V2I-Based Platooning Design with Delay Awareness
11 V2I-Based Platooning Design with DelayAwareness
Lifeng Wang, Yu Duan, Yun Lai, Shizhuo Mu, and Xiang Li
Abstract
This paper studies the vehicle platooning system based on vehicle-to-infrastructure (V2I) commu-nication, where all the vehicles in the platoon upload their driving state information to the roadsideunit (RSU), and RSU makes the platoon control decisions with the assistance of edge computing. Byaddressing the delay concern, a platoon control approach is proposed to achieve plant stability and stringstability. The effects of the time headway, communication and edge computing delays on the stabilityare quantified. The velocity and size of the stable platoon are calculated, which show the impacts of theradio parameters such as massive MIMO antennas and frequency band on the platoon configuration. Thehandover performance between RSUs in the V2I-based platooning system is quantified by consideringthe effects of the RSU’s coverage and platoon size, which demonstrates that the velocity of a stableplatoon should be appropriately chosen, in order to meet the V2I’s Quality-of-Service and handoverconstraints.
Index Terms
Vehicle platooning, V2I, edge computing, massive MIMO.
I. I
NTRODUCTION
The commercially-used adaptive cruise control (ACC) enables vehicles to maintain safe inter-vehicle distance, which can avoid the collision and achieve autonomous driving through followingthe vehicle ahead [1]. To obtain the inter-vehicle distance and relative velocity, such an intelligenttransportation system (ITS) fully depends on the vehicle’s radar sensing capability [2]. However,the drawback of radar sensor is that its efficacy could be degraded by the obstructions or badweather. More importantly, ACC system is susceptible to the string instability, which results in
Authors are with the Department of Electrical Engineering, Fudan University, Shanghai, China (E-mail: { lifeng . wang , lix } @fudan . edu . cn ). a r X i v : . [ c s . M A ] D ec phantom traffic jams [3]. Cooperative adaptive cruise control (CACC) is a promising approach todeal with these issues [4, 5]. As the extension of ACC, CACC allows vehicles to communicatewith each other for sharing their driving state information (DSI) such as position, spacing,velocity, acceleration/deceleration rate, and time headway etc. Compared to the ACC, CACCcan provide earlier collision avoidance, traffic jam mitigation, aerodynamic drag force reduction,and extended sensors [6–9].Vehicle-to-everything (V2X) communications allow vehicle-to-vehicle (V2V) or vehicle-to-infrastructure (V2I) connectivity in the CACC systems. The V2V-based CACC systems havebeen widely studied in the literature [10–17]. These works have shown that V2V communicationsimprove the stability and reduce the time headway in ITS systems, which means that highertraffic throughput and fuel efficiency can be achieved. However, the connectivity configurationsin CACC systems are various (See Fig. 10 in [18] and Fig. 2 in [19]), which may result inhigh complexity of the control design. Existing research contributions have pointed out thatconnections between the leader vehicle and following vehicles may be more critical in theplatooning system [12], which is the typical CACC scenario. Moreover, more V2V connectionsin the CACC systems may not necessarily improve the robustness if the control gains areinappropriately selected [18]. The V2V transmission rate needs to be large enough, in orderto mitigate the detrimental effects of communication delay on the system stability [4, 20, 21]. Inaddition, interference in the V2V-based CACC systems could deteriorate the V2V’s Quality-of-Service (QoS) and should be properly managed [11, 16], however, such interference managementproblem is challenging in practical dense traffic scenario [22]. The V2I-based ITS systems havealso attracted much attention [23–28]. It is known that V2I provides high-reliable and low-latencycommunication compared to the V2V, and the edge and central cloud computing resources [29]can be utilized in the V2I-based ITS systems. Therefore, V2I can ensure that the traffic flow ismanaged more efficiently and message dissemination is cost-effective [23, 26, 28], particularlyin dense traffic scenario with multi-platoons [25].In the CACC systems, vehicle platooning enables following vehicles to autonomously reachthe leader vehicle’s moving speed and keep the desired inter-vehicle distance while guaranteeingthe safety and stability. Such maneuver control functionality can improve the road throughputand disengage the following vehicles from driving tasks. The aforementioned works mainly focuson the V2V-based platooning systems. Due to its distributed feature, following vehicles undergodifferent levels of communication delays and different numbers of V2V links in