PDE Backstepping Control Traffic Congestion Control: of Congested Traffic A PDE Backstepping Perspective Miroslav Krstic (with Huan Yu)
Mud pump From Mud Pit Harder: PT Oil Drilling Mud-assisted drilling u c helps take cuttings away + gas ingestion into casing To Mud Pit Control PT increases penetration choke control = choke valve Sea Bed Oil & Gas Highway (single-lane) for gas, water, oil, mud, rock cuttings Choke valve = ramp metering Florent Di Meglio
Huan Yu
Control of Flow Dynamics in CONGESTED Traffic Traffic congestion problems Stop-and-go traffic oscillations Suppressing STOP-AND-GO oscilla7ons (coupled PDEs) Moving shock Moving traffic shockwave Keeping SHOCKS from driAing upstream (ODE with delays) Downstream traffic bottleneck Maximizing flow through BOTTLENECK (extremum seeking)
Control of Flow Dynamics in CONGESTED Traffic Traffic congestion problems Stop-and-go traffic oscillations Suppressing STOP-AND-GO oscilla7ons (coupled PDEs) Moving shock Moving traffic shockwave Keeping SHOCKS from driAing upstream (ODE with delays) Downstream traffic bottleneck Maximizing flow through BOTTLENECK (extremum seeking)
Control of Flow Dynamics in CONGESTED Traffic Traffic congestion problems Stop-and-go traffic oscillations Suppressing STOP-AND-GO oscilla7ons (coupled PDEs) Moving shock Moving traffic shockwave Keeping SHOCKS from driAing upstream (ODE with delays) Downstream traffic bottleneck Maximizing flow through BOTTLENECK (extremum seeking)
Boundary control framework under traffic management system Ramp metering, flow rate actuated Varying speed limit, velocity actuated
Boundary control framework under traffic management system Ramp metering, flow rate actuated Varying speed limit, velocity actuated
PDE Backstepping Control of Stop-and-Go Traffic
Aw-Rascle-Zhang model ⇢ ( x, t ) = traffic density, v ( x, t ) = traffic speed LWR model @ t ⇢ + @ x ( ⇢ v ) = 0 @ t ( v + p ( ⇢ )) + v @ x ( v + p ( ⇢ )) = V ( ⇢ ) − v ⌧ second-order , macroscopic , nonlinear hyperbolic PDEs
Freeway traffic open-loop simulation ⇢ ? = 120 vehicles / km , v ? = 10 m / s , 22 mph L = 500 m , ⌧ = 60 s Constant incoming and outgoing flow rate at q ? eigenvalues in RHP!
Control objective : ˜ ⇢ ( x, t ) → 0 , ˜ v ( x, t ) → 0 ⇢ ? , v ? ⇢ ( x, t ) , v ( x, t ) Uniform distributed vehicles on the road, constant density, velocity and flow rate. ⇢ ( x, t ) = ⇢ ( x, t ) − ⇢ ? Density disturbance: ˜ v ( x, t ) = v ( x, t ) − v ? ˜ Speed disturbance:
Linearized ARZ model (˜ ⇢ , ˜ v ) around uniform ( ⇢ ? , v ? ) ⇢ ( x, t ) = density disturbance, ˜ ˜ v ( x, t ) = speed disturbance ⇢ t + v ? ˜ ⇢ x = − ⇢ ? ˜ ˜ v x v x = − ˜ v + ˜ p v t − ( � p ? − v ? )˜ ˜ ⌧ • density disturbance ˜ ⇢ “propagates” at traffic speed setpoint v ? • speed disturbance ˜ v “ counter -propagates” at � p ? − v ?
