Longitudinal Vehicle Guidance in Networks with changing Communication Topology

6th IFAC Symposium Advances in Automotive Control Munich, Germany, July 12-14, 2010

Longitudinal Vehicle Guidance in Networks with changing Communication Topology Jan P. Maschuw ∗ D. Abel ∗ ∗ Institute of Automatic Control, 52056 Aachen, Germany (Tel: +49 (241) 8027516; e-mail: [email protected]).

Abstract: This paper presents a novel approach to control layout for vehicle guidance with changing communication topology. The control adapts to possible communication failure and varying access to other vehicle’s information including stability conditions for arbitrary switching between different topologies. On the one hand the control layout addresses deficiencies in vehicleto-vehicle communication, it further accounts for the reduction of spacing errors and a limitation in velocities and accelerations of following vehicles to avoid saturations. All objectives are formulated as one set of linear matrix inequalities that are solved for each of the adaptive vehicle controls. The optimization is presented for different information topologies of a platoon with four vehicles and the effectiveness of preserving the control performance in case of communication failure is shown through simulations. Keywords: Vehicle-to-vehicle communication, Networked control systems, Switching systems, Driver assistance systems, Automated guided vehicles, Decentralized control, Linear matrix inequalities 1. INTRODUCTION Information exchange between single vehicles (vehicle-tovehicle: V2V) and communication with surrounding infrastructure (vehicle-to-roadside: V2R) have attracted a lot of interest in research and industry in recent years. The topic is addressed in programs like “eSafety” (European commission) or “Car2Car Communication Consortium” (industry) and research projects directly using intervehicle communication for longitudinal or lateral control actuation, e.g. “PATH” (USA) or “Promote Chauffeur” (EU). The research field of longitudinal vehicle guidance has published lots of results over the last decades showing the importance and benefits of V2V communication. To cite just some of the important work we refer to Swaroop and Hedrick (1999) and Lu et al. (2004). On highways, one could also see individual traffic as (ad-hoc) platoons and apply the results and control concepts to a vehicle group especially in critical situations where the driver would be too slow to react. On the other hand, in industry (production) the use of V2V communication for longitudinal control seems to be further away. The integration of inter-vehicle information into a closed-loop control of longitudinal dynamics still lacks of guarantees for the quality of service (QoS) of a communication network. Possible deficiencies in security, latencies and reliability of radio communication (i.e. possible failure or change of the availability of a node and it’s information) limit the application of the information in vehicle dynamics. This paper focuses on a control layout for vehicle platoons that explicitly accounts for the limited QoS in radio communication networks. The authors present a control reconfiguration that changes with the amount of available information to overcome deficiencies from communication 978-3-902661-72-2/10/$20.00 © 2010 IFAC

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failure. A decentralized control layout is done for each vehicle exploiting all available information to optimize the overall platoon performance. The paper is structured as follows: In section 2 we will describe the models for both the vehicle dynamics and the expression of information topology. A control design based on linear matrix inequalities (LMIs) to stabilize the platoon for changing information access is presented in section 3. Simulation results for different information and switching conditions are given in section 4. 2. MODEL FOR PLATOON AND COMMUNICATION NETWORK 2.1 Vehicle Dynamics The longitudinal dynamics of a vehicle platoon is mainly governed by the drivetrain dynamics of each vehicle. Although complexity of drivetrain models is well known in literature, it was shown that by lower level controls the dynamics of vehicle acceleration can be approximated by a linear first order filter, see Ha et al. (1989) and Lu et al. (2004). According to Fig. 1, the time behavior of the vehicle platoon can then be described by the dynamics of distance errors and vehicle accelerations. For each following vehicle i the error ei is defined as the difference between the actual distance to the predecessor and a (fixed) reference distance: ei (t) = di (t) − dref,i . Together with the approximated drivetrain dynamics the resulting platoon model is given by e¨i = ai−1 − ai ,

(1)

a˙ i = −1/Ti · ai + 1/Ti · ui .

(2)

10.3182/20100712-3-DE-2013.00049

AAC 2010 Munich, Germany, July 12-14, 2010

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Fig. 1. Vehicle platoon with leading and following vehicles sharing their data via radio communication. We assume, that each vehicle is equipped with on-board sensors measuring the distance error ei , relative velocity ∆v = e˙ i and acceleration ai . All measurement data is exchanged among the following vehicles via radio communication as depicted in Fig. 1. For convenience we will express the platoon dynamics in state space form and summarize all vehicle states in the state vector x = [· · · ei , e˙ i , ai · · · ]T while the leading vehicle’s acceleration aL enters the dynamics as a disturbance variable. Typically, the time constant of the drivetrain Ti varies depending on the engine’s operating point, the current gear and finally the different dynamics of brakes and engine. Hence, the state space model has an affine dependence on the varying parameters ξi = 1/Ti and is given as x˙ = A(ξ)x + B1 aL +

X

B2,i (ξ)ui .

