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This paper proposes the new speed control method of electric vehicles by generating the longitudinal speed pattern for improving the ride comfort against the longitudinal acceleration/deceleration. Since the longitudinal acceleration/deceleration of EVs causes the discomfort of passengers, reducing such bad effect of acceleration/deceleration acting on passengers is very important for not only the ride comfort of passengers but also vehicle running safety. The proposed method is applying the general optimal control theory and it generates the speed pattern for improving the passenger ride quality based on evaluating the variations of acceleration and the jerk which is time derivative of the acceleration. The effectiveness of the present method is demonstrated through numerical experiments compared with some conventional methods.

In recent years, with the wide spread of internal-combustion engine vehicles (ICEVs) all over the world, the environment and energy problems are going severely. Therefore, the development of next-generation vehicles such as hybrid vehicles (HVs) and electric vehicles (EVs) has been focused. Especially, EVs have attracted great interests as a powerful solution against the problems. EVs are automobiles which are propelled by electric motors, using electrical energy stored in batteries or another energy storage device. Electric motors have several advantages over ICEs [

Energy efficient. Electric motors convert 75% of the chemical energy from the batteries to power the wheels―ICEs only convert 20% of the energy stored in gasoline.

Environmentally friendly. EVs emit no tailpipe pollutants, although the power plant producing the electricity may emit them. Electricity from nuclear-, hydro-, solar-, wind-powered plants causes no air pollutants.

The input/output response is faster than for gasoline/diesel engines. It is said that the motor torque response is 2 orders of magnitude faster than that of the engine. E.g., if engine torque response costs 500 ms, the response time of motor toque will be 5 ms.

The torque generated in the wheels can be detected relatively accurately. For engine, the output torque varies along with the temperature and revolutions, even it has high-nonlinearity. Consequently, the value of torque is too difficult to be measured accurately. However, the value of motor torque is surveyed easily and accurately from the view of current control.

The motor can be made small enough, then the vehicles can be made smaller by using multiple motors placed closer to the wheels. The drive wheels can be controlled fully and independently. E.g., it becomes easily achievable to control the differences of driving force developed between the left and right wheel.

From these good points of EVs, we can realize the superior running of vehicle with the good ride quality by using appropriate speed control. Such superior speed control is very useful for autonomous vehicle and some types of autonomous vehicle can also be applied, for example, PRT (Personal Rapid Transit) [

From the point of view of preventing a motion sickness and an accident, the ride comfort of the car is also very important [

As well as the vertical vibration, it is important to reduce the influence on lateral speed change for improving overall ride quality. However, unfortunately, there are few studies on ride discomfort due to longitudinal acceleration/dece- leration. Therefore, in this paper, we develop the speed control method by generating the longitudinal speed pattern using the jerk which is time derivative of the acceleration and the acceleration as the evaluation index, for improving the ride comfort against the longitudinal acceleration/deceleration. The method is applying the general optimal control theory and also based on the techniques proposed in [

The rest of this paper is organized as follows: Section 2 states the evaluation of ride comfort. In Section 3, the proposed generation method of speed pattern is presented. The extended proposed method is shown and simulation results are discussed in Section 4. Finally, we end the paper with some conclusions and future work in Section 5.

There are various factors which have an influence on ride quality, but the index of the ride quality depends on individuals. About the railroad carriage, various research about the evaluation of the ride comfort with respect to the frequency of vibrations in vertical and horizontal directions have been reported in [

However, most of these deal with the evaluation of ride quality with respect to the sustained vibration of the steady run or vibration in the vertical direction. In the case of EVs, it is important to evaluate the ride comfort with respect to lateral speed change. In relation to this point, there are [

d ( t ) = β 0 + β 1 a p + ( t ) + β 2 a p − ( t ) + β 3 j r + ( t ) + β 4 j r − ( t ) + ϵ ( t ) (1)

where parameters in (1) are defined in

a p + ( t ) = { max t ∈ T a ( t ) , | max t ∈ T a ( t ) | ≥ | min t ∈ T a ( t ) | 0 , | max t ∈ T a ( t ) | < | min t ∈ T a ( t ) | (2)

a p − ( t ) = { 0 , | max t ∈ T a ( t ) | ≥ | min t ∈ T a ( t ) | min t ∈ T a ( t ) , | max t ∈ T a ( t ) | < | min t ∈ T a ( t ) | (3)

