Robust Stability Analysis for Connected Vehicle Systems

Transcription

1 Proceedings of the th IFAC Workshop on Time Delay Systems, Istanbul, Turkey, June -, 6 TA. Robust Stability Analysis for Connected Vehicle Systems Dávid Hajdu, Linjun Zhang, Tamás Insperger, and Gábor Orosz Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, H- hajdu@mm.bme.hu, insperger@mm.bme.hu). Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 89 USA linjunzh@umich.edu, orosz@umich.edu). Abstract: For implementation in real traffic, connected systems should be designed to be robust against uncertainties arising from human-driven s. Assuming that the bounds of uncertainties are known, we propose a frequency domain approach to guarantee robust stability and to select optimal control parameters. The method is demonstrated by two case studies. Keywords: Connected systems, plant and stability, robustness, time delays. INTRODUCTION Nowadays, the emerging wireless -to- VV) communication enables s to monitor the motion of distant s, even those beyond the line of sight. Such technology has great potentials for improving traffic efficiency, reducing fuel consumption, and enhancing safety. Cooperative adaptive cruise control CACC) is a typical application of VV communication, where each automatically responds to the immediately ahead based on radar while also monitoring the motion of the designated leader via VV communication; see Kianfar et al. ), Alam et al. ), and di Bernardo et al. ). However, implementing CACC in real traffic is challenging since it requires s of high level of autonomy to travel together, which rarely occurs in practice due to the low penetration of such s. To overcome this limitation, the concept of connected cruise control CCC) was proposed by Zhang and Orosz 6); Ge and Orosz ). CCC allows the host to monitor the motion of multiple s ahead without requiring all s to be equipped with range sensors or VV communication devices, making practical implementations feasible. Mixing CCC s into traffic of human-driven s leads to connected systems CVSs) where neither a designated leader nor prescribed connectivity structure is needed. To reduce the complexity for design and analysis of CVSs, Zhang and Orosz 6) proposed a motif-based approach that allows one to design CVSs modularly and is scalable for large networks. The effects of acceleration feedback on the dynamics of CVSs were investigated by Ge and Orosz ) while the effects of stochastic communication delays were studied by Qin et al. ). The aforementioned studies on CVSs assumed that dynamic models of all s including the human-driven s) were known. However, in practice, there may exist uncertainties in dynamic models, especially for humandriven s. How these modeling uncertainties affect the performance of CVSs is still an open problem. In this paper, we investigate the robustness of CVSs against modeling uncertainties. The structure of the paper is as follows. In Sec. a generalized model for connected cruise control is presented and the head-to-tail transfer function is calculated. The robust stability analysis is presented in detail in Sec., including the evaluation of robust stability. Two examples are presented in Sec., where the results are summarized using robust stability diagrams. The results are concluded in Sec... STABILITY IN CONNECTED VEHICLE SYSTEMS In this section, we provide a framework for connected cruise control CCC) that includes information delays while allowing a large variety of connectivity structures. Then, plant stability and head-to-tail stability are defined that can be used to evaluate the performance of connected systems CVSs).. Connected Cruise Control In Fig. the CCC i at the tail) monitors positions s j and velocities v j of s j = p,..., i, where p denotes the furthest within the effective communication range of i. The symbol l j represents the length of j while ξ i,j denotes the information delay between s i and j. The short-range link solid arrow) can be realized by human perception resulting in the human reaction time ξ j,j. [s], by radar with sensing delay ξ j,j.. [s], or by wireless communication with delay ξ j,j.. [s] that occurs due to intermittencies and packet drops. The long-range Copyright 6 IFAC 6

