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Procedia Social and Behavioral Sciences 17 (2011) 678–697

19th International Symposium on Transportation and Tra?c Theory

Characterization of Tra?c Oscillation Propagation under Nonlinear Car-Following Laws

Xiaopeng Li and Yanfeng Ouyang1

Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801

Abstract Unlike linear car-following models, nonlinear models generally can generate more realistic tra?c oscillation phenomenon, but nonlinearity makes analytical quanti?cation of oscillation characteristics (e.g, periodicity and amplitude) signi?cantly more di?cult. This paper proposes a novel mathematical framework that accurately quanti?es oscillation characteristics for a general class of nonlinear car-following laws. This framework builds on the describing function technique from nonlinear control theory and is comprised of three modules: expression of car-following models in terms of oscillation components, analyses of local and asymptotic stabilities, and quanti?cation of oscillation propagation characteristics. Numerical experiments with a range of well-known nonlinear car-following laws show that the proposed approach is capable of accurately predicting oscillation characteristics under realistic physical constraints and complex driving behaviors. This framework not only helps further understand the root causes of the tra?c oscillation phenomenon but also paves a solid foundation for the design and calibration of realistic nonlinear car-following models that can reproduce empirical oscillation characteristics. Keywords: Tra?c oscillation; Describing function; Nonlinear; Car-following law

1. Introduction Tra?c oscillations, also known as the “stop-and-go” tra?c, refer to the phenomenon that vehicle movement in congested tra?c tends to alternate cyclically between “stop” (or slow movements) and “go” (or fast movements) patterns. Tra?c oscillations lead to a range of adverse consequences including safety hazards, travel delay, extra fuel consumption, air pollution and driving discomfort. In the 1980s, empirical studies used loop detector data as solid evidences of periodically oscillating patterns in congested tra?c [1, 2, 3]. Later, in the synchronized ?ow context [4, 5, 6], Helbing et al. [7] and Kerner [8] categorized observed oscillations into di?erent patterns. Methods to extract oscillation characteristics (e.g., frequency and amplitude) from tra?c data have been proposed in the time domain [9, 10, 11] and the frequency domain [12]. Empirical studies have also related tra?c oscillations to highway capacity drops [13, 14, 15], lane changes near merges and diverges [10, 15, 16, 17, 18, 19, 20], and roadway geometric features [21]. Motivated by these empirical ?ndings, intensive theoretical research has been conducted to investigate oscillation formation and propagation mechanisms. Early studies on linear car-following models can be traced back to the 1950s

1 Corresponding

author. Tel.: 217-333-9858, E-mail: yfouyang@illinois.edu.

1877–0428 ? 2011 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2011.04.538

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[22, 23]. Later, various non-linear models (e.g., [24, 25]) were developed in hope to better reproduce tra?c evolution. For example, Bando et al. ([26, 27]) developed a nonlinear optimal velocity (OV) model to study the stop-and-go tra?c, which became the building block of a set of extended models [28, 29, 30, 31, 32]. Treiber et al. [33] proposed an intelligent driver model (IDM) to qualitatively reproduce observed tra?c oscillations on German freeways. The IDM model has been revisited in a number of following studies, e.g., relating it to a macroscopic model [34] and adjusting it to match observed patterns in more data sets [35, 36]. In spite of numerous attempts, however, few car-following models are able to quantitatively explain propagation mechanisms of the observed tra?c oscillation phenomenon. Although the oscillation behavior of linear car-following models can be easily analyzed by frequency-domain analysis tools [22, 23], the results have very limited capabilities of explaining real-world tra?c oscillation evolution, primarily because of its exclusion of physical constraints (e.g., speed bounds) and nonlinear driving behaviors. For example, without imposing speed bounds, the magnitude of oscillation may grow to in?nity at an exponential rate. The hope for a better explanation of the stop-and-go phenomenon lies on the development of more complex nonlinear car-following models. For example, recent studies that try to explain oscillation propagation with nonlinear car-following behavior include [37, 38, 39, 40, 41, 42, 43, 44]. See [45] for a comprehensive review on this topic. However, due to the complexity from nonlinearity, these studies are mainly based on either numerical simulations or linearization of models. It remains a challenge to analytically quantify the global oscillation propagation properties of nonlinear car-following models. Without a clear connection between the car-following models’ structure (and parameter setting) and their oscillation behavior, it is generally very di?cult to calibrate a suitable car-following model that matches the observed oscillation characteristics. This paper aims to ?ll some of these gaps by proposing a mathematical approach that, for the ?rst time, can analytically quantify oscillation characteristics of general nonlinear car-following laws based on frequency response of nonlinear systems. This framework starts with a novel transformation scheme that expresses a general car following law in terms of pure oscillation components. Then the describing function technique from nonlinear control theory [46, 47] is applied to analyze the local and asymptotic stability properties and the propagation of oscillations. This technique approximates the output of a nonlinear system by the fundamental frequency component (which is computationally easy to characterize), and it allows us to derive a compact frequency response function of a nonlinear car-following law. We illustrate the application of this analytical framework with a set of nonlinear car-following laws, and the analytical predictions are compared with the results from numerical simulations. Numerical experiments show that the proposed method provides an accurate prediction of oscillation propagation in a vehicle platoon. The proposed framework can potentially enable the development of a guideline for designing and calibrating car-following models that can reproduce empirically observed oscillation characteristics. The remainder of the paper has the following structure. Section 2 introduces notation and proposes a new formulation scheme that expresses a general class of car-following laws in terms of pure oscillation components. Section 3 describes the analytical mathematical framework, including the nonlinear car-following stability analysis and oscillation propagation quanti?cation. Section 4 illustrates the application of this framework to a number of well-known car-following models; the performance of the proposed method is examined with numerical examples. Section 5 concludes this paper and brie?y discusses possible future research directions. 2. Car-following Law Representation Generally, vehicle trajectories exhibit both macroscopic and microscopic characteristics. Macroscopic characteristics are speci?ed by nominal states (e.g., average spacing, velocity and ?ow volume) that shall be consistent with a tra?c fundamental diagram [48]. Microscopic characteristics describe how actual vehicle trajectories deviate from the nominal states as a result of car-following dynamics, and such characteristics can often be approximately speci?ed by oscillation properties (e.g., period and amplitude). The coupled oscillation and nominal state components make it di?cult to analyze and quantify tra?c oscillation properties. Inspired by the “detrending” operations in tra?c and supply chain analysis (e.g., [12, 49]), this section proposes a decomposition method that extracts pure oscillation components from vehicle trajectories, which further allows us to represent a general class of non-linear car-following models in terms of only oscillation components. As such, the interference from nominal states is eliminated.