the V2V-based platooning systems, which makes the platooning design challenging [4, 20, 30]. The V2V-basedplatooning also has to bear the extra burden of the heterogeneous control mechanisms andhardware resulted from different types of vehicles. To address these issues, this paper proposesan V2I-based platooning design. Compared to the conventional vehicle platooning systems withV2V communications, the advantages of the proposed design are: i) The majority of existingvehicle platooning schemes highly depend on the V2V links, which cannot support long-rangecommunications and are subject to the blockages and severe interference in the dense trafficscenarios. Moreover, following vehicles that cannot directly communicate with the leader vehicleor other vehicles have to let other vehicles relay the vehicles’ DSI, which makes the reliabilitycompromised and inevitably results in high-latency. The proposed design only requires V2Iconnections, which are usually line-of-sight (the roadside units (RSUs) could be sites on thelamp posts); ii) By putting the platoon controller at the RSU with edge computing capability,the proposed design disengages following vehicles from making maneuver control decisions andenables simultaneous maneuver among vehicles in a platoon through sending control commandsto the vehicles’ actuators at the same time, in contrast to the V2V-based designs that differentvehicles receive vehicles’ DSI and carry out control decisions at the different time; iii) In existingplatoon systems, any changes involving targeted inter-vehicle distance and vehicle’s velocity haveto be known by all the vehicles in a platoon, in order to change their states for new formation.In the proposed design, such changes only need to be known at the RSU, which will update thecontrol commands accordingly. Therefore, the proposed design is more efficient and scalable forplatoon management.The main contributions of this paper are concluded as follows: • V2I-based Platooning Control Design:
In the considered system, all the vehicles’ DSI areuploaded to the RSU via massive multiple-input multiple-output (MIMO), and RSU makesthe platooning control decisions including the targeted velocity of the platoon based on theproposed control design. After computing the control inputs of all the following vehicles,RSU sends them to the following vehicles at the same time and frequency band. • Plant and String Stability for the Proposed Platooning Solution:
In light of the com-munication and computing delay concern, the feasible control gain regions for meeting theplant stability and string stability are presented, respectively. We show that the control gainsof the proposed platooning solution can be easily determined by using the D-subdivisionmethod, in order to achieve plant stability. The effects of time headway on the stability are
Fig. 1. An illustration of V2I-based platooning system with edge computing. quantified. • Relationships between Platoon’s Velocity, Radio Parameters and Handover:
To achievethe required QoS of the V2I and avoid frequent handover, the platoon’s velocity needs tobe appropriately chosen. With the assistance of massive MIMO, we provide a tractableapproach to explicitly quantify the relationships between platoon’s velocity, handover andradio parameters including the number of massive MIMO antennas and frequency band. Asimple solution with the help of dual connectivity has been proposed to achieve the seamlessplatooning control when handover occurs. The results are useful guidelines for fast radioresource allocation and handover management. • Design Insights:
Our results show that different control gains have a big impact on thetime of reaching the system stability. Different external disturbances and delays give rise todramatic variations in the vehicles’ traveling speeds and spacing errors, but have negligibleeffect on the disturbance time period before reaching the system stability. The effect ofplatoon size on the platooning stability and efficiency is marginal, which confirms thescalability of the proposed design.The rest of this paper is organized as follows. In Section II, the considered system model isdescribed and the platooning control design is proposed. The stability of the proposed controldesign is analyzed in Section III. The platoon’s velocity and handover are determined in SectionIV. Section V provides the simulation results. Finally, some concluding remarks are presentedin Section VI. II. S
YSTEM D ESCRIPTIONS