j r + ( t ) = { 1 3 ∫ t − 3 t j 2 ( τ ) d τ , j ( T ) ≥ 0 0 , j ( T ) < 0 (4)

ride comfort index at time t | |
---|---|

peak value of acceleration in T = (t − 3,t) (3 seconds just before time t) | |

peak absolute value of deceleration in T | |

average value of jerk in T | |

effective value of jerk in the case of positive value of j(T) | |

effective value of jerk in the case of negative value of j(T) | |

error term | |

constant term | |

partial regression coefficient |

j r − ( t ) = { 0 , j ( T ) ≥ 0 1 3 ∫ t − 3 t j 2 ( τ ) d τ , j ( T ) < 0 (5)

Unfortunately, it is not possible to derive the speed pattern by using d(t) directly since the d(t) is ride comfort index at the specific time t derived based on the acceleration and the jerk in the real time. It can’t show overall evaluation. Then we need to consider another index for speed control for overall ride comfort. In [

Therefore, it is important to suppress both of the acceleration/deceleration and the jerk to improve the ride quality and we construct the speed control method on these facts.

It’s important for speed control method to consider the speed pattern. The speed pattern is defined as the ideal speed plan to satisfy various demands/limits for ride comfort, energy efficiency at acceleration, position, time and so on. In [

J 0 = ∫ 0 t f ( d a d t ) 2 d t (6)

where a is the acceleration and t f is the terminal time, and where the state- space vehicle model is as

( v ˙ a ˙ ) = ( 0 1 0 0 ) ( v a ) + ( 0 1 ) u (7)

However, the acceleration, which is one of the most important factor influenced to the ride quality as I mentioned in Section 2, does not include directly in this evaluation function. In addition, a vehicle position is not included in this model. It’s important factor for realizing the automatic driving in the near future. From these facts, the state space model and the evaluation function in this paper is defined as follows.

( x ˙ v ˙ a ˙ ) = ( 0 1 0 0 0 1 0 0 0 ) ( x v a ) + ( 0 0 1 ) u (8)

J 1 = ∫ 0 t f [ ( d a d t ) 2 + ( q a ) 2 ] d t (9)

where (8) is written as x ˙ = A x + b u with;

x = ( x v a ) , A = ( 0 1 0 0 0 1 0 0 0 ) and B = ( 0 0 1 ) .

It’s also possible to control the vehicle position to add x in the state space variables for automatic driving. Furthermore, we can derive the speed pattern which emphasized the ride comfort against the acceleration and deceleration by adding the weighted a to evaluation function. (q is the weighting constant.)

Then, by using the generalized optimal control theory, we can derive the following Hamiltonian H from (8) and (9).

H = 1 2 ( u 2 + a 2 ) + λ T ( A x + B u ) = 1 2 ( u 2 + x T Q x ) + λ T ( A x + B u ) (10)

where λ is the Lagrange multiplier and where

Q = ( 0 0 0 0 0 0 0 0 q 2 ) (11)

The solution minimized J 1 is obtained from ∂ H ∂ u = 0 as

u = − b T λ (12)

From x ˙ = ∂ H ∂ λ and λ ˙ = − ∂ H ∂ x , we can obtain

( x ˙ λ ˙ ) = ( A − B B T − Q − A T ) ( x λ ) (13)

This equation is expressed as

( x ˙ v ˙ a ˙ λ ˙ 1 λ ˙ 2 λ ˙ 3 ) = ( 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 − q 2 0 − 1 0 ) ( x v a λ 1 λ 2 λ 3 ) (14)

From (14),

λ ˙ 2 = − C ″ (15)

is obtained. Where C ″ is given constant. Furthermore, the following state space equation is derived.

( a ˙ λ ˙ 2 λ ˙ 3 ) = ( 0 0 − 1 0 0 0 − q 2 − 1 0 ) ( a λ 2 λ 3 ) + ( 0 − C ″ 0 ) (16)

The time response of this system expressed as (16) is

x 1 ( t ) = e A 1 t x 1 ( 0 ) + ∫ 0 t e A 1 ( t − τ ) B 1 u 1 ( τ ) d τ (17)

where

x 1 = ( a ˙ λ ˙ 2 λ ˙ 3 ) , A 1 = ( 0 0 − 1 0 0 0 − q 2 − 1 0 ) , B 1 = ( 0 − C ″ 0 ) and u 1 = 1

Then, from (17), we can derive x 1 ( t ) as follows.