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2.1. Trajectory decomposition As shown in Figure 1, we consider a platoon of vehicles in a single lane, l = 0, 1, ..., L, indexed from downstream to upstream. In an in?nite time horizon t ∈ R, let xl (t) ∈ R denotes the location of vehicle l at time t. The actual trajectory of vehicle l can be denoted by a curve xl = {xl (t)}t∈R .2 Since vehicles normally do not move backwards, xl (t) shall be monotonically non-decreasing with t.

Actual Trajectories Nomial Components Oscillatory Components

x0

Location Location

x0

x1 x2

v

s1 s2 x2

Location

x1

? ? x0 x1

Time

? x2

Time

xL

=

Time

xL

+

...

? xL

...

De?nition 1. We call y = {y(t)}t∈R an oscillatory time series if ?P ∈ R+ , ?∞ < y(t) = y(t + P) < +∞, ?t ∈ R and P y(t) = 0, or equivalently y is comprised of a set of sinusoids (or frequency components) whose frequencies are all 0 multiples of 2π/P. De?nition 2. We say that a time series contains periodic patterns if it is a superposition of a time series of nominal states (which represents the trend) and an oscillatory time series (which captures oscillations). De?nition 3. For two given time series y1 := {y1 (t)}t∈R and y2 := {y2 (t)}t∈R , their di?erence is de?ned as y1 ? y2 := {y1 (t) ? y2 (t)}t∈R . [12] observed that a vehicle trajectory with well-developed oscillations demonstrates very salient periodicity and can be approximated by a narrow band of frequency components after detrending (i.e., removing the nominal series). ? ? As illustrated in Figure 1, we assume that trajectory xl is a superposition of a nominal series xl = { xl (t)}t∈R that dictates the underlying macroscopic tra?c characteristics (e.g, trend speed and average spacing) and an oscillatory ? ? series xl = { xl (t)}t∈R that results from car-following dynamics. Since the macroscopic characteristics of vehicle trajectories usually remain relatively stable in a short period of time, the linear regression line of a trajectory could potentially be considered as its nominal series3 and the remaining components (i.e., by subtracting the nominal series from the original trajectory) can be treated as the oscillatory part. In general, we use the average speed v and a set ? ? ? of average spacings {sl }?l (i.e., sl := xl?1 (t) ? xl (t), ?t) to denote the macroscopic characteristics. For each vehicle l, ? ? ? xl (t) = x0 (0) + vt ? ll =1 sl and hence xl (t) satis?es ?

l

Even for non-stationary tra?c where macroscopic characteristics vary over time (e.g., transition from a free-?ow state to a congestion state), the macroscopic characteristics usually have a much slower evolving pace than tra?c oscillations. The above-mentioned decomposition scheme can be easily adapted to handle such cases by allowing {? l }?l to be non-stationary but slowly varying. The other decomposition steps remain unchanged. x

realistic trajectory with ?nite length can be transformed into an in?nite trajectory by padding its own copies or zeros. case the macroscopic characteristics vary along the trajectories, we can either use the polynomial ?tting method proposed by [12] to extract ? xl , or simply divide xl into several segments by its macroscopic states so that each segment has relatively steady macroscopic characteristics.

3 In 2A

...

Figure 1: Decomposition of trajectories

xl (t) = xl (t) ? x0 (0) ? vt + ? ? ?

l =1

sl , ?t ∈ R, l = 0, 1, · · · , L.

(1)

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The above decomposition, albeit simple, can facilitate the analysis of tra?c oscillations. For example, vehicle trajectories can be plotted into oblique coordinates to remove nominal components and preserve oscillatory components, so that we can easily use frequency analysis tools to measure oscillation characteristics. In addition, car-following models can be calibrated in the oblique coordinates by ?tting the oscillatory components only. 2.2. Car-following model Let v? and v? respectively denote the minimum and maximum possible vehicle speeds.4 In our oscillation analysis, ? ? ? we are only interested in the non-trivial case where v? < v < v? , because v = v? or v = v? implies that {xl }?l are a set of parallel straight lines without any oscillations. Recall that v and sl re?ect macroscopic tra?c characteristics in ? ? stationary tra?c (i.e., when xl = {0}t∈R ); for each l, we assume that they follow a velocity function Fl : R → [v? , v? ], such that v = Fl (sl ) (which can also be interpreted by a ?ow-density fundamental diagram from a macroscopic per? spective). Here, we allow di?erent vehicles to have di?erent velocity functions so as to accommodate heterogeneous ? ? driving behavior.5 When tra?c is not stationary (i.e., when xl {0}t∈R ), Fl (sl ) may slightly deviate from v. We consider a class of car-following laws in the following form: dxl (t) dt = Gl [{Fl (xl?1 (t) ? xl (t))}t∈R ], ?l = 1, · · · , L

t∈R

(2)

where function Fl (xl?1 (t) ? xl (t)) is a target speed (based on the actual spacing) and Gl is an arbitrary linear operator (which might include di?erential, integral and time shift operations). In our analysis, we assume that Fl satis?es the following properties. (i) Fl (s) increases over s ∈ R. (ii) Fl (s) is Lipschitz continuous; i.e., there exists a scalar Kl ∈ R+ such that |Fl (s1 )? Fl (s2 )| ≤ Kl |s1 ? s2 |, ?s1 , s2 ∈ R. (iii) Fl (s) is di?erentiable and strictly increasing when Fl (s) is in the open set (v? , v? ), and for all v ∈ (v? , v? ) there exists one and only one s such that Fl (s) = v (or s = Fl?1 (v)). We de?ne s = limv→v+ Fl?1 (v) and ? s = limv→v?? Fl?1 (v). Property (i) ensures that in congested tra?c a lower vehicle density generally corresponds to a higher nominal speed. Property (ii) is satis?ed by all continuous fundamental diagrams. Property (iii) re?ects on the fact that during congestion a vehicle’s speed is normally sensitive to its spacing changes. Many well-known continuous fundamental diagrams satisfy these three properties. For example, the Greenshield’s fundamental diagram [50] can be speci?ed by letting Fl (s) = max(v? ? v? s0 /s, 0) where s0 is the stopping distance and the Lipschitz scalar Kl = v? /s0 . The triangle l l l fundamental diagram [51, 52] can be speci?ed by letting Fl (s) = λl (s ? s0 ) l · b := mid(a, b, ·) and the Lipschitz scalar Kl is equal to λl . a We further assume that Gl satis?es the following two properties

v? 0

where λl is a sensitivity coe?cient,

(iv) The integral Gl is a low-pass ?lter; i.e., among all frequency components in its input time series, low-frequency components more than high-frequency components. (v) For any constant c, Gl ({c}t∈R ) = {c}t∈R .