As illustrated in Fig. 1, we consider an V2I-based platooning system with massive MIMO,where each RSU equipped with N antennas has edge computing capability [29], and there are M + 1 single-antenna vehicles in a platoon with the leader vehicle and follower vehicle i ( i = 1 , · · · , M ). In such a system, each vehicle simultaneously sends its DSI involving positionand moving speed to the RSU , which shall be processed by RSU for determining platooningcontrol decisions. After edge cloud processing, RSU sends the control commands (i.e., desiredacceleration values) to the corresponding follower vehicles’ actuators. A point-mass model isconsidered to describe the longitudinal vehicle dynamics, which is given by [18, 20, 33] ˙ x i ( t ) = v i ( t ) , ˙ v i ( t ) = u i ( t ) , (1)where x i ( t ) , v i ( t ) , and u i ( t ) are the position, velocity, and control input (acceleration) of thevehicle i at time t , respectively. The spacing error is defined as e i ( t ) = x i ( t ) − x i − ( t ) + hv o + l, (2)where h is the time headway, v o is the targeted platoon’s velocity (The selection of v o valuewill be illustrated in Section IV), and l is standstill distance, hv o + l is the desired inter-vehicledistance. As the leader vehicle travels at the constant speed of v o , the platooning rule is lim t →∞ e i ( t ) = 0 , lim t →∞ v i ( t ) = v o . (3)Since all the vehicles undergo the identical communication delay and the processing delaywith the assistance of massive MIMO and edge computing, the platooning control law at theRSU is designed as u i ( t ) = − K x ( x i ( t − τ ) − x i − ( t − τ ) + hv i ( t − τ ) + l ) − K v ( v i ( t − τ ) − v i − ( t − τ )) − K v o ( v i ( t − τ ) − v o ) − K x o ( x i ( t − τ ) − x o ( t − τ ) + ihv o + il ) , (4)where K x , K v , K v o , and K x o are positive control gains, τ is the total amount of the delayresulted from the communication and edge cloud processing. To guarantee the platoon stability,the control gains need to be chosen appropriately. Remark 1:
The proposed control law only utilizes the DSI of the leader vehicle and the followervehicle i for determining the vehicle i ’s control input. Although existing V2V-based platooningcontrol designs [4, 10, 14, 16] have attempted to make the most of these DSI, the effects of time Note that vehicles’ positions could be evaluated at RSU by applying positioning techniques [31, 32], in this case, delay willbe further cut because of less DSI uploaded to the RSU. headway [4] or communication delay [10, 14] may be ignored for tractability, or some quiteconservative conditions are required [16]. Another benefit of the proposed V2I-based platooningdesign is that the control gains for system stability can be easily calculated, which is illustratedin the next section. III. S
TABILITY A NALYSIS
In this section, the control gains in (4) are determined from the perspective of plant stabilityand string stability. To facilitate the stability analysis, a frequency-domain approach is adopted.According to (1), we have ¨ x i ( t ) − ¨ x i − ( t ) = u i ( t ) − u i − ( t ) . (5)Substituting (4) into (5), after mathematical manipulations, (5) is rewritten as ¨ x i ( t ) − ¨ x i − ( t ) = − λ ( x i ( t − τ ) − x i − ( t − τ ))+ K x ( x i − ( t − τ ) − x i − ( t − τ )) − η ( v i ( t − τ ) − v i − ( t − τ ))+ K v ( v i − ( t − τ ) − v i − ( t − τ )) − K x o ( hv o + l ) , (6)where λ = K x + K x o and η = K x h + K v + K v o . Let E i ( s ) = L { e i ( t ) } denote the Laplacetransform of the spacing error e i ( t ) , taking the Laplace transform of (2) yields L { x i ( t − τ ) − x i − ( t − τ ) } = e − τs E i ( s ) − e − τs hv o + ls . (7)Based on (7), the Laplace transform of (6) is given by E i ( s ) = ( K v s + K x ) e − τs Θ ( s ) E i − ( s )+ s + ( η − K v ) e − τs + ( e − τs − K xo s Θ ( s ) ( hv o + l ) , (8)where Θ ( s ) = s + ηse − τs + λe − τs is referred to as characteristic function. Therefore, in theproposed platooning design, the spacing error transfer function is calculated as H i ( s ) = ( K v s + K x ) e − τs Θ ( s ) . (9) A. Plant Stability
Plant stability is achieved when the platooning rule given by (3) is met. As such, the necessaryand sufficient condition for satisfying the plant stability is
Re ( s ) < , ∀ Θ ( s ) = 0 , (10)which means that for an arbitrary characteristic root of Θ ( s ) , it has negative real part. Thecomplexity of solving (10) depends on the specific spacing error transfer function, which isdetermined by the platooning control law. The use of the Routh-Hurwitz criterion with Pad´eapproximation requires that the spacing error transfer function for the frequency range of interestcan be well approximated [16, 21, 34], which may bring in more complexity. Considering theproposed platooning law given by (4), we show that the control gains for achieving plant stabilitycan be easily and precisely obtained by leveraging the D-subdivision method [35]. Based on (10),we have the following theorem: Theorem Plant stability can be guaranteed if and only if ( λ, η ) belongs to the feasibleregion: G ( τ ) = (cid:40) ( λ, η ) : λ ≤ w cos ( τ w ) ,η ≤ w sin ( τ w ) , w ∈ (cid:16) , π τ (cid:17) (cid:41) . (11) Proof See Appendix A.
