Published on: **Mar 3, 2016**

- 1. Analysis and design of controllers for cooperative and automated driving Jeroen Ploeg
- 2. Analysis and design of controllers for cooperative and automated driving
- 3. The research reported in this thesis was supported by TNO and its program on Adaptive Multi-Sensor Networks, and by the High Tech Automotive Systems (HTAS) program of the Dutch Ministry of Economic Aﬀairs through the Connect & Drive project (grant HTASD08002) and the Connected Cruise Control project (grant HTASD09002). A catalog record is available from the Eindhoven University of Technology library. ISBN: 978-94-6259-104-2. Typeset by the author using LATEX 2ε. Cover Design: Jeroen Ploeg and Liesbeth Ploeg–van den Heuvel. Reproduction: Ipskamp Drukkers B.V., Enschede, The Netherlands. Copyright c 2014 by J. Ploeg. All rights reserved.
- 4. Analysis and design of controllers for cooperative and automated driving PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven op gezag van de rector magniﬁcus prof.dr.ir. C.J. van Duijn, voor een commissie, aangewezen door het College voor Promoties, in het openbaar te verdedigen op woensdag 9 april 2014 om 16.00 uur door Jeroen Ploeg geboren te Velsen
- 5. Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof.dr.ir. C.J. van Duijn promotor: prof.dr. H. Nijmeijer copromotor: dr.ir. N. van de Wouw leden: prof.dr.ir. B. van Arem (Technische Universiteit Delft) prof.dr.ir. P.P.J. van den Bosch dr. S. Shladover (University of California, Berkeley) prof.dr. H.J. Zwart (Universiteit Twente)
- 6. Summary Analysis and design of controllers for cooperative and automated driving The limited capacity of the road network has become an important factor in meeting the road transport demand. In addition, an increasing societal demand exists to reduce fuel consumption and emissions, and to improve traﬃc safety. Road capacity can be increased by improving the traﬃc eﬃciency, as an overall measure for throughput and time of travel, which is determined by the interaction between vehicles rather than by the characteristics of the individual vehicle and its driver. Although fuel eﬃciency and traﬃc safety can still be enhanced by optimizing the individual vehicle, acknowledgement of the fact that this vehicle is part of a traﬃc system creates new possibilities for further improvement of these aspects. To address this traﬃc-system approach, the ﬁeld of Intelligent Transportation Systems (ITS) emerged in the past decade. A promising ITS application is provided by Cooperative Adaptive Cruise Con- trol (CACC), which allows for automatic short-distance vehicle following using intervehicle wireless communication in addition to onboard sensors. The CACC system is subject to performance, safety, and comfort requirements. To meet these requirements, a CACC-equipped vehicle platoon needs to exhibit string-stable be- havior, such that the eﬀect of disturbances is attenuated along the vehicle string, thereby avoiding congestion due to so-called ghost traﬃc jams. The notion of string stability is, however, not unambiguous in the literature, since both stability- based and performance-based interpretations for string stability exist. Therefore, in this thesis, a novel string stability deﬁnition of nonlinear cascaded systems is proposed, based on the notion of input–output stability. This deﬁnition is shown to characterize well-known string stability conditions for linear cascaded systems as a special case. Employing these conditions, the string stability properties of a CACC system using information of the directly preceding vehicle are analyzed. Motivated by the proposed conditions for string stability of linear systems, a controller synthesis approach is developed that allows for explicit inclusion of the string stability requirement in the design speciﬁcations, thus preventing a v
- 7. vi SUMMARY posteriori controller tuning to obtain string-stable CACC behavior. The potential of this approach is illustrated by its application to the design of controllers for CACC for a one-vehicle and a two-vehicle look-ahead communication topology. As a result, string-stable platooning strategies are obtained in both cases, also revealing that the two-vehicle look-ahead topology is particularly eﬀective at a larger communication delay. To validate the theoretical analysis, a prototype CACC system has been devel- oped and installed in a platoon of six passenger vehicles. Experiments performed with this setup clearly show that the practical results match the theoretical anal- ysis, thereby illustrating the practical feasibility for automatic short-distance ve- hicle following. At the same time, however, the experiments clearly indicate the need for graceful degradation mechanisms, due to the fact that wireless commu- nication is subject to impairments such as packet loss. To address this need, a control strategy for graceful degradation is proposed to partially maintain the string stability properties of CACC in case of a failure of the wireless link. The development of driver assistance systems, among which CACC, is sup- ported by hardware-in-the-loop experiments. In such experiments, a test vehicle is placed on a roller bench, whereas wheeled mobile robots (WMRs) represent other traﬃc participants. These WMRs are overactuated, due to the fact that they have independently driven and steered wheels. Consequently, the WMRs can be regarded as automated vehicles, albeit with features far beyond those of nowadays road vehicles. To achieve a high degree of experiment reproducibility, in this thesis, focus is put on the design of an accurate position control system for the overactuated WMRs, taking the tire slip into account. A position controller based on input–output linearization by static state feedback is presented, using the so-called multicycle approach, which regards the robot as a set of identical unicycles. The controller thus aims for motion coordination of the four driven and steered wheels, such that a shared objective, i.e, trajectory tracking of the WMR body, is satisﬁed. In this sense, the control problem is conceptually similar to the aforementioned platoon control problem, in which also coordinated behavior of multiple systems is pursued. Practical experiments with the designed controller indicate that the WMR is capable of accurately following a desired spatial tra- jectory, thus allowing reproducible testing of intelligent vehicles in a controlled environment. Summarizing, this thesis focusses on the analysis and the design of controllers for cooperative and automated driving, both theoretically and experimentally. As an important result, it can be concluded that short-distance vehicle following by means of CACC is technically feasible, due to, ﬁrstly, the availability of low- latency wireless communication technologies, and, secondly, fundamental insight into the mechanism of disturbance propagation in an interconnected vehicle string. A prerequisite, however, is that graceful degradation strategies are implemented to cope with wireless communication impairments such as packet loss. Consequently, safety-critical cooperative driving applications require a thorough development process, to which end advanced hardware-in-the-loop test facilities are currently available.
- 8. Contents Summary v Nomenclature xi 1 Introduction 1 1.1 Cooperative driving . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The nature of cooperative driving . . . . . . . . . . . . . . 1 1.1.2 Technical aspects . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Development process . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Challenges in dynamics and control of cooperative vehicle-following systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Research objectives and contributions . . . . . . . . . . . . . . . . 10 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 String stability of cascaded systems: Application to vehicle platooning 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 String stability review . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Platoon dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Control problem formulation . . . . . . . . . . . . . . . . . 20 2.3.2 CACC design . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Homogeneous platoon model . . . . . . . . . . . . . . . . . 22 2.4 String stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Lp string stability . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 String stability conditions for linear systems . . . . . . . . . 26 2.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 String stability of vehicle platoons . . . . . . . . . . . . . . . . . . 32 2.6 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . 39 2.7.1 Vehicle model validation . . . . . . . . . . . . . . . . . . . . 40 2.7.2 String stability experiments . . . . . . . . . . . . . . . . . . 41 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vii
- 9. viii CONTENTS 3 Controller synthesis for string stability of vehicle platoons 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 CACC synthesis review . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Control problem formulation . . . . . . . . . . . . . . . . . . . . . 49 3.4 String stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 The H∞ control problem . . . . . . . . . . . . . . . . . . . . . . . 56 3.6 Controller synthesis for string-stable platooning . . . . . . . . . . . 57 3.6.1 One-vehicle look-ahead topology . . . . . . . . . . . . . . . 57 3.6.2 Two-vehicle look-ahead topology . . . . . . . . . . . . . . . 61 3.6.3 Performance comparison . . . . . . . . . . . . . . . . . . . . 66 3.7 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . 68 3.7.1 Frequency response experiments . . . . . . . . . . . . . . . 68 3.7.2 Time response experiments . . . . . . . . . . . . . . . . . . 72 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 Graceful degradation of Cooperative Adaptive Cruise Control subject to unreliable wireless communication 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Control of vehicle platoons . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 String stability of a vehicle platoon . . . . . . . . . . . . . . 79 4.2.2 Cooperative Adaptive Cruise Control . . . . . . . . . . . . . 81 4.3 Graceful degradation under communication loss . . . . . . . . . . . 83 4.3.1 Object tracking . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2 CACC fallback scenario . . . . . . . . . . . . . . . . . . . . 87 4.4 String stability of degraded CACC . . . . . . . . . . . . . . . . . . 88 4.5 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.1 Frequency response experiments . . . . . . . . . . . . . . . 92 4.5.2 Time response experiments . . . . . . . . . . . . . . . . . . 94 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 Implementation of Cooperative Adaptive Cruise Control 99 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Real-time control system for CACC . . . . . . . . . . . . . . . . . 100 5.2.1 The Real-Time CACC Platform . . . . . . . . . . . . . . . 101 5.2.2 The Vehicle Gateway . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Vehicle instrumentation . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 Position control of a wheeled mobile robot 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Moving Base characteristics . . . . . . . . . . . . . . . . . . . . . . 117 6.3 Control concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Unicycle modeling and control . . . . . . . . . . . . . . . . . . . . 121 6.4.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.4.2 Controller design . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5 Observer design for the MB . . . . . . . . . . . . . . . . . . . . . . 129
- 10. CONTENTS ix 6.5.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.5.2 Motion observer . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.3 Slip observers . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.6 Multicycle controller design . . . . . . . . . . . . . . . . . . . . . . 135 6.7 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7 Conclusions and recommendations 143 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography 149 Samenvatting 161 Dankwoord 165 Curriculum vitae 167
- 11. Nomenclature Acronyms and abbreviations ADAS Advanced Driver Assistance System (C)ACC (Cooperative) Adaptive Cruise Control CAN Controller Area Network CC Cruise Control dCACC degraded Cooperative Adaptive Cruise Control EGNOS European Geostationary Navigation Overlay Service ETSI European Telecommunications Standards Institute GCDC Grand Cooperative Driving Challenge GPS Global Positioning System HMI Human–Machine Interface I2V Infrastructure-to-Vehicle communication ITS Intelligent Transport System ITS-G5 Set of protocols and parameters speciﬁed in the ETSI Standard ES 202 663 LFT linear fractional transformation LMI Linear Matrix Inequality LQR Linear-Quadratic Regulator MIMO Multiple-input multiple-output MB Moving Base MPC Model Predictive Control P(I)D Proportional–(Integral–)Derivative SISO Single-input single-output SSCS String Stability Complementary Sensitivity TCP/IP Transmission Control Protocol/Internet Protocol UDP User Datagram Protocol UTM Universal Transverse Mercator V2X Vehicle-to-Vehicle (V2V) or Vehicle-to-Infrastructure (V2I) communication VeHIL Vehicle Hardware-In-the-Loop WGS84 World Geodetic System 1984 WMR Wheeled Mobile Robot xi