Gl ampli?es

Since an integral operation itself is a low-pass ?lter, property (iv) can be easily satis?ed if Gl is not dominated by a di?erential operation. This is the case for most existing car-following laws. Property (v) explains the system’s nominal behavior; i.e., it ensures that the macroscopic characteristics of the trajectories generated from (2) are consistent with those predicted by the fundamental diagram Fl .

minimum speed v? is usually equal to 0 in the real world, but in our framework it may take any value. function Fl can be further generalized into a backlash nonlinear system [47] that contains two speed-spacing functions, one for deceleration and the other for acceleration. For the illustration of the proposed framework, this paper only focuses on the simple function form of Fl .

5 Actually, 4 The

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Plugging (1) into (2) yields d xl (t) ? dt = Gl [{Fl ( xl?1 (t) ? xl (t) + sl ) ? v}t∈R ] , ?l = 1, · · · , L ? ? ?

t∈R

(3)

? ? v ? If we de?ne a new function Fl (s) := Fl [s + Fl?1 (? )] ? v (note that Fl (0) = 0), then (3) can be normalized as follows: d xl (t) ? dt ? ? = Gl {Fl ( xl?1 (t) ? xl (t) + sl )}t∈R , ?l = 1, · · · , L. ? ?

t∈R

(4)

Here, sl = sl ? Fl?1 (? ) is an unknown variable that denotes the deviation of the actual spacing from what the funda? v mental diagram would predict. Before calculating the value of sl , we ?rst introduce the following proposition. ? Proposition 1. Suppose F is continuous, increasing over (?∞, +∞) and strictly increasing over (a, c) for some given a < c ∈ R, and y = {y(t)}t∈R is an oscillatory series. Then for any b ∈ (a, c), there exists one and only one scalar s such that {F(y(t) + s) ? F(b)}t∈R is an oscillatory series. Proof. See Appendix A ? If a < 0 < c, F(0) = 0, and y is an oscillatory time series, then we de?ne a mapping s := S (y, F) such that ? (y, F) analytically. For example, {F(y(t) + s)}t∈R is a nominal time series. For some special y and F, we can compute S ? if F is an odd function and y is a pure sinusoid, then S (y, F) = 0. In general, however, there might not exist ? (y, F). Rather, based on the monotonicity of F, we can obtain S (y, F) from an ? an analytical method to compute S e?cient bisecting search method, as follows Step C0: Initialize s = 0, and return s if 0 [F(y(t) + s)]dt = 0; otherwise, let s? = a and s+ = 0 if 0, or let s? = 0 and s+ = c otherwise. Specify a small positive error tolerance . Step C1: Let s := (s+ + s? )/2. If |s+ ? s? | < , return s; otherwise, go to Step C2. Step C2: Let s? = s if

P [F(y(t) 0 P P [F(y(t) + s)]dt 0

>

+ s )]dt < 0, or let s+ = s otherwise. Go to Step C1.

? ? Now we discuss how to solve sl . According to (4), since xl is an oscillatory series, so is d xl (t) ? dt t∈R . Property (v) ? l ( xl?1 (t) ? xl (t) + sl )}t∈R shall also be purely oscillatory. We know that xl?1 ? xl is an oscillatory ? ? of Gl dictates that {F ? ? ? ? v v series, function F is continuously increasing over (?∞, +∞) and strictly increasing over s ? Fl?1 (? ), s ? Fl?1 (? ) , ?1 ? ?1 (0) = 0. Hence, for any given xl?1 ? xl and Fl , Proposition 1 indicates that sl = S (? l?1 ? xl , Fl ) ? ? x ? ? ? ? v ? s < Fl (? ) < s, and Fl can be obtained by Algorithm C0-C2. Then formula (4) can be equivalently expressed in terms of the oscillatory series only, as follows:

? xl =

? x ? ? ? ? Gl Fl xl?1 (t) ? xl (t) + S (? l?1 ? xl , Fl ) ?

t∈R

dt, ?l = 1, · · · , L.

(5)

Equation (5) can be used to express several well-known car-following models. The linear models [53, 22, 23] can be obtained by letting v? → ?∞, v? → +∞ and Fl be an identity function. The fundamental diagram based models [54, 55, 56] and the OV models [26, 27] are also special cases of (5), with Gl being an identity mapping and an integral operation, respectively (see Section 4 for more detailed discussion). 3. Oscillation Characteristics Analysis This section proposes a mathematical framework that analyzes stability properties and oscillation propagation characteristics for a class of nonlinear car-following laws (5). Stability analyses, including local and asymptotic stabilities, qualitatively explain whether a car following law will amplify or dampen a small trajectory perturbation over time and space. Local stability pertains to whether a perturbation at present will induce future ?uctuations on the same trajectory [23]. Asymptotic stability concerns whether perturbations in the leading trajectory will amplify