Remark 2:
As shown in Fig. 2, the size of the feasible region G ( τ ) decreases as delay increases.Based on Theorem 1 , we see that η < π τ . Therefore, for a specific η ∗ ∈ (cid:0) , π τ (cid:1) , the criticalvalue w ∗ for η ∗ = w ∗ sin ( τ w ∗ ) can be efficiently calculated by using one-dimension search since w sin ( τ w ) is the increasing function of w ∈ (cid:0) , π τ (cid:1) . Then, we can obtain the corresponding λ ∗ = ( w ∗ ) cos ( τ w ∗ ) . In light of the point ( λ ∗ , η ∗ ) on the D-curve (See Appendix A), the plantstability requires λ ∈ (0 , λ ∗ ) for a specific η ∗ ∈ (cid:0) , π τ (cid:1) . B. String Stability
In the platooning systems, unstable vehicle strings give rise to phantom traffic jams [3]. Stringstability ensures that the spacing error is not amplified in the traffic flow upstream [14, 21],namely the magnitude of the spacing error transfer function H i ( s ) needs to satisfy |H i ( jw ) | < .As such, we have the following theorem: Fig. 2. The plant stability region G ( τ ) for different levels of delay with different corner points. Theorem String stability can be guaranteed when ( λ, η ) belongs to the feasible region: S ( τ ) = (cid:40) ( λ, η ) : λ ≤ K v K v o , η ≤ τ (cid:41) . (12) Proof See Appendix B.
Remark 3:
From (12), we see that the size of the feasible region S ( τ ) decreases as delayincreases. The time headway satisfies h < (cid:0) τ − K v − K v o (cid:1) /K x . Compared to the platooningmethod of [30] with ACC where the time headway has to be larger than τ for string stability,our design can keep the time headway at a minimum required level by selecting the propercontrol gains based on (12), hence the road throughput can be significantly improved.IV. P LATOON ’ S V ELOCITY AND H ANDOVER
The previous section has provided the stability regions of the proposed platooning design givena targeted velocity of the stable platoon. In practice, the targeted velocity of a stable platoon has tobe chosen appropriately, which has a big impact on the inter-vehicle distance, platoon size/lengthand QoS of the V2X communications. Unfortunately, such concern has not been paid enoughattention yet. Existing works such as [11] have shown that inappropriate inter-vehicle distancein a platoon could deteriorate the message dissemination in the V2V links. Research effortshave focused on how to obtain the optimal inter-vehicle distance under QoS constraint [25].