2 l i s i v i ξ i,i l i ξ i,i v i s i ξ i,p l i... v i s i Fig.. A CCC i at the tail) monitors the motion of multiple s ahead, where ξ j,j are communication or sensing delays, while s j, l j, v j denote the position, length and velocity of j, respectively. Arrows denote the direction of information flow. The solid link can be realized by human perception, range sensors or VV communication, while the dashed links can only be realized by VV communication. links dashed arrows) can only be realized by wireless communication since those s are beyond the line of sight. Here, we neglect the air-drag and rolling resistance in the physics-based model given by Ulsoy et al., ) and Orosz, ), so that the acceleration of i is directly determined by the controller. We use the CCC framework presented by Zhang and Orosz 6) ṡ i t) = v i t), i [ v i t) = γ i,j α i,j V hi,j t ξ i,j ) ) v i t ξ i,j ) ) j=p + β i,j vj t ξ i,j ) v i t ξ i,j ) )], where α i,j, β i,j are control gains and the connectivity structure is represented by the adjacency matrix Γ = [γ i,j ] such that {, if i uses data of j, γ i,j = ), otherwise. In ), the range policy function V h) gives a desired velocity based on h. Here, we use the nonlinear range policy function, [ )] if h h st, v V h) = max cos π h hst h go h st, if h st < h < h go, v max, if h h go, ) which indicates that the tends to stop for small distances and aims to maintain the preset maximum speed v max for large distances. In the middle range, the desired velocity monotonically increases with the distance. The nonlinearity in ) ensures the smooth change of acceleration at h = h st and h = h go and hence improves the driving comfort. According to the data collected in real traffic Orosz et al., ), parameters in ) are set to be h st = [m], h go = [m], v max = [m/s]. Finally, in ) the quantity h i,j t) = i ) s j t) s i t) l k i j k=j l p s p v p ) ) denotes the average distance between s i and j; cf. Fig. Such averaging is used such that the desired velocity V h i,j ) is independent of the link length i j and also comparable for different j s. In this paper we assume that the networks contain human-driven s that are not equipped with the VV communication devices and only perceive the behavior of the immediately ahead. When j is driven by a human driver, we assume that its dynamics is governed by the model ṡ j t) = v j t), v j t) = α j,j V hj,j t ξ j,j ) ) v j t ξ j,j ) ) + β j,j vj t ξ j,j ) v j t ξ j,j ) ), ) which can be obtained from ) by setting γ i,i = while γ i,j = for all j < i. For human-driven s, the parameters α, β, ξ are uncertain. Note that the CCC framework ) ensures a unique uniform flow equilibrium i s i t) = v t ih l k, v i t) v = V h ), 6) k= for all i, which is independent of network size, connectivity structures, information delays, and control gains. Here the constant h denotes the equilibrium distance between any pair of consecutive s while the constant v is the equilibrium velocity.. Head-to-Tail Transfer Function To evaluate the performance of CVSs, we define the perturbations s i t) = s i t) s i t), ṽ i t) = v i t) v 7) about the uniform flow equilibrium 6) and investigate the dynamics in the vicinity of the uniform flow equilibrium. Linearizing ) about the equilibrium 6) and transforming the result to the Laplace domain with zero initial conditions, we obtain i Ṽ i s) = T i,j s)ṽjs), 8) j=p where Ṽ s) denotes the Laplace transform of ṽt) and γ i,j sβ i,j + ϕ i,j )e sξi,j T i,j s) = s + i k=p γ 9) i,ksκ i,k + ϕ i,k )e sξ i,k is called the link transfer function that acts as a dynamic gain along the link between s i and j. Here, γ i,j is defined in ) while the constants ϕ i,j and κ i,j are given by ϕ i,j = α i,jv h ), κ i,j = α i,j + β i,j, ) i j for j = p,..., i, where the prime denotes the derivative of function V h) in ) with respect to h. To evaluate plant stability and head-to-tail stability of a network, one needs the head-to-tail transfer function HTTF) G n, s), which represents the dynamic relationship between the head and the tail n such that Ṽ n s) = G n, s)ṽs). ) Note that G n, s) is determined by link transfer functions 9) and includes the dynamics of all s between 66