v ) around uniform ( ⇢ ? , v ? ) Linearized ARZ model (˜ ⇢ , ˜ ⇢ ( x, t ) = density disturbance, ˜ ˜ v ( x, t ) = speed disturbance ⇢ t + v ? ˜ ⇢ x = − ⇢ ? ˜ ˜ v x v x = − ˜ v + ˜ p v t − ( � p ? − v ? )˜ ˜ ⌧ ⇢ “propagates” at traffic speed setpoint v ? • density disturbance ˜ v “ counter -propagates” at � p ? − v ? • speed disturbance ˜
Coupled 2 × 2 hyperbolic PDE model ¯ w r 1 r 0 U ( t ) ¯ v x = 0 x = L w + v ? @ x ¯ @ t ¯ w =0 v − ( � p ? − v ? ) @ x ¯ @ t ¯ v = c ( x ) ¯ w w (0 , t ) = − r 0 ¯ ¯ v (0 , t ) ¯ v ( L, t ) = r 1 ¯ w ( L, t ) + U ( t ) Positive feedback throughout the domain
green/red light Full-state feedback control design at downstream ramp Theorem. Full-state feedback control law U out ( t ) = − r 0 ⇢ ? ( v ( L, t ) − v ? ) ✓ ⇠ Z L ◆� + r 0 ⇢ ? ( v ( ⇠ , t ) − v ? ) d ⇠ M ( L − ⇠ ) − r 0 K ( L, ⇠ ) exp ⌧ v ? 0 ✓ ⇠ Z L ◆ ( ⇢ ( ⇠ , t ) v ( ⇠ , t ) − ⇢ ? v ? ) d ⇠ , + k 0 0 K ( L, ⇠ ) exp ⌧ v ? makes equilibrium ¯ w ≡ ¯ v ≡ 0 exponentially stable in L 2 and equilibrium reached in finite time t = t f .
Full-state feedback control design: backstepping w t = − v ? ¯ ¯ w x v t =( � p ? − v ? )¯ ¯ v x + c ( x ) ¯ w w (0 , t ) = − r 0 ¯ ¯ v (0 , t ) ¯ v ( L, t ) = r 1 ¯ w ( L, t ) + U ( t ) Backstepping transformation: ↵ ( x, t ) = ¯ w ( x, t ) Z x Z x � ( x, t ) =¯ v ( x, t ) − 0 M ( x − ⇠ )¯ v ( ⇠ , t ) d ⇠ − 0 K ( x, ⇠ ) ¯ w ( ⇠ , t ) d ⇠ Kernel equations ( � p ? − v ? ) K x − v ? K ⇠ = c ( ⇠ ) K ( x − ⇠ , 0) K ( x, x ) = − c ( x ) � p ? where { 0 ≤ ⇠ ≤ x ≤ L } M ( x ) = − K ( x, 0)
Full-state feedback simulation ⇢ ? = 120 vehicles / km , v ? = 10 m / s , L = 500 m , ⌧ = 60 s
Boundary observer for traffic estimation Boundary measurement (of velocity and flow): Y ( t ) = ¯ w ( L, t ) Observer w t = − v ? ˆ ˆ w x + r ( x )( Y ( t ) − ˆ w ( L, t )) v t =( � p ? − v ? )ˆ ˆ v x + c ( x ) ˆ w + s ( x )( Y ( t ) − ˆ w ( L, t )) w (0 , t ) = − r 0 ˆ ˆ v (0 , t ) ˆ v ( L, t ) = r 1 Y ( t ) + U ( t ) Output injection gains r ( x ) and s ( x ) need to be designed
Density and velocity estimates estimation completed in 75 sec
Data validation of boundary observer Next Generation Simulation (NGSIM) traffic data I-80 in California
Traffic density and velocity for 5 : 15 pm - 5 : 30 pm mean velocity 12 mph traffic field data in RUSH HOUR traffic estimator converges states estimation in 3-5 minutes
Estimation errors of boundary observer 50 velocity errors percentage (%) density 40 30 20 10 0 0 2 4 6 8 10 time (min)
Two-lane and Two-class traffic congestion control
Two-lane and Two-class traffic congestion control Lane changing segregates drivers Mixed vehicle types induce creeping into the more ”risk-tolerant” ones in effect where the slow and bulky ve- the fast lane and more ”risk-averse” hicles block the traffic and small and ones in the slow lane . fast vehicles getting through.
