jack Haddad Technion Sustainable Mobility and Robust Transportation (T-SMART) Lab Faculty of Civil and Environmental Engineering Technion - Israel Institute of Technology webpage: haddad.net.technion.ac.il Nov 11, 214 Technion Sustainable Mobility and Robust Transportation Laboratory Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 1 / 3

2 6.2 Model Classification 57 Macroscopic ρ (x,t) Model Classification of Traffic Flow Models (aggregation level) Microscopic Model v (t) α n=1 Cellular n= 6.2 Model Automaton Classification (CA) 57 Macroscopic Pedestrian Model Model (v xα(t), v yα(t)) ρ (x,t) Fig. 6.2Microscopic Comparison of various model categories (with respect to the way they represent reality) including Model typical model equations Macroscopic v α (t) network model n=1 Cellular n= Macroscopic Automaton models (CA) describe traffic flow analogously to liquids or gases in motion. Hence they are sometimes called hydrodynamic models.thedynamicalvariablesare Pedestrian Model (v xα(t), v yα(t)) locally aggregated quantities such as the traffic density ρ(x, t), flow Q(x, t), mean speed V (x, t),orthespeedvarianceσv 2 (x, t).becausetheaggregationislocal,these quantities generally vary across space and time, i.e., they correspond to dynamic fields. Thus,macroscopicmodelsareabletodescribecollective phenomena such Fig. 6.2 Comparison of various model categories (with respect to the way they represent reality) including as the evolution typical model of congested equations regions or the propagation velocity of traffic waves. Furthermore, macroscopic model are useful, if effects that are difficult to describe macroscopically need not to be considered Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 2 / 3

3 6.2 Model Classification 57 Macroscopic ρ (x,t) Model Classification of Traffic Flow Models (aggregation level) Microscopic Model v (t) α n=1 Cellular n= 6.2 Model Automaton Classification (CA) 57 Macroscopic Pedestrian Model Model (v xα(t), v yα(t)) ρ (x,t) Fig. 6.2Microscopic Comparison of various model categories (with respect to the way they represent reality) including Model typical model equations Macroscopic v α (t) network model n=1 u 21 (t) Cellular n= q 21 (t) Macroscopic Automaton models (CA) describe traffic flow analogously to q 12 liquids (t) u 12 (t) or gases in motion. Hence they are sometimes called hydrodynamic models.thedynamicalvariablesare Pedestrian Model (v xα(t), v yα(t)) 2 1 locally aggregated quantities such as the traffic densityqρ(x, t), flow Q(x, t), mean speed V (x, t),orthespeedvarianceσv 2 11 (t) (x, t).becausetheaggregationislocal,these quantities generally vary across space and time, i.e., they correspond to dynamic fields. Thus,macroscopicmodelsareabletodescribecollective phenomena such Fig. 6.2 Comparison of various model categories (with respect to the way they represent reality) including as Aggregated the evolution typical model of network-level congested equations regions approach or theto propagation large-scale velocity urbanofmodeling traffic waves. Furthermore, macroscopic model are useful, if effects that are difficult to describe macroscopically need not to be considered Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 2 / 3

4 Fundamental diagram for a link i Three traffic regimes: undersaturated, saturated, oversaturated (if flow is restricted). Link i Trip completion rate [veh/sec] Saturated Oversaturated Undersaturated Accumulation [veh] Density[veh/km] Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 3 / 3

5 Fundamental diagram for a link i Three traffic regimes: undersaturated, saturated, oversaturated (if flow is restricted). Link i Single Detectors 1 Trip completion rate [veh/sec] Saturated.75 Oversaturated flow qi /max {qi }.5 Undersaturated.25 Detector #: 1-3D Detector #: T7-5D Accumulation [veh] Density[veh/km] o i (%) occupancy Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 3 / 3