- 12. xii NOMENCLATURE Roman symbols A system matrix a acceleration B input matrix C output matrix; covariance; cornering slip stiﬀness D delay transfer function d distance e error F discrete-time system matrix; force G vehicle transfer function g gravitational constant H spacing policy transfer function h time gap I identity matrix; moment of inertia j imaginary number K controller gain vector; controller transfer function; slip stiﬀness k controller gain; discrete time L vehicle length; observer gain matrix l length ℓ number of outputs M frequency-domain magnitude m platoon length; vehicle mass N number of time samples; lower linear fractional transformation n number of states; discrete frequency P probability P(s), P plant model Q state weighting matrix q position; number of inputs; state vector R input weighting matrix; radius r standstill distance; control signal; input vector S sensitivity transfer function s Laplace variable; generalized position T time period; torque; observer transfer function t time u vehicle input; lateral velocity V Lyapunov candidate function v controller input; (longitudinal) velocity; noise W weighting factor; vehicle width w exogenous input; disturbance; noise x state vector; position coordinate y output; position coordinate z exogenous output
- 13. NOMENCLATURE xiii Greek symbols α class K function; reciprocal maneuver time constant; lateral slip β class K function; road slope angle Γ string stability complementary sensitivity γ string stability complementary sensitivity impulse response; threshold value ∆ interval δ steering angle Θ transfer function relevant to semi-strict string stability θ communication delay κ longitudinal slip ν new system input ξ new vehicle input σ singular value; standard deviation; tire relaxation length τ time constant φ vehicle delay ψ orientation angle ω frequency; rotational velocity Subscripts cp control point max maximum d derivative action; drive line min minimum dd double derivative action nom nominal fb; ﬀ feedback; feedforward p proportional action; process h host vehicle r reference; desired i; j; k indices rl road load lat lateral s sampling; steering system long longitudinal t target vehicle m measurement z vertical direction Miscellaneous C set of complex numbers sup supremum F(·) Fourier transform xt transpose of x f(·) real-valued function ˙x (¨x) (second) time derivative of x h(·) real-valued function ¯x equilibrium; maximum of x k(·) real-valued function ˆx estimated value of x L (·) Laplace transform · , · inner product max maximum | · | absolute value N set of positive integer numbers · vector norm Rn×m set of real n × m matrices · H∞ H∞ system norm Re(·) real part · Lp Lp signal norm, p ∈ {1, 2, ∞} Sm set of all vehicles in a platoon of length m
- 14. Chapter 1 Introduction Cooperative driving is a promising ﬁeld of research in view of overall improvement of traﬃc eﬃciency and safety. The technology necessary to implement coopera- tive driving involves partial or full vehicle automation, supported by information retrieved through wireless communications between vehicles and/or between ve- hicles and roadside units. Before focussing on the control-relevant aspects of a closed-loop cooperative driving system in the coming chapters of this thesis, Section 1.1 of this chapter ﬁrst brieﬂy introduces some important aspects of co- operative driving in general. Section 1.2 then identiﬁes the main challenges in the ﬁeld of controller design for cooperative vehicle following, being a promising real- time closed-loop cooperative driving application. Section 1.3 further addresses these challenges by formulating the objectives of this thesis and by summarizing the speciﬁc contributions, after which Section 1.4 presents an outline of the thesis. 1.1 Cooperative driving This section introduces the typical features of cooperative driving, identiﬁes im- portant technical aspects of real-time closed-loop applications in this ﬁeld, and brieﬂy describes the development process of cooperative driving control systems, the latter focussing on the use of hardware-in-the-loop simulations in this process. 1.1.1 The nature of cooperative driving Vehicle automation by means of Advanced Driver Assistance Systems (ADASs) is believed to reduce the risk of accidents, improve safety, increase road capacity, reduce fuel consumption, and enhance overall comfort for drivers (Vahidi and Es- kandarian, 2003). Based on all these potential beneﬁts of automation, research on automating some or all aspects of the driving task has been pursued for several decades now. Initially aiming for increased driver comfort, this research resulted in Cruise Control and Adaptive Cruise Control (Piao and McDonald, 2008), both 1
- 15. 2 1 INTRODUCTION of which are widespread in nowadays commercially available vehicles. The new generation of ADASs is primarily directed towards increasing safety of both pas- sengers and vulnerable road users such as pedestrians and bicyclists. While this type of systems is still under development, the ﬁrst safety-oriented ADASs already entered the market some time ago, focussing on collision mitigation and collision avoidance (Lu et al., 2005). In parallel to the aforementioned developments, it is recognized that the cur- rent generation of advanced driver assistance systems still aims for optimizing the individual vehicle, which is an inherent consequence of the fact that onboard sensors such as radar, scanning laser (lidar), and vision systems, only provide information about other road users in the line-of-sight of the sensor. To optimize the road traﬃc system as a whole, either by centralized, decentralized, or dis- tributed control systems, real-time information on other road users beyond the immediate line-of-sight is generally required. In addition, real-time information may be required that cannot be obtained through onboard sensors, or requires state observers, thereby introducing inaccuracy and/or phase lag. With wireless communications, local information from cooperating vehicles signiﬁcantly extends the driver’s perception and the detection range of onboard sensors, thus creating a wider information horizon. Vehicle automation systems utilizing wireless com- munications are generally referred to as cooperative driving systems (Shladover, 2012), although the mere fact of employing wireless communications does not necessarily imply true cooperation. Cooperative driving may be described as inﬂuencing the individual vehicle behavior, either through advisory or automated actions, so as to optimize the collective behavior with respect to road throughput, fuel eﬃciency and/or safety, although a formal deﬁnition of cooperative driving does not exist to the best of the author’s knowledge. Within the ﬁeld of cooperative driving systems, numer- ous applications are under investigation, ranging from warning systems, support- ing the driver in potentially dangerous situations, to partial or even full vehicle automation (Piao and McDonald, 2008). These applications are not limited to intervehicle wireless communications, but also include infrastructure-to-vehicle communications (and vice-versa) to even further extend the information horizon. Besides the technical aspects of cooperative driving systems, which will be further addressed in the next subsection, the market introduction is of special importance since this involves a variety of stakeholders. In the European project DRIVE C2X1 , which currently runs under the 7th Framework Programme, three groups of stakeholders are identiﬁed, being the user, public bodies, and private companies, as schematically depicted in Figure 1.1. Furthermore, two types of co- operative driving systems are distinguished: cooperative vehicular functions and services, which involve the user and private companies, and traﬃc services, involv- ing the user and public bodies. The ﬁgure further indicates the main prerequisites for a successful market introduction of both types of systems, relating to techni- cal maturity, impact assessment, user awareness and acceptance, standardization, business models, and legislation. These topics are currently under investigation on a world-wide scale. 1http://www.drive-c2x.eu/project
- 16. 1.1 COOPERATIVE DRIVING 3 C2X Awareness; knowledge about C2X benefits Harmonized standards Validated business models (Joint) Business initiatives; new regulations Deployment of cooperative vehicular functions and services Successful tests of technical systems; user acceptance Deployment of traffic services Proven benefits on safety, efficiency, and environment Private Public User © DRIVEC2Xconsortium Figure 1.1: Requirements for successful market introduction of cooperative driving sys- tems (source: http://www.drive-c2x.eu/road-to-market). 1.1.2 Technical aspects2 Of special interest, from a control perspective, are cooperative driving systems that employ wireless communications in a closed-loop conﬁguration, thus consti- tuting a networked control system (Heemels and Wouw, 2010), being subject to time delay and sampled data eﬀects. An example of such a system, employing lon- gitudinal vehicle automation, is Cooperative Adaptive Cruise Control (CACC). CACC is a vehicle-following system that allows for short-distance following while maintaining a high level of safety, thus potentially improving road throughput and fuel eﬃciency due to reduced aerodynamic drag (Shladover et al., 2012; Ramak- ers et al., 2009). CACC can in principle be extended so as to incorporate lateral automation as well. Real-time closed-loop cooperative driving applications are characterized by a number of key technical aspects, as explained below. Robust fail-safe real-time control CACC can be regarded as the main real-time closed-loop cooperative driving ap- plication that has been under investigation in academia and industry. A frequently adopted approach is based on well-deﬁned vehicle platoons, i.e., a platoon leader is present and all platoon members are known. As opposed to this structured en- vironment, the application of automated vehicle following in everyday traﬃc will most likely be characterized by an unstructured environment consisting of vehicles of various types and instrumentation. Moreover, in practice, a natural platoon leader need not be present and the platoon may not be well-deﬁned, in the sense that all members of the platoon may not have been explicitly identiﬁed as such. 2This section is based on Ploeg et al. (2012).
- 17. 4 1 INTRODUCTION This situation can be handled by either implementing a negotiation mechanism to determine the platoon leader and the platoon members, thereby increasing the communication load, or an ad hoc vehicle-following approach, which is character- ized by a cluster of cooperative vehicle followers that are not necessarily aware of all members and do not rely on a leader. Note that these considerations not only apply to longitudinally automated vehicle systems, but just as well to systems con- sisting of fully automated vehicles, i.e., including lateral automation. Research into these implementation-relevant aspects has emerged only recently (compared to the system-theoretical aspects), see, e.g., Shaw and Hedrick (2007b) and Naus et al. (2010). Furthermore, application in everyday traﬃc of automated vehicle-following systems, either incorporating longitudinal automation or lateral automation or both, requires a high level of reliability and safety. Especially when wireless communications are employed, careful network planning and message handling is required to achieve the necessary reliability. A high level of redundancy may not be the a priori solution since this also increases system costs. Consequently, the actual challenge which has yet to be solved, is to design a system that achieves a suﬃcient level of reliability and includes mechanisms for graceful degradation, capable of coping with ﬂawed or missing data from other vehicles, to ensure safety, while keeping system costs to a minimum. Finally, the impact of automation on the driver necessitates a very fundamen- tal understanding of human factors in relation to (semi-)automated driving control or assistance systems. Hence, user acceptance and driver behavior are important aspects to be addressed before a safety-critical cooperative driving application, such as CACC, can be employed (Shladover et al., 2009). Distributed real-time information structures Cooperative driving technologies rely to a large extent on information exchange between traﬃc participants and/or between traﬃc participants and roadside units. To cooperate successfully, communicating nodes must have a common understand- ing of the exchanged information, which involves standardization of message for- mats, and communication and interaction protocols (Russo et al., 2008). Road users and roadside units fuse data from their own sensors and from communicated information to construct their local view of the world or “world model.” A world model includes a representation of the local traﬃc situation and the status of neighboring vehicles and roadside units and provides the input for control (Brignolo et al., 2008). On a traﬃc level, however, road users and roadside units have to maintain some level of consistency in the distributed world view to support cooperative (and safe) behavior. Consequently, a complex large- scale information ﬂow arises, exchanging motion data and events on a real-time basis. This requires a well-deﬁned information architecture, achieving a high level of reliability and scalability.