3 s and n; cf. 8). Zhang and Orosz 6) provided a systematic approach to calculate the HTTF, by using the dynamic coupling matrix Fs) = R ) Ts) + I n+ R T, ) where Ts) = [T i,j s)] with i, j =,..., n, R = [ n, I n ] and I n denotes the n-dimensional identity matrix while n is an n-by- zero vector. Indeed, Fs) can be obtained by deleting the first row and last column of the matrix Ts) + I n+. Then, the HTTF is given by G n, s) = n σ i S n i= F i,σi s) = Ns) Ds), ) where the sum is computed over all permutations σ i of the set S n = {,,..., n}. Formula ) is the same as the Leibniz formula for the determinant but does not include sign changes. It can be evaluated in a symbolic or numerical mathematical software based on the determinant.. Plant Stability and Head-to-Tail String Stability Plant stability and head-to-tail stability are two crucial properties for CVSs. Plant stability indicates that the equilibrium of each in the CVS is asymptotically in absence of external disturbances. When disturbances are imposed on the head, a network is said to be head-to-tail if the disturbances are attenuated when reaching the tail. Note that head-to-tail stability allows disturbances to be amplified by some s in the network. Such flexibility is particularly useful for CVSs that include human-driven s, for which the dynamics cannot be designed. Indeed, plant stability is a fundamental requirement for CVSs to avoid collisions. Assuming plant stability, head-to-tail stability is used for congestion mitigation. Here, we present conditions for plant stability and head-to-tail stability of the linearized system by using the HTTF ). Note that the delays ξ i,j result in infinitely many poles s i i=,,... ) that satisfy the characteristic equation Ds) = ; cf. ). Since the negative real parts of poles indicate the decay of initial perturbation, a network is plant if and only if Res i ) <, i =,,.... ) Based on the HTTF ), a sinusoidal disturbance with frequency ω from the head is amplified by the ratio G n, when reaching the tail. Thus, a CVS is head-to-tail if and only if G n, <, ω >, ) see Zhang and Orosz, 6). In the following part, the robustness of CVSs against model uncertainties will be investigated by using the conditions ) and ).. ROBUST HEAD-TO-TAIL STRING STABILITY ANALYSIS If the link transfer functions between s i and j are uncertain, then it can be expressed as T i,j s) = T i,js) + T i,j s), 6) where T i,j s) is the transfer function and T i,j s) represents the uncertainties due to parameter variations. a) ImT, ) c) T, Uncertain link transfer functions ReT, ) T *, b) ImT, ) * T, + Nominal link transfer function T, T *, r, ReT, ) ω T, Nominal Uncertain Envelope Fig.. Nyquist diagram a,b) and Bode plot c) of the link transfer function T, s), where the parameters are α, =.8 [/s], β, = [/s], ξ, =. [s] with uncertainties %, %, %, respectively. We remark that the time-varying and stochastic nature of the uncertainties are not taken into account during the analysis. For bounded perturbations for any link transfer function there exists an uncertainty radius r i,j s) such that T i,j s) r i,j s). 7) Thus, the range of uncertainties is represented by a complex disk on the Nyquist plots. Note, that by the definition of the link transfer functions 9), T i,j ) = always holds, therefore r i,j ) =. For example, consider a - scenario where the human-driven reacts to. Then the transfer function is written as T, s) = T,s) + T, s), 8) where T,s) = β,s + ϕ, ) e sξ, s + κ, s + ϕ, ) e sξ, 9) is the transfer function cf. 9)) and T, s) is the uncertainty. The Nyquist diagram in Fig. a) shows a bundle of transfer functions corresponding to the gain parameters α, =.8 [/s], β, = [/s], and delay ξ, =. [s] with uncertainties %, %, and %, respectively. The Nyquist plot in panel b) highlights the transfer function T, and a circle with the radius r, represents the uncertainty radius, which encompasses the perturbation T,. The Bode diagram in Fig. c) plots the magnitude of the transfer functions, where the envelope corresponding to the uncertainty radii is highlighted by gray shading.. Robust stability As mentioned above, the control gains and delays of other s are uncertain when those are driven by human 67