6 Macroscopic Fundamental Diagram (MFD) for an urban region MFD links space-mean flow, density, and speed of a large urban area. G(n) [veh/sec] Trip completion rate MFD i Trip completion rate for link i ncr njam n [veh] Accumulation Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 4 / 3

7 Macroscopic Fundamental Diagram (MFD) for an urban region MFD links space-mean flow, density, and speed of a large urban area. G(n) [veh/sec] Trip completion rate MFD i Trip completion rate for link i ncr njam n [veh] Accumulation Average network flow and trip completion rate Average network flow F F = i f i l i i l i where: f i (veh/s) - flow in link i, l i (m) - length of link i. Trip completion rate/average network flow constant G/F constant. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 4 / 3

8 Macroscopic Fundamental Diagram (MFD) for an urban region MFD links space-mean flow, density, and speed of a large urban area. G(n) [veh/sec] Trip completion rate MFD i Trip completion rate for link i ncr njam n [veh] Accumulation San Francisco (simulation) 15 Geroliminis and Daganzo (27) Tr. Res. Board Yokohama (experiment) 45 Geroliminis and Daganzo (28) Tr. Res. Part B 12 T r a v e l P r o d u c VKT t i o n 9 6 Average flow q u (vhs/5min) 3 15 A1 B1 C1 D1 A2 B2 C2 D Vehicle Accumulation 6 A ti o u (%) Average occupancy Other recent MFD studies (UC Berkeley, EPFL, TU Delft, Northwestern, INRETS, TUC, Penn State, UCL, Technion, etc.) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 4 / 3

9 Perimeter Traffic Flow Control for an Urban Region 2 u 21 (t) q 12 (t) u 12 (t) 1 q 11 (t) q 21 (t) Trip completion flow G1(n1(t)) (veh/s) n 1 n 1,jam Accumulation, n 1(t) (veh) Literature survey: perimeter control for a single MFD system Daganzo (27): the optimal control policy was presented for a single MFD system (bang-bang control). Explicit proof in Haddad (214) based on Modified Krotov-Bellman sufficient conditions of optimality. Keyvan-Ekbatani et al. (212): a classical feedback control approach. Haddad and Shraiber (214): Robust perimeter control design for an urban region, Transportation Research Part B, 68, pp , 214. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 5 / 3

10 Effect of Perimeter Control No Control With Control Trips Ended Trips Ended Time Time Video Video (Videos provided by LUTS Laboratory, EPFL, Switzerland) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 6 / 3

11 Well-defined MFD? Mazloumian, Geroliminis, and Helbing (21): an urban region with small variance of link densities has well-defined MFD. homogeneous distribution of congestion Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 7 / 3

12 Well-defined MFD? Mazloumian, Geroliminis, and Helbing (21): an urban region with small variance of link densities has well-defined MFD. homogeneous distribution of congestion uneven distribution of congestion Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 7 / 3

13 Perimeter Traffic Flow Control for Two Regions Region 2 Region 1 Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 8 / 3

