String and robust stability of connected vehicle systems with delayed feedback

String and robust stability of connected vehicle systems with delayed feedback Gopal Krishna Kamath, Krishna Jagannathan and Gaurav Raina Department of Electrical Engineering Indian Institute of Technology Madras 14 th IFAC Workshop on Time Delay Systems, 2018

Overview Problem Insight into control laws that stabilise traffic flow String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 2 / 19

Overview Problem Insight into control laws that stabilise traffic flow Implications Autonomous vehicles (with technology) Design guidelines to stabilize traffic flow Increase resource utilization Human-driven vehicles (without technology) Phenomenological insights into phantom jams [1] Explains back-propagating congestion waves [1] J. Kowszun, Jamitons: phantom traffic jams, School Science Review, vol. 350, pp. 53 61, 2013 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 2 / 19

Scenario Platoon size : N +1 Our setting Single-lane, infinitely long highway String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 3 / 19

Scenario Platoon size : N +1 Our setting Single-lane, infinitely long highway Car-following models Class of dynamical models Describe temporal variation of acceleration, velocity, position Mimic human drivers decisions String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 3 / 19

Scenario Platoon size : N +1 Our setting Single-lane, infinitely long highway Car-following models Class of dynamical models Describe temporal variation of acceleration, velocity, position Mimic human drivers decisions Basic philosophy Maintain acceptable headway Synchronise velocity with vehicle directly ahead String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 3 / 19

Scenario Platoon size : N +1 Our setting Single-lane, infinitely long highway Car-following models Class of dynamical models Describe temporal variation of acceleration, velocity, position Mimic human drivers decisions Basic philosophy Maintain acceptable headway Synchronise velocity with vehicle directly ahead Existing models Classical car-following model Optimal velocity model String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 3 / 19

Model representations Pictorial 0 1 2 N y 1 y 2 y i i i+1 x 0 x i Reference String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 4 / 19

Model representations Pictorial 0 1 2 N y 1 y 2 y i i i+1 x 0 x i Reference Symbolic x i : position of i th vehicle from fixed reference ẋ i : velocity of i th vehicle ẍ i : acceleration of i th vehicle y i : distance between i th and (i 1) th vehicle; y i = x i 1 x i v i : velocity of i th vehicle relative to (i 1) th vehicle; v i = ẋ i 1 ẋ i String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 4 / 19

Model representations Pictorial 0 1 2 N y 1 y 2 y i i i+1 x 0 x i Reference Symbolic x i : position of i th vehicle from fixed reference ẋ i : velocity of i th vehicle ẍ i : acceleration of i th vehicle y i : distance between i th and (i 1) th vehicle; y i = x i 1 x i v i : velocity of i th vehicle relative to (i 1) th vehicle; v i = ẋ i 1 ẋ i Mathematical ẋ(t) = f ( x(t),x(t τ 1 ),...,x(t τ N ) ), x R N, f C k String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 4 / 19

Classical Car-Following Model (CCFM) Original model [1] ẍ i (t) = α i (ẋ i (t)) m (ẋ i 1 (t τ) ẋ i (t τ)) (x i 1 (t τ) x i (t τ)) l, i = 1,2,...,N α i : sensitivity coefficient of i th driver τ : common reaction delay m [ 2,2],l R + : model parameters [1] D.C. Gazis, R. Herman and R.W. Rothery, Nonlinear follow-the-leader models of traffic flow, 1961 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 5 / 19

Classical Car-Following Model (CCFM) Original model [1] ẍ i (t) = α i (ẋ i (t)) m (ẋ i 1 (t τ) ẋ i (t τ)) (x i 1 (t τ) x i (t τ)) l, i = 1,2,...,N α i : sensitivity coefficient of i th driver τ : common reaction delay m [ 2,2],l R + : model parameters Transformed model [2] b i : equilibrium spacing v i (t) = β i 1 (t)v i 1 (t τ i 1 ) β i (t)v i (t τ i ) ẏ i (t) = v i (t), i = 1,2,...,N β i (t) = α i (ẋ 0 (t) v 0 (t) v i (t)) m (y i (t)+b i ) l [1] D.C. Gazis, R. Herman and R.W. Rothery, Nonlinear follow-the-leader models of traffic flow, 1961 [2] X. Zhang and D.F. Jarrett, Stability analysis of the classical car-following model, 1997 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 5 / 19