(3)

T (θ) = I + R(θ)T

(5)

which in turn depends on the binary parameter vector θ. All information accessed by node i is now given by the i-th row vector of the topology matrix which will be denoted by ti . The available state information can be represented by the output vector yi as yi = (diag(ti (θ)) ⊗ In )x = Cy,i (θ)x.

(6)

Since the diagonal matrix is extended by the Kronecker product with the Identity matrix of dimension n according to the number of vehicle states (here n=3), the multiplication with the full state vector x leads to a vector listing exactly the available states. The resulting matrix will be denoted by Cy,i (θ) in the following. Equation (3) and (6) build the full system description for the platoon, with the parameters ξ (continuously) and θ (event-based) changing in time. All variables introduced so far refer to the nomenclature given in Table 1. It should be noted that only communication between following vehicles is addressed in this work – the above representation and the following control design can be extended to communication with the leading vehicle, though.

i

Table 1. System variables and parameters. Symbol di dref,i ei ai vi ui ξi θi

Fig. 2. Topology of information transmitted between different vehicles. Since the available information (and hence the edges) vary in time, we express the adjacency matrix as R(θ) depending on a binary parameter vector θ with θi ∈ {0, 1}. The above graph only shows the transmitted state information. In addition, each vehicle has access to it’s own states by on-board sensors, this information we assume as fixed since it does not rely on radio communication. In order to describe all information, i.e. local and transmitted states, the adjacency matrix will be extended by ones in the diagonal representing the additional local information. Hence, we introduce the topology matrix T as

Description distance of vehicle i to predecessor reference distance for vehicle i distance error between vehicle i − 1 and i acceleration of vehicle i longitudinal velocity of vehicle i control signal of vehicle i variable of varying drivetrain dynamics variable of varying received information

2.2 Information Topology

3. CONTROL DESIGN Due to deficiencies in radio communication we can not assume that each following vehicle always has access to all other vehicles’ state information. A possible way of expressing the information received via radio communication is a directed graph as in Fig. 2. Each directed edge between two vehicle nodes i and j expresses that all state information of vehicle i is received by vehicle j. We assume that either all state information of another vehicle is available or none, also we don’t take latencies of the data transfer into account and only focus on the information topology. An algebraic representation of the above graph is the adjacency matrix R with  1, if an edge from node i to j exists, (4) ri,j := 0, otherwise. 786

3.1 Control Objectives The goal of platoon control typically is the stabilization of platoon dynamics and the minimization of distance errors (especially the avoidance of collisions) while keeping the control effort low. This so far can be considered as a problem of linear optimal control or an equivalent weighting of the H2 -norms for errors and control signals. Further structural conditions are string stability in terms of non-amplifying errors along the platoon but also the minimization of upper bounds on velocity and acceleration overshoots at followers with respect to the leading vehicle. This is of major importance since linear theory does not cover the existing saturation effects in velocity and especially acceleration that arise e.g. in emergency braking

AAC 2010 Munich, Germany, July 12-14, 2010

due to force limitation between road and tire, see also Kang and Hedrick (2004). The corresponding transfer functions for the velocity related to the leading vehicle vi (t) − vL (t) and the acceleration ai (t) are given as Fv,i (s) =

(vi − vL )(s) aL (s)

and Fa,i (s) =

ai (s) . aL (s)

(7)

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V1

For each vehicle the response in terms of the maximum values is bounded by the impulse response of the corresponding transfer function (we only give the relation for accelerations here): ||ai (t)||∞ ≤ ||fa,i (t)||1 · ||aL (t)||∞ .

(8)

The objective can now be formulated as minimizing or limiting the 1-norm of the impulse responses with an upper bound γ. The same bounding condition (although this is only necessary) can be given for the H∞ -norms of the corresponding transfer functions. Apart from the minimization of H2 /H∞ -norms we seek a control that leads to an asymptotically stable closed loop for all unknown or changing parameters ξ (drivetrain) and θ (information topology). Thus, a further objective is robustness with respect to changing drivetrain dynamics and changing information topology. Further constraints regard the controller dynamics and occur as bounds on the pole spectrum. They either result from the time discrete realization as digital controller to account for sampling limits. The same constraints also result from avoiding an amplification of measurement noise. The objective can be formulated as a set D where the eigenvalues of the ¯ should lie. controller or those of the closed loop λ Taking all conditions together, we can formulate the control objective as the following H2 /H∞ minimization problem: min α · ||F ||2∞ + β · ||H||22

s.t.