- 18. 1.1 COOPERATIVE DRIVING 5 Wireless communication in real-time environments It is well known that wireless and mobile communications are subject to failure by their very nature. Examples of phenomena, impeding ﬂawless communications, are varying signal strengths due to varying propagation conditions, multi-path fading including intersymbol interference, Doppler shifts due to station mobility, and many types of interference signals, such as man-made noise and intermodula- tion (Tse and Viswanath, 2005). In ad hoc networks, including vehicle-to-vehicle networks, where stations communicate without the use of ﬁxed infrastructure, additional problems arise. For instance, transmitting stations may cause mutual interference at a receiver, known as the hidden terminal problem (Hekmat, 2006). The latter problem actually becomes more dominant in the typical real-time set- ting, in which vehicles exchange motion data at relatively high update rates (10 Hz or higher) and require low latencies (signiﬁcantly less than 100 ms). Despite a plethora of mitigation strategies found in modern wireless communi- cation systems, none of these are fail-safe. The control system should therefore be robust against wireless communication impairments such as latency, fading, frame and packet loss, and limited range and bandwidth. A careful balance is needed between the use of and dependency on information obtained through wireless communications and the use of onboard sensors to obtain the required situation awareness and to ensure safety at all times. Finding this balance is an important objective in wireless communications research in view of large-scale deployment, which is more important than the communication technologies by themselves. Figure 1.2: The GCDC participants with the organization’s lead vehicle. Research activities into the technical aspects of cooperative driving are still on- going. These activities also include exper- imental evaluation of cooperative driving applications, a unique example of which is provided by the Grand Cooperative Driv- ing Challenge (GCDC), see Figure 1.2, which exclusively focussed on CACC. The GCDC took place in May 2011 in The Netherlands and involved nine interna- tional teams, each one having its own spe- ciﬁc implementation of control algorithms, while employing a common message set for the wireless intervehicle communica- tion (Nunen et al., 2012). The GCDC contributed in identifying directions for further research, one of the most impor- tant being the implementation of mech- anisms for fault tolerance and graceful degradation. Although not addressed in the GCDC, it may be expected that cy- bersecurity will also be an important topic in this context.
- 19. 6 1 INTRODUCTION {G} {L} road load simulation vehicle 2 vehicle 1 GPS emulation vehicle 1: Moving Base wireless communication real-time simulation Gx2 Gw1v2 Gx1 Gx2 Lx1 chassis dyno Tr ω,T vehicle 2: test vehicle radar Figure 1.3: Schematic representation of the VeHIL test bed. 1.1.3 Development process The development process of cooperative driving systems as performed by the automotive industry follows the so-called V-cycle, which identiﬁes the steps in the design ﬂow at diﬀerent hierarchical levels, with every speciﬁcation or design step having a corresponding validation or veriﬁcation step (Naus, 2010). This process is supported by hardware-in-the-loop testing of components. Especially for safety-critical systems, it is desired to ﬁrst test the entire vehicle in a hardware- in-the-loop setting before commencing road tests. To this end, a test bed called VeHIL (Vehicle Hardware-In-the-Loop) has been developed, which allows to test ADAS-equipped vehicles in a controlled (indoor) environment, while emulating other traﬃc by wheeled mobile robots (Gietelink et al., 2006). VeHIL is especially useful to test closed-loop vehicle control systems, among which cooperative driving systems, that aim to support the driver based on the detection of other traﬃc participants. The working principle of VeHIL is schematically depicted in Figure 1.3 by means of an example involving a platoon consisting of a lead vehicle (vehicle 1) and a follower vehicle (vehicle 2) equipped with CACC. The core of VeHIL is a real-time simulation model, describing the dynamic behavior of relevant traﬃc participants. Starting from an equilibrium situation, in this case deﬁned by a constant velocity of both vehicles whereas vehicle 2 follows at a desired distance, the simulated lead vehicle performs a maneuver, such as accelerating or braking, which is captured by its “state” vector G x1. This state vector contains the position
- 20. 1.1 COOPERATIVE DRIVING 7 and orientation of vehicle 1, and possibly the ﬁrst and second time derivatives, expressed with respect to the origin of a global ﬁxed coordinate system {G}3 . The current measured state G x2 of the test vehicle, which is placed on a chassis dynamometer, is then used to perform a coordinate transformation of G x1 to a coordinate frame {L} which is attached to the test vehicle, yielding L x1. This relative motion vector is subsequently sent as a control command to a wheeled mobile robot, the so-called Moving Base, which represents vehicle 1. The Moving Base motion is detected by the test vehicle’s environmental sensor, e.g., a radar, upon which the test vehicle velocity response is measured through the chassis dynamometer rotational velocity ω and subsequently forwarded as the vehicle velocity v2 to the simulation model. Finally, the test vehicle state vector G x2 is determined based on v2 through numerical integration and diﬀerentiation. The chassis dynamometer is torque controlled, where the desired torque Tr stems from the vehicle inertia and the estimated friction forces caused by aerodynamic drag and the tires. In addition, the test vehicle GPS signal and the wireless message G w1 from the lead vehicle can be emulated as well, e.g. to perform tests with vehicles equipped with CACC. In summary, VeHIL simulates the motion of traﬃc participants with respect to the test vehicle, which is safe and space eﬃcient, thus allowing for indoor testing under controlled and reproducible circumstances. The application of VeHIL in the development process of ADASs is extensively described in Gietelink (2007), which also contains a case study regarding CACC system validation. Of particular interest, from a control perspective, is the Moving Base. This wheeled mobile robot (WMR) is equipped with four independently driven and steered wheels to be able to independently control the orientation and the direc- tion of motion, the reason for which is shown in Figure 1.4. Figure 1.4(a) shows an example manoeuver, consisting of a test vehicle (vehicle 2) with velocity vector G v2 and a preceding vehicle (vehicle 1) that performs a cut-in, thus having a veloc- ity vector G v1 and a yaw rate G ˙ψ1. As a result of the relative motion principle of VeHIL, the Moving Base, representing vehicle 1, has a velocity vector L v1, whereas its yaw rate L ˙ψ1 is equal to G ˙ψ1, as shown in Figure 1.4(b). Consequently, the orientation of the Moving Base no longer corresponds to the direction of motion, hence requiring an all-wheel steered vehicle. The Moving Base platform thus constitutes an overactuated system, consisting of eight actuators (four steering and four driving motors), whereas the motion control objective only incorporates the three degrees of freedom in the horizontal plane. As a result, the motion of the four wheels needs to be coordinated, in order to impose the required moment and forces on the center of gravity of the robot platform, such that the desired trajectory in the horizontal plane is tracked. Consequently, the control problem can be characterized as actuator-level motion coordination. In this respect, a conceptual link exists with the vehicle-following problem, which, in fact, also requires motion coordination of dynamic systems, in this case vehicles in a platoon. For this reason, and because of the relevance of hardware-in-the-loop validation of cooperative driving systems, the motion control 3Since VeHIL involves various coordinate systems, the particular coordinate system in which a certain variable is expressed, is explicitly indicated by a left superscript.
- 21. 8 1 INTRODUCTION Gv2 Gψ1 Gv1 . vehicle 1 vehicle 2 {G} (a) vehicle 1: Moving Base Lv1 Gψ1=Lψ1 ‒Gv2 Gv1 . . vehicle 2: test vehicle {L} (b) Figure 1.4: Vehicle motion with respect to (a) a global ﬁxed coordinate frame {G} and (b) a local coordinate frame {L}, attached to the test vehicle. of the Moving Base, or, in general, an overactuated WMR, is incorporated in the challenges that are identiﬁed in the ﬁeld of cooperative vehicle-following systems, as presented in the next section. 1.2 Challenges in dynamics and control of cooperative vehicle-following systems Ad hoc cooperative vehicle following by means of longitudinal vehicle automation (i.e., CACC) is a closed-loop real-time cooperative driving application which has the potential to increase road throughput (Shladover et al., 2012) and reduce fuel consumption (Ramakers et al., 2009). Nevertheless, a number of challenges in the ﬁeld of controller design for CACC still exist, as will be explained hereafter. The concept of automated vehicle following with road vehicles has been well known for decades. One of the ﬁrst control-oriented publications on the subject dates back to 1966 (Levine and Athans, 1966). Since then, a large amount of relevant research has been published, see, e.g., Sheikholeslam and Desoer (1993), Swaroop and Hedrick (1996), and Vahidi and Eskandarian (2003). This research invariably takes string stability into account, which can be roughly described as the attenuation along the string of vehicles of the eﬀects of disturbances, such as initial condition perturbations or unexpected velocity variations of vehicles in the string (Seiler et al., 2004). String stability is generally considered a prereq- uisite for safety, driver comfort, and scalability with respect to platoon length.
