Robust Estimation and Control of Tire Traction Forces

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Robust Estimation and Control of Tire Traction Forces Tesheng Hsiao, Member, IEEE

Abstract—Due to increasingly demanding requirements for driving safety, precise and robust control of tire traction forces is essential to im-plement high-performance vehicle-control systems. However, traditional slip-ratio-based methods cannot guarantee achievement of desired trac-tion forces due to a lack of tire force feedback. This paper proposes an observer-based traction force control scheme that is robust with respect to variations of road conditions and tire model uncertainties. Moreover, achievement of desired traction forces is guaranteed. Simulation results verify the robust and satisfactory performance, even under suddenly changing road conditions, as well as parametric and nonparametric tire model uncertainties.

Index Terms—Tire force control, tire force estimation, traction control.

I. INTRODUCTION

Driving safety and performance can be significantly improved if tire traction forces are under careful control. However, control of tire traction forces is challenging in many aspects. First, the tire–road interaction is complicated, highly nonlinear, and uncertain. Many tire models have been proposed [1], but each of them considers only parts of tire behavior (e.g., speed dependence). Moreover, it is difficult to know in advance or to estimate online the parameters associated with the tire models. Therefore, tire-model-based control schemes inevitably suffer from considerable uncertainties from both model parameters and model structures. Robust control methods such as sliding-mode control and/or fuzzy logic control were investigated in [2]–[4]. Variants of sliding-mode control such as the second-order sliding-mode control [5], [6] and the dynamic surface control [7] were also explored for their good robustness and reduced chattering phenomenon.

The second challenge is that the operational conditions of tires are time varying and may substantially change. In particular, traction forces significantly depend on road conditions. To deal with this problem, some traction force control systems estimated the tire–road friction coefficient online using the recursive least square algorithm [2] or the sliding-mode observer [5], [8].

The third challenge comes from immeasurability of tire traction forces; thus, traction force control is indirectly accomplished by slip ratio control. Consequently, the control objectives of slip-ratio-based methods focus on tracking the desired slip ratio [2], [6], attaining maximum traction forces [2], [3], [5], [8], or stabilizing the slip ratio under all road conditions [7].

However, the ever-demanding requirements for driving safety in-spire the need for precise traction force control. Recent studies have shown that coordinating traction forces of all tires in an optimal way improves yaw stability of the vehicle in severe situations [9]–[12]. Unfortunately, perfect slip ratio control does not guarantee achieve-ment of desired traction forces because nonparametric tire model

Manuscript received November 17, 2011; revised March 29, 2012; July 24, 2012; and October 4, 2012; accepted October 30, 2012. Date of publication November 30, 2012; date of current version March 13, 2013. The review of this paper was coordinated by Dr. S. Anwar.

The author is with the Department of Electrical and Computer Engi-neering, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: tshsiao@cn.nctu.edu.tw).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2012.2230656

uncertainties hamper correct transfer of the desired traction force to the target slip ratio. To solve the problem, tire force estimators were investigated [13]–[16]. However, observer-based traction force control needs further exploration for guaranteed robust stability and performance. On the other hand, working prototypes of tire force sensors were utilized for firing different phases of the slip ratio control law [17], instead of directly controlling the traction force.

This paper proposes an observer-based traction force control scheme that guarantees achievement of the desired traction force and is robust with respect to tire model uncertainties, as well as variations in road conditions. The proposed controller accepts the traction force command from an “upper-level” vehicle-control system [9]–[12] and delivers the required wheel torque to attain the desired traction force. In addition, maximum traction force is estimated online to prevent the controller from tracking an unachievable force command.

The robust performance of the proposed controller was verified by simulations and compared with those of an “ideal” slip-ratio-based controller and a simple traction force estimator in [15]. The results show that the proposed method achieves superior performance, regardless of sudden changes in road conditions, as well as parametric and nonparametric tire model uncertainties.

The rest of this paper is organized as follows: Section II intro-duces the tire models used in this paper. Section III presents the robust traction force estimator. The control algorithm is proposed in Section IV. Simulation results are presented in Section V, and Section VI concludes this paper.

