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Performance analysis of Vehicular Platoon considering V2V Communication 基于车车通信的车辆队列性能分析 Keqiang LI 李克强 State Key Lab. of Automotive Safety and Energy 汽车安全与节能国家重点实验室 Tsinghua University 清华大学 1 Outline Background Problem Statements Performance Analysis considering V2V communication Conclusions State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 2 Outline Background Problem Statements Performance Analysis considering V2V communication Conclusions State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 3 Background History of Adaptive Cruise Control (ACC） Bosch Ltd, developed the first prototyping ACC Mercedes Benz first to offer ACC, called Distronic system State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 4 Background Next Generation of ACC V2V/V2I communication Multi-vehicle cooperative control (e.g., Vehicular Platoon, CACC) Potential Benefits Potential Benefits ACC Short following distance for better traffic capacity Faster reaction to leader fluctuation to reduce potential congestion Reduced aerodynamic resistance for better fuel economy and less emission More reliable automation for comfortable mobility Economic State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 5 Background Platoon: Topics and Researchers Research topics 1) Selection of spacing policies; 2) Communication delay; 3) Vehicle dynamic uncertainty; etc. Researchers 1) U.S.: J. Hedrick, P. Seiler, S. Darbha, Huei Peng etc. 2) Europe: Jeroen Ploeg, N. van de Wouw, H. Nijmeijer, etc. 3) China: Keqiang Li, Feng Gao, etc. Platoon: Experiments US: PATH Europe: SARTRE Japan: Energy ITS State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 6 Background Platoon: Emerging Topics Different Information Flow Topologies Typical Information Flow ... (a) ... ... (b) Connected Vehicle by V2V (c) ... (d) ... (e) ... (f) N N-1 N-2 Platoon: Key Tasks 2 1 0 ... 1. Model and design of a platoon system with a broad types of topologies. 2. Performance analysis of different topologies on platoons. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 7 Outline Background Problem Statements Performance Analysis considering V2V communication Conclusions State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 8 Problem Statements Modeling of Platoons from the viewpoint of Networks of Dynamical Systems From Control Perspective 1. Dynamics + Communication 2. Control Theory + Graph Theory Research topics 1. Dynamic: single integrator, double integrator, linear dynamic, nonlinear dynamic Applications 2. Communication: data rate, switching topology, time-delay A vehicular platoon can be viewed as a one-dimensional network of dynamical system State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 9 Problem Statements Modeling of Platoons under the Four-component Framework Information Flow Topology Distributed Controller Controller Ci CN uN ui ... N node i Vehicle Dynamics C1 Ci-1 dr,i ddes,i u1 v0 ui-1 t ... i-1 LV 1 0 Formation Geometry Vehicle Dynamics: The vehicle dynamics describe the behavior of each node; Formation Geometry: The formation geometry dictates the desired distance between any two successive nodes. Distributed Controller: The distributed controller implements feedback control for each vehicle; Information Flow Topology: The information flow topology defines how nodes exchange information with each other; State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 10 Problem Statements Model for Vehicle Longitudinal Dynamics Linear Dynamics 𝑥𝑖 𝑡 = 𝐴𝑥𝑖 𝑡 + 𝐵𝑢𝑖 𝑡 0 𝑝𝑖 0 𝑥𝑖 𝑡 = 𝑣𝑖 , 𝐴 = 𝑎𝑖 0 0 0 1 0 , 𝐵 = 1 1 0 − 𝜏 𝜏 Feedback linearization technique is used to convert the nonlinear model into a linear one; The vehicle dynamics is assumed to be homogeneous. 