- 22. 1.2 CHALLENGES IN DYNAMICS AND CONTROL OF COOPERATIVE VEHICLE SYSTEMS 9 Especially in case of ad hoc vehicle following (see Section 1.1.2), the scalability property is of the essence since the platoon length is unknown. Nevertheless, from the literature review on string stability as presented in Chapter 2 of this thesis, it appears that this notion is not unambiguous, since both stability and perfor- mance interpretations exist. Consequently, to allow for rigorous analysis of string stability properties of a vehicle platoon, a unifying (or at least general) deﬁnition of string stability is required ﬁrst. Despite the existing ambiguity of the notion of string stability, various types of controllers that realize a speciﬁc form of string-stable behavior have been pro- posed, see, e.g., Rajamani and Zhu (2002) for a PD-like controller, employing a one-vehicle look-ahead communication topology, and Swaroop et al. (2001), describing a sliding-mode controller that uses communicated lead vehicle infor- mation in addition. These controller synthesis methods, however, do not take the string stability requirement explicitly into account. Consequently, string-stable behavior has to be realized through a posteriori controller tuning. Moreover, in the scope of ad hoc vehicle following, communication of lead vehicle information is not possible since no vehicle is explicitly classiﬁed as such. Due to its capability of including constraints in the controller design, Model Predictive Control can en- force the attenuation of the L∞ signal norm of the disturbance responses; this is investigated in Dunbar and Caveney (2012), but only for a one-vehicle look-ahead communication topology. Furthermore, a mixed H2/H∞ problem formulation is applied in Maschuw et al. (2008), resulting in string-stable behavior of the vehicle platoon, albeit with a centralized controller that requires the states of all pla- toon vehicles to be available. From the aforementioned literature references and other relevant publications as thoroughly reviewed in Chapter 3 of this thesis, it appears that systematic controller design methods in which the string stability re- quirement is a priori included as a design speciﬁcation, are investigated to a very limited extent. This conclusion especially holds when considering ad hoc vehicle following for the general multiple-vehicle look-ahead communication topology. As already mentioned in Section 1.1.2, fail-safety of CACC is of the essence due to wireless communication impairments such as (varying) latency and packet loss. Current research in the ﬁeld of networked control systems, focussing on the ef- fects of varying latency (Öncü, 2014), contributes to rigorous knowledge regarding the maximum allowable latency and, consequently, to fail-safe controller design. However, also when the wireless link completely fails, it is necessary to guarantee safety, driver comfort, and platoon scalability. Consequently, it is required to design a controller which is robust to the loss of wireless communication, in the sense that performance degradation with respect to string stability is predictable and limited. This requirement constitutes yet another challenge for controller design, which is currently not addressed in the literature. The aforementioned challenges all relate to various aspects of string stability of controlled vehicle platoons. To establish the practical feasibility of developed the- oretical solutions, practical experiments are necessary. Although vehicle-following control systems have been evaluated in practice (see, for instance, Gehring and Fritz (1997) and Shladover et al. (2009)), such practical experiments in general do not focus on string stability. Vice versa, theoretical work on string stability,
- 23. 10 1 INTRODUCTION as described in the literature, is rarely validated in practice. Consequently, the execution of real-life tests of CACC technology, to validate theoretical results regarding string stability properties, can be considered a challenge in itself. Finally, the development of real-time safety-critical cooperative driving appli- cations, such as CACC, can be strongly supported by hardware-in-the-loop tests that allow for safe and reproducible evaluation of these systems. As described in Section 1.1.3, VeHIL establishes such a testing environment. However, VeHIL heavily relies on accurate motion control of the wheeled mobile robots (WMRs), which simulate other traﬃc participants. Due to the relative-motion principle of VeHIL, these WMRs are equipped with four independently driven and steered wheels to independently control the orientation of the body and the direction of motion. Consequently, the WMR itself can be regarded as an automated vehicle, albeit with features far beyond those of nowadays road vehicles. When neglecting tire slip, this type of robot falls within the classiﬁcation system as developed for wheeled mobile robots in Campion et al. (1996) and control solutions are readily available, see, e.g., Canudas de Wit et al. (1996), Bendtsen et al. (2002), and Ploeg et al. (2006). However, the added value of VeHIL is to be able to simulate safety- critical maneuvers, which inherently implies high decelerations and accelerations of the WMRs. Hence, tire slip plays an important role in the dynamic behavior of the WMR and needs to be taken into account in the control design, thus requiring a new controller design approach, incorporating the motion coordination on the actuator level, i.e., the four wheels. 1.3 Research objectives and contributions Following the aforementioned challenges, this thesis aims to contribute to the analysis and control of the dynamics of a cooperative vehicle-following control system, in particular CACC, which requires a form of motion coordination of the consecutive vehicles in a platoon. In addition, the controller design for a wheeled mobile robot is pursued; This robot not only serves as a component in a hardware- in-the-loop test facility for cooperative road vehicles, but can also be regarded as an autonomous vehicle in itself, the trajectory control of which requires motion coordination of the four actuated wheels. Consequently, the following objectives are deﬁned: • Rigourously formulate the notion of string stability and subsequently inves- tigate the possibilities for controller synthesis such that the string stability requirement is explicitly incorporated in the design speciﬁcations, while also taking into account fail-safety with respect to wireless communication im- pairments; • Experimentally validate the theoretical results regarding (controller design for) string stability, using a CACC-equipped vehicle platoon; • Develop and experimentally validate a trajectory controller for an over- actuated wheeled mobile robot, thereby illustrating actuator-level motion coordination.
- 24. 1.3 RESEARCH OBJECTIVES AND CONTRIBUTIONS 11 These objectives are addressed by means of the following contributions, cor- responding to the challenges as described in Section 1.2. 1. String stability deﬁnition and analysis String stability basically involves stability in the time domain and in the spatial domain, the latter referring to stability “over the cascaded vehicles”. Based on a review of the various string stability deﬁnitions and criteria as published in the literature, a novel generic deﬁnition regarding Lp string stability for nonlinear cascaded systems is proposed in Chapter 2 of this thesis. This deﬁnition is applied to derive string stability criteria for lin- ear systems, while introducing semi-strict and strict Lp string stability as special (stronger) forms of Lp string stability. These theoretical contribu- tions are used to analyze the string stability properties of vehicle platoons, both with and without the application of wireless communication to retrieve information about the preceding vehicle(s). 2. Controller synthesis for string stability Next to string stability analysis, this thesis also contributes to controller synthesis aiming for string stability. A literature review regarding controller synthesis methods for vehicle platooning and other applications that require string-stable behavior reveals that the vast majority of applied controllers realizes string-stable behavior by ad hoc tuning of the controller parameters. Therefore, in Chapter 3 a structured method is proposed, employing H∞ optimal control, which allows for explicit inclusion of the L2 string stability requirement in the controller synthesis speciﬁcations. The potential of this approach is illustrated by its application to the design of controllers for CACC for a one- and a two-vehicle look-ahead communication topology. 3. Fail-safe controller design for CACC Fail-safety by means of graceful degradation is an important prerequisite for a successful market introduction of CACC. Therefore, Chapter 4 presents a method for graceful degradation in case the wireless link fails due to, e.g., packet loss over an extended period of time. This method is based on esti- mation of the preceding vehicle’s acceleration using onboard sensors and has a clear advantage in terms of L2 string stability compared to the alternative fallback scenario consisting of conventional Adaptive Cruise Control (ACC). 4. Experimental evaluation using a platoon of passenger vehicles A platoon of CACC-equipped passenger vehicles has been developed to ex- perimentally validate all theoretical results. This may be considered an important contribution to the maturity level of CACC technology since practical implementation not only involves the platooning controller itself, but also requires mechanisms for graceful degradation, algorithms for ob- ject tracking, and human–machine interfacing. To structure this variety of algorithms, a layered control system architecture is developed in Chapter 5, consisting of a perception layer, containing observers for both the host vehi- cle motion pattern and that of target vehicles, a control layer, which includes
- 25. 12 1 INTRODUCTION the control algorithms for CACC, and a supervisory layer that coordinates controller settings received from the driver or from roadside equipment, and ensures safe operation of the vehicle. 5. Controller design for an overactuated wheeled mobile robot Finally, this thesis contributes to the design of position controllers for all- wheel steered mobile robots employing the so-called multicycle approach as described in Chapter 6. Such robots are used to emulate traﬃc participants in the VeHIL hardware-in-the-loop setup to test entire vehicles equipped with ADASs, among which CACC. A distributed position controller based on input–output linearization by state feedback appears to allow for highly dynamic maneuvers of the wheeled mobile robots, thus realizing an agile fully automated vehicle. 1.4 Outline This thesis is organized as follows. Chapter 2 introduces a new generic Lp string stability deﬁnition, based on which string stability conditions for linear systems are derived. Subsequently, the string stability properties of a vehicle platoon equipped with CACC are analyzed, both in theory and in practice. Note that this chapter is based on Ploeg et al. (2014b), extended with experimental results taken from Ploeg et al. (2011). Next, Chapter 3, which is directly based on Ploeg et al. (2014a), presents a controller design method that allows for explicit inclusion of the L2 string sta- bility requirement in the controller synthesis speciﬁcations. The potential of this approach is illustrated by its application to the design of controllers for CACC for a one- and a two-vehicle look-ahead communication topology. Chapter 4 fully focusses on the fact that CACC relies on communicated in- formation, thus inherently causing a certain vulnerability to communication im- pairments such as packet loss. To cope with this property, a graceful degradation mechanism is presented that involves estimation of the preceding vehicle’s accel- eration. The resulting string stability properties are analyzed and experimentally validated. This chapter is based on Ploeg et al. (2013), with additional experi- mental results. It is noted that, due to the fact that this chapter and the previous two chapters are based on literature publications, the introductory parts of these chapters are redundant, to a limited extent. As a part of the validation process, the previous three chapters all contain experimental results obtained with a CACC-equipped platoon of passenger vehi- cles. This test setup is described in more detail in Chapter 5, both with respect to vehicle instrumentation and to software, i.e., the algorithms that appear to be necessary to implement CACC in practice, next to the control algorithm itself. Having explored the theoretical and practical aspects of the control of cascaded systems in general and vehicle platoons in particular, Chapter 6 is concerned with the position control of the wheeled mobile robots used in the VeHIL test site for cooperative systems such as CACC. This chapter is, in fact, a revised version of Ploeg et al. (2009).
- 26. 1.4 OUTLINE 13 Finally, Chapter 7 summarizes the main conclusions of this thesis regarding string stability, vehicle platooning, and vehicle automation, and presents recom- mendations for future research into these and related topics.