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across the following trajectories [22]. These traditional stability analysis methodologies are essentially the same for both linear and nonlinear car-following models, except that nonlinear models are usually linearized before these analyses. If a car following law is both locally and asymptotically stable, it will dampen any perturbations from the leading vehicle and therefore all following vehicles will always move smoothly. This, however, is not consistent with empirical observations. Unstable models will amplify certain perturbations in the leading trajectory and certain oscillation patterns will propagate across trajectories. We will propose an analytical approach that quantitatively predicts the propagation of oscillation characteristics (e.g., periodicity and magnitude) in a vehicle platoon for an unstable nonlinear car-following law in the form of (2). This approach is built on the describing function method [47] from the nonlinear control literature, which is often used to quantify oscillation responses of a nonlinear system for a given sinusoidal input. With this method as a building block, the proposed approach is able to handle the nonlinearity in (2) from a frequency domain perspective and yield an accurate analytical prediction of tra?c oscillation propagation. 3.1. Stability analysis This section introduces methods to analyze the local and asymptotic stability properties of car-following law (5). The local stability pertains to whether the following vehicle’s trajectory can stabilize to its nominal state over time, despite a small perturbation from its immediate preceding vehicle [23]. The asymptotic stability describes whether perturbations from the leading vehicle’s trajectory will be ampli?ed across vehicles upstream [22]. It shall be noted that asymptotic stability is only well de?ned for car-following laws that are locally stable. For the convenience of the notation, we denote the value of Gl (·) at t by Gl (·, t), i.e., Gl (·) = {Gl (·, t)}t∈R . Local stability analysis is generally based on the linearization of car-following law (5). De?ne the Laplace transform of the linear operation Gl GlL (r) := lim

T 0

Gl ({e?rt }, t)dt

T 0

T →∞

e?rt dt

, ?r ∈ C,

(6)

then the linearized characteristic equation of (5) in the Laplace space is de?ned as ? GlL (r) d Fl (s) r ds + 1 = 0, ?r ∈ C,

s=0

(7)

Equation (7) is the denominator of the close-loop transfer function for (5) (see [57] for the introduction to a close-loop system). Car-following law (5) is locally stable if every solution r to (7) (which is a pole of the close-loop transfer function) is within the left half complex plane; i.e., the real part of the solution is negative. For a locally stable car-following law, asymptotic stability can be analyzed based on the frequency response gain of the linearized car-following law (5). De?ne a complex function Gl (ω) := GlL ( jω) = j π

π ?π

Gl ({e?ωt }, t)e? jωt dωt, ?w ∈ R+ .

(8)

√ where j = ?1 and Gl (ω, t) is the value of Gl ({sin(ωt)}t∈R ) at time t. Function (8) is the Fourier transform [58] of the linear system Gl , which is also called the frequency transfer function. The measure for asymptotic stability can be de?ned as follows ? Gl ( jω) d Fl (s) ds s=0 (9) , ?ω ∈ R+ . ? Gl ( jω) d Fl (s) + jω ds

s=0

Car-following law (5) is asymptotically stable if the value of (9) is uniformly no larger than 1 for all ω ∈ R+ . 3.2. Oscillation characteristic quanti?cation 3.2.1. Limit cycle analysis for locally unstable car-following laws For a locally unstable nonlinear car-following law (5), if the value of function Fl is bounded, a leading vehicle’s perturbation shall just lead to bounded oscillation (or a limit cycle [47]) in the following vehicles’ trajectories

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(rather than increasing toward in?nity). In this section we will show how to calculate the oscillation propagation characteristics for nonlinear car-following law (5). Let xl?1 (t) = 0, ?t, and then (5) becomes ? ? xl = ? ? Gl Fl (? xl (t) + sl ) ?

t∈R

dt, ?l = 1, · · · , L

(10)

? Assume time series xl can be approximated by a sinusoid {Al sin(ωt + φl )}t∈R with amplitude Al ∈ R+ , frequency ? ? ? ω ∈ R+ and phase angle φl ∈ [0, 2π). Hence {Fl (? xl (t) + sl )}t∈R shall also include frequency ω, but due to nonlinearity, it may contain sinusoidal components of higher frequencies. Fortunately, these higher frequencies, if any, will likely be dampened by the low-pass ?lter Gl . Thus, the describing function method [47] just considers the fundamental si? ? nusoidal component of {Fl (? xl (t)+ sl )}t∈R . Then, it can be derived that equation (10) can be approximately represented ? in the frequency domain as follows, ? jω ≈ 0, (11) Fl (Al ) ? Gl (ω) where Fl (A) : = =

π ?π π ?π

? ? ? Fl A sin(t) + S ({A sin(t)}t∈R , Fl ) e? jt dt

π ?π

A sin(t)e? jt dt 0∈C

(12) (13)

? ? ? Fl (A sin(t) + S ({A sin(t)}t∈R , Fl ) e? j(t?π/2) dt, ?A

See Appendix B for the detailed derivation of (11). Solving this complex-valued equation (11) (which is equivalent to two real-valued equations) yields the candidate ? frequency ω and amplitude Al for the limit cycle in xl . In cases (11) cannot be solved analytically, we can solve it numerically in the following way. Note that {Fl (Al )}Al ∈R and {? jω/Gl (ω)}ω∈R are two curves on the complex plane. Since |Fl (·)| ∈ [0, Kl ], {Fl (Al )}Al ∈R shall lie within a circle which has radius Kl and is centered at the origin. If {? jω/Gl (ω)}ω∈R lies outside this circle, there is no solution to (11) and the car-following law should be locally stable. Otherwise, we just need to ?nd the intersection(s) of these two curves. For a given ω, it is easy to evaluate the transfer ? ? function Gl (ω). For a given Al , it is also easy to evaluate Fl (Al ) (from Algorithm C0-C2) to obtain S ({Al sin(t)}t∈R , Fl ) and then calculating (12). Hence, we can ?nd the intersection(s) by enumerating a reasonable range of Al and ω values.6 After obtaining such an intersection, we need to verify its stability; i.e., stable solution (ω, Al ) shall satisfy ? jω ? jω |Fl (Al? )| > | Gl (ω) | for an Al? slightly smaller than Al and |Fl (Al+ )| < | Gl (ω) | for an Al+ slightly greater than Al . Only the stable solution(s) is suitable for quanti?cation of the oscillation characteristics for the limit cycle. 3.2.2. Oscillation propagation analysis for locally stable car-following laws However, many car-following models capable of reproducing tra?c oscillations are locally stable but asymptotically unstable. For such car-following laws, we will propose a describing-function-based method to quantify oscillation characteristics for each generated trajectory. These oscillation characteristics quantitatively predict how small perturbations of vehicle 0 are ampli?ed into fully-grown oscillations across the following vehicles l = 1, 2, · · · , L. ? Suppose that x0 can be approximated by a single sinusoid of frequency ω. In car-following law (5), suppose that ? ? the input xl?1 can be approximated by a sinusoid {Al?1 sin(ωt)}t∈R where Al?1 is the amplitude.7 Since xl is generated ? ? from xl?1 , it shall also preserve the same periodicity. The low-pass property of Gl says that xl shall also follow ? a sinusoidal shape (although the phase angle might have changed). This means that all xl , ?l = 1, 2, · · · , L, can be approximated by sinusoids of the same frequency ω. Suppose xl (t) is approximated with Al sin(ωt + φl ) where Al and ? ? φl are, respectively, the amplitude and the phase angle of xl . Then (5) can be represented as {Al sin(ωt + φl )}t∈R ≈