4 drivers. Such uncertainties affect not only the link transfer functions, but also the head-to-tail stability of a network, and therefore they influence the strategies to select appropriate control gains for the s in the. If the head-to-tail transfer function is uncertain, then it can be expressed by G n, s) = G n,s) + G n, s), ) where G n, s) represents the complex perturbation due to parameter uncertainties around the system G n,s). Robust control parameters must guarantee stability for any perturbation included in G n, s). We assume that the CCC is autonomous such that the corresponding delays are known while control gains can be designed. Therefore, there exist no uncertainties in the CCC. Note that plant stability is only determined by the parameters of the CCC see ), α i,j, β i,j, j = p,..., i ), that is, the uncertainties arising from other s have no impact on the plant stability of the CCC. Thus, we focus on the robustness of head-totail stability against uncertain parameters of other s. Condition ) for stability including uncertainties defined by ) reads as G n, + G n, <, ω >, ) which must be satisfied for any perturbation G n,. The uncertainty G n, depends on the propagation of the perturbation along the of s. Let us assume that the bound of uncertainties is given by a disk on the complex plane with the radius R, that is G n, + G n, G n, + G n,, ) }{{} R cf. Fig.. An upper estimation of the uncertainty of the HTTF can be given by applying the triangular inequality. Thus, the uncertainty radius is defined as R := n F i,σi + r i,σi ) σ i S n i= σ i S n i= n Fi,σ i, ) where F = RT + I n+ )R T, T = [T i,j ] and T i,j r i,j ; cf. 7). If the uncertainty radius r i,j of the link transfer functions are known, then ) can be evaluated. Based on )-) the condition for stability can be written as G n, + R <, ω >. ) Note that G n,) = and R) = hold for every configuration of connected systems. Arranging the inequality and dividing by R >, one can obtain the condition for robust stability in the form G n, < S := min, ) ω> R where S is called the safety factor. The contour curves at S = correspond to the robust stability boundary and S = gives the boundary of the system. Robustness can be guaranteed for any perturbation bounded by the uncertainty radius SR.. CASE STUDIES In this section two examples and the corresponding robust stability diagrams are presented. It is assumed that certain s in the system are driven by human drivers which utilize information only from the immediately ahead. Since each human driver has a different driving strategy, their link transfer functions are uncertain. Here uncertainties are modeled by perturbations in the control gains α j,j, β j,j and the reaction time ξ j,j. Throughout the examples the generated perturbations satisfy ) ) α j,j β j,j + + ɛ α,j,j α j,j ɛ β,j,j β j,j ) ξ j,j, 6) ɛ ξ,j,j ξ j,j where α j,j, βj,j and ξ j,j are the perturbations of parameters, moreover ɛ α,j,j, ɛ β,j,j and ɛ ξ,j,j are the weights of the perturbations. For instance if ɛ α,j,j =., then α j,j is perturbed by % of its value. Therefore the perturbed parameters must be inside a three-dimensional ellipsoid. The uncertainty radii r j,j is then determined such that all possible transfer functions are covered by the envelope. The measurement of real transfer functions of human drivers is a challenging task but can be done using empirical data Safety Pilot Model Deployment, ).. Case ) The configuration of Case ) is presented on Fig., where it is assumed that the in the middle is driven by a human driver and therefore its transfer function T, s) is uncertain. The head-to-tail transfer function can be calculated using ), where the dynamic coupling matrix and its perturbation matrix read F s) = [ T, s) T,s) T,s) ], Fs) = [ T, s) ], 7) and the link transfer functions are T,s) = β,s + ϕ, ) e sξ, s, 8) sξ, + κ, s + ϕ, ) e T,s) β, s + ϕ, ) e sξ, = s + κ, s + ϕ, ) e sξ, +κ, s + ϕ, ) e, sξ, T,s) = β, s + ϕ, ) e sξ, s + κ, s + ϕ, ) e sξ, +κ, s + ϕ, ) e sξ,, cf. 9,). Therefore the HTTF is G,s) = T,s)T,s) + T,s). 9) Introducing perturbation in the human driver model, i.e., T, s) = T,s) + T, s), the modified HTTF reads G, s) = G,s) + T,s) T, s). ) 68