- 27. Chapter 2 String stability of cascaded systems: Application to vehicle platooning1 Abstract Nowadays, throughput has become a limiting factor in road transport. An eﬀective means to increase the road throughput is to decrease the intervehicle time gap. A small time gap, however, may lead to string instability, being the ampliﬁcation of velocity disturbances in upstream direction. String-stable behavior is thus considered an essential requirement for the design of automatic distance control systems, which are needed to allow for safe driving at time gaps well below 1 s. However, the formal notion of string stability is not unambiguous in literature, since both stability interpretations and performance interpretations exist. Therefore, a novel deﬁnition for string stability of nonlinear cascaded systems is proposed, using input–output properties. This deﬁnition is shown to result in well-known string stability conditions for linear cascaded systems. Employing these conditions, string stability is obtained by a controller that uses wireless intervehicle communication to provide information of the preceding vehicle. The theo- retical results are validated by implementation of the controller, known as Cooperative Adaptive Cruise Control, on a platoon of six passenger vehicles. Experiments clearly show that the practical results match the theoretical analysis, thereby indicating the practical feasibility for short-distance vehicle following. 2.1 Introduction Limited highway capacity regularly causes traﬃc jams, which tend to increase over the years with respect to both the number of traﬃc jams and their length. An eﬀective means to increase road capacity is to decrease the intervehicle distance while maintaining the same velocity. This would, however, seriously compro- mise traﬃc safety. Moreover, human drivers are known to overreact to velocity variations, thereby amplifying these variations in upstream direction (Sugiyama 1This chapter is based on Ploeg et al. (2014b), extended with experimental results taken from Ploeg et al. (2011). 15
- 28. 16 2 STRING STABILITY OF CASCADED SYSTEMS et al., 2008). Consequently, vehicle automation in longitudinal direction is re- quired. To this end, the application of Adaptive Cruise Control (ACC) can be beneﬁcial. ACC automatically adapts the velocity of a vehicle so as to realize a desired distance to the preceding vehicle, or, in the absence of one, a desired velocity (Venhovens et al., 2000; Corona and Schutter, 2008). The intervehicle distance and the relative velocity are measured by means of a radar or a scanning laser (lidar). However, ACC is primarily intended as a comfort system. There- fore, relatively large intervehicle distances are adopted (Vahidi and Eskandarian, 2003), with a standardized minimum of 1 s time gap (International Organization for Standardization, 2010), the latter referring to the geometric distance divided by the vehicle velocity2 . Decreasing the ACC time gap to a value signiﬁcantly smaller than 1 s is ex- pected to yield an increase in traﬃc throughput (Santhanakrishnan and Rajamani, 2003; Arem et al., 2006). Moreover, a signiﬁcant reduction in the aerodynamic drag force is possible in case of heavy-duty vehicles, thereby decreasing fuel con- sumption and emissions (Bose and Ioannou, 2003a; Shladover, 2005; Alam et al., 2010). It has however been shown that the application of ACC ampliﬁes disturbances in upstream direction at small time gaps, see, e.g., Yanakiev and Kanellakopoulos (1998) and Naus et al. (2010), similar to the disturbance ampli- ﬁcation in case of human drivers. These disturbances may be induced by velocity variations of the ﬁrst vehicle in a string of vehicles, for instance. As a result, fuel consumption and emissions increase, and so-called ghost traﬃc jams may occur, negatively inﬂuencing throughput, whereas safety might be compromised as well. Note that this type of disturbance ampliﬁcation becomes even worse when apply- ing a constant-distance spacing policy instead of the common constant time gap policy, as assumed above (Swaroop and Hedrick, 1999). Disturbance attenuation along the vehicle string is therefore an essential re- quirement, to be achieved by appropriately designed vehicle-following controllers. The disturbance evolution along a string of vehicles, or, in general, along a num- ber of interconnected systems, is covered by the notion of string stability, where string-stable behavior can be loosely deﬁned as the attenuation of the eﬀect of disturbances in upstream direction. Automatic vehicle following based on data exchange by means of wireless communication, in addition to the data obtained by radar or lidar, is commonly referred to as Cooperative ACC (CACC), and is known to achieve string stability at time gaps signiﬁcantly smaller than 1 s. A vast amount of literature on string stability is available, where the ﬁrst application to vehicle following systems probably dates from 1966 (Levine and Athans, 1966). In Swaroop and Hedrick (1996), Wang et al. (2006), and Cur- tain et al. (2009), for instance, several types of string stability deﬁnitions are given, focussing on a speciﬁc type of perturbations, or on speciﬁc interconnection topologies or other characteristics such as inﬁnite string length. Consequently, a variety of string stability deﬁnitions exists. In addition, publications that fo- cus on controller design for linear interconnected systems, in particular vehicle 2In literature, this measure is regularly referred to as the “time headway.” In this thesis, how- ever, the term “time gap” is adopted, in accordance with the ISO standard 15622 (International Organization for Standardization, 2010).
- 29. 2.2 STRING STABILITY REVIEW 17 strings, tend to interpret string stability as a performance criterion, rather than a stability property (Sheikholeslam and Desoer, 1993; Rajamani and Zhu, 2002; Naus et al., 2010), the advantage being that control design is directly supported. As a result, however, the notion of string stability has become rather ambiguous over the years. This chapter, therefore, ﬁrst aims to formally deﬁne string stability, provid- ing a rigorous basis for often-used string stability criteria for linear systems, thus including and generalizing existing results. To this end, the well-known notion of input–output stability is utilized. Second, it is shown that, employing these criteria, controller design for string stability is not only theoretically, but also practically feasible, using an experimental setup consisting of six passenger vehi- cles equipped with CACC, that has been speciﬁcally developed for the purpose of evaluating string stability properties. The outline of this chapter is as follows. Section 2.2 ﬁrst provides an overview of existing string stability concepts. Section 2.3 derives a model of a string of vehicles that are interconnected through their vehicle-follower control laws (here- after shortly referred to as “platoon”). Adopting this model as a general model for an interconnected system, it forms the basis for the deﬁnition of string sta- bility in Section 2.4 and the analysis thereof for vehicle platoons in Section 2.5. Next, Section 2.6 introduces the test vehicles and their instrumentation, after which Section 2.7 presents experimental results, obtained with the test vehicles. Finally, Section 2.8 summarizes the main conclusions. 2.2 String stability review As opposed to conventional stability notions for dynamical systems, which are essentially concerned with the evolution of system states over time, string stabil- ity focusses on the propagation of system responses along a cascade of systems. Several approaches exist in the literature regarding the notion of string stability, as reviewed below. Probably the most formal approach can be characterized as a Lyapunov- stability approach, of which Sheikholeslam and Desoer (1992a) provide an early description, which has been comprehensively formalized later in Swaroop and Hedrick (1996) and applied for controller design and analysis in Swaroop and Hedrick (1999). In this approach, the notion of Lyapunov stability is employed, which focusses on initial condition perturbations. Since, however, initial condition perturbations can be randomly distributed across the interconnected systems, a disturbance propagation in a clear direction cannot be distinguished anymore, thereby abandoning the original idea behind string stability to a certain extent. Instead, string stability is interpreted as asymptotic stability of an arbitrary num- ber of interconnected systems, of which Yadlapalli et al. (2006) provide an elegant analysis. Recently, new results appeared in Klinge and Middleton (2009a), regard- ing a one-vehicle look-ahead control architecture in a homogeneous vehicle pla- toon. Herein, the response to an initial condition perturbation of a single vehicle in the platoon is considered, thereby conserving the disturbance-propagation idea
- 30. 18 2 STRING STABILITY OF CASCADED SYSTEMS behind string stability. The drawback of this approach, however, is that only this special case is regarded, ignoring the eﬀects of initial condition perturbations of other vehicles in the platoon, as well as the eﬀects of (possibly persistent) external disturbances to the interconnected system. Consequently, the practical relevance of this approach is limited, since external disturbances, such as velocity variations of the ﬁrst vehicle in a platoon, are of utmost importance in practice. Summariz- ing, although the string stability deﬁnitions in the Lyapunov-stability approach are rigorous, they only capture the notion of disturbance propagation along a set of interconnected systems to a limited extent. An attempt to overcome this ap- parent limitation has been made in Wang et al. (2006), which extends the focus on initial condition perturbations of the states by including external system in- puts, ultimately leading to input-to-state string stability. The presented analysis, however, assumes that the interconnected system can be described by a singular perturbation model, and is therefore limited to weakly coupled interconnected systems. The perspective of inﬁnite-length strings of interconnected systems also gave rise to a clear mathematical formulation of string stability, described in Melzer and Kuo (1971) in the context of a centralized control scheme and then in Chu (1974) for a decentralized controller. Various applications regarding optimal controller design for interconnected systems such as seismic cables and vehicle platoons are reported in El-Sayed and Krishnaprasad (1981), Barbieri (1993), and Liang and Peng (1999), whereas Bamieh et al. (2002) and Curtain et al. (2009) provide extensive analyzes of the properties of inﬁnite-length interconnected systems. In this approach, the model of such a system is formulated in an inﬁnite-dimensional state space form and subsequently transformed using the bilateral Z-transform. The Z-transform is executed over the vehicle index instead of over (discrete) time, resulting in a model formulated in the “discrete spatial frequency” domain (Bamieh et al., 2002), related to the subsystem index, as well as in the continuous-time domain. String stability can then be assessed by inspecting the eigenvalues of the resulting system matrix as a function of the spatial frequency. In practice, however, vehicle platoons will be ﬁnite. Unfortunately, the stability properties of ﬁnite-length strings might not converge to those of inﬁnite-length strings as length increases. This can be understood intuitively by recognizing that in a ﬁnite- length platoon, there will always be a ﬁrst and a last vehicle, whose dynamics may signiﬁcantly diﬀer from those of the other vehicles in the platoon, depending on the controller topology. Consequently, the inﬁnite-length platoon model does not always serve as a useful paradigm for a ﬁnite-length platoon as it becomes increasingly long (Curtain et al., 2009). Finally, a performance-oriented approach for string stability is frequently ap- plied, since this appears to directly oﬀer tools for synthesis and analysis of linear vehicle-following control systems. An early application to vehicle platooning is presented in Peppard (1974), focussing on the analysis of a PID-controlled sys- tem. In Sheikholeslam and Desoer (1993) and in Lu and Hedrick (2004), the performance-oriented approach is employed to investigate the control of a vehicle platoon with and without lead vehicle information, whereas Naus et al. (2010) and Rajamani and Zhu (2002) apply intervehicle communication to obtain in-
- 31. 2.2 STRING STABILITY REVIEW 19 formation of the directly preceding vehicle only. In Stanković et al. (2000), a decentralized optimal controller is designed by decoupling the interconnected sys- tems to a certain extent using the so-called inclusion principle, and in Khatir and Davison (2004) optimal decentralized control is pursued as well, resulting in nonidentical controllers. Furthermore, Middleton and Braslavsky (2010) ex- tensively investigate the limitations on performance, whereas Chien and Ioannou (1992) compare the performance of a vehicle-following controller with three types of human driver models. In González-Villaseñor et al. (2007), a controller design methodology is presented, adopting the performance-oriented approach. This ap- proach is basically also used in Chakravarthy et al. (2009) to investigate a warning system for preventing head-tail collisions, taking mixed traﬃc (i.e., controlled and uncontrolled vehicles) into account. In the performance-oriented approach, string stability is characterized by the ampliﬁcation in upstream direction of either dis- tance error, velocity, or acceleration, the speciﬁc choice depending on the design requirements at hand. Let the signal of interest be denoted by yi for vehicle i, and let Γi(jω) denote the frequency response function, with the frequency ω ∈ R, describing the relation between the scalar output yi−1 of a preceding vehicle i − 1 and the scalar output yi of the follower vehicle i. Then the interconnected system is considered string stable if sup ω |Γi(jω)| ≤ 1, 2 ≤ i ≤ m, (2.1) where m is the string length; the supremum of |Γi(jω)| is equal to the scalar version of the H∞ norm. Since the H∞ norm is induced by the L2 norms of the respective signals (Zhou et al., 1996), this approach in fact requires the L2 norm yi(t) L2 to be nonincreasing for increasing index i (i.e., in upstream direction) for string stability. Because of its convenient mathematical properties, the L2 gain is mostly adopted, according to (2.1); nevertheless, approaches that employ the induced L∞ norm are also reported (Gehring and Fritz, 1997; Eyre et al., 1998; Klinge and Middleton, 2009b). Regardless of the speciﬁc norm that is employed, the major limitation of the performance-oriented approach is that only linear systems are considered, usually without considering the eﬀect of nonzero initial conditions. Moreover, (2.1) should be considered as a criterion for string stability of linear systems, rather than a deﬁnition of this notion. Summarizing, string stability appears to be deﬁned in various ways, focusing on speciﬁc system properties. Building on these earlier results, a novel generic deﬁnition of string stability is proposed, based on the notion of input–output sta- bility, which is applicable to both linear and nonlinear systems, while taking both the eﬀects of initial conditions and external disturbances into account. Further- more, for the special case of linear systems, a rigorous basis is obtained for the frequency-domain string stability conditions discussed above and it is proven that (2.1) indeed serves as a condition for string stability for a certain class of linear interconnected systems. To this end, the next section will ﬁrst introduce a model of a homogeneous vehicle platoon, which motivates the formal deﬁnition of string stability as proposed in Section 2.4.
