Time Headway Requirements for String Stability of Homogeneous Linear Unidirectionally Connected Systems

Transcription

1 Joint 48th IEEE Conference on Decision an Control an 8th Chinese Control Conference Shanghai, PR China, December 6-8, 009 WeBIn53 Time Heaway Requirements for String Stability of Homogeneous Linear Uniirectionally Connecte Systems Steffi Klinge an Richar H Mileton Hamilton Institute, NUI Maynooth Maynooth, Co Kilare, Irelan mail@steffi-klingee an RicharMileton@nuimie Abstract This paper investigates string stability issues in homogeneous strings of strictly proper feeback control systems with uniirectional nearest neighbour communications, using only linear systems with two integrators in the loop We show uner which conitions the inuce L -norm of the isturbance to error transfer function is boune inepenently of the string length when using a constant time heaway an erive a formula for the infimal time heaway to guarantee string stability I INTRODUCTION One control objective in the fiel of coorinate systems is formation control In formation control a group of vehicles shoul follow a given group trajectory an in aition every vehicle nees to maintain a prescribe istance to the surrouning vehicles Increasing commercial an private vehicle traffic motivates a growing interest in the one imensional version of this problem which is often calle platooning In this case we focus on a linear string of automobiles riving in a column In its simplest form platoon control requires a constant istance between the vehicles an the lea vehicle follows a given trajectory, eg [] [5] To simplify communication requirements we consier the case where the automobiles are equippe with a local controller base on sensing the istance to the preceing vehicle We call the string homogeneous if the ynamics of the vehicle an controller are inepenent of location in the string If every controller only uses the information of the separation to its preecessor the system structure will be triangular Hence, stuying the stability of the system is relatively easy In other wors, for a fixe string length, an appropriately esigne local controllers, asymptotic an input-output stability can be guarantee Unfortunately, in some cases, these forms of stability are not uniform with respect to string length, an as the string length grows, the isturbance response may grow without boun This effect is referre to as string instability In the past, ifferent efinitions of string stability have been utilise While most researchers work with input-output formulations, efinitions involving the initial conitions an state space formulations can also be foun, [6] Due to the ease of working with the Eucliean norm [7], [8], it is often preferre to the maximum norm, [9] The authors woul like to thank the Science Founation of Irelan for supporting this work with grant 07/RPR/I77 It has been shown that it is not possible to achieve string stability in a homogeneous string of strictly proper feeback control systems with nearest neighbour communications when using only linear systems with two integrators in the open loop an constant inter-vehicle spacing, [3], [0] This result is inepenent of the particular linear controller esign, [7], [0] The problem was also stuie using partial ifferential equations, [], [] from the perspective of the slowest close loop eigenvalue for problems with biirectional control However, string stability can be guarantee with a spee epenent inter-vehicle spacing policy also calle time heaway policy, [3] Other research was one on heterogeneous strings, ie the particular controller epens on the position within the string, [8], [4] an on nonlinear spacing policies, [5] We woul like to present a precise iscussion of string stability of a homogeneous system with two integrators in the open loop of the subsystem an uniirectional