5 . Vehicle with Connected Cruise Control T,. T, T, Human-driven. Communicating Fig.. Configuration of Case ), where the transfer function T, s) is uncertain, i.e., T, s) = T,s) + T, s). a) α, [/s] Validated boundary un robust β, [/s] b) α, [/s] Robust method un robust S=. S=. S=. S= β, [/s] Fig.. Robust stability diagram of Case ): a) green curve indicates the results obtained from simulations b) black curve indicates the results of the robust method for ɛ α, = ɛ β, = ɛ ξ, =.. Red curves indicate the stability boundary of the system. The last term expresses how the uncertainty propagates in the system. According to ), the uncertainty radius becomes R = T, r,. ) The robust stability boundaries are then given by the contour lines of S = according to ). The most robust control gains are the ones where the safety factor is maximal. Fig. presents the stability diagram in the β,, α, ) parameter plane with the parameters α, = α =.6 [/s], β, = β, =.7 [/s], ξ, = ξ, =. [s], ξ, =. [s], h = [m] and v = [m/s]. In the perturbed model the following uncertainties were assumed ɛ α, = ɛ β, = ɛ ξ, =. % perturbation). In panel a) the shaded robust stability domain was determined from the intersection of stability diagrams obtained from independent calculations, where the perturbed parameters were taken from the boundary of the domain defined by 6). The resulting robust stability boundary is indicated by the green curve. The boundary of the system with no perturbation is denoted by the red curve. Panel b) shows the stability diagram determined using the proposed method. For comparison the and the validated boundaries are reproduced as green and red curves respectively, while the boundary of the calculated robust domain is indicated by solid black curve for safety factor S =. The difference between the green and black curves is negligible, although the latter one is in fact a conservative estimate since the uncertainty radius r, is based on an upper estimation of the perturbation. Note, that the computation time in panel a) significantly depends on the number of the generated stability diagrams, while panel b) is obtained only from a single computation. Also note, that the conservativity depends on how well the envelopes cover the transfer functions. In other words, larger uncertainties can give a more conservative estimate on the uncertainty radii and finally gives a more conservative stability diagram. Some contour curves for different values of the safety factor S are also shown in Fig. b). Larger S indicates more robust system. In this case study the maximal safety factor is S =.. It can be seen that the most robust parameter point is not located in the center of the area but is shifted to larger values of α, and β,.. Case ) The second example presents the robust stability diagram for the configuration shown in Fig.. The head-totail transfer function can be calculated using ), where the dynamic coupling matrix and its perturbation matrix become T,s) F s) = T,s), ) T,s) T,s) T, s) Fs) = T, s). ) The HTTF therefore reads G n,s) = T,s) + T,s)T,s)T,s). ) Introducing perturbations for the human-driven s as T, s) = T,s)+ T, s) and T, s) = T,s)+ T, s), the modified HTTF can be written as G n, s) = G n,s) + T,s) T,s) T, s)+ + T, s)t,s) + T, s) T, s) ). ) Utilizing ), an upper estimation of the uncertainty can be given and the robust boundaries can be calculated, yielding R = T, T, r, + + r, T, + r, r, ). 