- 32. 20 2 STRING STABILITY OF CASCADED SYSTEMS di di ‒1 vi +1 di +1i+1 vi vi ‒1 wireless communication radar i–1i Figure 2.1: CACC-equipped vehicle platoon. 2.3 Platoon dynamics In order to arrive at a model that describes the dynamics of a vehicle platoon, the control problem is formulated ﬁrst. Next a controller is designed, which then allows for the formulation of a homogeneous (closed-loop) platoon model. 2.3.1 Control problem formulation Consider a platoon of m vehicles, schematically depicted in Figure 2.1, with di being the distance between vehicle i and its preceding vehicle i − 1, and vi the velocity of vehicle i. The main objective of each vehicle is to follow its preceding vehicle at a desired distance dr,i. The ﬁrst vehicle in the platoon (with index i = 1), not having a preceding vehicle, can be velocity controlled, for instance. Another option, adopted here, is to make the lead vehicle follow a so-called virtual reference vehicle (with index i = 0), which has the advantage that the lead vehicle will employ the same controller as the other vehicles in the platoon. The model for the virtual reference vehicle will be further explained in Section 2.3.3. The desired distance dr,i is deﬁned according to a constant time gap spacing policy, formulated as dr,i(t) = ri + hvi(t), i ∈ Sm, (2.2) where h is the time gap, and ri is the standstill distance. Herein, Sm = {i ∈ N | 1 ≤ i ≤ m} is the set of all vehicles in a platoon of length m ∈ N. The spacing policy (2.2) is not only known to improve string stability (Rajamani and Zhu, 2002; Naus et al., 2010), but also contributes to safety (Ioannou and Chien, 1993). This is essentially due to the fact that (2.2) represents a diﬀerential feedback of the vehicle position, contributing to a well-damped behavior of the vehicle, as will be further explained in Section 2.4. Furthermore, a homogeneous platoon is assumed, which is why the time gap h is the same for all i. The spacing error ei(t) can now be deﬁned as ei(t) = di(t) − dr,i(t) = qi−1(t) − qi(t) − Li − ri + hvi(t) , i ∈ Sm, (2.3) with qi being the rear-bumper position of vehicle i and Li its length. The pla- toon control problem now encompasses two requirements: one being the vehicle-
- 33. 2.3 PLATOON DYNAMICS 21 following control objective, to be formulated as limt→∞ ei(t) = 0 ∀ i ∈ Sm, and the other being the string stability requirement. The next subsection will focus on the ﬁrst requirement, whereas the second requirement will be addressed in Section 2.4. 2.3.2 CACC design As a basis for controller design, the following vehicle model is adopted, omitting the time argument t for readability: ˙di ˙vi ˙ai = vi−1 − vi ai − 1 τ ai + 1 τ ui , i ∈ Sm, (2.4) where ai is the acceleration of vehicle i, ui the external input, to be interpreted as desired acceleration, and τ a time constant representing driveline dynamics. This model is in fact obtained by formulating a more detailed model and then applying a precompensator, designed by means of input–output linearization by state feedback (Sheikholeslam and Desoer, 1993; Stanković et al., 2000). Also note that the time constant τ is assumed to be identical for all vehicles, corresponding to the above mentioned homogeneity assumption. With diﬀerent types of vehicles in the platoon, as suggested by Figure 2.1, homogeneity may be obtained by low- level acceleration controllers so as to arrive at identical vehicle behavior according to (2.4). A suitable method to arrive at a controller for CACC is based on formulation of the error dynamics. Deﬁne to this end the error states e1,i e2,i e3,i = ei ˙ei ¨ei , i ∈ Sm (2.5) with ei deﬁned by (2.3). Then, obviously, ˙e1,i = e2,i and ˙e2,i = e3,i. The third error state equation is obtained by diﬀerentiating e3,i = ¨ei, while using (2.3) and (2.4), eventually resulting in: ˙e3,i = − 1 τ e3,i − 1 τ ξi + 1 τ ui−1, i ∈ Sm (2.6) with ξi := h ˙ui + ui, (2.7) which can be regarded as the new input to vehicle i. From (2.6), it is immediately clear that the input ξi should be used so as to stabilize the error dynamics while compensating for the (original) input ui−1 of the preceding vehicle in order to obey the vehicle-following control objective. Hence, the control law for ξi is chosen as follows: ξi = K e1,i e2,i e3,i + ui−1, i ∈ Sm (2.8)
- 34. 22 2 STRING STABILITY OF CASCADED SYSTEMS with K := kp kd kdd . Note that the feedforward term ui−1 is obtained through wireless communication with the preceding vehicle and, therefore, is the reason for the employment of a wireless communication link in the scope of CACC. Due to the additional controller dynamics (2.7), the error dynamics must be extended, to which end the input deﬁnition (2.7) can be employed, while substi- tuting the control law (2.8): ˙ui = − 1 h ui + 1 h (kpe1,i + kde2,i + kdde3,i) + 1 h ui−1, i ∈ Sm. (2.9) As a result, the 4th -order closed-loop model reads ˙e1,i ˙e2,i ˙e3,i ˙ui = 0 1 0 0 0 0 1 0 − kp τ −kd τ −1+kdd τ 0 kp h kd h kdd h − 1 h e1,i e2,i e3,i ui + 0 0 0 1 h ui−1, i ∈ Sm. (2.10) This error model has an equilibrium in the origin for ui−1 = 0. Applying the Routh-Hurwitz stability criterion while using the fact that the system matrix in (2.10) is lower block-triangular, it follows that this equilibrium is asymptotically stable for any time gap h > 0, and with any choice for kp, kd > 0, kdd + 1 > 0, such that (1 + kdd)kd − kpτ > 0, thereby fulﬁlling the vehicle-following control objective. The second objective, being string stability, is not necessarily fulﬁlled yet. Note that the stability of the dynamics (2.10) is sometimes referred to as individual vehicle stability (Swaroop and Hedrick, 1999; Rajamani, 2006). 2.3.3 Homogeneous platoon model Since string stability is commonly evaluated by analyzing the ampliﬁcation in upstream direction of either distance error, velocity, and/or acceleration, a platoon model is formulated in terms of these state variables. Using the spacing error (2.3), the vehicle model (2.4), and the control law (2.9), the following homogeneous platoon model is obtained: ˙ei ˙vi ˙ai ˙ui = 0 −1 −h 0 0 0 1 0 0 0 − 1 τ 1 τ kp h −kd h −kd − kdd(τ−h) hτ −kddh+τ hτ ei vi ai ui + 0 1 0 0 0 0 0 0 0 0 0 0 0 kd h kdd h 1 h ei−1 vi−1 ai−1 ui−1 , i ∈ Sm (2.11)
- 35. 2.4 STRING STABILITY 23 or, in short, ˙xi = A0xi + A1xi−1, i ∈ Sm (2.12) with state vector xt i = ei vi ai ui , and the matrices A0 and A1 deﬁned ac- cordingly. Note that (2.11) is not similar to (2.10), since the former encompasses two controlled vehicles i and i − 1, with external input ui−1 as a result, whereas the latter describes a controlled vehicle i, using the state of the vehicle i − 1 as external “input”. Based on the vehicle model (2.4) and the input dynamics (2.7), the virtual reference vehicle i = 0 may be formulated as ˙e0 ˙v0 ˙a0 ˙u0 = 0 0 0 0 0 0 1 0 0 0 − 1 τ 1 τ 0 0 0 − 1 h e0 v0 a0 u0 + 0 0 0 1 h q0 (2.13) or, in short, ˙x0 = Arx0 + Brur (2.14) with state vector xt 0 = e0 v0 a0 u0 , input ur = q0, being the external in- put to the platoon, and the matrices Ar and Br deﬁned accordingly. Here, the state variables are chosen in accordance with the real vehicles in the platoon. Consequently, (2.13) represents a nonminimal realization, in which e0 (where e0(t) = e0(0)) is a dummy state variable, having no further inﬂuence since the ﬁrst column of both Ar and A1 are equal to the zero column. In the remainder of this chapter, e0(0) = 0 is chosen. The equilibrium state of (2.13) is then equal to ¯xt 0 = 0 ¯v0 0 0 for ur = 0, where ¯v0 is a constant velocity. Note that this equilibrium is only marginally stable since the virtual reference vehicle is in fact an uncontrolled vehicle model. In Kim et al. (2012), the same virtual reference vehicle concept is applied, but using a velocity-controlled vehicle model as virtual reference vehicle instead. This is considered unnecessary in the scope of the cur- rent application, since the virtual reference vehicle does not involve uncertainties or unknown disturbances. Returning to the homogeneous platoon model (2.11), it can be easily estab- lished that xi = ¯x0, with i = 1, 2, . . ., m, is an equilibrium of the vehicle platoon for x0 = ¯x0 and ur = 0; in other words, the platoon equilibrium is characterized by a constant velocity ¯v0 of all vehicles. This equilibrium is asymptotically stable under the same conditions as mentioned for the error dynamics (2.10), which can be easily understood by replacing the ﬁrst state ei in (2.11) by a newly deﬁned state zi := −ei − hvi. As a result, the system matrix A0 transforms into the system matrix of the error dynamics (2.10). 2.4 String stability Having derived a homogeneous platoon model, this section will ﬁrst generalize this model to a nonlinear cascaded state-space system, after which a new string stability deﬁnition is proposed. It is then shown that this deﬁnition serves as
- 36. 24 2 STRING STABILITY OF CASCADED SYSTEMS a rigorous basis for L2 and L∞ string stability conditions commonly used in the performance-oriented approach for string stability (see Section 2.2), and the relation to the other string stability notions is brieﬂy discussed. 2.4.1 Lp string stability The homogeneous platoon model (2.12), (2.14) is a special, linear case of the following cascaded state-space system: ˙x0 = fr(x0, ur) (2.15a) ˙xi = fi(xi, xi−1), i ∈ Sm (2.15b) yi = h(xi), i ∈ Sm, (2.15c) representing a general, possibly nonlinear, heterogeneous interconnected system with the same interconnection structure as (2.12), (2.14). Here, ur ∈ Rq is the external input, xi ∈ Rn , i ∈ Sm ∪ {0}, is the state vector, and yi ∈ Rℓ , i ∈ Sm, is the output. Moreover, fr : Rn × Rq → Rn , fi : Rn × Rn → Rn , i ∈ Sm, and h : Rn → Rℓ . In the scope of vehicle platooning, the state is typically de- ﬁned as xt i = ei vi ai . . . , i ∈ Sm ∪ {0}, indicating a possible extension with additional states, due to, e.g., controller dynamics or spacing policy dynamics, see Section 2.3. Note that the heterogeneity property usually refers to the un- controlled interconnected systems having diﬀerent dynamical properties. In very rare cases, the (decentralized) controllers are nonidentical (Khatir and Davison, 2004). Furthermore, although the majority of platooning applications is based on linear models, nonlinear models will arise due to, e.g., nonlinear spacing poli- cies (Yanakiev and Kanellakopoulos, 1998). Using the model (2.15), the following string stability deﬁnition is now proposed. Deﬁnition 2.1 (Lp string stability). Consider the interconnected system (2.15). Let xt = xt 0 xt 1 . . . xt m be the lumped state vector and let ¯xt = ¯xt 0 . . . ¯xt 0 denote a constant equilibrium solution of (2.15) for ur ≡ 0. The system (2.15) is Lp string stable if there exist class K functions3 α and β, such that, for any initial state x(0) ∈ R(m+1)n and any ur ∈ Lq p, yi(t) − h(¯x0) Lp ≤ α( ur(t) Lp ) + β( x(0) − ¯x ), ∀ i ∈ Sm and ∀ m ∈ N. If, in addition, with x(0) = ¯x it also holds that yi(t) − h(¯x0) Lp ≤ yi−1(t) − h(¯x0) Lp , ∀ i ∈ Sm{1} and ∀ m ∈ N{1}, the system (2.15) is strictly Lp string stable with respect to its input ur(t). Here, · denotes any vector norm, · Lp denotes the signal p-norm (Desoer and Vidyasagar, 2009), and Lq p is the q-dimensional space of vector signals that are bounded in the Lp sense. 3A continuous function α : R≥0 → R≥0 is said to belong to class K if it is strictly increasing and α(0) = 0.