14 . and u Macroscopic modeling and 21(t) 1, control are the ratio of the transfer state of the model, see Fig. 2. The open-loop optimization control Optimal perimeter control Numerical verifications Conclusions flow that transfers from R 1 to R 2 and R 2 to R 1 at time t, problem yields a sequence of optimal control variables after respectively. several iterations of solving nonlinear programming, and the The criterion is to maximize the output of the traffic first control in this sequence is applied to the plant, then the network, i.e. the number of vehicles that complete their trips procedure is carried out again. and reach their destinations. Therefore, the two-region MFDs Optimal perimeter control for two urban regions with Macroscopic Fundamental Diagrams Horizon 1 Horizon 2 control problem with four state variables is formulated as follows (similarly to [28]): tf [ ] J = max u12(t),u21(t) M11(t) + M22(t) dt (1) t subject to u12(t) R1 dn11(t) q12(t) = q11(t) + u21(t) q21(t) M21(t) u21(t) M11(t) (2) Region 2 dt Region 1 R2 dn12(t) = q12(t) u12(t) q22(t) M12(t) (3) dt dn21(t) = q21(t) u21(t) q11(t) M21(t) (4) dt dn22(t) = q22(t) + u12(t) M12(t) M22(t) (5) dt n11(t) + n12(t) (6) n21(t) + n22(t) (7) n11(t) + n12(t) n1,jam (8) Prediction n21(t) + horizon n22(t) n2,jam (9) umin u12(t) umax (1) umin u21(t) umax (11) Time tkc 1 tkc tkc+np 1 n11(t) = n11, ; n12(t) = n12, (12) n21(t) = n21, ; n22(t) = n22, where t f [sec] is the final time, n ij,, i, j = 1, 2 are the initial Time tkc 1 tkc tkc+np 1 accumulations at t, n 1,jam and n 2,jam [veh] are the accumulations at the jammed density in R 1 and R 2, respectively, u min and u max are the lower and upper bounds for u 12(t), u 21(t), respectively. Recall that M ij = (n ij/n i) G i(n i(t)), i, j = 1, 2. The equations (2) (5) are the conservation of mass equations for n ij(t), while the equations (6), (7) and (8), (9) are the lower and upper bound constraints on accumulations in RFundamental 1, R 2, respectively. Diagrams. Main contributions This scheme of feedback control, i.e. the feedback loop of states from the plant to the model as initial states for the optimization, can handle the errors between the prediction model and the plant. MPC for perimeter control Fig. 2. q(t) q(t) Two-region MFDs plant dñ(t) dt = f(ñ(t), u(k), q(t), ε(k)) u (kc) tkc 1 t ñ(tkc ) tkc G1(n1) G2(n2) ε(kc) Two-region MFDs prediction model dn(t) dt = f(n(t), u(k), q(t)) tk 1 t tk, k = kc,, kc + Np 1 The MPC controller obtains the optimal control sequence III. MODEL PREDICTIVE CONTROL FOR TWO-REGION for the current horizon by solving an optimization problem OptimalMFDS Perimeter PROBLEM Control Synthesis for Two Urbanformulated Regions with withboundary prediction Queue model, Dynamics see bottom of Fig / 3 u(kc ) u(kc + 1) u(kc + Np 1) G1(n1) MPC controller G2(n2) Maximizing the number of trips ended (Open-loop optimization problem) kc = kc + 1 n(tkc 1) = ñ(tkc 1) Model predictive control scheme for two-region MFDs system. Formulation the perimeter control problem of two urban regions by the Macroscopic A. Two-region MFDs prediction model and optimization problem Solving the control problem by Model Predictive Control.

15 4 GC Accu MPC Accu Macroscopic modeling control Optimal perimeter control without errors (αand α2 = ) small errors (α1 1 = 3 eh] GC [veh] MPC-GC [veh sec] MPC GC (%2.6) (%2.7) Example: different levels of (%3.3) 8 x 14 2 MFD2 Flow [veh/sec] G(n) [veh/sec] % demand 4 MPC GC Time [sec] (b) 84% demand errorsconclusions (α1 = α GC MP MFD Cumulative trip completion [veh] Time [sec] (a) 1% demand 1 2 Time [sec] (c) Cumulative trip completion [veh] verifications = α2numerical =.2) large MPC-GC MPC (%2.6) (%2.7) demand (%3.3) x 1 q q q 21 MPC GC q MPC GC Time [sec] 1(d) Time [sec] Time [sec] Accumulation u u [veh] u 21MPC 12GC 21GC (b) e 1: small errors themfd differences between the total8delays are related to(a) the congestion level, 1 in 8 the greedy controller performs similar to the MPC controller in MFD uncongested situation. 1 MFD1 MFDtrip completion for (a) example 2 Fig. 7. The cumulative MFD Journal paper u 12MPC 6 u [ ] h/sec] 5 ) [veh/sec] and (c) example 4. N. Geroliminis, J. Haddad, and M. Ramezani, Optimal perimeter control for two urban regions with Macroscopic Fundamental Diagrams: A model predictive approach, IEEE Transactions on Intelligent Transportation Systems, vol. 14, no. 1, pp , Optimal Perimeter Control Synthesis for Two Queue Dynamics 8 nurbannregions n withn Boundary n n 13 1 u/12mpc