6) Fig. 6 presents the results corresponding to the parameters α, = α, = α, =.6 [/s], β, = β, = β =.7 [/s], ξ, = ξ = ξ, =. [s], ξ, =. [s], h = [m] and v = [m/s]. In the perturbed model the uncertainties were assumed as ɛ α, = ɛ α, = ɛ β, = ɛ β, = ɛ ξ, = ɛ ξ, =.. In panel a) the validated robust stability domain was determined from the intersection of 6 stability diagrams, so that the uncertainty of the parameters of both transfer functions T, s) and T, s) satisfy the equality in 6). Panel b) shows the level curves corresponding to different safety factors, including S = that gives the robust boundary. The largest safety factor S =.7) is shifted to the upper boundary. Again it can be seen that the lower boundary curve is more sensitive to parameter uncertainties. That is, in order to achieve robustness for large safety factor, CCC shall use data received directly from the lead as the short links contain uncertainties. However if the gains on the long link are much larger than 69

6 . Vehicle with CCC T, T T,,. Human-driven. T, Human-driven. Communicating Fig.. Configuration of Case b), where the transfer functions T, s) and T, s) are uncertain, i.e., T, s) = T,s) + T, s) and T, s) = T,s) + T, s). a) 7 6 α, [/s] Validated boundary robust un β, [/s] b) 7 6 α, [/s] Robust method S=.7 S=. S=.6 S= robust un β, [/s] Fig. 6. Robust stability diagram of Case ), the same notation is used as in Fig.. those of the short link the system may become more prone to collisions. Therefore in practice, we need to consider the trade-off between safety and mobility.. CONCLUSION In this paper we considered bounded uncertainties arising from human-driven s, and proposed a frequencydomain method to design connected cruise control that ensures robust stability. To make the calculation feasible for large networks, we derived a method that gives an estimate of robust stability boundaries. The method is presented for two case studies. The results show that the robustness of control parameters is not trivial. The safest point, where the safety factor is maximal, is not necessarily located at the center of the area, but may be close to the boundaries. The other advantage of the technique is that it can be applied to measured frequency response functions without parameter estimation, as only an uncertainty radius must be determined. di Bernardo, M., Salvi, A., and Santini, S. ). Distributed consensus strategy for platooning of s in the presence of time-varying heterogeneous communication delays. IEEE Transactions on Intelligent Transportation Systems, 6),. Ge, J.I. and Orosz, G. ). Dynamics of connected systems with delayed acceleration feedback. Transportation Research Part C, 6, 6 6. Kianfar, R., Augusto, B., Ebadighajari, A., Hakeem, U., Nilsson, J., Raza, A., Tabar, R.S., Irukulapati, N.V., Englund, C., Falcone, P., Papanastasiou, S., Svensson, L., and Wymeersch, H. ). Design and experimental validation of a cooperative driving system in the grand cooperative driving challenge. IEEE Transactions on Intelligent Transportation Systems, ), Orosz, G. ). Connected cruise control: modeling, delay effects, and nonlinear behavior. Vehicle System Dynamics, submitted. Orosz, G., Wilson, R.E., and Stépán, G. ). Traffic jams: dynamics and control. Philosophical Transactions of the Royal Society A, 6898), 79. Qin, W.B., Gomez, M.M., and Orosz, G. ). Stability and frequency response under stochastic communication delays with applications to connected cruise control design. In IEEE Transactions on Intelligent Transportation Systems, accepted). Safety Pilot Model Deployment ). University of Michigan Transportation Research Institute. Ulsoy, A.G., Peng, H., and Çakmakci, M. ). Automotive Control Systems. Cambridge University Press. Zhang, L. and Orosz, G. 6). Motif-based design for connected systems in presence of heterogeneous connectivity structures and time delays. IEEE Transactions on Intelligent Transportation Systems, published online. ACKNOWLEDGEMENTS This work was supported by the Hungarian National Science Foundation under grant OTKA-K and by the National Science Foundation Award No. 6). REFERENCES Alam, A., Mårtensson, J., and Johansson, K.H. ). Experimental evaluation of decentralized cooperative cruise control for heavy-duty platooning. Control Engineering Practice, 8,. 7