- 37. 2.4 STRING STABILITY 25 Remark 2.1. The interconnected system formulation (2.15) could be further gen- eralized with respect to the interconnection structure (or “topology”), so as to include multiple-vehicle look-ahead, or even bidirectional interconnections. This has no principal consequences for Deﬁnition 2.1 since output responses are con- sidered due to external inputs or initial condition perturbations. Clearly, Deﬁnition 2.1 takes the external disturbance ur into account, imposed by the virtual reference vehicle, through the class K function α( ur(t) Lp ), as well as initial condition perturbations through the class K function β( x(0) − ¯x ). It should be mentioned, that, as a consequence of the latter, only initial condition perturbations are considered for which the norm x(0) − ¯x exists, which limits the allowable class of perturbations in view of the fact that x will be inﬁnite- dimensional for m → ∞. Studies of string stability speciﬁcally focussing on initial condition perturbations can be found in, e.g., Yadlapalli et al. (2006) and Klinge and Middleton (2009a). Furthermore, Deﬁnition 2.1 obviously applies to both linear and nonlinear systems, and includes homogeneous as well as heterogeneous strings, providing a rigorous basis for the string stability analysis of heterogeneous strings pursued in Liang and Peng (2000) and Shaw and Hedrick (2007b). It is important to note that Deﬁnition 2.1 closely resembles the common input– output or Lp stability deﬁnition (Khalil, 2000) as far as (nonstrict) Lp string stability is concerned, except for the fact that the norm requirements must hold for all string lengths m ∈ N. Consequently, if an interconnected system is Lp string stable, it is also Lp stable. The reverse statement, however, does not hold since Lp stability of a string with a given ﬁnite length m does not imply Lp stability for all m ∈ N, i.e., Lp string stability. The latter is essential to string stability, indicating that a string-stable system is scalable in terms of the number of subsystems (Yadlapalli et al., 2006). The notion of strict Lp string stability, for which not only the ﬁrst inequality but also the second inequality in Deﬁnition 2.1 must hold, dictates that the Lp norm of the outputs of the interconnected systems must be nonincreasing along the string, in the direction of increasing system index. As such, it is a stronger require- ment than Lp string stability per se, which only requires the outputs to be bounded in response to a bounded input and a bounded initial condition perturbation. This notion has been introduced to accommodate the requirement of upstream distur- bance attenuation as mentioned before. The deﬁnition of strict string stability diﬀers from the one introduced in Bose and Ioannou (2003a), in that the latter explicitly excludes the possibility that yi(t) − h(¯x0) Lp = yi−1(t) − h(¯x0) Lp . Section 2.5, however, shows that in case of platooning, the equality is the best possible result. This is due to the vehicle following objective, which implies that with a constant velocity v0 of the virtual reference vehicle, all velocities should asymptotically converge to v0. Note that the system (2.15a), which may be re- ferred to as the virtual reference system, does not have an output associated with it, since a “virtual output” is considered practically irrelevant. Therefore, i = 1 has been excluded in the norm requirement for strict string stability. The proposed string stability deﬁnition provides a rigorous basis for the often- used frequency-domain string stability conditions for linear interconnected sys- tems, as shown hereafter.
- 38. 26 2 STRING STABILITY OF CASCADED SYSTEMS 2.4.2 String stability conditions for linear systems In order to derive string stability conditions for linear systems, (2.15) is assumed to describe a linear homogeneous system, i.e., fi is a linear function of the states and considered to be independent of the vehicle index i. Consequently, (2.15) can be reformulated into a linear state-space model, which, in lumped form, is denoted by ˙x0 ˙x1 ... ˙xm = Ar O A1 A0 ... ... O A1 A0 x0 x1 ... xm + Br 0 ... 0 ur (2.16) or, in short, ˙x = Ax + Bur (2.17) with xt = xt 0 xt 1 . . . xt m , and the matrices A and B deﬁned accordingly. The matrices A0, A1, Ar, and Br can, e.g., be chosen identical to those used in (2.12) and (2.14). In addition, consider linear output functions according to yi = Cxi = Cix, i ∈ Sm (2.18) with output matrix C and Ci = 0ℓ×n(i−1) C 0ℓ×n(m−i) . Also, the equilibrium state ¯xt = ¯xt 0 . . . ¯xt 0 = 0 is chosen, hence h(¯x0) = Ci ¯x = 0 ∀ i ∈ Sm. This choice is without loss of generality because there is always a coordinate transfor- mation possible such that ¯x = 0. The model (2.17), (2.18) can then be formulated in the Laplace domain as follows: yi(s) = Pi(s)ur(s) + Oi(s)x(0), i ∈ Sm (2.19) with outputs yi(t) ∈ Rℓ and exogenous input ur(t) ∈ Rℓ , whose Laplace trans- forms are denoted by yi(s) and ur(s), with s ∈ C, respectively. Note that, with a slight abuse of mathematical notation, ·(s) denotes the Laplace transform of the corresponding time-domain variable ·(t) throughout this chapter; if the argument is omitted, then the domain is either irrelevant or can be easily determined from the context. x(0) ∈ R(m+1)n denotes the initial (time-domain) condition. Pi(s) and Oi(s), i ∈ Sm, are the complementary sensitivity transfer function and the initial condition transfer function, respectively, according to Pi(s) = Ci(sI − A)−1 B Oi(s) = Ci(sI − A)−1 . (2.20) Pi(s) is thus assumed to be square, i.e., dim(ur) = dim(yi) = ℓ; this property is adopted in view of the upcoming analysis. Since (2.16) describes a controlled system, the matrix A0 is typically Hurwitz. However, this may not be the case for the matrix Ar, related to the virtual ref- erence vehicle in case of vehicle following. As indicated by (2.13), for instance, Ar has a marginally stable mode associated with v0 (besides the mode associ- ated with the dummy state e0). Hence, the system matrix A in (2.17) is not
- 39. 2.4 STRING STABILITY 27 Hurwitz in case of the vehicle following control problem. In the remainder of this section, however, it is assumed that the pair (Ci, A) is such that unstable (including marginally stable) modes are unobservable by a speciﬁc choice of Ci. Consequently, it suﬃces to only analyze the output response to the external input in view of string stability (or, equivalently, to choose x(0) = ¯x = 0), in accordance with the following remark. Remark 2.2. If (2.15a), (2.15b) represents a linear system, the existence of α implies that β exists, provided that unstable and marginally stable modes are unobservable. This can be shown as follows. Consider the system (2.15a), (2.15b) for a ﬁxed but otherwise arbitrary index i = k. Then this system is Lp stable if yk(t) − h(¯x0) Lp ≤ αk( ur(t) Lp ) + βk( x(0) − ¯x ) with class K functions αk and βk (Khalil, 2000). If (2.15a), (2.15b) represents a linear system, with possible unstable or marginally stable modes being unobserv- able, the existence of αk implies that βk exists (Hespanha, 2009). Because this statement holds for any k ∈ Sm, it also applies to α and β in Deﬁnition 2.1. In order to arrive at conditions for L2 string stability, the H∞ norm is intro- duced ﬁrst, being deﬁned as Pi(s) H∞ := sup Re(s)>0 ¯σ Pi(s) . (2.21) Here, ¯σ(·) denotes the maximum singular value, which, according to the maximum modulus theorem (Zhou et al., 1996), can also be computed by evaluation of ¯σ Pi(s) along the imaginary axis, i.e., supRe(s)>0 ¯σ Pi(s) = supω∈R ¯σ Pi(jω) , provided that Pi(s) represents a causal and stable system. It can then be shown (Zhou et al., 1996, p. 101) that Pi(s) H∞ is equal to the L2 induced system norm related to the input ur(t) and the output yi(t): Pi(s) H∞ = sup ur=0 yi(t) L2 ur(t) L2 , (2.22) where the L2 norm is deﬁned on the interval t ∈ [0, ∞). Consequently, from (2.19) it follows that, with x(0) = 0, yi(t) L2 ≤ Pi(s) H∞ ur(t) L2 ≤ max i∈Sm Pi(s) H∞ ur(t) L2 , ∀ i ∈ Sm. (2.23) It is important to note that, due to (2.22), (2.23) is not conservative, in the sense that there is always a subsystem i ∈ Sm and a speciﬁc signal ur(t) for which the right-hand sides in (2.23) are equal and become arbitrarily close to yi(t) L2 . Therefore, according to Deﬁnition 2.1 and under the assumptions as mentioned in Remark 2.2, the existence of maxi∈Sm Pi(s) H∞ , for all m ∈ N, is a necessary and suﬃcient condition for L2 string stability of the interconnected system (2.17), (2.18). For further analysis, a speciﬁc type of interconnection topology will be adopted, as mentioned in the following remark.