However, the study of the relationships between platoon’s velocity, time headway, handoverand radio parameters is still in its infancy. Some critical concerns in the early works such asmassive information exchange for centralized formation control [10] can be easily addressednow, since the radio technologies have developed faster than ever before. In this section, weseek a low-complexity approach to answer the following questions: • How to quantify the relationship between the RSU coverage and platoon size/length? • How to allocate the radio resources given a platoon configuration? • How to manage the handover between RSUs given a platoon configuration?It is de facto challenging to find a generic solution for these questions. As such, we considerthe platooning systems with the massive MIMO aided V2I communications. Massive MIMO isone of key 5G radio technologies and enables communications with dozens of users at the sametime and frequency band [36]. Moreover, it can achieve high-speed transmission rate, combatthe co-channel interference, and facilitate resource allocation [29, 37].We adopt a linear massive MIMO processing method for V2I communication, i.e., zero-forcing(ZF) detection is implemented at RSU. The achievable communication rate (bps) of the vehicle i is given by [38] R i = B log (cid:18) P v i ( N − M − βd i − α σ (cid:19) , (13)where B is the platoon system bandwidth, P v i is the vehicle i ’s transmit power, β is the constantparameter commonly-set as ( c πf c ) with c = 3 × m / s and the carrier frequency f c , d i is thecommunication distance, α is the path loss exponent, and σ is the noise power. Note that dueto the “channel hardening” feature of massive MIMO [29, 37], the small-scale fading effectsare averaged out. Therefore, given a minimum communication rate threshold R th (namely QoSconstraint), the radius of the RSU coverage is d th = P v i ( N − M − βσ (cid:16) R th B − (cid:17) /α . (14)Let r o and h o denote the perpendicular distance and the absolute antenna elevation differencebetween the platoon vehicle and the RSU, respectively, based on (14), the maximum longitudinalcoverage range of the RSU is (cid:96) th = (cid:113) d − r o − h o . (15) For a specific targeted velocity of the stable platoon v o , the platoon size/length is calculatedas D platoon = M hv o + M l. (16)The traveling time for a stable platoon in an RSU coverage area before undergoing handover is T stay = 2 (cid:96) − D platoon v o , (17)where (cid:96) is calculated by using (15) with P v i = P v , due to the fact that the leader vehicle isthe first to leave an RSU’s coverage area. Let f handover denote the maximum allowable handoverfrequency between RSUs, in other words, the minimum traveling duration for a platoon in anRSU coverage area is /f handover . It is obvious that T stay should be greater than /f handover .Thus, by considering (16) and (17), we have the following condition: v o ≤ (cid:96) − M lM h + 1 /f handover . (18) Remark 4:
It is indicated from (18) that given the radio resources and handover frequency,platoon’s velocity decreases when time headway increases, i.e., there is a tradeoff betweenplatoon’s velocity and time headway. Given a platoon configuration, the minimum requirednumber of massive MIMO antennas or bandwidth under the QoS and handover constraintscan be easily evaluated based on (18). Therefore, (18) is useful for the fast radio resource
TABLE IR
ESULTS BASED ON (18) f c R th (Mbps) f handover (times/s) maximum v o (m/s)3.5GHz 75 1/30 2475 1/20 3575 1/10 655.9GHz 75 1/30 1475 1/20 2075 1/10 38N.B.: In the table, τ = 0 . s, h = 0 . s, N = 64 , M = 9 , ML = 15 m, r o = 10 m, h o = 6 m, α = 2 , β = ( c πf c ) , B = 5 MHz, P v = 20 dBm, σ = −
174 + 10 log ( B ) dBm. allocation and handover management in the platoon systems. As shown in the Table I, higherplatoon’s velocity results in more handovers for the same frequency band, and higher frequency Fig. 3. A platoon can be seamlessly served by RSUs as the platoon is in the dual-connectivity range during the handover. band reduces the level of the maximum allowable platoon’s velocity for a fixed number ofantennas and bandwidth( N = 64 and B = 5 MHz in the Table I). To keep the desired levelsof platoon’s velocity and QoS, more numbers of antennas and bandwidths are demanded in thehigher frequencies.The aforementioned has shown how to manage the platoon’s velocity and radio resources inorder to avoid frequent handover and meet the QoS requirement. In practice, it is important thatthe V2I-based platoon can be seamlessly controlled by RSUs when handover occurs. We realizethat dual connectivity has been adopted in 4G and 5G systems [39, 40], to enhance the mobilityrobustness in cellular networks. Since dual connectivity allows a user to communicate with mul-tiple network nodes at the same time, the QoS constraint can be guaranteed during the handover.As shown in Fig. 3, the inter-site longitudinal distance (ISLD) should be kept at a certain levelto ensure that the platoon is in the dual-connectivity range during the handover. Based on (14)and (15), we can easily calculate the maximum allowable ISLD for dual connectivity as (cid:96) maxISLD = 2 (cid:96) − D platoon = 2 P v ( N − M − βσ (cid:16) R th B − (cid:17) /α − r o − h o / − D platoon . (19)From (19), we see that by using dual connectivity, the V2I-based platooning systems can beseamlessly served by RSUs when the ISLD is below (cid:96) maxISLD . It should be noted that such V2I-based platooning handover approach is flexible, for instance, by managing the radio resourcessuch as transmit power and the number of massive MIMO antennas in (19), ISLD can be easilytailored to meet various circumstances. TABLE IIS
IMULATION PARAMETERS IN F IGS . 4
AND τ K v K v o K x K x o V. N
UMERICAL R ESULTS
In this section, numerical results are provided to demonstrate the efficiency of the proposedV2I-based platooning design and validate our analysis. In addition, the effects of different controlgains, external disturbances, platoon sizes and delays on the performance are illustrated.