II. TIREMODELS

In this paper, we assume that the vehicle is traveling along a straight line without cornering. Neglecting the rolling resistance and deformation of the tire, we have Iwω = T˙ − FxR, where T is the wheel torque, which is the sum of the braking torque and the driving torque from the engine. T is positive for acceleration and negative for braking. Iw, R, ω, and Fx are the moment of inertia, radius, rotational velocity, and traction force of the wheel, respectively. ω is positive when the vehicle is moving forward, and Fxis positive during acceleration.

The traction force Fx is closely related to the tire’s slip ratio, which is defined as λ = (Rω− vw)/ max{Rω, vw}, where vwis the translational velocity of the wheel. One of the most popular tire models characterizing the relation between Fxand λ is the magic formula [18]

Fx= fM(λ, μp, Fz)

= μpDMsinCMtan−1BMλ− EM

×BMλ− tan−1(BMλ)

(1) where μpis the tire–road friction coefficient, and Fzis the normal tire force. Fzis not a constant due to the weight transfer effect. BM, CM, DM, and EMare parameters associated with the magic formula whose values vary with Fz. Another widely used tire model is the Dugoff model [19]

Fx= fD(λ, μp, Fz) = Cx λ

1− |λ|k(σ) (2)

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 3, MARCH 2013 1379

The Fx−λ relations based on (1) and (2) for various μpand Fz are shown in Fig. 1, and we can observe that (1) and (2) are similar for small slip ratios but significantly different near the limitation of the traction force.

In a general slip-ratio-based control system, the desired traction force Fxd is converted to the target slip ratio λdthrough an inverse nominal tire model, and the controller tracks the target slip ratio. Although the slip ratio tracking is perfect, i.e., λ = λd, the desired traction force may not be attained due to tire model uncertainties. Parametric model uncertainties can be compensated for by online estimation of the tire–road friction coefficient [2], [5] and tire model parameters [20], [21]; however, compensation for the nonparametric model uncertainties remains unsolved. To emphasize the robustness of the proposed control scheme with respect to both parametric and nonparametric tire model uncertainties, we choose the Dugoff model with fixed Cx, μp, and Fz as the nominal tire model for controller design and assume that the actual tire model is unknown. In the simulation, we use the magic formula as the actual tire model.

III. TRACTIONFORCEESTIMATION A. Design of the Estimator

We assume that Fx= f (λ, μp, Fz), where f is an unknown function satisfying ∂f (λ, μp, Fz)/∂λ = (1 + Δ)(∂f0(λ)/∂λ), and

|Δ| ≤ ¯Δ for all λ, μp, and Fz, where f0(λ) = fD(λ, μp0, Fz0) is the Dugoff model evaluated at fixed constants μp0and Fz0. Δ is the model uncertainty and is assumed to be bounded by a known constant

¯

Δ. To avoid the potential numerical problem introduced by the large slope ∂Fx/∂λ for small λ, we use the normalized traction force μ = Fx/Fz0 in the subsequent derivation. μp is regarded as quasi-static; hence, we have ˙μ = (1/Fz0)[(∂f /∂λ) ˙λ + (∂f /∂Fz) ˙Fz]

Δ = (1 + Δ)B(μ, λ, vw, ω, T )+(∂f /∂Fz)( ˙Fz/Fz0), where B(μ, λ, vw, ω, T ) = (∂f0/∂λ)((1−|λ|)/Fz0)[((T−μFz0R)/ωIw)−( ˙vw/vw)].

Assume that ω, vw, ˙vw, and T are available, then the proposed traction force estimator is

˙ˆ μ = ˆB + η(φ, ˆω, ˆωf) ˙ φ = Φ ˙ˆ ω =T− ˆμFz0R Iw + a (ω− ˆω) ˙ˆ ωf =T− ˆμFz0R Iw + k1(ω− ˆωf)− k2 RFz₀ Iw

where ˆB = B(ˆμ, λ, vw, ω, T ), and ˆμ is an estimate of μ. φ is an auxiliary state variable. Both ˆω and ˆωf are estimates of ω. The third feedback term in ˙ˆωf is used to stabilize the estimator, whereas ˆω is used to estimate an upper bound of μ− ˆμ. Detailed discussion will be presented shortly. To complete the design, we have to determine the update law Φ, the feedback correction term η(φ, ˆω, ˆωf), and the constant gains a, k1, and k2.