1 0 Model for Formation Geometry &lim 𝑣𝑖 𝑡 − 𝑣0 𝑡 𝑡→∞ = 0, 𝑖 = 1,2, ⋯ 𝑁 &lim 𝑝𝑖−1 𝑡 − 𝑝𝑖 𝑡 − 𝑑𝑖−1,𝑖 = 0 ,& 𝑡→∞ Constant distance (CD) policy Constant time headway (CTH) policy Nonlinear distance policy State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 11 Problem Statements Model for information flow topology ... The V2V communication can generates various information flow topologies. (a) a) Predecessor following topology; PF topology ... ... (b) b) Predecessor-leader following topology; PLF topology (c) c) Bidirectional topology; BD topology ... (d) d) Bidirectional-leader topology; BDL topology ... (e) e) Two predecessors following topology TPF topology f) Two predecessor-leader following topology TPLF topology ... (f) N N-1 N-2 2 1 0 ... State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 12 Problem Statements Model for information flow topology Algebraic Graph Theory Viewed as a directed graph G, and use Pinning matrix, Adjacent matrix and Laplacian matrix to model the connections. The communication is assumed as to be perfect. There is no delay, data loss etc. Definitions • Pinning Matrix • Adjacent Matrix To model the information flow from the leader to followers To model the information flow among followers 𝒫= 𝓅1 • Laplacian Matrix An induced matrix from adjacent matrix ℒ = 𝑙𝑖𝑗 ∈ ℝ𝑁×𝑁 &𝒜𝑁 = 𝑎𝑖𝑗 ∈ ℝ𝑁×𝑁 ⋱ 𝓅𝑁 &𝓅𝑖 = 1 , 𝑖𝑓&& 𝛼0 , 𝛼𝑖 ∈ 𝐸 &𝑎𝑖𝑗 = 1, 𝑖𝑓&& 𝛼𝑗 , 𝛼𝑖 ∈ 𝐸 &𝑎𝑖𝑗 = 0, 𝑖𝑓&& 𝛼𝑗 , 𝛼𝑖 ∉ 𝐸 𝑙𝑖𝑗 = State Key Lab of Automotive Safety and Energy· Tsinghua Univ. &−𝑎𝑖𝑗 &&&&&&&, &&𝑖 ≠ 𝑗 & 𝑁 𝑘=1 𝑎𝑖𝑘 , &𝑖 = 𝑗 . 13 Problem Statements Model for information flow topology Example 1: Bidirectional Topology 1 0 𝒫= ⋱ 0 &𝒜𝑁 = 0 1 3 1 0 1 1 0 ⋱ 。。。 N ⋱ ⋱ 1 1 0 1 2 3 。。。 N 1 2 ℒ= 1 −1 −1 2 −1 −1 2 ⋱ ⋱ ⋱ −1 −1 1 ⋱ −1 1 Example 2: Predecessor-leader following Topology N-1 N N-2 1 0 ... 1 𝒫= 1 &𝒜𝑁 = ⋱ 1 0 1 0 1 0 ⋱ ℒ= ⋱ 1 0 −1 0 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 1 −1 1 ⋱ 14 Problem Statements Model for Distributed Controller Linear State Feedback Controller 𝑢𝑖 𝑡 = − 𝑘𝑝 𝑝𝑖 − 𝑝𝑗 − 𝑑𝑖,𝑗 + 𝑘𝑣 𝑣𝑖 − 𝑣𝑗 + 𝑘𝑎 𝑎𝑖 − 𝑎𝑗 𝑗∈&𝕀𝑖 The local controller in node 𝑖 only uses its neighborhood information specified by&𝕀𝑖 . The controller is assumed to be linear for the convenience on theoretical analysis. The controller in each node is assumed to be homogeneous. Formulation for the Closed-loop Dynamics of Platoons Tracking error 𝑥𝑖 𝑡 = 𝑥𝑖 𝑡 − 𝑥0 𝑡 − 𝑑𝑖 , &&& 𝑑𝑖 = 𝑑0,𝑖 , 0,0 Collective state vector 𝑋 = 𝑥1𝑇 , 𝑥2𝑇 , ⋯ , 𝑥𝑁𝑇 𝑇 ∈ ℝ3𝑁×1 , Collective input vector 𝑈 = 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑁 𝑇 ∈ ℝ𝑁×1 , Controller 𝑈 = − ℒ + 𝒫 ⊗ 𝑘𝑇 ∙ 𝑋 Closed-loop Dynamics of platoon 𝑇 𝑇 𝑘 = 𝑘𝑝 , 𝑘𝑣 , 𝑘𝑎 &,⊗&is Kronecker product. 