Practical String Stability for Longitudinal Control of Automated Vehicles XIAO-YUN LU
∗
and
J. KARL HEDRICK
∗
SUMMARY This paper considers string stability for vehicle longitudinal control using linear control (PID and a type of sliding mode) from a practical implementation viewpoint. A different parameterization is used compared to previous work. Two following strategies are considered. i.e. Adaptive Cruise Control (ACC ) and vehicle platooning with inter-vehicle communication. A transfer function approach is used for analysis. It is shown that ACC cannot achieve string stability with linear control applied to feedback linearized error dynamics and while the latter can, which agree with practical test.
1 INTRODUCTION String stability describes the dynamic interaction between vehicles in short inter-vehicle distance following in longitudinal motion. Previous work in [8] defines the string stability as an asymptotic stability of the overall system which is composed of a finite number of inter-connected sub-systems with the same or similar dynamics. Necessarily, each closed-loop sub-system must be asymptotically stable. This is the ideal case for the dynamic behavior of a series of sub-system inter-connected in a string. For practical implementation, it is necessary to analyze the string stability under nearly real circumstances which would involve the following factors: (a) Time lag in sensor and actuators (b) Pure time delays in sensor measurement and signal processing (c) Model mismatch (d) Measurement noises The time delays will naturally cause measurement and actuation discrepancies. Due to such discrepancies in addition to model mismatch, measurement noises and external disturbances, each sub-system (a single vehicle) can only achieve ultimate boundedness in stability [2, 4], which coincides with experimental work. Thus to require strict attenuation of tracking error down stream (direction from the first vehicle to the last vehicle) along the platoon is too restrictive. Practically, string stability in vehicle following can only require that distance and speed tracking error will not propagate or has limited propagate 1 PATH, U. C. Berkeley, Richmond Field Station Building 452, 1357 S. 46th Street, Richmond, CA 94804-4648, Email:
[email protected] [email protected]
down stream in a platoon. However, for theoretical analysis, the definition for string stability in [8] is reasonable. In ([9]), string stability for many vehicle following strategies has been considered. Time lag is involved but not pure time delays. ([3]) considered both time lag caused by actuators and pure time delay caused by inter-vehicle communication. Due to the complication of the problem formulated, it is impossible to consider arbitrary design parameters. i.e. It only show that the system is string stable/unstable when some particular control parameters are chosen. [1] simplified the mathematical model as a finite number of linear spring-damper systems connected in a string fashion. The problem is that the spring and damping effects is expected to be produced from the closed controller instead of being assumed a priori. In this paper, two time delays are taken into consideration while model mismatch and measurement noise are ignored for simplicity. The control strategies are feedback linearization plus linear (PID and a type of sliding mode) control. A different parameterization approach is adopted compared to those in [3, 9]. It greatly simplifies the problem for both theoretical analysis and practical implementation. Since the second order vehicle dynamics is feedback linearizable and PID control represents all linear control based on the error dynamics, the results here suitable for all linear control applied to feedback linearized error dynamics. String stability mainly depends on following strategies as discussed in [5] while the latter on information available from the preceding vehicles. If each vehicle follows its immediate preceding vehicle only, it is Adaptive Cruise Control (ACC) (implicitly, no communication). It will be shown mathematically that ACC mode cannot achieve string stability for all linear controller. However, if certain amount of information from leader vehicle (passed over with communication ) is used, string stability can be achieved irrespective of the time delays. In fact, the key to achieve string stability in vehicle following is to appropriately incorporate leader vehicle information. The rest of the paper is organized as follows. Section 2 introduces string stability in vehicle following and its mathematical criteria. Section 3 is for string stability analysis using transfer functions for two typical following strategies in practice. Section 4 presents test results using 4 automated full size passenger cars with brief discussion. Section 5 present some concluding remarks. Basic Notations xi (t) or simply (xi ) − position of vehicle i in longitudinal direction. All the vehicles are with respect to a inertia frame. vi (t), ai (t)− speed and acceleration of vehicle i hp1 − time delay for obtaining front range hp2 − time delay for obtaining preceding vehicle’s speed and acceleration hl − time delay for on-car sensor measuring and for communication system to pass the leader vehicle’s distance, speed and acceleration to other vehicles Li is the desired inter-vehicle distance with vehicle length accounted for l− subscript for the leader vehicle 2 STRING STABILITY FOR VEHICLE FOLLOWING This section will provide mathematical criteria for string stability in vehicle following.