nearest neighbour communication First we will clarify the notation use an erive the isturbance-to-error-transfer function in Section II Thereafter we will show that string instability can be avoie using a time heaway policy only if the time heaway is sufficiently large In particular, we erive a formula for the infimal time heaway to guarantee L -string stability in Section III In Section IV string stability in the L sense will be prove using a sufficiently large time heaway Examples in Section V illustrate the results II PRELIMINARIES We wish to iscuss the stability of a simple chain of N vehicles where all but the first shoul keep a fixe istance x to their preecessor The first car follows a given trajectory We will choose the same vehicle moel with transfer function Ps an the same linear controller Cs for every subsystem, ie every car The open loop transfer function Ls has exactly two poles at the origin, Ls PsCs s Ls with L0 0 The position of the ith vehicle x i epens on the isturbance i an the actuator signal of the ith controller u i The local control objective is to force the separation error e i to zero Measurement noise is neglecte for simplicity x i Psu i + i u i Cse i e i x i x i x /09/$ IEEE 99 Authorize license use 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2 WeBIn53 x i x 0 i e i u i x i C h s Ps Qs Fig : Block iagram of the linear system with time heaway with the vector of error signals et T e e e N an the isturbances t T N It is known that the absolute value of the complementary sensitivity function Ls +Ls of a single subsystem, Ts, is greater than one for a range of frequencies, +, an that the system therefore will be string unstable for constant spacing x const, [3], [7] We consier the following efinition of L -string stability: Definition L -String Stability: Consier a string of N ynamic systems with the local error signal e i an the isturbance i The error signals et epen on the isturbances t in the following manner: et H e, s t 4 where e, R N, N N an H e, s : R N R N The system 4 is L -string stable if given any ǫ > 0 there exists a δ > 0 such that i < δ e i < ǫ where δ is inepenent of the string length N Since using a constant spacing policy the system is string unstable, a linear time heaway h is incorporate in the feeback path In aition to a fixe vehicle separation, a velocity v i epenent istance is require between the vehicles, x supremum over frequency of its largest singular value, σ max : λmax A A Aj i ess sup σ max Aj ess sup x 0 +hv i To simplify the following erivations an because we are intereste in the isturbance to error behaviour we 8 shall set x 0 0 below The complementary sensitivity function of the new subsystem shown in Fig is Γs ajoint with A i,j Ā PsC h s +PsC h sqs PsCs Qs +PsCs with C hs Cs Qs an j,i Qs hs + III INDUCED NORM OF H e, FOR Γ > Where Ā is the complex conjugate of A an A its Hermitian Since the output of the ith subsystem position x i Lemma String instability for Γ > : Suppose the is the reference signal for the ith system with the output x i, isturbance to error performance of an interconnecte system we can write the transfer function H xi,x i s Γs is escribe by 7, where Γs PsCs Qs +PsCs an Consier a isturbance i s that enters the ith subsystem between the controller C h s an the plant Ps It affects the output of the ith subsystem with H xi, i s C h sγs x i s Γsx i s + C h sγs is 5 The error signal e i for i N can be expresse as e i s x i s Qsx i s Γsx i s Qsx i s + ΓsC h s is Qs i s Γse i s + ΓsC h s is Qs i s In vector form, we can write 0 0 Γs 0 e e 0 Γs 0 Qs 0 Qs + ΓsC h s 0 Qs 0 Γs 0 Γs }{{} Γ 0 Q s ΓsC s 7 } 0 Q s {{ } Q with H e, Γ QΓsC s We wish to iscus string stability accoring to Definition That is, we require L boune error signals inepenent of the string length N for any vector of L boune isturbances Thus, the inuce L -norm of the operator H e, must be boune inepenently of N The inuce L -norm of a matrix operator Aj is the Qs hs + an the controller Cs internally stabilises the plant Ps Suppose also that there exists a frequency 0 such that Γj 0 >, then there exists a τ 0 > 0 such that H e, i Γ QΓsC s i Γj 0 N τ 0 Proof: The over all isturbance-to-error-transfer function H e, is H e, Γ QΓC 0 Q Γ ΓQ Γ Q Γ Γ N Q Γ Γ N3 Q Γ ΓC Authorize license use limite to: The Library NUI Maynooth Downloae on February 9, 00 at 08:36 from IEEE Xplore Restrictions apply