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 15 Problem Statements Unified Closed-loop Dynamics of Platoons 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋 Information Flow Topology Distributed Controller Controller Ci CN uN ui ... N node i Vehicle Dynamics C1 Ci-1 dr,i ddes,i u1 v0 ui-1 t ... i-1 LV 1 0 Formation Geometry Questions Q1. What’&the&stabilizing&region&of&controller&gain&𝑘 under different information flow topologies? Q2. How to choose topology and design controller to improve stability? State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 16 Outline Background Problem Statements Performance Analysis considering V2V communication Conclusions State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 17 Performance Analysis considering V2V communication Performance Definition Closed-loop Stability: A platoon with linear time-invariant dynamics is said to be closed-loop stable if and only if the closed-loop system has eigenvalues with strictly negative real parts Stability Margin: The stability margin of a platoon is defined as the absolute value of the real part of the least stable eigenvalue, which characterizes the convergence speed. Im Stability Margin dmin R State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 18 Performance Analysis considering V2V communication 1. Stability Region Analysis Unified Closed-loop Dynamics for vehicular platoons 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∈ ℝ3𝑁×3𝑁 Requirements of Closed-loop Stability Re 𝜎𝑖 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 large-scale matrix, which is difficult to analyze directly. <0 Dynamic decouple by using similarity transformation Key step: to decompose the large-scale vehicular platoon into multiple subsystems, which is easier to handle the closed-loop stability 𝑁 𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 = &&&& 𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 𝑖=1 𝑆 ∙ is the spectrum of a matrix. 𝜆𝑖 is the eigenvalue of ℒ + 𝒫 Multiple small-scale matrix, whose size is equal to that of node dynamics (n=3). State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 19 Performance Analysis considering V2V communication 1. Stability Region Analysis Dynamic decouple by using similarity transformation 𝑁 𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 = 𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 𝑖=1 Routh–Hurwitz stability criterion 𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 s3& s2 𝑠1 𝑠0 1 𝜆𝑖 𝑘𝑎 + 1 𝜏 𝜆𝑖 𝑘𝑣 𝜆𝑖 𝑘𝑎 + 1 − 𝜆𝑖 𝑘𝑝 𝜏 𝜏 𝜆𝑖 𝑘𝑎 + 1 𝜆𝑖 𝑘𝑝 𝜏 = 𝑠3 + 𝜆𝑖 𝑘𝑣 𝜏 𝜆𝑖 𝑘𝑝 𝜏 𝜆𝑖 𝑘𝑎 +1 2 𝑠 𝜏 + 𝜆𝑖 𝑘𝑣 𝑠 𝜏 + 𝜆𝑖 𝑘𝑝 𝜏 . Real coefficient polynomials. It needs 𝜆𝑖 to be positive real number. &𝑝>0 𝑘 &𝑘𝑣 > 𝑘𝑝 𝜏 𝜆𝑖 𝑘𝑎 + 1 &𝑘𝑎 > − 1 𝜆𝑖 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 20 20 Performance Analysis considering V2V communication 1. Stability Region Analysis Consider a homogeneous platoon with linear controllers given by 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 (1.1) If graph&𝐺 satisfies certain conditions (all the eigenvalues of ℒ + 𝒫&are positive real numbers), the platoon is asymptotically stable if and only if &𝑝>0 𝑘 &𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1 &𝑘𝑎 > − 1 max 𝜆𝑖 Remarks 𝜆𝑖 is the eigenvalue of ℒ + 𝒫 (𝜆𝑖 &need to be positive real number). This result can cover a lot of information flow topologies, including all the aforementioned topologies The influence of information flow topology on stability is mainly reflected by 𝜆𝑖 . State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 