(9)

||Fi ||∞ < γ

∀i,

(10)

asymp. stab. of cl. loop ∀ξ(t), θ(t), ¯∈D λ

(11) (12)

where the transfer function F (s) stands for both the velocities and the accelerations, the transfer function H(s) expresses the mapping from the disturbance aL to distance errors and control signals. The two objectives are weighted by the factors α and β. 3.2 Control Structure If each vehicle receives the information of all others a centralized control layout with full state feedback can be done. In this case, the topology matrix T (θ) only includes ones and hence Cy,i (θ) becomes the Identity or similarly yi becomes the state vector x of the platoon. If all states are available the best overall performance of the closed loop in terms of the H2 or H∞ conditions can already be achieved by static state feedback, i.e. no further improvement will be given by dynamic controllers, see Boyd et al. (1994). For this case, we write the control for each vehicle as u i = Ki x

(13)

or for the whole platoon as u = Kx. For any other case, when certain state information is not known at some 787

V4

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Fig. 3. Decentralized control layout on the basis of available information, here for vehicle 4. vehicle i, the above statement does not hold. The missing information can be expressed by enforcing zeros in Ki at each position where no information is received. Even reconfiguration of the remaining parameters in Ki will lead to suboptimal performance of the platoon as compared to full state feedback. To avoid deficiencies from communication failure, we will present a control reconfiguration on the basis of dynamic output feedback for each vehicle i. One approach taken in literature is the design of observers (as part of the controller) to reconstruct missing data. Stankovic et al. (2007) use Luenberger observers for platoon guidance based on pure on-board data. In this paper we don’t determine the controller dynamics by an observer model but by direct optimization of problem (9)-(12) presented in the last section. To account for the (potentially) different individual access to information we follow a decentralized control layout for each vehicle i that only uses the information given by the current topology as sketched in Fig. 3. Since the objective function still covers the overall performance of the full platoon we assume a cooperative control of the other vehicles. It will be shown that with dynamic output feedback the performance (achieved with full state feedback) can in principle be recovered. Hence, a good approximation of the other vehicles’ behavior is given by the optimal solution of full state feedback Kr . At vehicle i, dynamic output feedback is applied using only the available information. The corresponding plant is then given as x˙ = A(ξ)x + B1 aL + B2,r (ξ)Kr x + B2,i (ξ)ui ,

(14)

yi = Cy,i (θ)x.

(15)

The control command ui is determined by a dynamic output feedback controller of the form x˙ k = Ak (θ)xk + Bk (θ)yi , ui = Ck (θ)xk + Dk yi .

dim(xk ) = dim(x) (16) (17)

For each information topology T (θ) the controller matrices Ak , Bk and Ck are optimized and hence, depend itself on the current topology given by θ. Only the static gain matrix Dk will be fixed to avoid high gains of unfiltered disturbances on the measured and transmitted information yi . We choose Dk as the i-th row of the optimal state feedback matrix, i.e. in case of full information access at vehicle i the control equals full state feedback if just Ck (θ) is set to zero. A control layout accounting for robustness with respect to changing drivetrain parameters ξ was presented in prior works of the authors already, see Maschuw et al.

AAC 2010 Munich, Germany, July 12-14, 2010

(2008). Therefore, the authors will focus on robustness with respect to changing information topology in the following section of this paper.

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3.3 LMI Formulation

0.2

All objectives from problem (9)-(12) can be cast as LMI conditions and hence lead to a convex optimization problem that can be solved very efficiently. The closed-loop system is given by inserting (16)-(17) into (14)-(15) such that

(18)

x˙ k = Bk (θ)Cy,i (θ)x + Ak (θ)xk .

z∞ = C1 x + D1,1 aL + D1,2 ui ,

(20)

z2 = C2 x + D2,1 aL + D2,2 ui .

(21)

If in the above equations the control ui is also substituted by the dynamic output feedback (17) the generalized plant can be formulated in terms of the state vector [x, xk ]T and the system matrices of the closed loop that will ¯ ¯1 , C¯1 , C¯2 (θ) and D. ¯ The mixed be denoted by A(θ), B H2 /H∞ optimization can now be given as s.t.

