Supervisory Control

After detecting road environment and determining the operation modes, we estimate the desired velocity by a two-stages approach. The desired acceleration of each operation mode is determined in the first stage, and then is converted to the desired velocity in the second stage.

This velocity is then issued to vehicle-body control. We here use a desired velocity, instead of acceleration, due to that the subject-vehicle’s velocity can be measured directly. Moreover, high-frequency noise problems from sensed velocity differential can be avoided.

In the first stage, we adopt the sliding mode control (SMC) technique to design the corresponding control laws for various driving modes. A proper surface S is defined as the function of the system states, and the sliding manifold S=0 defined on the closed-loop system is asymptotically stable. To satisfy the reachable and sliding condition, we choose S such that

0

SS< while S ≠ 0 (S =C S( ) is a continuous function of S.). One conventional method of the sliding surface design is as [47]

1 where n is the relative degree of the system, λ is the chosen sliding surface gain, and e presents system error which is desired to drive to zero. In this way the state trajectory is forced to a stable manifold state (S=0) in a finite time interval, and then slides toward equilibrium state. We define the control law as the first derivative of the surface S.

In the CC mode, the desired acceleration is determined such that the subject-vehicle can track any assumed velocity under comfort situation. We define the velocity tracking error as eV = Vf − Vdes . The sliding surface is chosen as SCC = eV and the control law is defined as

CC CC CC

S = −K S (4-19) with constant parameter KCC > 0 for SCC to approach zero. This control law satisfies globally asymptotical stable and S S_CC_CC <0 criteria. We note that in SMC-related literatures the control law always includes a discontinuous sign function; for example, C(SCC) = −KCCSCC + k0sgn(SCC). This will result in unexpected chattering phenomena and high-frequency noise.

Besides, the proposed continuous function in (4-19) is a feasible choice for implementation due to its simplicity. Furthermore, we constrain the acceleration command within one afmax for ride-comfort consideration, especially when the initial value of eV is large. Therefore, the sliding surface is modified as

max and the desired acceleration is

max The jerk in (4-23) can be neglected in practice since it may constrain the achievement of the desired acceleration. So we have aCC = ± afmax. The proposed control laws in (4-21) and (4-23) are simple, and contains vehicle velocity only. Therefore, the implementation and the realization of ride comfort are achievable.

In the ACC mode, we propose the control law based on the desired acceleration such that a safety following-distance from the preceding-vehicle is maintained. The constant headway time policy is commonly suggested as a safe practice for human drivers and is frequently used in ACC systems design. In this policy, the desired inter-vehicle range is Rdes = σVf + L, and the spacing error between the desired and relative headway distances is defined as

( )

R des f

e = −R R = −R σV +L , (4-24) where the relative distance R = Xp − Xf .

The sliding surface is chosen as

max with KACC > 0, the desired acceleration can be derived as

1( )

ACC ACC R

a K e R

=σ +  . (4-27)

For |a_f +e_R|≥a_fmax, the sliding surface becomes

max

ACC f f

S = − ±a a , (4-28) and the desired acceleration is

max

Remark 1: For ride-quality consideration, the estimated acceleration of the subject-vehicle in conventional approaches is constrained. If |aACC| ≥ afmax , then aACC = ±afmax. However, this can not be guaranteed for the real acceleration of a vehicle. We use the saturation function of the sliding surface to constrain the actual acceleration from reaching the desired value. In other words, our control law can limit the real acceleration of controlled vehicle. With respect to afmax , the value of 2.0 m/s² and -3.0 m/s² can be set for acceleration and deceleration, respectively, in accordance with the specification [43].

As far as the supervisory control is concerned, the transport lag in the servo-loop should be analyzed in the performance of ACC design. The servo-loop can be approximated by a first-order lag system where ai is the subject-vehicle’s actual acceleration, ai,des is the desired acceleration from the ACC mode design (4-27), and τlag is the servo-loop time constant. The vehicle’s velocity and position can be obtained through integration.

By considering a platoon of identical ACC-controlled vehicles running on highway in a string, the string stability refers to a property in which spacing errors are guaranteed not to amplify as they propagate toward the tail of the platoon [2]. The errors eRi and eRi-1 are denoted as the spacing deviation from the desired safety-distance as in (4-24) for the subject-vehicle and the preceding-vehicle, respectively, and the objective for the string stability can be stated as follows [2]:

1 1 1 1

After considerable algebraic computation, the string stability conditions of (4-31) can be guaranteed as long as

2 _lag

σ > τ (4-34) where τlag is the time constant of the servo-loop as in (4-30).

In other words, the string stability condition in (4-34) means that the headway time must be chosen above the critical value of 2τlag . The time constant τlag arises due to lag in actuators, the bandwidth of the lower level control, and filtering of the range finder. Typically, experimental work shows that τlag has a value which varies between 0.5 and 1.0 s. In the specification of [43], the spacing policy allows a headway time between 1.0 and 2.2 s. At highway the vehicle velocity can be in the range of 60–100 km/h. In our system, the headway time is set as 1.0 s for ACC mode and thus the spacing between cars can be translated into 17–28 m which is adequate to achieve the safe car-following.