Define eμ= μ− ˆμ, eI=−(Iw/RFz0)(ω− ˆω), and ef =−(Iw/ RFz0)(ω− ˆωf). Then

˙eμ= (1 + Δ)A0eμ+ Δ ˆB +

∂f ∂Fz ˙ Fz Fz0 − η (3) ˙eI= eμ− aeI ˙ef = eμ− k1ef− k2Φ (4)

where A0=−(∂f0/∂λ)(R/ω)((1− |λ|)/Iw). Assume that there ex-ist Eμ(t) and a positive constant Γ such that

|eμ(t)| ≤ Eμ(t) ∀t ≥ 0 ∂f ∂Fz ˙ Fz Fz0 ≤ Γ ∀λ, μp, and Fz. (5)

Since|Fx| ≤ Fz, a trivial choice of Eμ(t) is (Fz(t)/Fz0) +|ˆμ(t)|. We will investigate a tighter upper bound Eμ(t) in Section III-B. Note that (5) does not imply boundedness of eμ because Eμ(t) may go unbounded as t increases.

Consider the following Lyapunov function: VE(eμ, eI, ef, ˜φ) = (1/2) γ1e2μ+ (1/2)γ2e2f+ (1/2)γ3φ˜2+ (1/2)e2I, where ˜φ = φ− eI and γi, i = 1, 2, 3, are positive constants. Given a positive constant βE, if we choose k1≥ (βE/2) > 0, a > (βE/2) > 0, and α1> (βEγ1/2) +

γ1(1 + ¯Δ)|A0| > 0, then ˙

VE= γ1eμ˙eμ+ γ2ef˙ef+ γ3φ˜

_˙ φ− ˙eI+ eI˙eI < (γ1(1 + Δ) A0− α1) e2μ − γ2k1e2f− ae 2 I− βEγ3 2 ˜ φ2+ q(eμ)

where q(eμ) = α1e2μ+ beμ+ c, b = γ2ef− γ3φ + eI˜ − γ1η, and

c = γ1(| ˆB| ¯Δ+Γ)Eμ+ (γ3φ˜− γ2k2ef)Φ + γ3aeIφ + (β˜ Eγ3/2) ˜φ2. If q(eμ)≤ 0 for all |eμ| ≤ Eμ, then ˙VE<−βVE. This implies that VE converges to zero exponentially with a time constant 1/βE. Because q(eμ) is a convex parabolic function of eμ, one sufficient condition for q(eμ)≤ 0, ∀|eμ| ≤ Eμ is b≤ 0 and q(−Eμ)≤ 0. Rearranging these two inequalities yields

γ2ef−γ3φ+eI˜ ≤γ1η ≤−1 Eμ α1E2μ− γ2ef−γ3φ+eI˜ Eμ+γ1BˆΔ+Γ¯ Eμ +γ3φ˜− γ2k2ef Φ + γ3aeIφ +˜ βEγ3 2 ˜ φ2 . (6)

If we choose η = 1/γ1{γ2ef− γ3φ + eI˜ + K/2} for some K > 0, then the first inequality of (6) is satisfied. Next, we consider the following three cases to choose k2 and Φ such that the second inequality of (6) is satisfied.

Case I (| ˜φ| ≥ ε for some ε > 0): Let k2= 0 and ˙ φ = Φ =−βE 2 ˜ φ− aeI−α1E 2 μ+ γ1BˆΔ + Γ¯ Eμ+ KEμ γ3φ˜ . Case II (| ˜φ| < ε and |ef| > ε/k1): Let k2= 2γ3k1/γ2and

˙ φ = Φ =−α1E 2 μ+γ1BˆΔ+Γ¯

Eμ+γ3aeIφ+˜ βE₂γ3φ˜2+KEμ γ3φ−γ˜ 2k2ef

.