6 For

Gl |Al |Fl (|Al |)| sin ωt + ∠(Al ) + ∠(Fl (|Al |))

t∈R

dt.

(14)

7 This

most car-following laws in the literature, there is usually no more than one intersection. expression does not include a phase angle because we can always shift the time axis to remove it.

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where Al = Al?1 ? Al e jφl . The frequency domain representation of (14) is Al e jφl ≈ or Al e jφl 1 + Gl (ω) Fl (|Al |)Al . jω (15)

Gl (ω) Gl (ω) Fl (|Al |) ≈ Fl (|Al |)Al?1 . jω jω

(16)

? ? If the oscillatory component of xl?1 is given as xl (t) = A sin(ωt), the oscillation characteristics Al and φl for xl can be quanti?ed using Equation (16). If the analytical solution is di?cult to obtain, it can be solved numerically as follows: Step A0: Initialization. Let F 0 = Kl , k = 0. Step A1: Let Ak = l Gl (ω)F k Al?1 . jω + Gl (ω)F k

Step A2: Let F k+1 = βFl (|Al?1 ? Ak |) + (1 ? β)F k with a proper scalar β ∈ (0, 1). l Step A3: Stop if {Ak } converges, and output Al = |Ak+1 | and φl = ∠(Ak+1 ); otherwise, k = k + 1 and go to Step A1. l l l For any given ω and Al?1 , the above approach can be used to obtain Al . De?ne the amplitude ampli?cation ratio R(Al?1 , ω) := Al /Al?1 . We can create a surface for the ampli?cation ratio {R(Al?1 , ω)}Al?1 ,ω∈R+ for all possible ω and Al?1 values. If Fl is linear, the surface will no longer depend on Al and shall degrade to a single curve (i.e., the Bode plot), which is exactly the frequency response (9). So we call this surface the generalized frequency response, which ? can be used to quantify oscillation propagation for a given leading trajectory x0 . For a linear car-following model, propagation (and ampli?cation) of each frequency component is independent, ? and hence x0 can be decomposed into a set of individual sinusoids and the propagation of each sinusoid can be ? independently quanti?ed. As such, xl can be obtained by superposition of all these sinusoids. However, for a nonlinear model in the form of (5), di?erent frequency components may signi?cantly interfere with each other during propagation. The propagation and growth of tra?c oscillations can be quanti?ed as follows. ? ? In case x0 is a pure sinusoid, i.e., x0 (t) = A0 sin(ωt), with a ?xed frequency ω ∈ R+ and a very small amplitude A0 ∈ R+ , we can look up the corresponding ampli?cation ratio at frequency ω in the generalized frequency response surface and calculate the oscillation amplitude of the next vehicle trajectory. This can be repeated for all L vehicles to obtain values for A1 , · · · , AL . We can also repeat this for all ω ∈ R+ , and we obtain an oscillation propagation surface {Al }ω∈R+ ,l=0,··· ,L . ? ? In real data, x0 is likely to include random perturbations rather than a pure sinusoid [12]. Since function Fl is di?erentiable around the origin, we can ?rst treat the car-following law as a linear law when the oscillation magnitude is small (e.g., for the ?rst few downstream vehicles), and we use the above described decomposition-superposition approach to quantify the oscillation propagation. As a result, the frequency components that result in highest values on the generalized frequency response surface for small Al?1 will be ampli?ed the most. We call these frequency components “dominating.” Once the oscillation magnitude grows larger so that the nonlinear e?ect of the car-following ? law is signi?cant, we will approximate xl with a pure sinusoid (which shall be of one of the previous dominating frequencies) and only analyze this frequency component for all the following trajectories. 4. Numerical Examples The modeling framework proposed in Section 3 can obviously be applied to a wide range of car-following laws (i.e., with di?erent fundamental diagram function Fl and operator Gl ). For illustration purposes, we will consider a few well-known examples and compare the analytical oscillation propagation predictions with those observed in numerical simulations.

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4.1. Examples of Fl and Gl We will consider the following two types of Fl . Newell’s Model. We ?rst consider the case where function Fl is the velocity-spacing representation of a triangular ?ow-density fundamental diagram [59]; see Figure 2.8 Parameters v? and s0 are the free-?ow speed (or the posted l speed limit) and the stopping distance, respectively. Scalar λl is a sensitivity factor that re?ects the aggressiveness of the driver [61]. The mathematical expression for Fl is Fl (s) = λl (s ? s0 ) l

v? 0

(17)

Note that expression (17) satis?es Properties (i)-(iii) with v? = 0 and Kl = λl . It is easy to show that ? Fl (s) = λl s

v? ?? v ?? v

.

(18)

Fl (s)

v?

λl

0

s0 l

s

Figure 2: Triangular fundamental diagram based Fl .

Then complex function Fl can be derived as follows, Fl (A) := 2 Aλl ? j(2φ+π/2) ? e? j(2σ+π/2) + (v? ? v ? λl sl )e? jφ ? ? e Aπ 4 +(? + λl sl )e? jσ + (v? ? 2? )e jπ/2 + v ? v where σ= φ= sin?1 π/2 sin?1 ?π/2

? sl ?? /λl ? v A

λl (φ ? σ) , ?A ∈ R+ , 2

(19)

? if ? A < sl < A ? v/λl ; ? , ? if A ? v/λl ≤ sl < A. ? ? ? if ? A + (v? ? v)/λl < sl < A; ? if ? A < sl ≤ ?A + (v? ? v)/λl ?