( a λ 2 λ 3 ) = 1 2 ( ( e q t + e − q t ) 1 q 2 ( − 2 + e q t + e − q t ) 1 q ( − e q t + e − q t ) 0 1 0 q ( − e q t + e − q t ) 1 q ( − e q t + e − q t ) ( e q t + e − q t ) ) × ( a ( 0 ) λ 2 ( 0 ) λ 3 ( 0 ) ) + − C ″ 2 ( 1 q 2 ( − 2 t + 1 q e q t − 1 q e − q t ) 2 t 1 q ( 2 q − 1 q e q t − 1 q e q t ) ) (18)

Also a(t) is obtained as follows.

a ( t ) = ( 1 2 a ( 0 ) + 1 2 q 2 λ 2 ( 0 ) − 1 2 q λ 3 ( 0 ) − C ″ 2 q 3 ) e q t + ( 1 2 a ( 0 ) + 1 2 q 2 λ 2 ( 0 ) + 1 2 q λ 3 ( 0 ) + C ″ 2 q 3 ) e − q t + C ″ q 2 t + λ 2 ( 0 ) q 2 (19)

where q , a ( 0 ) , λ 2 ( 0 ) , λ 3 ( 0 ) , C ″ are given constants. Expressing constant terms in (19) as C ″ j ( j = 0 , 1 , 2 , 3 ) , (19) is described as follows.

a ( t ) = C ″ 0 e q t + C ″ 1 e − q t + C ″ 2 t + C ″ 3 (20)

Finally we can derive the following speed pattern after deformation of equations with state space equation: x ˙ = ∂ H ∂ λ , co-state space equation: λ ˙ = − ∂ H ∂ x and stationarity equation: 0 = − ∂ H ∂ u .

x ( t ) = C 0 e q t + C 1 e − q t + C 2 t 3 + C 3 t 2 + C 4 t + C 5 v ( t ) = q C 0 e q t − q C 1 e − q t + 3 C 2 t 2 + 2 C 3 t + C 4 t a ( t ) = q 2 C 0 e q t + q 2 C 1 e − q t + 6 C 2 t + 2 C 3 (21)

where C j ( j = 0 , 1 , 2 , 3 , 4 , 5 ) are constant coefficients. These values can be decided from initial and terminal conditions of t, x, v and a.

This speed pattern includes the method by [

One example is shown. Let’s consider the following initial and terminal conditions.

{ t 0 = 0 : x 0 = 0 , v 0 = 10 , a 0 = 1 t f = 10 : x f = 100 , v f = 0 , a f = 0 (22)

These conditions show the situation that the vehicle runs from a state running in speed 10 m/s and acceleration 1 m / s 2 to the 100 m spot ten seconds later, and to stop.

In

both methods (the proposed and the method in [

The proposed method is extended to the flexible generation method which can cope with the change of terminal conditions in the way of the run for practical use. For example, the method is extended to be able to deal with the situation that it is necessary to shorten a stop spot by some kind of factors such as other vehicles getting into the way. In such situation, the speed pattern should be re-generated flexibly in real-time accordance with the change of conditions.

As the result of many simulations, we see that the remaining run time after the pattern re-generated greatly influenced the quality of ride comfort. Since the change of the run time brings the sudden change of the jerk. Therefore, the following evaluation function ( J 2 ) is introduced to decide appropriate remaining run time.

J 2 = r x + s y (23)

where x is the absolute value of the difference of the value of the jerk just before and after the pattern re-generated, y is the absolute value of the difference of the derivative value of the jerk just before and after the pattern change, and r, s are constant weights. Because it was a problem that the jerk suddenly changes in before and after the pattern re-generated, appropriate values of the weight q of J 1 in (9) and the remaining run time are decided by using this evaluation function J 2 in (23) for the change of jerk consecutively and smoothly as much as possible. A flow of rough processing of this method is shown in

In addition, a search range at remaining run time is set. For example, the search range is set as not exceeding the whole running time set beforehand. If the running distance becomes long, remaining run time is able to be increased and search the best values of q and the remaining run time in the enlarged range.