3 WeBIn53 Note that H e, i ess sup H e, j i maxh e, i,j ij Γ N Q ΓΓC ess sup Γ N Q Γ C 9 The last equality hols because Γ, Q an C are scalar transfer functions Uner the assumption that there exists a non zero frequency 0 for which Γj 0 >, [6], [9], the absolute value of Q Γ an C cannot be zero at 0 as we now emonstrate First, suppose C j 0 0 So Cs has two poles at s ±j 0 Since a marginally stable pole zero cancellation woul contraict internal stability of the loop Pj 0 cannot be zero Hence, Pj 0 Cj 0 Γj 0 Qj 0 + Pj 0 Cj 0 Qj 0 C j 0 Pj 0 + Qj 0 0 an thus Γj 0 Q j 0 < which contraicts the first assumption that Γj 0 > Also, the magnitue of Q j 0 Γj 0 cannot be zero because Q < for all frequencies greater than zero an Γj 0 > Therefore the inuce L -norm of H e, will grow exponentially with the string length N an the system will be string unstable with τ 0 C j 0 Q j 0 Γj 0 Thus, one necessary conition for string stability is that Γj for all Note that Γj Lj + h + Lj Hence the infimal time heaway essential to permit string stability since otherwise Γ i > is h 0 h 0 : Lj +Lj max Since the maximum in can be attaine at 0 or at at least one 0 0, we will istinguish between two cases: a The maximum in is attaine at 0 only Using L Hôpital s Rule an the fact that L0 L0 L0 conition becomes h 0 lim Since Γ0, 0 0 Lj +Lj / L0 3 Hence, choosing h / L0 guarantees that Γ an Γ only at 0 In fact, this conition has a simple geometric interpretation For h / L0 the secon erivative of Γ at 0 is zero, Γ 0 Since Γ is equal to at the 0 origin, it woul be greater than for some frequency > 0 if its secon erivative at the origin woul be greater or equal to zero b The maximum in is attaine at at least one 0 0 In that case Γ an Γ only at 0 an 0 Conition becomes Lj 0 +Lj 0 h IV INDUCED NORM OF H e, FOR Γ As we have seen that string stability cannot be achieve for a system with a time heaway less than h 0 we will now choose a time heaway of h > h 0 Lemma String stability for h > h 0 : Suppose the isturbance to error performance of an interconnecte system is escribe by 7, where Γs Qs PsCs +PsCs an Qs hs + Suppose the time heaway h is strictly greater than h 0 as efine in an the controller Cs internally stabilises the plant Ps Then there exists a τ 0 such that H e, i Γ QΓsC s i τ 0 Proof: H e, as Using the structure of Γ an Q we can write H e, Γ QΓC 0 0 I + Γ Q Γ 0 ΓC Using the triangle inequality we can boun the inuce L - norm of H e, as H e, i + Γ Q Γ i Γ i C i 6 Since the norms of Γ an C o not epen on the string length, the norm of Γ Q Γ can be use to boun H e, i Γ Q Γ ess sup σ min Γ Q Γ i 7 Using the Gersgorin-Theorem see eg [6], we can estimate the minimal Eigenvalue of a matrix λ min A max min j min i a jj a ii n i,i j n j,j i a ij a ij, Authorize license use limite to: The Library NUI Maynooth Downloae on February 9, 00 at 08:36 from IEEE Xplore Restrictions apply