21 Performance Analysis considering V2V communication 1. Stability Region Analysis– Simulation Results 𝑘𝑝 = 1, 𝑘𝑣 = 2, 𝑘𝑎 = 1, 𝜏=0.5 (a) (d) (b) (e) &𝑝>0 𝑘 &𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1 &𝑘𝑎 > − 1 max 𝜆𝑖 Stable (c) (f) State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 22 Performance Analysis considering V2V communication 1. Stability Region Analysis– Simulation Results 𝑘𝑝 = 1, 𝑘𝑣 = 0.2, 𝑘𝑎 = 1, 𝜏=0.5 (a) (d) (b) (e) &𝑝>0 𝑘 &𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1 Unstable &𝑘𝑎 > − 1 max 𝜆𝑖 (c) (f) State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 23 Performance Analysis considering V2V communication 2. Scaling trend of Stability Margin with increasing platoon size Consider a homogeneous platoon with linear controllers given by 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 (2.1) if the graph 𝐺 is in Bidirectional topology, then the stability margin of platoons decays to zero as 𝑂 1 𝑁 2 (2.2) if the graph 𝐺 is in BDL topology, then the stability margin of platoons is always bounded away from zero. N N-1 N-2 1 ... 0 Decay to zero with increasing size; Independent with controller gains. Bidirectional (BD) topology N N-1 N-2 1 ... 0 Will not decay to zero as platoon size increases. Bidirectional-leader (BDL) topology State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 24 Performance Analysis considering V2V communication 2. Scaling trend of Stability Margin with increasing platoon size - Proof Step. 1 Closed-loop Dynamics 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 Step. 2 Similarity transformation 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∈ ℝ3𝑁×3𝑁 𝑁 𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 = 𝑖=1 Step. 3 Eigenvalue Analysis 𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 Im 𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 = 𝑠3 + 𝜆𝑖 𝑘𝑎 +1 2 𝑠 𝜏 + 𝜆𝑖 𝑘𝑣 𝑠 𝜏 Stability Margin + 𝜆𝑖 𝑘𝑝 𝜏 . dmin R Stability Margin dmin = Re smin = O 𝜆min . For Bidirectional topology dmin = O 𝜆min = 𝑂 1 𝑁 2 For Bidirectional-leader topology dmin = O 𝜆min =constant number State Key Lab of Automotive Safety and Energy· Tsinghua Univ. decay to zero Independent with size 25 Performance Analysis considering V2V communication 2. Scaling trend of Stability Margin with increasing platoon size - Simulations N-1 N N-2 1 0 ... 𝑘𝑝 = 1, 𝑘𝑣 = 2, 𝑘𝑎 = 1, 𝜏=0.5 Bidirectional topology 0 10 0 10 -1 10 -1 Stability Margin 10 Stability Margin -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 -5 10 10 100 The number of vehicles:N -5 10 10 100 The number of vehicles:N 500 500 N N-1 N-2 1 0 ... Bidirectional-leader topology State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 26 26 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Topology Selection Consider a platoon with linear controllers given by 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 (3.1) if the graph 𝐺 is undirected, to maintain bounded stability margin, it needs at least lots of followers (i.e. Ω(N) = O(N)) to obtain the leader’s information. Remarks i+2 i i+1 i-1 i-h i-2 ... ... ... i+h ... BD&topology&is&a&special&case,&i.e.,&Ω(N)=1, for which the stability margin decays to zero as the platoon size increases; N N-1 N-2 1 ... 0 It implies that the information flow from the leader is more important than that among the followers. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 27 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Topology Selection To maintain bounded stability margin, the tree depth of graph G should be a constant number and independent of the platoon size N. Tree dpeth&𝑐 = max 𝑛1 , 𝑛2 − 𝑛1 , ⋯ , 𝑛𝑝 − 𝑛𝑝−1 , 𝑁 − 𝑛𝑝 + 1 , where& 𝑛1 , 𝑛2 , ⋯ , 𝑛𝑝 , 1 ≤ 𝑛1 ≤ • ⋯ ≤ 𝑛𝑝 ≤ 𝑁&is&the&set&of&followers&pinned&to&the&leader ... n+1 n 2 1 n 0 ... N=c(k+1) ... ... ck+1 c(k-1)+1 ck ... c 1 0 c 0 N=2n Tree depth is N/2, increasing with platoon size; Stability margin decay to zero; c+1 ... ck+1 c(k-1)+1 ... c+1 1 ... n ... n-1 c ... n+2 2 ... 1 0 N=c(k+1) N=2n ... Tree depth is c, a constant number, independent with platoon size; Stability margin can be bounded away from zero State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 28 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Topology Selection 0 10 -1 0.25 10 -2 Stability Margin Stability Margin 0.2 0.15 0.1 0.05 0 0 10 -3 10 -4 10 Ω(N)=N/2,Tree Ω(N)=N/4,Tree Ω(N)=N/2,Tree Ω(N)=N/4,Tree -5 10 10 50 20 40 30 20 40 Tree Depth Parameter:c 50 30 10 0 Local Communication Range:h It is the tree depth 𝑐 rather than local communication range ℎ that dominates the stability margin. -6 10 10 depth: depth: depth: depth: c=2 c=4 c=N/2 c=3N/4 100 The number of vehicles:N 500 Extending information flow to reduce the tree depth is one major way to guarantee a bounded stability margin. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 29 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Asymmetric Control Consider a homogeneous platoon under the BD topology with the asymmetric controller architecture given by 𝑋 = 𝐼𝑁 ⊗ 𝐴 − ℒ𝐵𝐷 + 𝒫𝐵𝐷 𝜖 ⨂𝐵𝑘 𝑇 𝑋 (3.1) For any fixed&ϵ ∈ 0,1 , the stability margin is bounded away from zero and independent of the platoon size&𝑁 (N can be any finite integer). kfN kbN-1 N kfN-1 N-1 kbN-2 ... kf3 kb2 kf2 2 kb1 kf1 1 ... 0 LV Asymmetric control The controller is called asymmetric, if 𝑓 &𝑘𝑖 = 1 + 𝜖 𝑘, 𝑘𝑖𝑏 = 1 − 𝜖 𝑘&&&&𝑖 = 1, ⋯ , 𝑁 − 1 𝑓 &𝑘𝑁 = 1 + 𝜖 𝑘,&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& where&𝜖 ∈ 0,1 is called the asymmetric degree. Note that if 𝜖 = 0, then it is reduced to the symmetric case. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 30 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Asymmetric Control- Proof Eigenvalue Analysis 𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇 = 𝑠3 + Im 𝜆𝑖 𝑘𝑎 +1 2 𝑠 𝜏 + 𝜆𝑖 𝑘𝑣 𝑠 𝜏 + 𝜆𝑖 𝑘𝑝 𝜏 . Stability Margin dmin Stability Margin dmin = Re smin = O 𝜆min . R For asymmetric control dmin = O 𝜆min = constant number, For any fixed&ϵ ∈ 0,1 Independent with size Tradeoff between Convergence Speed and Transient Performance Benefit: bounded stability margin, good for convergence speed Cost: overshooting phenomena in transient process. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 31 Performance Analysis considering V2V communication 3. Stability Margin Improvement – Asymmetric Control 30 15 1 6 12 18 24 30 10 Spacing Error (m) 10 -1 -2 10 0 0 -10 -5 10 -10 -20 -30 0 50 -3 100 Time (s) 150 200 0 50 (a) 10 100 Time (s) 150 200 (b) 25 -5 10 1 6 12 18 24 30 30 20 10 100 The number of vehicles:N 500 The stability margin of a platoon with asymmetric controllers is indeed bounded away from zero and independent of with the platoon size. 10 0 -10 1 6 12 18 24 30 20 15 10 Spacing Error (m) -4 10 40 ε=0 ε=0.2 ε=0.4 ε=0.6 Spacing Error (m) Stability Margin 10 5 1 6 12 18 24 30 20 Spacing Error (m) 0 5 0 -5 -10 -20 -15 -30 -40 -20 -25 0 50 100 Time (s) 150 200 (c) 0 50 100 Time (s) 150 200 (d) Space errors for homogeneous platoon under BD topology with different asymmetric&degree&ϵ.