(22)

 ¯ + X A¯T B ¯1 X C¯1T AX  ¯T ¯T  < 0 B −I D 1 ¯ −γ 2 I C¯1 X D   Q C¯2 X >0 X C¯2T X ¯ L ⊗ X + M ⊗ (AX) + M T ⊗ (X A¯T ) < 0 

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Fig. 4. Platoon response to a velocity step with static feedback of full information from preceding vehicles. for switching information access. It should be mentioned that this condition is quite conservative, on the one hand because a CQLF is only sufficient and no necessary condition and second, because infinitely fast switching of the communication topology is very unlikely even for traffic applications. We further recall that the decentralized layout approximates the other vehicles’ behavior by full state feedback (irrespective of their information access) and hence, asymptotic stability of the whole platoon can not directly be guaranteed and has to be checked a posteriori. Nevertheless, the results presented in the following section show the effectiveness of the proposed design. 4. RESULTS

where the last LMI relates to the region for the closed-loop poles which is defined by the matrices L and M . These conditions can be transformed such that the optimization parameters γ 2 , Q, X along with the control parameters only appear linearly, see Chilali and Gahinet (1996) and Scherer et al. (1997) for an exhaustive description of LMI formulations. ¯ In the above set of LMIs the closed-loop matrices A(θ) ¯ and C2 (θ) depend on the current topology given by θ. To achieve an optimal performance for each control applied to the current communication topology the LMI formulations simply have to be repeated for each topology. A worst-case assumption for the variation of the parameters in θ is arbitrary switching. To achieve asymptotic stability for arbitrary switching a (sufficient) condition for the set of linear switched systems is the existence of a common quadratic Lyapunov function (CQLF), see Liberzon (2003). In this context it follows that the stability condition ¯ ¯ T < 0 with A(θ)X + X A(θ)

2

(19)

This system can now be extended to a generalized plant including the outputs z2 and z∞ that represent spacing errors, control signals, velocities and accelerations of the platoon to be evaluated by the H2 - and H∞ -norm:

min α · γ 2 + β · trace(Q)

0

i

+B1 aL ,

−0.2

v [m/sec]

x˙ = (A + B2,r Kr + B2,i Dk Cy,i (θ))x + B2,i Ck (θ)xk

0

(23)

has to be fulfilled for all θ with a single positive definite and symmetric matrix X. Hence, the above series of LMIs has to hold for a common X to achieve quadratic stability 788

For control optimization and simulation we analyze a platoon of four members with one leading and three following vehicles that are to keep a constant reference distance. For all vehicles a nominal time constant is assumed and was chosen as Ti = 0.5s, see Rajamani (2006) for references of drivetrain time constants. The numerical optimization of the semidefinite program (22) was done using the Matlab toolbox Yalmip, see L¨ofberg (2004). 4.1 Nominal Control For full access to information of all vehicles the optimal H2 /H∞ performance can be found by static state feedback which was done in prior works of the authors. To achieve low overshoots in velocities (and likewise accelerations) and a fast decline of distance errors a weighting of the H2 and H∞ criterion with α = 0.5 and β = 1 was chosen. The achievable performance depends on the amount of information from other platoon members, especially from preceding vehicles. For comparison, we first analyze the solution with full state feedback and missing information of successors, the corresponding entries in Ki are set to zero for simulation purposes. To analyze the quality of disturbance rejection a velocity step of the leading vehicle is chosen as test function, the resulting platoon response is shown in Fig. 4. It can be seen, that fast attenuation of spacing errors and low overshoots of velocities are achieved as long as full information of preceding vehicles is available. If additionally, information of preceding vehicles is missing due to communication failure performance deteriorates

AAC 2010 Munich, Germany, July 12-14, 2010

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heavily if no reconfiguration is used. The result in Fig. 5 shows the platoon response for the case when information of the first following vehicle is not accessed by the successors, the corresponding entries in Ki are set to zero again. It can be seen that not only the overshoots in velocities increase for the following vehicles but also the distance errors amplify – string stability in terms of declining errors along the platoon is lost. The performance becomes even worse when no communication information is available and only on-board data can be used.

Fig. 6. Reconfiguration with information topology switching between full and no information from preceding vehicles.

u [m/sec2]

Fig. 5. Effect of communication failure at the first following vehicle.

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As reconfiguration, the decentralized control structure is now used and the optimization problem (22) including all possible information topologies for each vehicle is solved. The communication of the vehicle platoon with three nodes for the following vehicles involves a maximum of six edges in it’s graph representation that can exist or disappear in time. Since the information of preceding vehicles is of major importance for disturbance rejection we can layout the control for a topology matrix with only varying information of predecessors, i.e. information of successors is not used. The topology matrix for follower 1 up to follower 3 is then given as

T (θ) =

"

1 0 0 θ1 1 0 θ2 θ3 1

#

with

θi ∈ {0, 1}.