Note that|γ3φ˜− γ2k2ef| > γ3ε since|γ2k2ef| = 2γ3k1|ef| > 2γ3ε.

Case III (| ˜φ| < ε and |ef| ≤ ε/k1): Let k2= 0 and ˙ φ = Φ =−βE 2 φ˜−aeI− α1Eμ2+γ1BˆΔ+Γ¯ Eμ+KEμ γ3ε sgn( ˜φ) . It is easy to show that (6) holds for both Cases I and II but not for Case III. In Cases I and II, we have ˙VE<−βEVE. Thus, all error signals converge toward zero exponentially until Case III is

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Fig. 1. Fx–λ relations of the magic formula and the Dugoff model for various μpand Fz. reached; then, the estimator stays in Case III forever. In Case III, we

have ˙eI= eμ− aeI and ˙ef = eμ− k1ef, where a > 0 and k1> 0. Because|ef| ≤ (ε/k1), eμis bounded, and so is eI. Furthermore, if the steady state is reached, i.e., ˙ef = 0, then |eμ| = k1|ef| ≤ ε. In other words, eμis confined to a predefined bound in the steady state.

B. Upper Bound of the Estimation Error

We want inequality (5) as tight as possible to alleviate the vibrations caused by large ˙φ. Discretizing (4) with sampling time Tsyields the following [22]: e_I[k] = adeI[k−1] + (1−ad)eμ[k] = gd[k]∗ eμ[k], where k∈ Z+_{, a}

d= e−aTs, eI[k] = aeI(kTs), eμ[k] = eμ(kTs), gd[k] = (1− ad)ak

d, and∗ is the discrete-time convolution. Let δdbe the discrete-time impulse signal. Then, we have

eμk∞− eIk∞≤ eμ− eIk∞

=(δd− gd)∗ eμ_k_∞≤ δd− gd₁eμ_k_∞ •1is the l1-norm, andeμk∞= maxΔ 0≤m≤k|eμ[m]|. If a>(ln 2/ Ts), then d= 1Δ − δd− gd1= 1− 2ad> 0, and

|eμ[k]| ≤ eμ_k_∞≤ 1 de

Ik_∞= a

deIk∞. (7) Equation (7) is an upper bound of|eμ(t)| at t = kTs. If Ts is sufficiently small, we can assume that |eμ(t)| ≤ (a/d)eIk∞ for kTs≤ t < (k + 1)Ts. Note that eIk_∞ is a nondecreasing func-tion of time; however, the estimafunc-tion error may exponentially decay. Equation (7) will become overconservative as t increases. Hence, we replaceeIk_∞ by the recent maximum value of |eI[k]| and allow this maximum value to exponentially decay with a time constant 1/β, 0 < β < βE. The flowchart in Fig. 2 illustrates the procedure for calculating Eμ. In Fig. 2, Δt defines the length of the time window within which the historical maximum value of|eI[k]| is preserved. Ei, i = 0, 1, 2, . . ., denotes the historical maximum value of (a/d)|eI(t)| that takes place at time ti. The initial values for t0 and E0 are set to be zeros.

Fig. 2. Flowchart of evaluating Eμ(t).

IV. TRACTIONFORCECONTROL

The proposed traction force control scheme is aimed at tracking a force command F_xd∗ from an upper-level control system. To prevent the proposed controller from tracking an unachievable force command, a maximum traction force estimator modified from [23] is included. On the other hand, the same tire force may correspond to two distinct slip ratios; however, it is desirable for the tire to operate under the slip ratio with a smaller magnitude. Therefore, the proposed controller will suppress excessive slip ratios first and then carry out the force tracking task in the “stable region” of slip ratios.