A. Efficiency of the Proposed Platooning Design
This subsection shows the efficiency of the proposed design. In the simulations, the timeheadway h = 0 . s, the number of follower vehicles is M = 4 , and the spacing error is zerobefore leader vehicle changes its velocity. The leader vehicle suffers an external disturbanceduring the time period ≤ t ≤ s, which is modeled by assuming that its acceleration variesas ˙ v ( t ) = − sin ( t ) . The other system parameters are summarized in the Table II.Fig. 4 shows the proposed platooning design can efficiently achieve plant stability and stringstability for different levels of delay. As mentioned in Theorem 2, the magnitude of the spacingerror transfer function is kept below 1 for an arbitrary frequency w and the spacing error decreasesin the traffic flow upstream(namely the vehicle index increases) since the control gains are chosenfrom the feasible region S ( τ ) given by (12). The spacing errors of the follower vehicles can bequickly diminished to zero when the leader vehicle’s external disturbance is gone at t > s,since the control gains belongs to the feasible region G ( τ ) given by (11) and thus plant stabilityis guaranteed.Fig. 5 shows the case when the control gains are chosen from the outside of S ( τ ) ( λ > K v K v o in the Table II). As analyzed before, the magnitude of the spacing error transfer function is largerthan 1 for certain w values, in this case, the spacing errors of the follower vehicles are amplifiedin the traffic flow upstream, i.e., string instability occurs. w (rad/s) M a gn it ud e o f t h e s p ac i ng e rr o r t r a n s f e r f un c ti on t (s) -0.400.40.81.21.6 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 (a) w (rad/s) M a gn it ud e o f t h e s p ac i ng e rr o r t r a n s f e r f un c ti on t (s) -0.8-0.400.40.81.2 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 (b) w (rad/s) M a gn it ud e o f t h e s p ac i ng e rr o r t r a n s f e r f un c ti on t (s) -0.6-0.20.20.611.4 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 (c)Fig. 4. The platooning performance of the proposed design. w (rad/s) M a gn it ud e o f t h e s p ac i ng e rr o r t r a n s f e r f un c ti on t (s) -10-50510 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4
Fig. 5. String instability result when control gains do not belong to the feasible region given by (12).
B. Effects of Control Gains
This subsection shows the effects of choosing different control gains. The leader vehicle’sacceleration varies as ˙ v ( t ) = − sin ( t ) at ≤ t ≤ (s), M = 4 , τ = 0 . s and h =0 . s. The control gain vectors in Fig. 6(a) and Fig. 6(b) are given by [ K v , K v o , K x , K x o ] =[1 . , . , . , . and [ K v , K v o , K x , K x o ] = [1 . , . , . , . , respectively, which are chosenfrom the the feasible regions in Section III.It is seen in Fig. 6(a) and Fig. 6(b) that both control gain vectors can achieve plant stabilityand string stability, since they belong to the feasible regions mentioned in Section III. Althoughthe platoon experiences the same external disturbance, the slightly different values of the controlgains may cause significantly different performance behaviors, i.e., the control gains used in Fig.6(a) make the follower vehicles’ space errors vary more drastically, and the platoon needs tospend more time on reaching the stability, compared to the case of control gains used in Fig.6(b). C. Effects of External Disturbance
This subsection shows the effects of different external disturbances imposed on the leadervehicle. Specifically, the external disturbances of the platoon for the simulations in Fig. 7(a) and t (s) -0.20.20.611.41.6 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 (a) t (s) -0.20.20.611.4 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 (b)Fig. 6. Effects of different control gains.