(24)

Hence, the LMI conditions in (22) involve two switching cases for the second follower control (depending on θ1 ) and four switching cases for the third follower control (depending on θ2 and θ3 ). The extreme cases are that all θi = 1 (full information of predecessors) and all θi = 0 (no communication) which means that all followers can only use their on-board data. At first, we analyze the decentralized control layout without strict pole constraints for controller dynamics, the inclusion of these constraints will be shown in the next section. For the case of fast dynamics (here, pole radii ¯ i | < 95 are allowed) the platoon response for with |λ switching information access is shown in Fig. 6. At first all information is available, at time t = 2s the topology starts switching between no information and full information 789

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Fig. 7. Control signals for information topology switching between full and no information from preceding vehicles. of predecessors in time sequences of ∆t = 0.1s. It can be seen that the performance formerly achieved with full information of preceding vehicles can be established again (compare Fig. 4 and Fig. 6). We also mention that exactly the same behavior of errors and velocities can be seen for each single information topology (with and without switching). The switching can actually not be seen in the dynamics of Fig. 6, it is reflected in the control commands of the following vehicles though. In Fig. 7 the transients after each switching of information can be seen clearly for the control commands of the second and third follower. For better resolution the three different controls are plotted as solid lines with decreasing thickness. While the switching is active for all times t > 2s the peaks of the transients settle down asymptotically. Only the command of the first follower is not affected since it does not depend on information transmitted via radio communication, see the topology matrix in (24). 4.3 Constraints on Controller Dynamics As a further step, limitations in terms of sampling (i.e. a maximum sampling rate) and measurement noise are analyzed. As an example, the refined model includes a sampling limit of 50ms for measurements and additive white noise on all data channels. To realize good perfor-

AAC 2010 Munich, Germany, July 12-14, 2010

The proposed control for longitudinal vehicle guidance shows, that in principle it is possible to preserve control performance up to a certain limit in case of communication failure and possible switching between different information topologies. The inclusion of deficiencies from the network’s QoS offers strategies to use communicated data for vehicle dynamics control despite possible failures. Currently, the authors investigate the proposed control with nonlinear vehicle models. Besides the more detailed dynamics including gear shifts, load variance and wheel slip we further analyze the effects of disturbances on sensor data. So far, the control layout included arbitrary switching as a very pessimistic scenario. Further work will include design criteria for vehicle guidance when margins of switching frequency are known and taken into account.

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REFERENCES

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Fig. 8. Comparison of effective vehicle accelerations with and without communication among platoon members. mance of the closed loop despite the above limitations, further constraints have to be placed on the dynamics of the controller. Otherwise, high frequency noise would get amplified and control dynamics would not be properly represented by it’s time discrete realization. The decentralized control layout now constrains the pole region of ¯i) < 0 the closed loop to a rectangle with −20 < Re(λ ¯ i )| < 5. So far, only test functions in terms of and | Im(λ unit steps have been used in simulations since they allow a good comparison of control performance for different layouts or information access. To further include practical vehicle scenarios, a simulation with an acceleration phase (with up to 1m/s2 ) and a deceleration phase (with up to −5m/s2 ) of the leading vehicle is analyzed. Although the reconfigured control again conserves performance of spacing errors and velocities for the case of communication failure, the performance in terms of accelerations is quite different when all information can be accessed or is totally missing. For both cases the platoon response is depicted in Fig. 8 where both controls obey the same pole constraints. The upper plot representing the case when all information from preceding vehicles is available shows good noise rejection and no amplification of accelerations even with the above pole constraints. If communication fails and only on-board data can be used by the reconfiguration controller, the slower control dynamics leads to an increase of accelerations for the following vehicles which can be seen in the lower plot. On the other hand, the acceleration profile is more affected by measurement noise. Hence, control dynamics can not be made faster without negatively affecting the influence of noise. As a result, performance of the reconfigured control is limited by the trade-off between acceleration attenuation and noise rejection. This limitation gradually increases with the amount of missing information from preceding vehicles. The analysis of this section shows that the presented control reconfiguration improves performance in case of (possibly switching) communication failure, but the achieved performance is limited the more the control dynamics is constrained by noise or sampling constraints. 5. CONCLUSION AND FUTURE WORK This paper presents a decentralized control architecture and an optimization based on LMI formulation that accounts for varying information access of platoon vehicles. 790

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