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A. Estimation of Maximum Traction Forces

According to the brush tire model [24], the traction force can be ex-pressed as Fx= ψTθ if|λ|≤(3 ¯Fx/Cx), and Fx= ¯Fxsgn(λ) if|λ|> (3 ¯Fx/Cx), where Cxis the longitudinal stiffness, ¯Fx= μpFz is the maximum tire force, θ = [θ1, θ2, θ3]T= [Cx, (Cx2/ ¯Fx), (Cx3/ ¯Fx2)]T, and ψ = [λ, (−λ2_{sgn(λ)/3), (λ}3_/27)]T_{. Note that the components of}

θ are not independent; instead, they satisfy the equality constraint

g(θ) = θ2

2−θ1θ3= 0. If Fxis replaced by its estimate ˆFx= ˆμFz0, then for the case of unsaturated tire forces, θ can be estimated by the gra-dient projection method [25]˙ˆθ =[I−P∇g(∇gT_P_∇g)−1_∇gT_]Pε_ψ, where ε= ˆFx−ψTθ and Pˆ ∈ R3×3 is a constant positive definite matrix. Then, the estimated maximum tire force is ˆ¯Fx= (ˆθ1θˆ2/ˆθ3), and the desired traction force that the proposed controller should follow is Fxd= min{|Fxd|, ˆ¯∗ Fx}·sgn(Fxd∗).

B. Controller Design

Consider the following wheel torque T = FxdR+ωIw[( ˙vw/vw)+ (∂f0/∂λ)−1(1/(1− |λ|))( ˙Fxd− ς)], where ζ is to be determined. Then, the closed-loop dynamics of the slip ratio is ˙λ = (∂f0/∂λ)−1 (A0et+ ˙Fxd− ς). Note that (∂f0/∂λ) is always positive. If |λ| exceeds a predefined threshold λ∗, the controller should suppress the slip ratio first. Hence, we choose

ζ = ˙Fxd+|A0| (|ea|+EμFz0) sgn(λ)+κ1

∂f0

∂λλ, if|λ|>λ ∗ ₍₈₎ where κ1> 0 is a constant. Under control law (8), we can show that the closed-loop slip ratio dynamics satisfies λ ˙λ≤ −κ1λ2< 0 for

|λ| > λ∗_{. In other words, λ will converge toward zero exponentially} with a time constant 1/κ1.

For|λ| ≤ λ∗, the proposed controller performs the force tracking task. Define the force tracking error et= Fx− Fxd. Then, straight-forward computation leads to

˙et= (1 + Δ) A0et+ Δ ˙Fxd+

∂f

∂FzFz˙ − (1 + Δ) ς. (9) Define ea= ˆFx−Fxd= et−eμFz0. Then, consider the following Lyapunov function: V = (1/2)e2

a+VE. From (3) and (9), we obtain ˙

V = ea(A0ea−ς + ηFz0)+ ˙VE. For any given constant βt, 0 < βt< βE, if we choose ς = κea+ηFz0, where κ > (βt/2)+A0, then

˙ V = (A0− κ)e2a+ ˙VE<− βt 2e 2 a − βEVE<−βt ₁ 2e 2 a+ VE =−βtV. Therefore, ea converges to zero exponentially with a time constant 1/βt. Since eμis bounded, we conclude that et= ea+ eμFz0is also bounded. Furthermore, we have|et| ≤ εFz0in the steady state.

V. SIMULATIONS A. Setting

We assume that the proposed traction force control scheme is implemented in each rear wheel of a rear-wheel-drive vehicle. A 14-degree-of-freedom nonlinear vehicle model [26] is used to simulate the vehicle dynamics. Let vxand axbe the longitudinal velocity and acceleration of the vehicle’s center of gravity. Since the vehicle is moving forward without cornering, we have vx= vw and ax= ˙vw. The radius and the moment of inertia of the wheel are R = 0.3 m and Iw= 2.03 N· m2_{, respectively. The magic formula with parameters} in [24] is used to simulate tire behavior. The nominal tire model is f0(λ) = fD(λ, μp0, Fz0), where μp0= 0.8, Fz0= 4263 N, and Cx= 111169 N.