? sl +(v? ?? )/λl ? v A

8 A triangular fundamental diagram is simple yet capable of explaining the constant backward wave speed observed in reality [60]. Note that it is not di?erentiable everywhere.

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? ? and sl = S ({A sin(t)}t∈R , Fl ) can be obtained by solving the following equation via Algorithm C0-C2, ?

π/2 ?π/2

? [F(A sin(t) + sl )]dt = ?? σ + ? v

π π + (v? ? v) ? φ + sl (φ ? σ) + A(cos(σ) ? cos(φ)) = 0. ? ? 2 2 v? /λl + 2A v? /λl 1? 2A

2? ?

(20)

Note that if 2? = v? , v v? /λl 2λl Fl (A) = sin?1 π 2A

? ? ?. ? ? ? ?

? ? and S ({A sin(t)}t∈R , Fl ) = 0. OV Model. Another popular form of Fl can be drawn from the OV model [26, 27], as illustrated in Figure 3. Its function form can now be speci?ed as follows Fl (s) = 2λl (s ? sm ) v? l )+1 , tanh( 2 v? (21)

where sm is a scalar and tanh(z) = (ez ? e?z )/(ez + e?z ), ?z ∈ R is the hyperbolic tangent function. Equation (21) also l satis?es Properties (i)-(iii) with v? = 0 and Kl = λl . The shape of (21) is similar to that of (17), although the boundary conditions of (21) are somehow unrealistic (e.g., equation (21) does not yield a positive stopping distance and the target speed can never reach v? for any s). However, the di?erentiability of this function are favorable for stability analysis [26].

Fl (s)

v?

λl

0

sm l

s

Figure 3: Hyperbolic tangent based Fl .

? The corresponding Fl function can be shown to be 2λl sl 2? v v? v? ? tanh ? v. ? Fl (s) = + tanh?1 ? ? 1 + 2 v? v 2 (22)

In this case, Fl does not have an analytical expression in terms of ω, and hence we will numerically solve sl = ? ? ? S ({A sin t}t∈R , Fl ) from the following equation via Algorithm C0-C2:

π/2 ?π/2

2λl A sin(t) v? v? tanh + tanh?1 v ? ? 2 v? 2

+

v? ? v dt = 0. ? 2

(23)

Benchmark: Linear Model. For comparison purposes, we de?ne a benchmark linear function Fl (s) as follows. Fl (s) = λl (s ? s0 ), l (24)

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which can be equivalently transformed into ? Fl (s) = λl s, and Fl (A) = λl . (26) Note that the slopes of (18), (22) and (25) are all the same at the origin. This implies that these three car-following laws shall lead to similar oscillation propagation when oscillation magnitudes are very small. We also consider the following two types of operator Gl : Speed Following. In some car following models [22, 23, 54], Gl is a simple time shift operator, Gl {y(t)}t∈R = {y(t ? τl )}t∈R , (27) (25)

where τl is the driver’s time lag. This operator implies each vehicle l will exactly follow the target speed based on the spacing observed τl time ago. Equations (6) and (9) now become GlL (r) = e?rτl , and Gl (ω) = e? jωτl . (28) (29)

Speed Target. Operator Gl may also be drawn from the OV model such that the acceleration of a vehicle is proportional to the di?erence of its actual speed and the target speed observed τl time ago. That is, if the following vehicle’s current speed {y (t)}t∈R = Gl ({y(t)}t∈R ), then dy (t) = α(y(t ? τl ) ? y (t ? τl )), dt where α is a positive scalar. Equation (7) and (9) can be derived easily as follows GlL (r) = and Gl (ω) = α , rerτl + α α jωe jωτl +α . (31) (30)

(32)

In the following subsections we will analyze the four possible combinations of Fl and Gl .9 For each case, we will ?rst examine the local and asymptotic stabilities, and then quantify oscillation propagation properties. 4.2. Case 1: Newell’s Fl and Speed Following Gl From (18), we obtain the slope ? d Fl (s) ds To obtain the root(s) of (7), we need to solve = λl .

s=0

λl e?rτl + 1 = 0. (33) r Obviously, for any root of (33), the real part is negative if and only if λl τl < π/2. Hence, as well known in the literature [23], this car-following law is locally stable if the sensitivity scalar λl < π/(2τl ). With regard to asymptotic stability, (9) becomes λl . jωe jωτl + λl

9 Note

(34)

that (5) becomes the OV car-following model in [27] if OV Fl and Speed Target Gl are combined.

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The maximum absolute value of (34) over ω ∈ R+ is smaller than 1 if and only if λl τl < 0.5. Hence, as shown in [22], this car-following law is asymptotically stable if λl < 1/(2τl ). It shall be noted that since (7) and (9) are based on a linearized car-following law, the stability results shall be the ? same for other forms of Fl (s) as long as the slope at s = 0 is preserved (e.g, (22) and (25)). For a locally unstable car-following law, we can quantify its limit cycle characteristics with the method proposed ? in Section 3.1. Figure 4 shows the numerical results for di?erent v, λl and τl values. We see that for a given v, the ? amplitude A increases with λl and τl . The oscillation period 2π/ω is around 4τl for all di?erent settings (it is exactly ? 4τl when v = 0.5v? ). These results are consistent with simulation outcomes.

250 200 150

τl =1 τl =1.5

v =0.4v ? ?

τl =0.5

150

v =0.5v ? ?

τl =0.5

100

τl =1 τl =1.5

150

v =0.6v ? ?

τl =0.5

100

τl =1 τl =1.5

A

50

A

50

1 2 3 4 5

100 50 0 1 2 3 4 5

A

0

0

1

2

3

4

5

10 8

10 8

10 8

2π/ω

2π/ω

4 2 0 1 2 3 4 5

4 2 0 1 2 3 4 5

2π/ω

6

6

6 4 2 0 1 2 3 4 5

λl

λl

λl

Figure 4: Limit cycle characteristics for Case 1 (v? = 50).