Then, it becomes possible to derive the speed pattern with best ride comfort, which minimize the total d(t) in whole running time, against the change of terminal conditions.

Let’s confirm the effectiveness of the proposed extended method in some simulations.

Firstly, let’s consider the situation that the stop spot is shortened after starting off. The first condition is as follows.

{ t 0 = 0 : x 0 = 0 , v 0 = 0 , a 0 = 0 t f = 10 : x f = 100 , v f = 0 , a f = 0 (24)

Then, at running 60 m spot, the stop spot is shortened from 100 m to 70 m as follows.

{ t 0 = 0 : x 0 = 60 , v 0 = v s , a 0 = a s t f = [ search ] : x f = 70 , v f = 0 , a f = 0 (25)

where v s is final speed value before re-generate the speed pattern and a s is final acceleration value before re-generate the pattern.

The results by the proposed method are shown as

stop spot at 5.61 s. But it’s natural response due to the sudden shortening of stop distance for safety. Therefore, we can say that the proposed method shows the good performance totally.

Let’s consider the situation that the stop spot is lengthened after starting off.

The first conditions as follows.

{ t 0 = 0 : x 0 = 0 , v 0 = 0 , a 0 = 0 t f = 10 : x f = 100 , v f = 0 , a f = 0 (26)

Then, at running 60 m spot, the stop spot is lengthened from 100 m to 130 m as follows.

{ t 0 = 0 : x 0 = 60 , v 0 = v s , a 0 = a s t f = [ search ] : x f = 130 , v f = 0 , a f = 0 (27)

The results by the proposed method are shown in

From

In this simulation, let’s consider the situation that the stop spot is changed two times.

The first conditions as follows.

{ t 0 = 0 : x 0 = 0 , v 0 = 0 , a 0 = 0 t f = 10 : x f = 100 , v f = 0 , a f = 0 (28)

Firstly, at running 60 m spot, the stop spot is lengthened from 100 m to 130 m as follows.

{ t 0 = 0 : x 0 = 60 , v 0 = v s , a 0 = a s t f = [ search ] : x f = 130 , v f = 0 , a f = 0 (29)

Then, at running 100 m spot, the stop spot is shortened from 130 m to 120 m as follows.

{ t 0 = 0 : x 0 = 100 , v 0 = v s , a 0 = a s t f = [ search ] : x f = 120 , v f = 0 , a f = 0 (30)

The results by the proposed extended method are shown in

From this figure, we can see that the method can cope with the situation of change the stop spot two time.

From

In this section, the proposed extended method is compared with the conventional method [

{ t 0 = 0 : x 0 = 0 , v 0 = 0 , a 0 = 0 t f = 10 : x f = 100 , v f = 0 , a f = 0 (31)

Then, at running 40 m spot, the stop spot is shortened from 100 m to 90 m as follows.

{ t 0 = 0 : x 0 = 40 , v 0 = v s , a 0 = a s t f = [ search ] : x f = 90 , v f = 0 , a f = 0 (32)

The result of comfort index d(t) is shown as

In this paper, we have proposed the speed control method based on general optimal control theory for improving the passenger ride comfort of electric vehicles.

The method is applying the general optimal control theory and also based on the conventional techniques. Furthermore, it includes the conventional method and extends the flexible speed-pattern-generation method which can cope with the change of terminal conditions in the way of the run. The method also aims to contribute to improving the beginner driver’s driving skill from the viewpoint of passenger’s comfortability by showing the ideal running pattern and checking the driving. The proposed method can expect to be also useful for the run which emphasized ride comfort of the automatic operation car which would come to practical use in the future. It can also be applied to some types of autonomous vehicle, transit system, for example, PRT, and so on.

In future work, the suitability of the method must be studied not only the longitudinal run but also for overall driving situations. Furthermore, it is necessary to verify the effectiveness by actual experiments. Let’s confirm the effectiveness.

Fuse, H., Kawabe, T. and Kawamoto, M. (2017) Speed Control Method of Electric Vehicle for Improving Passenger Ride Quality. Intelligent Control and Automation, 8, 29-43. https://doi.org/10.4236/ica.2017.81003