16 Macroscopic modeling and control Optimal perimeter control Numerical verifications Conclusions n1 (t)[veh] Stability Analysis of Perimeter Control n1,jam 2 III 18 IV n1,cr n1(t) [veh] MFD for region 1 14 unstable node saddle point I II γ1 G1 (n1 ) [veh/sec] stable node 4 saddle point n2,cr n2,jam n2 (t)[veh] MFD for region n2(t) [veh] γ2 G2 (n2 ) [veh/sec] Main contributions in stability analysis analysis of the dynamic equations. stability characterization algorithm. a state-feedback control strategy. Figure 7: Numerical example 1 demonstrates case a: the trajectories are in green and the red curve is the boundary. 4. if n2,b > µ2, then calculate trajectory from point B to the unstable equilibrium point (n2,eq, n1,eq )IV direction according to (A.3), (A.4), and (A.5) in Appendix A.1, with initial state point B and t = trajectory B-(n2,eq, n1,eq)iv does not enter the state region III, i.e. does not intersect the line n2 (t) = µ case a, otherwise it is case b: case a: draw a horizontal line stars from the unstable equilibrium point (n2,eq, n1,eq )IV moves saddle point (n2,eq, n1,eq )III, and ends at n2 (t) =. The line is horizontal according to the cor (a) Numericaleigenvalue example 6: RAs for surface boundaries in three-state point two-region eigenvector of the negative the saddlesurfaces equilibrium insystem, state region III. RAumax and RAumin surface boundaries are drawn in red and cyan, respectively, and stable and Journal paper 2 5. if n2,b = µ2 then it is case c. Calculate trajectory from points B to C and C to D in reverse way a Appendix A.3. J. Haddad and N. Geroliminis, On the stability of perimeter traffic control in two-region urban cities, Transportation Research Part B, 46, pp , 212. Note that the RA boundary curve is4 combined from several trajectories some of them are calculated numeri 2 other trajectories are calculated analytically, see Appendix A.1, Appendix A.2, and Appendix A.3. Optimal Perimeter Control Synthesis Twoof Urban Regions with / 33, a Boundary Thefor region attraction boundaries for cases a,queue b, anddynamics c are demonstrated by examples 1, 11 2, and n (t) [veh] unstable trajectories correspond to umax are drawn in green. case b: calculate trajectories from points B to C and C to D in reverse way, according to Appe

17 Cooperative Control for Mixed Urban-Freeway Networks Urban network Freeway MUF netwok Freeway (3) Region (2) Freeway (3) Region 2 Region 1 + Region (1) two-region MFDs dynamics asymmetric cell transmission model (ACTM) MUF dynamics Different levels of coordination Region 2 Region 1 C-MPC: Centralized MPC (network delay). CD-MPC: Cooperative Decentralized MPC (network delay). D-MPC: Decentralized MPC (freeway and urban delays). ALINEA: ALINEA control for theregion freeway (2) and u max for urban network. Freeway (3) Region (1) Journal paper J. Haddad, M. Ramezani, and N. Geroliminis, Cooperative Traffic Control of Mixed Urban and Freeway Networks, Transportation Research Part B, 54, pp , 213. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 12 / 3

18 Main contributions in this talk modeling and integrating the boundary queue dynamics, perimeter control policy taking into account the maximum and minimum boundary queue constraints. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 13 / 3