We can also quantify the oscillation propagation in a vehicle platoon. For illustration purposes, we set τl = λl = 1 and v? = 50. Now τl = 1/λl , which is consistent with the conjecture that the backward shock wave speed equals s0 /τl l [61]. Since 0.5 < λl τ = 1 < π/2, the car-following law is locally stable but asymptotically unstable. Figure 5 plots ampli?cation ratio {R(Al?1 , ω)}Al?1 ,ω∈R+ and oscillation magnitude {Al }ω∈R+ ,l=0,··· ,L that are obtained from Algorithm A0-A3. We see that for a given frequency, the oscillation amplitude grows to a certain bound value and then ?attens out, which is consistent with empirical observations [12]. ? The results in Figure 5 give us a way to predict oscillation propagation for any leading vehicle trajectory x0 ; some examples are shown in Figure 6. We conduct the simulation with the car-following model for a certain given ? ? leading vehicle trajectory x0 (which is speci?ed by x(t) and v in Figure 6), and we obtain a platoon of trajectories of the following vehicles. Then we decompose these trajectories into nominal and oscillatory components (see Figure 1), and plot the magnitudes of the oscillatory components from downstream to upstream as the blue solid curves in Figure 6. Then we apply the proposed analytical approach to predict the oscillation magnitudes with the same input x0 , which are plotted as the green dashed curves. For comparison, we also plot the predictions from the corresponding linear model as red dot-dashed curves. Note that in Figure 6, the oscillation magnitude is measured by the standard ? ? ? deviation (STD) of xl (rather than Al ) to accommodate non-sinusoidal xl . As we discussed in Section 3.2, when x0 ? ? is a pure sinusoid, the frequency of xl , ?l will remain the same; when x0 is a random time series, we analyze the ?rst 5 vehicles with the decomposition-superposition approach across all frequency components and then focus on the most dominating frequency component for the rest of following vehicles. We see that for di?erent v values and input ? patterns, the predicted oscillation magnitudes generally are very close to those obtained in simulations; especially, they

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Figure 5: {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L surfaces for Case 1.

are almost overlapping when v = v? /2. Note how, in contrast, the linear model yields unbounded oscillation growth, ? while the proposed approach has successfully produced the growing-and-?attening pattern of oscillation propagation resulted from nonlinear car-following laws.

50 45 40 35 30

v =0.4v ? ? Simulated Predicted Linear model

50 45 40 35 30

v =0.5v ? ? Simulated Predicted Linear model

50 45 40 35 30

v =0.6v ? ? Simulated Predicted Linear model

STD

STD

25 20 15 10 5 0 0 5

x 0(t) =2sin(1t) ? x 0(t) random process ? x 0(t) =2sin(1.6t) ?

25 20 15 10 5

STD

x 0(t) =2sin(1t) ? x 0(t) random process ? x 0(t) =2sin(1.6t) ?

25 20 15 10 5

x 0(t) =2sin(1t) ? x 0(t) random process ? x 0(t) =2sin(1.6t) ?

l

10

15

0

0

5

l

10

15

0

0

5

l

10

15

Figure 6: Prediction of oscillation propagation for Case 1.

4.3. Case 2: Newell’s Fl and Speed Target Gl Equation (7) becomes

λl α 2 erτl + r

αr

+ 1 = 0. (35)

or r2 erτl + αr + λl α = 0.

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Equation (35) does not have closed-form analytical solutions and shall be solved numerically. For illustration purposes, we set τl = 0.10 The solution to (35) is r=? α ± 2 α2 ? 4αλl , 2

which always has a negative real part. This implies this car-following model is always locally stable when τl = 0. Regarding asymptotic stability, Equation (9) becomes ?ω2 e jωτl + jαω +1 λl α

?1

.

(36)

The maximum value of (36) has to be numerically solved too. When τl = 0, the maximum value of (36) over ω ∈ R+ is greater than 1 if α < 2λl (which is consistent with the results from [26]). We now quantify the oscillation propagation when τl = 0, αl = 1, λl = 1 and v? = 50. The car-following law under these parameters is also locally stable but asymptotically unstable. Figure 7 shows the ampli?cation ratio {R(Al?1 , ω)}Al?1 ,ω∈R+ and oscillation propagation {Al }ω∈R+ ,l=0,··· ,L . The dominating frequencies in Figure 7 are generally

Figure 7: {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L surfaces for Case 2.

smaller than those in Figure 5, which implies that Speed Target Gl tends to generate a larger oscillation period. Figure ? 8 plots the predicted oscillation magnitudes for di?erent v and x0 . Again, we see the predicted and simulated results ? match each other and converge to a ?nite bound. 4.4. Case 3: OV’s Fl and Speed Following Gl Since OV’s Fl has the same slope λl at the origin as that in Case 1, the stability analysis results shall be the same as well. Again, we set τl = 1, λl = 1 and v? = 50. Figure 9 plots {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L . All surfaces in Figure 9 are smoother than those in Figure 5 because OV’s Fl is smoother than Newell’s Fl . Figure 10 predicts

10 Zero

time lag has also been assumed in other oscillation analysis such as [26].

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50 45 40 35 30

v =0.4v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

50 45 40 35

v =0.5v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

50 45 40 35

v =0.6v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

STD

STD

x 0(t) random process ?

STD

30 25 20

x 0(t) random process ?

30 25 20 15 10 5

x 0(t) random process ?

25 20 15 10 5 0 0 10 20 30 40 50 60 70

x 0(t) =2sin(0.9t) ?

15 10 5 0 0 10 20 30 40

x 0(t) =2sin(0.9t) ?

x 0(t) =2sin(0.9t) ?

l

l

50

60

70

0

0

10

20

30

l

40

50

60

70

Figure 8: Prediction of oscillation propagation for Case 2.

? oscillation propagation for di?erent v and x0 , which are consistent with simulation results. The magnitude growth ? also seems smoother.

Figure 9: {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L surfaces for Case 3.

4.5. Case 4: OV’s Fl and Speed Target Gl The car-following law in this case is exactly the OV model [26, 27], and the stability results shall be the same as those in Case 2. While the parameters are the same as those in Case 2 (i.e., τl = 0, αl = 1, λl = 1 and v? = 50), the {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L surfaces in Figure 11 are smoother than the counterparts in Figure 7. The oscillation magnitude growth predictions in Figure 12 are again consistent with simulation results, and they also seem smoother than those in Case 2.