Two-lane and Two-class traffic congestion control Lane changing segregates drivers Mixed vehicle types induce creeping into the more ”risk-tolerant” ones in effect where the slow and bulky ve- the fast lane and more ”risk-averse” hicles block the traffic and small and ones in the slow lane . fast vehicles getting through.
lane changes ~ heat exchanger Two-lane ARZ model Coupled 4 × 4 nonlinear first-order hyperbolic PDEs @ t ⇢ f + @ x ( ⇢ f v f ) = 1 ⇢ s − 1 ⇢ f T s T f ⇢ f v f + ⇢ f ( V ( ⇢ f ) − v f ) f ) − ( ⇢ f � p f ) @ x v f = 1 ⇢ s v s − 1 @ t ( ⇢ f v f ) + @ x ( ⇢ f v 2 T e T s T f f @ t ⇢ s + @ x ( ⇢ s v s ) = 1 ⇢ f − 1 ⇢ s T f T s s ) − ( ⇢ s � p s ) @ x v s = 1 ⇢ f v f − 1 ⇢ s v s + ⇢ s ( V ( ⇢ s ) − v s ) @ t ( ⇢ s v s ) + @ x ( ⇢ s v 2 T e T f T s s T e i relaxation time that reflects driver’s behavior adapting to the traffic equi- librium velocity in lane i . T i describe driver’s preference for remaining in lane i , which relates to local density and velocity.
Two-class ARZ model Coupled 4 × 4 nonlinear first-order hyperbolic PDEs @ t ⇢ 1 + @ x ( ⇢ 1 v 1 ) =0 @ t ( v 1 + p 1 ( AO )) + v 1 @ x ( v 1 + p 1 ( AO )) = V e, 1 ( AO ) − v 1 ⌧ 1 @ t ⇢ 2 + @ x ( ⇢ 2 v 2 ) =0 @ t ( v 2 + p 2 ( AO )) + v 2 @ x ( v 2 + p 2 ( AO )) = V e, 2 ( AO ) − v 2 ⌧ 2 • Vehicle class i density ⇢ i ( x, t ) and speed v i ( x, t ) with i = 1 , 2 • Dependence on “Area Occupancy” AO ( ⇢ 1 , ⇢ 2 ) = a 1 ⇢ 1 ( x,t )+ a 2 ⇢ 2 ( x,t ) W
Coupled 4 × 4 nonlinear first-order hyperbolic PDEs Two-lane Two-class ˜ w f ˜ w 1 ˜ ˜ VSL w 2 v f ˜ ˜ w 3 w s ˜ VSL v s ˜ v U ( t ) v i ( L, t ) = U i ( t ) ˜ x = 0 L x = 0 L 2 + 2 coupled PDE system 3 + 1 coupled PDE system
Two-lane traffic full-state feedback simulation slow lane fast lane
Two-class traffic full-state feedback simulation Class1 small and fast Class2 big and slow
Bilateral Boundary Control of Moving Shockfront Moving Traffic Shockwave
Model with state-dependent delays D f ( t ) D c ( t ) U in ( t ) U out ( t ) ˙ X ( t ) = − b ( U in ( t − D f ( X ( t ))) + U out ( t − D c ( X ( t ))) shock D f ( t ) = l ( t ) D c ( t ) = L − l ( t ) u , u Predictor feedback design is to compensate the input delays by feeding back the future states so that the inputs arrive at the moving interface as if without delays.
LWR traffic model ρ f − ρ 2 ! ∂ ∂ density of “FREE traffic” f ∂ t ρ f = − v m (upstream of shock) ∂ x ρ m ρ c − ρ 2 ! ∂ ∂ c ∂ t ρ c = − v m density of “CONGESTED traffic” (upstream) ∂ x ρ m dtl ( t ) = v m − v m d ( ρ c ( l ( t ) , t ) + ρ f ( l ( t ) , t )) shock front location ρ m
LWR traffic model ρ f − ρ 2 ! ∂ ∂ density of “FREE traffic” f ∂ t ρ f = − v m (upstream of shock) ∂ x ρ m ρ c − ρ 2 ! ∂ ∂ c density of “CONGESTED traffic” ∂ t ρ c = − v m (downstream of shock) ∂ x ρ m dtl ( t ) = v m − v m d ( ρ c ( l ( t ) , t ) + ρ f ( l ( t ) , t )) shock front location ρ m
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