19 Two-region MFD system Two Urban Regions with Boundary Queue Dynamics Traffic terminology: demands: q 11 (t), q 12 (t), q 21 (t), q 22 (t) (veh/s) accumulations: n 1 (t), n 2 (t), n 12 (t), n 21 (t) (veh) exit flows of MFDs: G 1 ( n1 (t) ), G 2 ( n2 (t) ) (veh/s) perimeter control inputs: u 1 (t) and u 2 (t) (-) u 1 (t), u 2 (t) 1 perimeter saturation flow: d (veh/s) q 12(t) u 1(t) Region 1 q 11(t) q 21(t) Region 2 q 22(t) u 2(t) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 14 / 3

20 Two-region MFD system Two Urban Regions with Boundary Queue Dynamics q12(t) u1(t) Region 1 q21(t) Region 2 q22(t) q11(t) u2(t) Traffic terminology: demands: q 11 (t), q 12 (t), q 21 (t), q 22 (t) (veh/s) accumulations: n 1 (t), n 2 (t), n 12 (t), n 21 (t) (veh) exit flows of MFDs: G 1 ( n1 (t) ), G 2 ( n2 (t) ) (veh/s) perimeter control inputs: u 1 (t) and u 2 (t) (-) u 1 (t), u 2 (t) 1 perimeter saturation flow: d (veh/s) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 15 / 3

21 Two-region MFD system Region 1 q11(t) q12(t) u1(t) q22(t) Two Urban Regions with Boundary Queue Dynamics q21(t) Region 2 u2(t) Dynamic equations Traffic terminology: demands: q 11 (t), q 12 (t), q 21 (t), q 22 (t) (veh/s) accumulations: n 1 (t), n 2 (t), n 12 (t), n 21 (t) (veh) exit flows of MFDs: G 1 ( n1 (t) ), G 2 ( n2 (t) ) (veh/s) perimeter control inputs: u 1 (t) and u 2 (t) (-) u 1 (t), u 2 (t) 1 perimeter saturation flow: d (veh/s) dn 1 (t) = q 11 (t) + q 12 (t) + u 2 (t) d G 1 (n 1 ), dt dn 2 (t) = q 21 (t) + q 22 (t) + u 1 (t) d G 2 (n 2 ), dt dn 12 (t) = q 12 (t) u 1 (t) d, dt dn 21 (t) = q 21 (t) u 2 (t) d. dt Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 15 / 3

22 Two-region MFD system Optimal perimeter control problem definition Given: time varying demands: q 11 (t), q 12 (t), q 21 (t), q 22 (t), the initial accumulation: n 1 (), n 2 (), n 12 (), n 21 (), the MFDs: G 1 (n 1 ), G 2 (n 2 ), accumulation (state) constraints: n 12 (t) n 12, control constraints: n 21 (t) n 21, u 1 (t), u 2 (t), u 1 u 1 (t) u 1, u 2 u 2 (t) u 2, u 1 (t) + u 2 (t) 1 Manipulate u 1 (t) and u 2 (t) to maximize: J = tf t (G 1 (n 1 ) + G 2 (n 2 ))dt. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 16 / 3

23 Two-region MFD system Brief description of Pontryagin s Maximum Principle Classical optimal control problem (OCP) T f (x, u)dt min (1) dx(t) = f(x, u) (2) dt x() = x, x(t ) = x T (3) u min u(t) u max (4) where: control variables u(t) R m, state variables x(t) R n, f(x, u) R n, and m n. According to PMP: H = p T f(x, u) f (x, u) (5) dp dt = H T x = f T x Hamiltonian = H, costate variables p(t) R n. If (x, u ) p such that: p + f T x (a) H(x, u, p ) H(x, u, p ), (b) x, p satisfy (2) and (6), (c) u satisfies (4), (d) the end conditions in (3) must hold. (6) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 17 / 3