4 WeBIn53 For Γ Γ we obtain λ min Γ Γ min { + Γ Γ, + Γ Γ, Γ } + Γ Γ Γ 9 Thus, the inuce L -norm of Γ Q Γ can be boune as Γ Q Q Γ Γ ess sup i Γ Q +L ess sup Q L an from 6 H e, i +L ess sup Q + L L + ess sup ess sup 0 Q + L L C Γ ess sup since Γ an C are boune inepenently of the string length N However, we nee to have a closer look at for 0, where Γ0 lim Q + L L lim lim h + L + L h + Using L Hôpital s Rule, becomes + L 4 L 4 L + L + L L lim Q + L L h lim 4 L + L + L h + + h + L + L L + L + L L 4 L + L + L L L0 h L0+ L0 L0 3 At zero frequency L0 L0 L0 Since h is strictly greater than h 0 an therefore greater than / L0, lim Q + L L is boune Hence, He, i is boune inepenently of N an the system is string stable accoring to Definition We have proven string stability for h > h 0, an string instability for h < h 0 It remains therefore to consier the case where h h 0 We will show that the inuce L -norm of H e, will grow at least as fast as the square root of the string length N First, we will analyse case b where h 0 is chosen accoring to 4 an Γj 0 Since the first element of He, H e, is N H e, H e, + Γ i Q Γ Γ C,, i0 4 Q j 0 Γj 0 0, an C j 0 0 the norm of He, H e, will grow with the string length N Hence, the largest Eigenvalue of He, H e, an therefore the square of the inuce L -norm of H e, will grow with the string length N The proof for case a is given in the appenix q Tj He, i V EXAMPLES 07 T T Frequency q Tj a for ifferent transfer functions N 00 N Time heaway h b Inuce L -norm of H e, for ifferent time heaways h Example Infimal Time Heaway h 0 : In orer to fin the infimal time heaway h 0, the maximum over all frequencies of Tj / must be evaluate For T s s+ s +s+ the maximum is achieve at 0 an 995 Authorize license use limite to: The Library NUI Maynooth Downloae on February 9, 00 at 08:36 from IEEE Xplore Restrictions apply

5 WeBIn53 h 0 is chosen accoring to 3 For T s s+ s +s+ it is achieve at 0 05 Thus, h 0 47 is chosen accoring to 4 In Fig a both cases are illustrate Example Inuce L -Norm of H e, : Fig b shows H e, i for ifferent time heaways h an string lengths N For time heaways less than h 0 ashe line the inuce L - norm of H e, grows exponentially with the string length N However, if the time heaway is sufficiently large, h > h 0, H e, i converges as the string length increases VI CONCLUSIONS AND FUTURE DIRECTIONS In this paper we have iscusse string stability for a homogeneous string of strictly proper feeback control systems with nearest neighbour communications when using only linear systems with two integrators in the open loop We have shown how the inuce L -norm of the isturbance to error transfer function H e, grows as the string length increases if no or a small time heaway is use A formula for the infimal time heaway has been erive We prove that using a sufficiently large time heaway bouns the inuce L - norm of H e, inepenently of the string length As for future irections, it woul be interesting to exten the results presente to more general cases That coul be analyzing heterogeneous systems, biirectional controller esigns, or using the L -norm APPENDIX We will prove that an interconnecte system where the maximum of is achieve at 0 is string unstable if the time heaway h is equal h 0 in 3 More precisely, we will show that there exists a τ > 0 an a c such that H e, i τn + c We assume that there exist a 0 0,], a l min an a l max such that 0 < l min Lj l max for all [0, 0 Then there exist α an β such that Γ +α 4 an Q Γ 4 β are satisfie for all frequencies < 0 Later, these inequalities will be use to prove string instability First, we will analyse Γ for this special case Γ L 4 h + L + L + L 4 L L0 + 4 L + L + L L0 L L L + L L0 L 4 L + L L L0 We want to fin an α such that Γ +α 4 for small frequencies < 0 Hence, α must satisfy α sup L0 L + L L + L L0 L L+ L + L L0 + 6 For all fixe frequencies 0 < < 0, there exists a α which satisfies 6 However, it also must be boune for 0 α lim L0 L + L L + L L0 L L+ L + L L0 + 7 To evaluate the last term in 7, we make use of the following facts: an L Hôpital s Rule: L a + bj 8 a a 0 + a + a L+ L + L L0 lim lim b b + b b lim lim L + L + L0 L L + L + L L0 L L + L L + L + L0 L L + 4 L + L 4a + a 0 4a0 a + b a 0 3 Since L0 a 0 0, the limit in 3 exists Therefore 7 is boune, an there exists an α that satisfies 6 for all frequencies < 0 an Γ + α 4 < Authorize license use limite to: The Library NUI Maynooth Downloae on February 9, 00 at 08:36 from IEEE Xplore Restrictions apply