Optimal velocity model Original model [1] ẍ 1 (t) =a ( V(x N (t τ) x 1 (t τ)) ẋ 1 (t τ) ) ẍ i (t) =a ( V(x i 1 (t τ) x i (t τ)) ẋ i (t τ) ), i = 2,3,...,N a : common sensitivity coefficient τ : common reaction delay V : optimal velocity function [1] M. Bando, K. Hasebe, K. Nakanishi and A. Nakayama, Analysis of optimal velocity model with explicit delay, 1998 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 6 / 19

Optimal velocity model Original model [1] ẍ 1 (t) =a ( V(x N (t τ) x 1 (t τ)) ẋ 1 (t τ) ) ẍ i (t) =a ( V(x i 1 (t τ) x i (t τ)) ẋ i (t τ) ), i = 2,3,...,N a : common sensitivity coefficient τ : common reaction delay V : optimal velocity function Models in literature Circular loop Do not capture effect of leader s profile [1] M. Bando, K. Hasebe, K. Nakanishi and A. Nakayama, Analysis of optimal velocity model with explicit delay, 1998 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 6 / 19

Optimal velocity functions Properties Monotonic increasing : y 1 > y 2 = V(y 1 ) > V(y 2 ) Upper bounded : V b such that V(y) V b y Continuously differentiable : V C 1 (R + ) [1] M. Batista and E. Twrdy, Optimal velocity functions for car-following models, 2010 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 7 / 19

Optimal velocity functions Properties Monotonic increasing : y 1 > y 2 = V(y 1 ) > V(y 2 ) Upper bounded : V b such that V(y) V b y Continuously differentiable : V C 1 (R + ) Examples [1] Function 1 Function 2 (tanh( ) V(y) = V y ym 0 ỹ V(y) = V 0 e 2ym y )) +tanh( ym ỹ V 0,y m,ỹ : model parameters [1] M. Batista and E. Twrdy, Optimal velocity functions for car-following models, 2010 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 7 / 19

Variant for straight road Modified optimal velocity model (MOVM) v i (t) =a ( V(y i 1 (t τ i 1 )) V(y i (t τ i )) v i (t τ i ) ) ẏ i (t) =v i (t), i = 1,2,...,N String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 8 / 19

Models Classical car-following model (CCFM) v i (t) = β i 1 v i 1(t τ i 1 ) β i v i(t τ i ) i, Equilibrium: v i = 0, y i = 0 i β i = α i(ẋ 0 ) m /(b i ) l Modified optimal velocity model (MOVM) v i (t) = du i 1 (t τ i 1 ) du i (t τ i ) av i (t τ i ), u i (t) = v i (t) Equilibrium: vi = 0, y i = V 1 (V (ẋ 0 )) i d = av (V 1 (ẋ 0 )); d = a d String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 9 / 19

Summary of known results Analysis CCFM MOVM Pairwise stability β i τ i < π/2 χτ i < tan 1 (χ/ d) Loss of pairwise stability Hopf Hopf Non-oscillatory convergence β i τ i 1/e m dτ i < ln ( a(m+1) m 2 d ) Rate of convergence β i τ i = 1/e No closed-form expression String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 10 / 19

String stability Pairwise stability = platoon stability [1] [1] L.E. Peppard, String stability of relative-motion PID vehicle control systems, IEEE Transactions on Automatic Control, 1974 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 11 / 19

String stability Pairwise stability = platoon stability [1] How? Ensure magnitude-squared Bode plot of pairwise transfer function less than unity! [2] sup H i (jω) 2 1 i ω 0 [1] L.E. Peppard, String stability of relative-motion PID vehicle control systems, IEEE Transactions on Automatic Control, 1974 [2] R. Sipahi et al., Chain stability in traffic flow with driver reaction delays, American Control Conference, 2008 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 11 / 19