&(a)&ϵ=0& (symmetric);&(b)&ϵ=0.2;&(c)&ϵ=0.4;&(d)&ϵ=0.6 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 32 Performance Analysis considering V2V communication 4. Linear Stable Controller Design - solving a Riccati equation Consider a homogeneous platoon with linear controllers given by 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋 Stability Region &𝑝>0 𝑘 &𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1 &𝑘𝑎 > − 1 max 𝜆𝑖 How to choose a specific controller gain? Controller design by solving a Riccati equation 𝐴𝑇 𝑃𝜀 + 𝑃𝜀 𝐴 − 𝑃𝜀 𝐵𝐵𝑇 𝑃𝜀 + 𝜀𝐼3 = 0, Choosing the controller gain as 𝑘 𝑇 = 𝛼𝐵𝑇 𝑃𝜀 Influence of Information flow topology Closed-loop Stability requirements 𝛼≥ 1 2 min 𝜆𝑖 𝑖∈𝒩 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 33 Performance Analysis considering V2V communication 4. Linear Stable Controller Design - Simulation 15 4.5 Spacing Error(m) 5 0 -5 1 3 5 7 10 4 3.5 3 Spacing Error(m) 1 3 5 7 10 10 1.2 0.8 2.5 2 1.5 1 -10 1 3 5 7 10 1 Spacing Error(m) 0.6 0.4 0.2 0.5 -15 0 20 40 60 Time (s) 100 -0.5 0 20 1 3 5 7 10 3 1 3 5 7 10 3 2.5 2 1.5 1 10 20 Time (s) 30 3 1 𝜀 0.5 0.5 22.5 0.5 0.5 0.5 1 1 1 1 1 1 1 3 5 7 10 3.5 2 40 𝛼 (c) 4 2.5 2 1.5 1 0 0.5 0.5 -1 0 -0.5 0 4.5 4 Spacing Error(m) 3.5 -0.2 60 (b) 5 4 40 Time (s) (a) 4.5 Spacing Error(m) 80 Spacing Error(m) -20 0 0 Topolo 𝜆𝑚𝑖𝑛 gy PF 1 PLF 1 BD 0.022 BDL 1 TPF 1 TPLF 1 0 20 40 Time (s) (d) 60 -2 0 0 20 40 Time (s) (e) 60 -0.5 0 20 40 60 Time (s) (f) State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 34 Outline Background Problem Statements Performance Analysis considering V2V communication Conclusions State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 35 Conclusions Vehicular platoon can bring many potential benefits, e.g., Improving traffic capacity; Enhancing highway safety; Reducing road congestion V2V communication can generate various types of topologies for platoon. For&vehicular&platoons&under&“homogeneity&+&linear&feedback” 1) Stability Region Analysis • Explicitly established the stabilizing thresholds of linear controller gains for platoons under different information flow topologies. 2) Stability Margin Scaling Trend • Obtained stability margin scaling trend for platoons under two typical topologies, i.e., Bidirectional Topology and Bidirectional-leader Topology. 3) Stability Margin Improvement • Proposed two basic ways to improve the stability margin, i.e., topology selection and controller adjustment. 4) Linear Stable Controller Design • Converted the platoon control problem to a parametric algebraic Riccati equation. The designed controllers can guarantee the internal stability for a variety of topologies State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 36 Thanks for your attention Q & A? 37 Appendix: Robust issue Robust Performance Disturbance with Finite Energy +∞ 𝑤𝑖 𝑡 1. First-to-last amplification factor (𝐴𝐹𝑓2𝑙 ) 𝐴𝐹𝑓2𝑙 𝑝𝑁 = sup 𝑤1 ℒ2 ℒ2 = 𝐺𝑓2𝑙 𝑠 ℋ∞ ℒ2 = 0 𝑤𝑖 𝑡 2 𝑑𝑡 < ∞ 2. All-to-all amplification factor(𝐴𝐹𝑎2𝑎 ) 𝐴𝐹𝑎2𝑎 = sup 𝑝 𝑊 ℒ2 ℒ2 = 𝐺𝑎2𝑎 𝑠 ℋ∞ Dynamics in Frequency domain 𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋 + 𝐵 ∙ 𝑊 Time domain Frequency domain 𝑌 =𝐶∙𝑋 ℒ 𝑌 𝐺 𝑠 = = 𝐶 𝑠𝐼3𝑁 − 𝐴𝑐𝑙 −1 𝐵 ℒ 𝑊 = 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2 −1 It’s&very&difficult&to&analysis&the&ℋ∞ norm of transfer function 𝐺 𝑠 under general information flow topology. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 38 Appendix: Robust issue 1. Scaling Trend of 𝓗∞ norms under PF topology Consider a homogeneous platoon under the PF topology with linear controllers given by −1 𝐺 𝑠 = 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2 (2.1) the amplification factors 𝐴𝐹𝑓2𝑙 and 𝐴𝐹𝑎2𝑎 satisfy the following conditions 𝛽2 𝛼 𝑁 − 1 𝑁−1 𝑁−1 𝑁−1 𝛽1 𝛼 ≤ 𝐴𝐹𝑎2𝑎 ≤ 𝛽1 𝛼 ≤ 𝐴𝐹𝑓2𝑙 ≤ 𝛽2 𝛼 𝛼−1 where, 𝛽1 , 𝛽2 is constant real number, 𝛼&>1. N N-1 N-2 1 ... 0 Fig. Predecessor-following (PF) topology Remarks The amplification factors will exponentially grow with increasing platoon size. These results are independent with controller gains, which means this is a fundamental drawback of PF topology when using identical linear controller State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 39 Appendix: Robust issue 1. Scaling Trend of 𝓗∞ norms under PF topology-simulation results N-1 N N-2 1 0 ... 3 10 3 Bode Diagram 10 A f2l 80 N=22 N=18 N=14 N=10 N=6 N=2 Upper bound: β2*α N-1 60 Lowerbound: β1*α N-1 1 10 0 10 10 40 Magnitude (dB) a2a 2 Amplification factor:A Amplification factor： A f2l 2 10 1 10 0 A a2a 10 20 0 -20 Upper bound: β3*(α N-1) -1 10 Lower bound: β1*α N-1 -1 0 5 10 15 Platoon size: N 20 25 10 0 5 10 15 Platoon size: N 20 -40 25 -2 10 -1 10 0 10 1 10 Frequency (rad/s) The amplification factors&𝐴𝐹𝑓2𝑙 and 𝐴𝐹𝑎2𝑎 indeed exponentially grow with increasing platoon size. State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 40 Appendix: Robust issue 2. Scaling Trend of 𝓗∞ norms under PLF topology-simulation results N-1 N N-2 1 0 ... 0 Bode Diagram 1 10 10 0 A f2l Upper bound: β2*α1N-1 -1 10 -50 -100 10 -3 10 -4 10 Magnitude (dB) a2a Lower bound: γ 1*α1 -2 Amplification factor:A Amplification factor： A f2l N-1 0 10 -200 N=22 N=18 N=14 N=10 N=6 N=2 -250 -300 A a2a -5 10 -6 10 -150 -1 0 5 10 15 Platoon size: N 20 25 10 -350 Upper bound: γ3*(α1N-1) Lower bound: γ2 0 5 10 15 Platoon size: N 20 -400 -2 10 25 -1 10 0 10 1 10 Frequency (rad/s) If the leader broadcast its information to all the following vehicle, resulting in predecessor-leader following topology, then amplification factor will become better State Key Lab of Automotive Safety and Energy· Tsinghua Univ. 41 Appendix: Robust issue 3. Scaling Trend of 𝓗∞ norms under BD topology Consider a homogeneous platoon under the BD topology with linear controllers given by −1 𝐺 𝑠 = 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2 (4.1) the amplification factor 𝐴𝐹𝑎2𝑎 at least increase with the platoon size as 𝑂 𝑁 2 & N-1 N N-2 1 0 ... A a2a 40 N=22 N=18 N=14 N=10 N=6 N=2 12 2 10 1 10 10 20 Magnitude (dB) Amplification factor： A a2a 30 f2l Lower bound: N2/(k p*π2) 3 10 Amplification factor:A Bode Diagram 14 4 10 8 6 10 0 4 0 10 -1 10 -10 2 0 0 10 1 10 Platoon size: N 0 5 10 15 20 Platoon size: N 25 30 -20 -2 10 State Key Lab of Automotive Safety and Energy· Tsinghua Univ. -1 10 0 10 1 10 Frequency (rad/s) 42