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693

50 45 40 35 30

v =0.4v ? ? Simulated Predicted Linear model

50 45 40 35 30

v =0.5v ? ? Simulated Predicted Linear model

50 45 40 35 30

v =0.6v ? ? Simulated Predicted Linear model

STD

STD

20 15 10 5 0 0 5 10

x 0(t) =2sin(1t) ?

25 20 15

STD

25

x 0(t) =2sin(1t) ? x 0(t) random process ?

25 20 15 10

x 0(t) =2sin(1t) ? x 0(t) random process ?

x 0(t) random process ? x 0(t) =2sin(1.6t) ?

15 20

10 5 0 0 5 10

x 0(t) =2sin(1.6t) ?

15 20

5 0 0 5 10

x 0(t) =2sin(1.6t) ?

15 20

l

l

l

Figure 10: Prediction of oscillation propagation for Case 3.

Figure 11: {R(Al?1 , ω)}Al?1 ,ω∈R+ and {Al }ω∈R+ ,l=0,··· ,L surfaces for Case 4.

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50 45 40 35 30

v =0.4v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

50 45 40 35

v =0.5v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

50 45 40 35

v =0.6v ? ? Simulated Predicted Linear model x 0(t) =2sin(0.5t) ?

x 0(t) random process ?

STD

30 25 20 15

x 0(t) random process ?

30

x 0(t) random process ?

STD

STD

25 20 15 10 5 0 0 10 20 30 40 50 60 70

25 20 15

x 0(t) =2sin(0.9t) ?

10 5 0 0 10 20 30 40

x 0(t) =2sin(0.9t) ?

10 5

x 0(t) =2sin(0.9t) ?

l

l

50

60

70

0

0

10

20

30

l

40

50

60

70

Figure 12: Prediction of oscillation propagation for Case 4.

5. Conclusion This paper proposes a mathematical framework that is capable of characterizing tra?c oscillation properties for a general class of car-following models, allowing for both linear and nonlinear dynamics. This framework starts with a new representation of car-following models using only oscillatory components in vehicle trajectories. A series of analytical methods to analyze local and asymptotic stabilities are discussed. In addition, we propose a novel systematic approach to quantify oscillation propagation period and magnitude across a platoon of vehicles for any given leading vehicle trajectory. Numerical experiments show that the proposed analysis framework can accurately quantify oscillation characteristics for a variety of car-following laws. In particular, our formulas can accurately analyze nonlinear car-following behavior and realistically predict the ampli?cation of oscillation magnitude, while the traditional analysis based on linear models often leads to very unrealistic results. This proposed framework provides a global and quantitative perspective of the e?ects of nonlinearity on tra?c oscillation’s growth. It serves as a methodological basis for the design of dynamics models that are able to capture actual oscillation propagation mechanisms and reproduce empirically observed oscillation characteristics. Furthermore, this framework lays a solid foundation for future development of proper control strategies to e?ectively dampen oscillation ampli?cation and mitigate tra?c congestion. This research can be extended in several directions. On the methodology side, this describing function technique can be extended to incorporate more frequency components in approximating an oscillatory process (see the harmonic balancing method in [46]). Such extension can possibly further enhance the accuracy of the predicted oscillation characteristics. The form of the car-following model may also be further generalized. For example, we can incorporate asymmetric driving behaviors and generalize function Fl into an asymmetric form [62]. This improvement is promising since the describing function method has been successfully used to quantify the oscillation response of an asymmetric nonlinear system [47]. On the application side, we are interested in applying this approach to empirical tra?c data (e.g. NGSIM trajectory data), in the hope of using this framework to explain oscillation patterns observed in the ?eld. The proposed framework may also serve as a building block to develop a guideline to map oscillation characteristics directly to the structures of nonlinear car-following models. With such a guideline, we may be able to e?ectively design or calibrate car-following models to reproduce any desired oscillation characteristics. This will possibly pave the foundation for developing e?ective countermeasures to tra?c oscillations. Acknowledgment This research was supported in part by the National Science Foundation through Grant CMMI #0748067.

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Appendix A. Proof for Proposition 1 Proof. Since y is an oscillatory series, let P denote its fundamental period. Then from De?nition 1 we have ?∞ < P y(t) = y(t + P) < +∞, ?t ∈ R and 0 y(t) = 0. Then there exist y? , y+ ∈ R such that mint∈R y(t) = y? and maxt∈R y(t) = y+ . Since F is continuous and increasing, F is strictly increasing in (a, c),

P [F(y(t) + s)]dt 0 ? P [F(y(t) 0 P [F(y(t) 0

+ s)]dt shall also be continuous and increasing with s. Sine

P [F(y(t) + s)]dt 0 ?

+ s)]dt shall be strictly increasing over (a ? y+ , c ? y? ). It is obvious that ≥ PF(c) > PF(b) for any s ≥ c ? y? . Thus

P P [F(y(t) + s 0

≤ PF(a) < PF(b) for any s ≤ a ? y+ and

)]dt = PF(b), or 0 (F(y(t) + s? ) ? F(b))dt = 0. Also, there exists a unique s ∈ (a ? y+ , c ? y? ) such that it is obvious that {F(y(t) + s) ? F(b)}t∈R is bounded and has period P. Hence {F(y(t) + s) ? F(b)}t∈R is an oscillatory series. This completes the proof. Appendix B. Derivation of equation (11) Property (ii) ensures that |Fl (A)| ∈ [0, Kl ] for some Kl . Note that Fl (Ae jφ ) = Fl (A), ?φ ∈ R, and hence we can only use Fl (A) with A ∈ R+ . The fundamental sinusoidal component equals ?Al |Fl (Al )| sin ωt + φl + ∠ (Fl (Al )) where function ∠(·) gives the phase angle of a complex variable. From (10) we obtain {Al sin(ωt + φl )}t∈R ≈ Gl ?Al |Fl (Al )| sin ωt + φl + ∠ (Fl (Al ))

t∈R

dt.

(B.1)

Xiaopeng Li and Yanfeng Ouyang / Procedia Social and Behavioral Sciences 17 (2011) 678–697

697

The frequency domain representation of (B.1) is Al e jφ ≈ which yields Fl (Al ) ? which is equation (11). Gl (ω) Fl (Al )(?Al )e jφ . jω ? jω ≈ 0, Gl (ω) (B.2)

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