2.1 String Stability of Vehicle Following Let εi (t) . εi (t) .. εi (t)
= xi (t) − xi−1 (t) + Li = vi (t) − vi−1 (t) = ai (t) − ai−1 (t)
Ei (s) is the Laplace transformation of εi (t). G(s) is the transfer function of the closed-loop dynamics g(t) of sub-system i, which is the same for each vehicle. Then Ei (s) G(s) = (1) Ei−1 (s) The string stability for a platoon of n vehicles requires that ε1 ∞ ≤ ε2 ∞ ≤ ... ≤ εn ∞ From linear system theory ∞ εi ∞ ≤ g(t)1 = 0 |g(τ )| dτ g ∗ εi ∞ ≤ g(t)1 εi ∞ G (s)∞ ≤ g(t)1
(2)
Thus the inter-connected system is string stable if g(t)1 < 1 and string unstable if G(s)∞ > 1. To practically check it, one needs to evaluate g(t)1 . 2.2 Following Strategy and String Stability There are two typical vehicle following strategies: (a) short distance following with inter-vehicle communication in Automated Highway Systems [7]; (b) Adaptive Cruise Control (ACC ) without inter-vehicle communication in normal highway system. However, in practice, the string stability of automated vehicle platooning can be achieved for 8 vehicles as tested in PATH, while string stability cannot be achieved for ACC with more than 2 vehicles. The reason for this is that the most important factor, time delay which is due to sensor estimation and actuators, has been largely ignored in vehicle following. Due to feedback linearizability of vehicle dynamics, it is sufficient to consider the following simplified model for string stability analysis .
xi = vi . v i = ui
(3)
where ui is the synthetic force. Two types of control strategies will be considered. 1. Sliding Model Control Let the sliding surface be defined as i . Si = α εi +αqεi + (1 − α) (vi − vl ) + (1 − α) q xi − xl + Lj j=2
where α ∈ [0, 1] is the interpolation parameter. Two extreme cases are: α = 1 which means that each vehicle follows the preceding vehicle only and no lead vehicle information is used; α = 0 which implies that each vehicle follows the leader vehicle only. However, the most interesting cases correspond to 0 < α ≤ 1. . One can choose general sliding reachibility condition S i = − γi (s) as in ??. For special case of γi (s) = λSi (λ > 0), the controller (synthetic force) is solved out as (d)
..
..
.
ui = α xi−1 + (1 − α) xl −α(q + λ) εi −αλqεi i −(1 − α)(q + λ)(vi − vl ) − λq (1 − α) xi − xl + j=2 Lj
(4)
The design parameters (q, λ, α) are to be chosen such that (a) The closed loop controller for each vehicle is stable; (b) The overall system which is composed of finite number of inter-connected similar sub-systems is string stable. 2. PID Control Now considering the interpolation of two PID controls. The errors are with respect to the preceding vehicle and the leader vehicle respectively. (d)
ui = αUi1 + (1 − α)Ui2 t d 1 Ui = KD dt vi−1 − KP (vi − vi−1 ) − KI 0 (vi (τ ) − vi−1 (τ )) dτ t d Ui2 = KD dt vl − KP (vi − vl ) − KI 0 (vi (τ ) − vl (τ )) dτ
(5)
The closed loop stability requires that KD ζ 2 + KP ζ + KI = 0 be Hurwitz, where one can always normalize KD = 1 for stability analysis. The following relationship shows that if the sliding reachability condition . S i = −λSi is used, the sliding mode control is a special case of PID control with: t i xi (t) = 0 vi (τ )dτ, xi (0) = j=2 Lj KP = (q + λ) , KI = λq, KD = 1 which is not true in general case ??, however. 2.3 Time Delays In practice, there are two types of time delays: time lag and pure time delay. A first order filter is inserted to represent the effect of time lag as .
τ ui +ui = uid which links the controller (4) and upper level vehicle model (3) with τ = 0.15[s]. There are two fundamentally different cases for Pure time delays. Case 1: With inter-vehicle communication Relative distance (xi − xi−1 ) is estimated from distance sensor reading, which causes pure time delay of hp1 ≈ 0.25[s] where 0.1[s] comes from radar/Lidar sensor delay (physical and radar internal signal processing) and 0.15[s] is due
to signal processing in feed-forward control. (vpre , apre ) is passed over by communication which causes time delay about hp2 ≈ 0.1[s] in which 0.02[s] is the communication cycling period and 0.08[s] is due to the speed and acceleration sensor delays. Assumption 1: The communication system passes information from the leader vehicle to each vehicle and from each vehicle to its follower simultaneously. The common time delays for each vehicle is 0.02[s] (time step used for control.). Assumption 2: Pure sensor time delay on vehicle i with respect to the preceding vehicle is the same for all the vehicles. All the sensor measurement discrepancies can be ignored. Case 2: Without inter-vehicle communication All the three elements in [(vi − vi−1 ), (ai − ai−1 ), (xi − xi−1 )] are estimated from measurement by Doppler radar, Lidar and video camera. In this case: hp1 ≈ 0.25[s], hp2 = 0.35[s]. Note that leader vehicle information is unavailable directly. 2.4 Transfer Function Expression To use frequency analysis approach to calculate the H∞ gain, the transfer function for the closed-loop system of each vehicle is calculated as follows. 3
..