24 Two-region MFD system Brief description of Pontryagin s Maximum Principle Classical optimal control problem (OCP) T f (x, u)dt min (1) dx(t) = f(x, u) (2) dt x() = x, x(t ) = x T (3) u min u(t) u max (4) where: control variables u(t) R m, state variables x(t) R n, f(x, u) R n, and m n. According to PMP: H = p T f(x, u) f (x, u) (5) dp dt = H T x = f T x Hamiltonian = H, costate variables p(t) R n. If (x, u ) p such that: p + f T x (a) H(x, u, p ) H(x, u, p ), (b) x, p satisfy (2) and (6), (c) u satisfies (4), (d) the end conditions in (3) must hold. (6) Advertisement: new course Optimal Control: Theory and Transportation Applications (undergraduate and graduate levels, civil and environmental engineering faculty). Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 17 / 3

25 Optimal control solution synthesis Optimal control solution via PMP The augmented Hamiltonian function, H, is formed as H =p n1 (t) [q 11 (t) + q 12 (t) + u 2 (t) d G 1 (n 1 ) ] +p n2 (t) [q 21 (t) + q 22 (t) + u 1 (t) d G 2 (n 2 ) ] +p n12 (t) [q 12 (t) u 1 (t) d ] + p n21 (t) [q 21 (t) u 2 (t) d ] + G 1 (n 1 ) + G 2 (n 2 ) λ 12 [n 12 (t) n 12 ] λ21 [n 21 (t) n 21 ] + λ12 n 12 (t) + λ 21 n 21 (t) λ 1 12 [q 12 (t) u 1 (t) d ] λ 1 21 [q 21 (t) u 2 (t) d ] λ 1 12 [ q 12 (t) + u 1 (t) d ] λ 1 21[ q21 (t) + u 2 (t) d ], where p n1 (t), p n2 (t), p n12 (t), p n21 (t) satisfy dp n1 (t) dt dp n2 (t) dt dp n12 (t) dt dp n21 (t) dt = H p n1 = (p n1 (t) 1) G 1(n 1 ) n 1 = H p n2 = (p n2 (t) 1) G 2(n 2 ) n 2 = H p n12 = λ 12 λ 12, = H p n21 = λ 21 λ 21. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 18 / 3

26 Optimal control solution synthesis Switching function S(t) max u1 (t), u 2 (t) H subject to the control constraints simple LP problem. If both coefficients for u 1 (t), u 2 (t) in the Hamiltonian are positive u 1 (t) + u 2 (t) = 1. Switching function S(t) (coefficient of u 2 (t)) S(t) = p n1 (t) p n21 (t) p n2 (t) + p n12 (t) λ λ λ 1 12 λ1 21. The optimal control solution obtained by max u1 (t),u 2 (t) H is u 2 (t) = u 2, u 1(t) = 1 u 2 S(t) >, u 2(t) = 1 u 1, u 1(t) = u 1 S(t) <, singular control S(t) =. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 19 / 3

27 Optimal control solution synthesis Optimal control cases Case 1: n 1 (t) < n 1 and n 2(t) < n 2 G 1 (n 1 ) n 1 >, G 2 (n 2 ) n 2 >. Case 2: n 1 (t) > n 1 and n 2(t) > n 2 n1(t)(veh) n1,jam G 1 (n 1 ) n 1 <, G 2 (n 2 ) n 2 <. Case 3.b Case 2 Case 3.a: n 1 (t) < n 1 and n 2(t) > n 2 G 1 (n 1 ) n 1 >, G 2 (n 2 ) n 2 <. MFD for region 1 n 1 Case 1 Case 3.a Case 3.b: n 1 (t) > n 1 and n 2(t) < n 2 G 1 (n 1 ) n 1 <, G 2 (n 2 ) n 2 >. G1(n1) (veh/s) n 2 MFD for region 2 n 2,jam n 2(t)(veh) G2(n2) (veh/s) subcases: (state) constrained trajectories. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 2 / 3