Condition for string stability is more stringent! Transfer function for pairwise interaction CCFM H i (s) = V i(s) V i 1 (s) = β i 1 e sτi 1 s+β i e sτ i MOVM H i (s) = U i(s) U i 1 (s) = de sτ i 1 s 2 +(as+d)e sτ i Sufficient condition for string stability CCFM β i 1 β i and β i τ i 1/2 MOVM a 2 2d and aτ i < 1/2 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 12 / 19

Condition for string stability is more stringent! Transfer function for pairwise interaction CCFM H i (s) = V i(s) V i 1 (s) = β i 1 e sτi 1 s+β i e sτ i MOVM H i (s) = U i(s) U i 1 (s) = de sτ i 1 s 2 +(as+d)e sτ i Sufficient condition for string stability CCFM β i 1 β i and β i τ i 1/2 MOVM a 2 2d and aτ i < 1/2 Note Two conditions: platoon wide and pairwise Conditions more stringent String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 12 / 19

CCFM: Is monotonicity of β i really necessary? Relative velocity (m/s) 0 0.005 0.01 v 1 (t) v 2 (t) v 3 (t) v 4 (t) 0 5 10 Time (s) String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 13 / 19

CCFM: String stability bears out in non-linear system! 0.01 Relative velocity (m/s) 0.005 v 1 (t) v 2 (t) v 3 (t) v 4 (t) 0 0 5 10 Time (s) String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 14 / 19

MOVM: String stability bears out in non-linear system! 0.002 Relative velocity (m/s) 0 v 1 (t) v 2 (t) v 3 (t) v 4 (t) 0.002 0 5 10 Time (s) String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 15 / 19

Robust stability Why? Parameters need to be estimated; not known exactly Ensure stability despite uncertainty = robust stable String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 16 / 19

Robust stability Why? Parameters need to be estimated; not known exactly Ensure stability despite uncertainty = robust stable What? For illustration, find bounds on reaction delay = bounds on delay guarantees by communication protocol Assume parameters uncertain in interval; not time varying Use sufficient condition for stability [1] & worst possible parameter values [1] V.L. Kharitonov et al., On delay-dependent stability conditions, Systems & Control Letters, 2000 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 16 / 19

Making entire platoon robust is harder! Uncertainty regions CCFM: β i [β i,β i] MOVM: a [a,a] & d [d,d] String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 17 / 19

Making entire platoon robust is harder! Uncertainty regions CCFM: β i [β i,β i] MOVM: a [a,a] & d [d,d] Pairwise robust stability CCFM: β i τ i < 1 MOVM: τ i a 2 +d 2 < 1 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 17 / 19

Making entire platoon robust is harder! Uncertainty regions CCFM: β i [β i,β i] MOVM: a [a,a] & d [d,d] Pairwise robust stability CCFM: β i τ i < 1 MOVM: τ i a 2 +d 2 < 1 Platoon robust stability CCFM: i β iτ i < 1 MOVM: ( a 2 +d 2 ) i τ i < 1 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 17 / 19

Summary of stability results Analysis CCFM MOVM Pairwise stability β i τ i < π/2 χτ i < tan 1 (χ/ d) Robust pairwise stability β i τ i < 1 τ i a 2 +d 2 < 1 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 18 / 19

Summary of stability results Analysis CCFM MOVM Pairwise stability β i τ i < π/2 χτ i < tan 1 (χ/ d) Robust pairwise stability β i τ i < 1 τ i a 2 +d 2 < 1 String stability βi 1 β i a 2 2d and and βi τ i 1/2 aτ i < 1/2 Robust platoon stability i β iτ i < 1 ( a 2 +d 2 ) i τ i < 1 String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 18 / 19

In a nutshell Models considered Classical Car-Following Model (CCFM) Modified Optimal Velocity Model (MOVM) String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 19 / 19

In a nutshell Models considered Classical Car-Following Model (CCFM) Modified Optimal Velocity Model (MOVM) Contributions: sufficient conditions for String stability Robust stability: pairwise & platoon Implications Insight into models: trade-offs among model parameters Autonomous vehicles: design guidelines for upper longitudinal control algorithm String and robust stability of connected vehicle systems with delayed feedback IFAC TDS 2018 19 / 19