Besides, for consideration of increasing highway capacity, the subject-vehicle may keep a fixed spacing from the preceding-vehicle, that is, σ = 0 and Rdes = L. Therefore, the desired acceleration can not be derived according to the sliding surface in (4-25) and the control law in (4-26). We choose the following sliding surface defined on the error and error-variation as

max The stability of the sliding surface in (4-36) is examined by setting SFT =0, and the error dynamics eR(t) =e^-λt is asymptotically stable for all positive λ. By differentiating (4-36), the acceleration of the subject-vehicle can be obtained by S_FT = +R λR. We then choose the control law as

FT FT FT

S = −K S , (4-37) where KFT > 0 is to guarantee the desired condition and asymptotic stability. By (4-36) and (4-37), the desired acceleration is derived as

( )

FT FT R R P

a =K e +λe +a + λR. (4-38)

The sliding surface is mode, only the fixed-distance tracking for two cars is discussed instead of the control for a platoon. Equation (4-37) ensures that the vehicle states converge to the sliding surface SFT , and the subject-vehicle can track the preceding-vehicle with a constant spacing. To verify the asymptotic stability for the car-following controller (4-33), a necessary condition [5] for such stability is where Vp(s) and Vf (s) are the Laplace transform of the velocity of the preceding-vehicle and the subject-vehicle, respectively. To analyze the stability of the system, the following transfer function can be obtained from (4-38)

2 The parameters KFT and λ are chosen so that the system can satisfy the condition (4-42) for asymptotic stability. By choosing

2

the control law (4-38) becomes

2 2

FT P n R n R

a =a + ξωe +ω e , (4-43) where the gain ξ can be regarded as a damping ratio and set at 1 for critical damping, and ωn

is the bandwidth of the controller.

That is, the control law in (4-43) is to ensure not only the spacing error eR to converge to zero, but also the spacing errors not to amplify down a two-car platoon [2, 5].

In the stop-and-go mode, the subject-vehicle has to keep spacing from the preceding-vehicle with a quicker response than that of ACC mode. Therefore, the desired control law in (4-27) which is derived from the sliding surface in (4-25) seems to be inadequate. We choose the following sliding surface defined on the error and error-variation

as The stability of the sliding surface in (4-45) is examined by setting SS&G =0, and the error dynamics is eR(t) =e^-λt − R^_des/λ . Thus, SS&G is stable but not asymptotically since

( ) ( ) /

R des

e t → R t λ as t → ∞ which is bounded and can be reduced by choosing larger positive value of λ. Moreover, for steady-state car-following, the subject-vehicle maintains a constant speed or stops as the same as the preceding-vehicle such that R_des =σa_f =0; the spacing error will approach zero, that is, eR → 0. We then choose the control law as

& & &

S G S G S G

S = −K S (4-46) where KS&G > 0 is to guarantee the desired condition and asymptotic stability.

By virtue of (4-45) and (4-46), the desired acceleration is derived as

& & We note that the jerk of the subject vehicle in (4-49) is still negligible (aS&G = ±afmax ). The acceleration of the preceding vehicle can be accessed by the double differential of the relative distance, i.e., a_p = +R a _f .

In the stop-and-go mode, the vehicle velocity is usually within 40 km/h. As a result, the headway time is chosen as 2.0 s such that the desired inter-vehicle distance will be sufficient to achieve the smooth response in maneuvering throttle and brake control.

In the second stage, the conversion from the desired acceleration into the desired velocity takes the form

( )

C f des t f C

V =a −k V −V , (4-50)

where kt > 0 is chosen to avoid sudden velocity-change from sensor noise in the measurement of spacing and vehicle velocity. To achieve real-time conversion, the differential equation in (4-50) is approximated by Euler’s method,

( 1) (1 ) ( ) ( ( ) ( )), 0

C s t C s t f f des

V k+ = −T k V k +T k V k +a k k≥ (4-51) The conversion (4-51) is able to adjust the aggressiveness of desired velocity from the desired acceleration by depending on the current velocity. A smooth reference trajectory of VC can be generated. Note that the set-point of VC is determined from the current velocity Vf and prohibited by the larger value of kt , therefore the exceeding error between VC and Vf can be prevented.

The autonomous switching scheme chooses the desired mode by min-operation

/ & car-following situation including both ACC and stop-and-go mode. The subject-vehicle drives in CC mode in general. It will automatically switch to ACC mode or stop-and-go mode once a valid-target vehicle is detected. This concept is identical to the Gipps model (see Appendix C) which is demonstrated to be capable of longitudinal human driving characteristics [48]. The Gipps model (C.1) determines the subject-vehicle’s velocity by two halves of the equation:

the first half tends to accelerate to the desired velocity, while the second half tends to keep a safe distance from the preceding-vehicle.