TABLE I DESIGNPARAMETERS

In practice, the uncertainty bounds ¯Δ and Γ should be determined from experimental data. In the simulations, they are obtained by numerically searching and comparing the values of the magic formula and the Dugoff model over presumed ranges of the dependent vari-ables. By definition, λ∈ [−1, 1], and we assume that μp∈ [0.2, 1], which represents the road conditions from icy roads to dry asphalt roads. In addition, we assume Fz∈ [1200 N, 7400 N], which corre-sponds to more than 1g longitudinal acceleration. The result suggests that ¯Δ = 32 and Γ = 0.1858. Other design parameters are listed in Table I. The initial velocity of the vehicle is vx(0) = 80 km/h, and the initial conditions of the estimator are ˆμ(0) = 0, φ(0) = 0, and ˆ

ω(0) = ˆωf(0) = ω(0). In addition, zero-mean Gaussian noises are added to the measurements of vx, ax, and ω such that the signal-to-noise ratios of all measurements are around 20 dB. We also assume that the tire–road friction coefficient suddenly changes. Let μp(t) = 0.9 for 0≤ t < 10 s, μp(t) = 0.2 for 10≤ t < 20 s, and μp(t) = 0.5 for 20≤ t ≤ 30 s.

To compare estimation performance with other tire force estimators, we also implement the estimator in [15]

˙ˆ ω∗= 1 Iw T− ˆFx∗R _ˆ Fx∗= 1 R[T− Iw( ˙ω + β ∗_(ω_{− ˆω}∗_{)] (10)}

where β∗ is the rate of convergence and is set to be equal to βE. ˙

ω in (10) is approximated by the backward difference method with a sampling time of 0.001 s. To alleviate high-frequency noise, ω is filtered by a second-order Butterworth low-pass filter with a cutoff frequency of 100 rad/s before (10) is applied.

B. Simulation Results

Let F_xd∗ = 900 N and λ(0) = 0. Then, the desired traction force is Fxd= F_xd∗ for μp= 0.9 and 0.5. However, Fxd∗ exceeds the maximum tire force for μp= 0.2; therefore, Fxd reduces to about 790 N for μp= 0.2. Since the two rear wheels are identical, only the results of the rear right wheel are shown in Fig. 3. The absolute tracking error|et| and the absolute estimation error |eμ|Fz0 is less than 5 N for all t except during the rapidly decayed transient caused by sudden changes of road conditions. This is expected since|et| ≈ |eμ|Fz0≤ εFz0= 21.32 N. On the other hand, ˆFx∗ given by (10) follows Fx robustly; however, the estimate is noisy due to the direct differentiation of ω. Stronger filtering of ω reduces the noise level but introduces additional phase lag. Moreover, there is no guaranteed error bound for the estimator (10). Fxd is also applied to the “ideal” slip-ratio-based control system, i.e., the actual slip ratio is exactly the same as the desired slip ratio, which is converted from Fxd based on the Dugoff tire model with exact tire–road friction coefficient and longitudinal stiffness. The resulting traction force is denoted by Fλ

x in Fig. 3(a). Due to the nonparametric tire model uncertainties, significant errors exist between Fλ

x and Fxd. For μp= 0.2, Fxdstill exceeds the maximum value of the corresponding Dugoff model; therefore, the desired slip ratio is set to 1 to attain the largest traction force (according to the Dugoff model). This is undesirable in practical applications.

Next, we apply F_xd∗(t) = Fxd(t) = 600 cos(2t)N to the proposed control system to demonstrate its tracking performance for both the accelerating and the braking cases. In addition, the initial slip ratio is set to 0.3, whereas λ∗= 0.2. The results of the rear right wheel are

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Fig. 3. (a) Force command from the upper controller (F_xd∗): Dashed line; desired traction force (Fxd): Solid line; estimated traction force by the proposed estimator ( ˆFx): Square marker; estimated traction force by (10) ( ˆFx∗): Dotted line; traction force attained by the proposed controller (Fx): Dash-dot line; traction force attained by the “ideal” slip-ratio-based controller (Fxλ): Diamond marker. (b) Tracking error (et): Solid line; estimation error (eμFz0): Dotted line; auxiliary error (ea): Dash-dot line.