6 WeBIn53 We will now show that there exists a β satisfying Q Γ 4 β Q Γ 4 h + 4 L + L + L 33 For small frequencies < 0, there exists a β such that h + h + β < 0 34 Furthermore, there exists a β satisfying L 4 + L + L β < 0 35 such that Hence, β + sup L + L + Q Γ 4 4 β β β L 36 < 0 37 Using the special structure of H e,, we can boun its inuce L -norm as follows: H e, i + ΓC H i e, + I N ΓC i sup H e, + I N ΓC v i v with a vector v of length v N Γ Γ H e, + I N ΓC v i 38 Γ Γ Γ N Γ N T 39 Using Γ, 3, 37 an 39, inequality 38 becomes H e, i + ΓC i Q Γ Γ C ++ Γ + N + + Γ + Γ + + Γ N C 4 Γ N + Γ N + Γ N + N β + Γ N + Γ N + Γ N + + Γ N C 4 N β Γ N + + N C 4 N β + α 4 N N NN 6 40 For any string length N, the maximum over all frequencies in 40 must be greater or equal to that obtaine by choosing N /4 : H e, i + ΓC i C + α N N NN 6β N N 4 for sufficiently large strings, N > 4 0 Since + α N N e α, 4 can be boune by H e, i + ΓC i C N NN eα 4 6β 6 C 3β eα N N 43 Thus, the inuce L -norm of H e, grows at least as far as the square root of the string length N an the system is not string stable accoring to Definition REFERENCES [] W Levine an M Athans, On the optimal error regulation of a string of moving vehicles, IEEE Transactions on Automatic Control, vol AC-, no 3, pp , 966 [] K C Chu, Decentralize control of high-spee vehicular strings, Transportation Science, vol 8, no 4, pp , 974 [3] S Sheikholeslam an C Desoer, Longituinal control of a platoon of vehicles, in Proceeings of the American Control Conference, 990, pp 9 97 [4] D Swaroop, String stability of interconnecte systems: An application to platooning in automate highway systems, PhD issertation, University of California, Berkeley, CA, 994 [5] S Klinge, Stability issues in istribute systems of vehicle platoons, Master s thesis, Otto-von-Guericke-University Mageburg, [6] D Swaroop an J Herick, String stability of interconnecte systems, IEEE Transactions on Automatic Control, vol 4, no 3, pp , 996 [7] P Seiler, A Pant, an K Herick, Disturbance propagation in vehicle strings, IEEE Transactions on Automatic Control, vol 49, no 0, pp , 004 [8] I Lestas an G Vinnicombe, Scalability in heterogeneous vehicle platoons, in Proceeings of the American Control Conference, 007, pp [9] J Eyre, D Yanakiev, an I Kanellakopoulos, A simplifie framework for string stability analysis of automate vehicles, Vehicle Systems Dynamic, vol 30, no 5, pp , 998 [0] P Barooah an J Hespanha, Error amplification an isturbance propagation in vehicle strings with ecentralize linear control, in Proceeings of the 44th IEEE Conference on Decision an Control, an the European Control Conference 005, 005, pp [] J H Davis an B Barry, A istribute moel for stress control in multiple locomotive trains, Applie Mathematics & Optimization, vol 3, no -3, pp 63 90, June 976 [] P Barooah, P Mehta, an J Hespanha, Decentralize control of vehicular platoons: Improving close loop stability by mistuning, 009, to appear in the IEEE Transactions on Automatic Control [3] C Chien an P Ioannou, Automatic vehicle following, in Proceeings of the American Control Conference, 99, pp [4] M Khatir an E Davison, Boune stability an eventual string stability of a large platoon of vehicles using non-ientical controllers, in Proceeings of the 43r Conference on Decision & Control, 004, pp 6 [5] D Yanakiev an I Kanellakopoulos, Nonlinear spacing policies for automate heavy-uty vehicles, IEEE Transactions on Vehicular Technology, vol 47, no 4, pp , 998 [6] R A Horn an C R Johnson, Matrix Analysis Cambrige University Press, Authorize license use limite to: The Library NUI Maynooth Downloae on February 9, 00 at 08:36 from IEEE Xplore Restrictions apply