Fig. 7(b) are given by ˙ v ( t ) = , ≤ t ≤ , , < t ≤ , − , < t ≤ , (20)and ˙ v ( t ) = , ≤ t ≤ , − , < t ≤ , (21) t (s) -6-4-201 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 t (s) V e l o c it y ( m / s ) Veh 0Veh 1Veh 2Veh 3Veh 4 (a) t (s) -7-5-3-11 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4 t (s) V e l o c it y ( m / s ) Veh 0Veh 1Veh 2Veh 3Veh 4 (b)Fig. 7. Effects of different external disturbances. respectively. The other basic simulation parameters are identical in the results of Fig. 7(a) andFig. 7(b), namely the control gain vector [ K v , K v o , K x , K x o ] = [0 . , . , . , . , M = 4 , τ = 0 . s and h = 0 . s.It is seen from Fig. 7(a) and Fig. 7(b) that although the platoon stability for these two typesof external disturbances are achieved at almost the same time, the external disturbance givenby (21) forces the follower vehicles to change their moving speeds more rapidly and results inlarger spacing errors, compared to the type of external disturbance given by (20). Such dramaticchanges of the vehicles’ driving states during the external disturbance may need to be properlyaddressed in practice, due to the fact that different vehicles may have velocity limitations under t (s) -0.6-0.300.30.60.91.21.4 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3 (a) t (s) -0.6-0.300.30.60.91.21.4 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4Veh 5Veh 6Veh 7Veh 8 (b)Fig. 8. Effects of different platoon sizes. hardware constraints.
D. Effects of Platoon Size
This subsection shows the effects of platoon size. In the simulations, there are two platoonsconsisting of three and eight follower vehicles, respectively, the control gain vector [ K v , K v o , K x , K x o ] =[0 . , . , . , . , the leader vehicle’s acceleration varies as ˙ v ( t ) = − sin ( t ) at ≤ t ≤ (s), τ = 0 . s and h = 0 . s.It is seen from Fig. 8(a) and Fig. 8(b) that when the control gains and other system parametersare fixed, changing the platoon size has negligible effect on the spacing errors of the follower t (s) -0.200.10.40.711.31.6 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4Veh 5Veh 6 (a) t (s) -0.6-0.300.30.60.91.21.4 S p ac i ng e rr o r ( m ) Veh 1Veh 2Veh 3Veh 4Veh 5Veh 6 (b)Fig. 9. Effects of different levels of delay. vehicles, which confirms the scalability of the proposed platooning design. In addition, platoonswith different sizes have nearly same disturbance time period before reaching the system stability.