Fig. 4. (a) Tracking error (et): Solid line; estimation error by the proposed estimator (eμFz0): Dash-dot line; auxiliary error (ea): Dashed line; estimation error by (10) (e∗= Fx_{− ˆ}F_x∗): Dotted line. (b) Slip ratio for the first 5 s.

shown in Fig. 4. The slip ratio converges to the stable region rapidly, and despite sudden changes of road conditions, the maximum relative tracking error is less than 4% on the icy road (μp= 0.2). On the other hand, ˆFx∗is robust with respect to road conditions; however, the estimate is noisy.

VI. CONCLUSION

Traction force control is usually accomplished by means of slip ratio control because the traction force is immeasurable. However,

perfect slip ratio control does not guarantee achievement of desired traction forces due to tire model uncertainties. To solve this problem, this paper proposed a direct control scheme that is based on robust estimates of traction forces. The desired traction force can be attained, even under suddenly changing road conditions, as well as parametric and nonparametric tire model uncertainties. In addition, excessive slip ratios were suppressed. Simulations verified the robust and satisfactory performance of the proposed estimator and controller. Integrating the proposed control scheme into an upper level vehicle-control system will be the future research topic.

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Sum of Discrete i.i.d. Random Variables and Its Application to Cooperative Spectrum Sensing

Chulhee Jang, Student Member, IEEE, and Jae Hong Lee, Fellow, IEEE

Abstract—In this paper, we derive the probability mass function (PMF) and the cumulative mass function (CMF) of the sum of discrete indepen-dent and iindepen-dentically distributed random variables. As an application of the PMF and CMF, we analyze the missed-detection, false-alarm, and overall error probabilities of cooperative spectrum sensing with any numbers of quantization levels and any quantization thresholds in a closed form. Computer simulation results match the analysis perfectly. In addition, centralized and distributed threshold selections are discussed.

Index Terms—Cooperative spectrum sensing, discrete random variable, sum of independent and identically distributed (i.i.d.) random variables.

I. INTRODUCTION

A theory of random variable is important for the design and per-formance analysis of wireless communication systems [1]. In par-ticular, the statics of the sum of several independent and identically distributed (i.i.d.) random variables are involved in many applications [2]. However, although the distribution of the sum of continuous i.i.d. random variables is widely investigated, e.g., [2] and [3], that of discrete i.i.d. random variables is not fully investigated. As the discrete random variables have many applications such as discrete memoryless networks [4] and feedback systems [5], the distribution of the sum of discrete i.i.d. random variables can also have many applications.

In a cognitive radio (CR) network with multiple secondary users (SUs), cooperative spectrum sensing is applied to detect the primary user (PU) [6]. Compared with the spectrum sensing with one SU, the cooperative spectrum sensing with multiple SUs achieves a lower overall error probability [7]–[11]. In the cooperative spectrum sensing, multiple SUs observe the signal from the PU, and the observed signals at the SUs are combined at the fusion center. Hard and soft combinations are the typical combination strategies. The whole observed signal is combined in the soft combination, whereas two-level quantized information (1-bit information) is combined in the hard combination. These two strategies have a tradeoff between com-plexity and performance. For a better tradeoff, the combination of

Manuscript received April 22, 2012; revised September 2, 2012; accepted November 12, 2012. Date of publication November 21, 2012; date of current version March 13, 2013. This work was supported by the National Research Foundation of Korea funded by the Korea government (Ministry of Education, Science and Technology) under Grant 2012-0000919 and Grant 2012-0005692. The review of this paper was coordinated by Prof. B. Hamdaoui.

The authors are with the Department of Electrical Engineering and the Institute of New Media and Communications, Seoul National University, Seoul 151-742, Korea (e-mail: jangch@snu.ac.kr; jhlee@snu.ac.kr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.