(d)
(d)
τ ddtε3i + εi = ui − ui−1 . = −αKP εi −αKI εi − KI (1 − α) εi .. − (1 − α) KP (vi (t) − vi−1 (t)) + α xi−1 (t − hp2 ) .. . . −α xi−2 (t − hp2 ) + αKP xi−1 (t − hp2 )− xi−2 (t − hp2 ) +αKI (xi−1 (t − hp1 ) − xi−2 (t − hp1 )) Using Laplace transformation on both side to get G(s) =
Ei (s) αKI e−hp1 s + αse−hp2 s (s + KP ) = Ei−1 (s) τ s3 + s2 + KP s + KI
(6)
3 STRING STABILITY ANALYSIS This section analyzes the string stability with respect to the two typical following strategies above. 3.1 Vehicle Following without Communication (ACC ) In (6), set α = 1 which is equivalent to using preceding vehicle information only the control law (4) becomes (d)
ui
..
.
=xi −KP εi −KI εi
It is obtained that G(s) =
KI + se−(hp2 −hp1 )s (s + KP ) −hp1 s e τ s3 + s2 + KP s + KI
(7)
Additionally, for the stability of the feedback dynamics, it is necessary and sufficient that D(s) = τ s3 + s2 + KP s + KI be Hurwitz, which is equivalent to the parameter constraints: KI > 0 KP − τ K I > 0
(8)
Because e−hp1 s does not effect the value of |G(jω)| and thus it is ignored. Let hp = hp2 − hp1 > 0 for simplicity. Thus G(jω) =
KI + jω (cos(ωhp ) − j sin(ωhp )) (jω + KP ) −τ jω 3 − ω 2 + KP jω + KI
Now it is necessary to evaluate G∞ = maxω |G(jω)|. For considering only very small ω > 0, the 3rd order terms are ignored to obtain G(jω)
≈ =
KI + jω (1 − jωhp ) (jω + KP ) ω 2 h p KP = 1 + −ω 2 + KP jω + KI (KI − ω 2 ) + KP jω 2 2 ω hp KP (KI − ω ) KP jω 1+ − (KI − ω 2 )2 + KP2 ω 2 (KI − ω 2 )2 + KP2 ω 2
One can observe that for feedback control law (7) with constraints (8), if ω > 0 is chosen very small such that ω 2 < KI arbitrarily, hp > 0 will lead to ω 2 hp KP (KI − ω 2 ) >0 (KI − ω 2 )2 + KP2 ω 2 which means that the real part is positive and greater than 1. This implies from (2) that G∞ g(t)1
=
max |G(jω)| > 1 ω
> 1
It is thus concluded that the system is string unstable under any feedback control law in (7) although each sub-system is stabilizable. This is summarized in the following theorem. Theorem 1. For vehicle following in ACC mode, inter-vehicle range and range-rate are measured by radar, the system is string unstable for any linear controller (PID) of the type (7). 3.2 Vehicle Following with Inter-vehicle Communication Now (6) is directly considered with some information from the leader vehicle passed over by communication. G(s) has the form
α KI e−hp1 s + se−hp2 s (s + KP ) G(s) = , 0 0(i = 1, 2, 3). Then M can be estimated as 1 (1) σ1 hp1 (2) σ2 hp1 (3) σ3 hp1 |g0 (t)| ≤ M = + a e + a e a e τ 1 + b(1) eσ1 hp2 + b(2) eσ2 hp2 + b(3) eσ3 hp3 τ 1 . which provides an upper bound for α < M Proof. Directly from the proof of the Lemma. ♦ Remark 3.1 In theory, when every vehicle follows the leader vehicle only, the overall system is always string stable. This following strategy sounds ideal but impractical. This is due to the following reasons: (a) Real-time distance estimation (xl − xi ) of each vehicle with respect to the leader vehicle is difficult to obtain. If radar distance with respect to the preceding vehicle is used for the estimation of (xl − xi ), error accumulation may increase down stream in the platoon even if inter-vehicle communication is available. (b) For safety, each vehicle must avoid conflict with its immediate front vehicle, which requires α to be as large as possible. This suggests to choose 1 α= M − ε with ε(> 0) sufficiently small.
4 TEST RESULTS For experimental work, 4 automated full size vehicle Buick Le Sabre have been used for experiment. The desired inter-vehicle distance is 6[m]. Delco radar is used for distance measurement with internal delay about 50[ms]. The above following strategy is used with α = 0.65. Test results show that reasonably good string stability has been achieved (Fig. 2). However, both distance tracking error and speed tracking error did not monotonically decrease down stream in the platoon, which is not exactly as predicted by the analysis. This is due to measurement error, external disturbances and differences between vehicles which are ignored here. 5 CONCLUDING REMARKS Automated vehicle short distance following is a hot research topic for both academic researchers and R&D of vehicle manufacturers. There are typically two following modes: Adaptive Cruise Control (ACC ) and platooning with inter-vehicle communication. This paper uses a different parameterization approach to consider the string stability using feedback linearization and linear (PID and a type of sliding mode) control to the two vehicle following modes.
car−1: red; car−2: magenta; car−3: green; car−4: blue
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Figure 2: String stability test of 4 cars Particularly, the main obstacle for vehicle following, i.e. the time lag and pure time delay are taken into consideration. It is shown that ACC cannot achieve string stability using any linear control to the feedback linearized error dynamics. Platooning, on the other hand, can achieve string stability regardless the time delay in the range sensing.
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