28 Optimal control solution synthesis Unbounded trajectories all Lagrange multipliers are equal to zero. one can choose p n12 (t) = p n21 (t) =, for t t t f. one can choose p n1 (t) >, p n2 (t) > and p n1 (t) > p n2 (t) S(t) >. Switching function S(t) (coefficient of u 2 (t)) S(t) = p n1 (t) p n21 (t) p n2 (t)+p n12 (t) λ λ λ1 12 λ1 21 >. the optimal solution is u 2 (t) = u 2, u 1 (t) = 1 u 2. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 21 / 3

29 Optimal control solution synthesis Unbounded trajectories all Lagrange multipliers are equal to zero. one can choose p n12 (t) = p n21 (t) =, for t t t f. one can choose p n1 (t) >, p n2 (t) > and p n1 (t) > p n2 (t) S(t) >. Switching function S(t) (coefficient of u 2 (t)) S(t) = p n1 (t) p n21 (t) p n2 (t)+p n12 (t) λ λ λ1 12 λ1 21 >. the optimal solution is u 2 (t) = u 2, u 1 (t) = 1 u 2. Define P n1 (t) = p n1 (t) 1 and P n2 (t) = p n2 (t) 1. recall that... dp n1 (t) dt dp n2 (t) dt = H p n1 = (p n1 (t) 1) G 1(n 1 ) n 1 = P n1 (t) G 1(n 1 ) n 1, = H p n2 = (p n2 (t) 1) G 2(n 2 ) n 2 = P n2 (t) G 2(n 2 ) n 2, Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 21 / 3

30 Optimal control solution synthesis Unbounded trajectories all Lagrange multipliers are equal to zero. one can choose p n12 (t) = p n21 (t) =, for t t t f. one can choose p n1 (t) >, p n2 (t) > and p n1 (t) > p n2 (t) S(t) >. Switching function S(t) (coefficient of u 2 (t)) S(t) = p n1 (t) p n21 (t) p n2 (t)+p n12 (t) λ λ λ1 12 λ1 21 >. the optimal solution is u 2 (t) = u 2, u 1 (t) = 1 u 2. Define P n1 (t) = p n1 (t) 1 and P n2 (t) = p n2 (t) 1. Therefore, ds dt = P n1(t) G 1(n 1 ) = P n1 (t) P n2 (t) G 2(n 2 ) n 1 n 2 [ ] G 1 (n 1 ) G 2(n 2 ) n 1 n 2 + S(t) G 2(n 2 ) n 2. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 21 / 3

31 Optimal control solution synthesis Therefore, S(t) = p n1 (t) p n2 (t) >, [ ds dt = P G 1 (n 1 ) n1(t) G 2(n 2 ) n 1 n 2 ] + S(t) G 2(n 2 ) n 2 <. choosing the initial values of the costate variables P n1 (t) and P n2 (t) to make ds/dt <. let us consider: G 1 (n 1 )/ n 1 > G 2 (n 2 )/ n 2. S(t) and P n1 (t), P n2 (t) will decrease. Singular solution: ds(t)/dt = holds if S(t) = and G 1 (n 1 ) n 1 = G 2(n 2 ) n 2. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 22 / 3

32 Optimal control solution synthesis Singular solution taking full time derivatives of G 1 (n 1 )/ n 1 and G 2 (n 2 )/ n 2, one gets ( ) d G 1 (n 1 ) = 2 G 1 (n 1 ) [q dt n 1 n 2 11 (t) + q 12 (t) + u 2 (t) d G 1 (n 1 ) ] = 1 ( ) d G 2 (n 2 ) = 2 G 2 (n 2 ) [q dt n 2 n 2 21 (t) + q 22 (t) + (1 u 2 (t)) d G 2 (n 2 ) ]. 2 Denoting a = 2 G 1 (n 1 ) n 2 1, b = 2 G 2 (n 2 ) n 2 2, c = q 11 (t) + q 12 (t) G 1 (n 1 ), e = q 21 (t) + q 22 (t) G 2 (n 2 ). The singular control inputs u 2 (t) = [b e + b d a c]/[(a + b) d], u 1 (t) = 1 u 2 (t). Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 23 / 3