- 40. 28 2 STRING STABILITY OF CASCADED SYSTEMS Remark 2.3. In the case of a look-ahead topology, such as described by (2.16), the interconnection is unidirectional, from which it directly follows that the dynamics of the ﬁrst n < m systems in a string of length m do not depend on the systems n+ 1, n+2, . . . , m. Consequently, if an inﬁnite-length unidirectionally-interconnected string of cascaded systems has a bounded output response to a bounded input, then all ﬁnite-length strings as a subset thereof have a bounded output response as well. Therefore, in order to assess string stability, not all values for the string length m ∈ N need to be evaluated, but only the case where m → ∞. This implies that the sets Sm, m ∈ N, can be reduced to a single set N. As a result, the interconnected system (2.17), (2.18) is L2 string stable if and only if supi∈N Pi(s) H∞ exists (under the assumptions as mentioned in Re- mark 2.2). The class K function α in Deﬁnition 2.1 can then be chosen as α( ur(t) L2 ) = sup i∈N Pi(s) H∞ ur(t) L2 . (2.24) Because of the linear form of α in (2.24), this type of string stability may be referred to as ﬁnite-gain L2 string stability, similar to the notion of ﬁnite-gain L2 stability (Khalil, 2000). The existence of the supremum of the L2 gain can be further analyzed by factorization, leading to the theorem below. As a preliminary to this theorem, the string stability complementary sensitivity is introduced ﬁrst. From (2.19), it directly follows that, with x(0) = 0, yi(s) = Γi(s)yi−1(s) (2.25) with the string stability complementary sensitivity Γi(s) := Pi(s)P−1 i−1(s), (2.26) assuming that Pi(s) is nonsingular4 for all i, thus guaranteeing the existence of P−1 i−1(s) in (2.26). The following theorem, formulating conditions for (strict) L2 string stability, can now be stated. Theorem 2.1. Let (2.17), (2.18) represent a linear unidirectionally-intercon- nected system of which the input–output behavior is described by (2.19), (2.20). Assume that the pair (Ci, A) is such that unstable and marginally modes are unob- servable and that Pi(s) is square and nonsingular for all i ∈ N. Then the system (2.17), (2.18) is L2 string stable if 1. P1(s) H∞ < ∞ and 2. Γi(s) H∞ ≤ 1, ∀ i ∈ N{1} with Γi(s) as in (2.26). Moreover, the system is strictly L2 string stable if and only if conditions 1 and 2 hold. 4A transfer function matrix P (s) is nonsingular if it is invertible for almost all s.
- 41. 2.4 STRING STABILITY 29 Proof. Using (2.19), (2.25), and (2.26), the input–output relation for a speciﬁc subsystem i ≥ 2 can be formulated as yi(s) = Pi(s)ur(s) = i k=2 Γk(s) P1(s)ur(s), i ∈ N{1}. Having factorized Pi(s) in this way, the submultiplicative property dictates that Pi(s) H∞ is subject to the following inequality: Pi(s) H∞ ≤ i k=2 Γk(s) H∞ P1(s) H∞ , i ∈ N{1}. Consequently, under the conditions 1 and 2 in Theorem 2.1, supi∈N Pi(s) H∞ exists. Because it is also assumed that the pair (Ci, A) is such that unstable and marginally stable modes are unobservable for all i ∈ N, it thus follows that the linear system is L2 string stable, according to Deﬁnition 2.1 and Remark 2.2, while using (2.24). Moreover, from (2.25) and condition 2, it follows that yi(t) L2 ≤ Γi(s) H∞ yi−1(t) L2 ≤ yi−1(t) L2 , ∀ i ∈ N{1}, which yields the interconnected system strictly L2 string stable. Note that i = 1 must be excluded here because y0, which would be the output of the virtual reference system, has not been deﬁned. Let us now show the necessity of condition 1 and 2 for strict L2 string stability. Clearly, condition 1 is necessary for both L2 string stability and strict L2 string stability. Moreover, if condition 2 is not satisﬁed, then there exists an i ∈ N{1} such that Γi(s) H∞ > 1 yielding yi(t) L2 > yi−1(t) L2 , which contradicts the strict string stability requirement in Deﬁnition 2.1. Therefore, condition 2 is also a necessary condition for strict L2 string stability. It is important to note that condition 2 in fact very closely resembles the well-known string stability criterion (2.1). As such, Deﬁnition 2.1 together with Theorem 2.1 provide a rigorous basis for this criterion. The fact that Theorem 2.1 only yields suﬃcient conditions for L2 string stability is basically due to the sub- multiplicative property. In speciﬁc cases, however, the submultiplicative property becomes an equality, upon which the L2 string stability conditions become not only suﬃcient but also necessary. The following corollary deals with such a spe- ciﬁc case, which nevertheless appears to be practically relevant, as described in Section 2.5 and 2.7. Corollary 2.2. Let (2.17), (2.18) represent a linear unidirectionally-intercon- nected system, with ur ∈ R and yi ∈ R ∀ i ∈ N, for which the input–output behavior is described by (2.19), (2.20). Assume that the pair (Ci, A) is such that unstable and marginally stable modes are unobservable and that Pi(s) is nonsin- gular for all i ∈ N. Furthermore, let the string stability complementary sensitivity
- 42. 30 2 STRING STABILITY OF CASCADED SYSTEMS Γ(s) = Pi(s)P−1 i−1(s), i ∈ N{1}, be independent of the vehicle index i. Then the system (2.17), (2.18) is L2 string stable if and only if 1. P1(s) H∞ < ∞ and 2. Γ(s) H∞ ≤ 1. Moreover, L2 string stability and strict L2 string stability are equivalent notions in this case. Proof. Referring to the factorization of Pi(s) used in the proof of Theorem 2.1, the following equalities hold for systems with scalar input and output: Pi(s) H∞ = sup ω |Pi(jω)| = sup ω Γ(jω)i−1 P1(jω) = sup ω |Γ(jω)|i−1 |P1(jω)| , i ∈ N, using the fact that the H∞ norm can be computed by evaluation along the imag- inary axis s = jω, as mentioned earlier. Hence, Pi(s) H∞ exists for all i ∈ N, and especially for i → ∞, if and only if |P1(jω)| < ∞ and |Γ(jω)| ≤ 1 for all ω ∈ R, rendering the interconnected system L2 string stable. Since |Γ(jω)| ≤ 1, the system is also strictly L2 string stable. From Corollary 2.2, it follows that for linear unidirectionally-interconnected systems with scalar input and output, L2 string stability and strict L2 string stability are equivalent in case of homogeneous strings, rendering the notion of (nonstrict) L2 string stability only relevant for heterogeneous strings of this type. Until now, only L2 string stability has been considered. Physically, this can be motivated by the requirement of energy dissipation along the string. Alternatively, it is also possible to use the induced L∞ norm instead, which then leads to L∞ string stability. In the scope of vehicle following, the motivation for using this norm would be traﬃc safety, since the L∞ norm is directly related to maximum overshoot. As will be shown below, the analysis of L∞ string stability is similar to that of L2 string stability. Let pi(t) denote the impulse response matrix, corresponding to the transfer function Pi(s). Then, the L1 signal norm pi(t) L1 is induced by the L∞ signal norms of input and output (Desoer and Vidyasagar, 2009), i.e., pi(t) L1 = sup ur=0 yi(t) L∞ ur(t) L∞ . (2.27) Consequently, the unidirectionally interconnected system is L∞ string stable if and only if supi∈N pi(t) L1 exists. The class K function α in Deﬁnition 2.1 can then be chosen as α( ur(t) L∞ ) = sup i∈N pi(t) L1 ur(t) L∞ . (2.28) This leads to the following theorem, formulating conditions for (strict) L∞ string stability.
- 43. 2.4 STRING STABILITY 31 Theorem 2.3. Let (2.17), (2.18) represent a linear unidirectionally-intercon- nected system for which the input–output behavior is described by (2.19), (2.20). Assume that the pair (Ci, A) is such that unstable and marginally stable modes are unobservable and that Pi(s) is square and nonsingular for all i ∈ N. Then the system (2.17), (2.18) is L∞ string stable if 1. p1(t) L1 < ∞ and 2. γi(t) L1 ≤ 1, ∀ i ∈ N{1}, where p1(t) and γi(t) are the impulse response functions corresponding to P1(s) and Γi(s), respectively, with Γi(s) as in (2.26). Moreover, the system is strictly L∞ string stable if and only if conditions 1 and 2 hold. Proof. Expressing the factorization of Pi(s), i ≥ 2, used in the proof of Theo- rem 2.1, in the time domain results in yi(t) = (pi ∗ ur)(t) = (γi ∗ γi−1 ∗ . . . ∗ γ2 ∗ p1 ∗ ur)(t), i ∈ N{1}, where ∗ denotes the convolution operator. Applying Young’s inequality for con- volutions, the following inequality is obtained: pi(t) L1 ≤ i k=2 γk(t) L1 p1(t) L1 , i ∈ N{1}, from which it follows that supi∈N pi(t) L1 exists, under the conditions 1 and 2 in Theorem 2.3. Since it is also assumed that the pair (Ci, A) is such that unstable and marginally stable modes are unobservable for all i ∈ N, it thus follows that the linear system is L∞ string stable, according to Deﬁnition 2.1 and Remark 2.2, while using (2.28). Moreover, using the L∞ gain deﬁnition of the system with impulse response γi(t) and condition 2 yields yi(t) L∞ ≤ γi(t) L1 yi−1(t) L∞ ≤ yi−1(t) L∞ , ∀ i ∈ N{1}, implying that the interconnected system is strictly L∞ string stable. The necessity of the conditions 1 and 2 for strict L∞ string stability can be proven with the same type of reasoning as used in the proof of Theorem 2.1. Again, Theorem 2.3 only provides suﬃcient conditions for L∞ string stability. In this case, however, the additional assumption that ur and yi are scalar, similar to the assumption used in Corollary 2.2, does not lead to necessary and suﬃcient conditions for L∞ string stability. Since the induced L2 norm is used far more often, in practice, than the induced L∞ norm, no further attention will be paid to this issue in the scope of this chapter. Note that a general treatment of the relation between γ(t) L1 and Γ(s) H∞ is given in Desoer and Vidyasagar (2009). Using a Lyapunov-stability approach