E. Effects of Delay
This subsection shows the effects of delay. In the simulations, we consider two delay cases,i.e., τ = 0 . s in Fig. 9(a) and τ = 0 . s in Fig. 9(b), the platoon consists of six follower vehiclesbesides the leader vehicle, the control gain vector [ K v , K v o , K x , K x o ] = [0 . , . , . , . ,the leader vehicle’s acceleration varies as ˙ v ( t ) = − sin ( t ) at ≤ t ≤ (s), and h = 0 . s. It is seen from Fig. 9(a) and Fig. 9(b) that when the control gains and other system parametersare fixed, different levels of delay have a big impact on the spacing errors during the disturbancetime period. However, the time of reaching the system stability is nearly unaltered for differentdelay cases. Through the comparison with the results in Fig. 8, it is again confirmed that platoonswith different sizes has negligible effect on the stability and efficiency of the proposed designwhen the rest of system parameters and external disturbance are identical.VI. C
ONCLUSIONS
This paper concentrated on the V2I-based platooning systems, where RSUs have the capa-bilities of massive MIMO and edge computing. By considering the effect of delay, an efficientplatooning control approach was developed. We demonstrated that the proposed platooning designcan achieve both plant stability and string stability by selecting control gains in the derivedfeasible regions. Moreover, we provided a tractable method to explicitly quantify the relationshipsbetween the platoon’s velocity, platoon size/length, radio resources and handover. By usingour derivations, the platoon’s velocity, radio sources and handover can be easily determined.Simulation results confirmed the efficiency of the proposed platooning design, and the effects ofdifferent control gains, external disturbances, platoon sizes and delays on the performance werecomprehensively illustrated. A
PPENDIX
A: P
ROOF OF T HEOREM s = ξ + jw , the characteristic equation Θ ( s ) = 0 can be decomposedinto real and imaginary parts, which are Re : ηξ cos ( τ w ) + ηw sin ( τ w ) + λ cos ( τ w ) = e τξ (cid:0) w − ξ (cid:1) , (A.1) Im : ηw cos ( τ w ) − ηξ sin ( τ w ) − λ sin ( τ w ) + 2 e τξ ξw = 0 . (A.2)By letting ξ = 0 , the D-curves can be expressed as Re : ηw sin ( τ w ) + λ cos ( τ w ) = w , (A.3) Im : ηw cos ( τ w ) = λ sin ( τ w ) . (A.4)The above equation can be equivalently written as λ = w cos ( τ w ) , η = w sin ( τ w ) . (A.5) Note that w > and τ w ∈ (cid:0) kπ, π + 2 kπ (cid:1) , k = 0 , , , · · · , since λ > and η > .To determine the crossing direction from stability to instability along the D-curves, we firsttake the first-order derivative of (A.1) and (A.2) with respect to η at ξ = 0 (along the D-curves),after mathematical manipulations, the first-order derivative of ξ at ξ = 0 is dξdη = w ( τη − τw ))(2 w + τλ sin( τw ) − τηw cos( τw ) − η sin( τw )) ( τηw sin( τw )+ τλ cos( τw ) − η cos( τw )) (2 w + τλ sin( τw ) − τηw cos( τw ) − η sin( τw )) + 1 . (A.6)From (A.6), we see that when η > τw ) τ , dξdη > , i.e., there exists the positive real partof the characteristic root, and the plant stability is violated as η increases. Based on (A.5), η = π τ + kπτ ( k = 0 , , , · · · ) as λ = 0 . Considering the fact that dξdη > with η = π τ , ( λ, η ) = (cid:0) , π τ (cid:1) is a corner point of the stability region, which means that τ w ∈ (cid:0) , π (cid:1) .Likewise, taking the first-order derivative of (A.1) and (A.2) with respect to λ at ξ = 0 (alongthe D-curves), after mathematical manipulations, we have dξdλ = τλ + η ( τλ cos( τw )+ τηw sin( τw ) − η cos( τw )) (2 w + τλ sin( τw ) − τηw cos( τw ) − η sin( τw )) ( τλ cos( τw )+ τηw sin( τw ) − η cos( τw )) + 1 . (A.7)From (A.7), we see that dξdλ > for arbitrary λ value, which means that the plant stability isviolated as λ increases. Thus, we can finally obtain the feasible region G ( τ ) given by (11).A PPENDIX
B: P
ROOF OF T HEOREM |H i ( jw ) | < . Based on (9), |H i ( jw ) | is given by |H i ( jw ) | = (cid:115) K v w + K x Ξ ( w ) + K v w + K x , (B.1)where Ξ ( w ) = w − η sin ( τ w ) w + (cid:0) K x h + 2 K x ( K v + K v o ) h + K v o + 2 K v K v o (cid:1) w − λ cos ( τ w ) w + K x o + 2 K x K x o . (B.2)Note that both the numerator and denominator of (B.1) have the positive term K v w + K x , thus |H i ( jw ) | < is equivalently transformed as Ξ ( w ) > , ∀ w ≥ . Considering the fact that sin ( τ w ) ≤ τ w and cos ( τ w ) ≤ , we have − η sin ( τ w ) w ≥ − ητ w , λ cos ( τ w ) w ≤ λw . (B.3) Based on (B.2) and (B.3), the following inequality is obtained as
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