33 Optimal control solution synthesis E.g. if the MFD shapes are approximated by second order polynomial functions, then the obtained singular curve n 1 (t) = θ(n 2 (t)) is linear: u 1(t) = 1 u 2, u 2(t) = u 2, n 1 (t) < θ(n 2 (t)), u 1 (t) = u 1, u 2 (t) = 1 u 1, n 1 (t) > θ(n 2 (t)), singular control n 1 (t) n 1 (t) = θ(n 2 (t)). G1(n1) n1 < G2(n2) n2 u 1(t) = u 1, u 2(t) = 1 u 1 Singular control G1(n1) n1 = G2(n2) n2 u 1(t) = 1 u 2, u 2(t) = u 2 G1(n1) n1 > G2(n2) n2 n 2 (t) Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 24 / 3

34 Optimal control solution synthesis Kelley condition Kelley condition: Second order necessary condition of optimality for the singular arc where q is a so-called degree of singularity. ( 1) q ( d 2q ) H u dt 2q, u H =S(t) = p n1 (t) p n2 (t) = P n1 (t) P n2 (t), u 2 d 2q dt 2q H u 2 = d2 S dt 2 = P n1(t) { 2 G 1 (n 1 ) [q n 2 11 (t) + q 12 (t) G 1 (n 1 ) + d u 2 (t) ] 1 2 G 2 (n 2 ) [q n 2 21 (t) + q 22 (t) G 2 (n 2 ) + (1 u 2 (t)) d ]}, 2 ( ) ( 1) q d 2q H u 2 dt 2q = P n1 (t) (a + b) d. u 2 a <, b <, and P n1 (t) is also negative Kelley condition is satisfied. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 25 / 3

35 Optimal control solution synthesis Switching from unbounded to (state-)constrained trajectories unbounded trajectories might switch to constrained trajectories, before entering to the singular arc, if the upper or lower state constraint becomes active. E.g. : the upper bound n 12 (t) = n 12 Switching function S(t) (coefficient of u 2 (t)) S(t) = p n1 (t) p n21 (t) p n2 (t) + p n12 (t) λ λ λ1 12 λ1 21. λ 1 12 (t) will become positive such that the switching function S(t) =. ds/dt will be held equal to zero by applying corresponding values of λ 1 12 (t), and the upper boundary singular control will be applied such that n 12 (t) = n 12 is satisfied. The upper boundary singular control inputs are calculated from dn 12 /dt =, i.e. u 1 (t) = q 12(t)/d, u 2 (t) = 1 u 1 (t). Switching to a lower state constraint can be analyzed in a similar way. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 26 / 3

36 Numerical verifications Unbounded trajectories (with singular solution) Accumulation [veh] 3 2 n 1 n 2 n 12 n Time [s] 3 u [ ] u 1 u Time [s] 3 n 2 [veh] G(n) [veh/s] Optimal trajectory in (n 1,n 2 ) plane n 1 [veh] MFD 1 MFD Accumulation [veh] Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 27 / 3

37 Switching to the upper state constraint n 12 = n 12 Accumulation [veh] u [ ] n 1 n 2 n 12 n Time [s] u 1 u Time [s] 6 n 2 [veh] Optimal trajectory in (n 1,n 2 ) plane n 1 [veh] 14 G(n) [veh/s] MFD 1 MFD Accumulation [veh] Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 28 / 3

38 Conclusions Conclusions the optimal perimeter control synthesis has been presented for different cases of initial accumulation conditions. the optimal control law is presented in analytical feedback form, as a function of current regional accumulations n 1 (t) and n 2 (t). Future research perimeter adaptive control based on MFD model with time delays (travel times). an application of these strategies in the field and/or in a micro-simulation environment. Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 29 / 3

39 Macroscopic modeling and control Optimal perimeter control Numerical verifications Conclusions T-SMART Monitoring System: Bluetooth sensors!! Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 3 / 3