CN109307064B - Locking clutch slip rotating speed control method for vehicles under different starting working conditions - Google Patents
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H61/00—Control functions within control units of change-speed- or reversing-gearings for conveying rotary motion ; Control of exclusively fluid gearing, friction gearing, gearings with endless flexible members or other particular types of gearing
- F16H61/02—Control functions within control units of change-speed- or reversing-gearings for conveying rotary motion ; Control of exclusively fluid gearing, friction gearing, gearings with endless flexible members or other particular types of gearing characterised by the signals used
- F16H61/0202—Control functions within control units of change-speed- or reversing-gearings for conveying rotary motion ; Control of exclusively fluid gearing, friction gearing, gearings with endless flexible members or other particular types of gearing characterised by the signals used the signals being electric
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Abstract
The invention discloses a slip rotating speed control method of a locking clutch under different starting working conditions of a vehicle, which comprises the following steps: step one, collecting the opening degree of an accelerator and the actual slip rotating speed, and determining the starting working condition and the starting working condition of a vehicleThe target slip speed thereof; the accelerator opening is divided into a small accelerator opening, a medium accelerator opening, a large accelerator opening and a full accelerator opening; step two, controlling the slip rotating speed of the locking clutch according to the opening degree of the accelerator; when the accelerator opening is at a middle accelerator opening or a large accelerator opening, parameters of a PID controller are adjusted through a fuzzy control model, and three parameters delta K are adjusted through the PID controller p 、ΔK i 、ΔK d The actual slip speed is adjusted.
Description
Technical Field
The invention relates to the field of lockup clutch control, in particular to a lockup clutch slip rotating speed control method used under different starting working conditions of a vehicle.
Background
The existing hydromechanical automatic transmission (AT for short) almost adopts the lockup clutch slip control technology, and under the starting working condition, compared with the traditional mechanical transmission, the automatic transmission vehicle with the AT can enable the automobile to start more stably and enable the engagement of an engine and a transmission system to be softer. However, this advantage is provided when the torque converter is in a purely hydrodynamic condition. However, at the same time, there is a bad consequence, namely a decrease in efficiency due to the pure hydraulic state, which directly affects the economy of the vehicle.
To improve vehicle economy, it is desirable to lock up the torque converter when appropriate. The latch-up allows the vehicle efficiency to be improved, but at the same time causes a decrease in riding comfort, automobile drivability, and the like. Therefore, to balance the contradiction between the two, the slipping technique has been developed.
In the prior art, a slip control mode based on a PID control mode is commonly applied. The control performance of the control mode depends on the adjustment of the actual slip rotation speed, i.e. whether the real-time and ideal target slip rotation speed are consistent. However, the existing slip rotating speed PID control technology is poor in parameter setting, poor in adaptability to operation conditions, high in hysteresis, incapable of meeting the requirement of being adjusted to a target slip rotating speed in real time, and easy to cause phenomena of overlarge deviation from the target slip rotating speed, overheating of a friction plate and the like when the friction plate slides.
Disclosure of Invention
The invention designs and develops a slip speed control method of a lockup clutch under different starting working conditions of a vehicle, and one of the purposes of the invention is to balance the contradiction among riding comfort, automobile operability and automobile economy by adopting a slip technology, combine fuzzy control and PID control, and further optimize the overall performance of the PID control.
The second purpose of the invention is to correct the actual slip rotating speed when the small accelerator opening, the large accelerator opening or the full accelerator opening is carried out, so that the slip rotating speed can be better controlled, and the overall performance of the clutch is further optimized.
The technical scheme provided by the invention is as follows:
a lockup clutch slip speed control method for vehicles under different starting conditions comprises the following steps:
step one, collecting the opening degree of an accelerator and the actual slip rotating speed, and determining the starting working condition of a vehicle and the target slip rotating speed of the vehicle;
the accelerator opening is divided into a small accelerator opening, a medium accelerator opening, a large accelerator opening and a full accelerator opening;
step two, controlling the slip rotating speed of the locking clutch according to the opening degree of the accelerator;
when the accelerator opening is at a middle accelerator opening or a large accelerator opening, parameters of a PID controller are adjusted through a fuzzy control model, and three parameters delta K are adjusted through the PID controller p 、ΔK i 、ΔK d The actual slip speed is adjusted.
Preferably, in the second step, the adjusting the PID controller parameter by the fuzzy control model includes:
setting an initial parameter K regulated by a PID controller p 、K i 、K d Then, determining a sampling period;
the difference e between the target slip rotation speed and the actual slip rotation speed and the slip rotation speed change rate are respectively calculatedΔK for adjusting PID controller parameters p 、ΔK i 、ΔK d Converting into quantization levels in the ambiguity domain; inputting the difference e between the target slip rotating speed and the actual slip rotating speed and the slip rotating speed change rate ec into a fuzzy controller, wherein the output of the fuzzy controller is delta K for adjusting the parameters of the PID controller p 、ΔK i 、ΔK d ;
Three parameters delta K are regulated by PID controller p 、ΔK i 、ΔK d And obtaining an adjusted actual slip speed, calculating the difference between the target slip speed and the adjusted actual slip speed, and if the difference is within the error allowable range, the adjusted actual slip speed is the controlled slip speed.
Preferably, when the accelerator opening is at the large accelerator opening, the actual slip speed adjusted by adjusting the PID controller parameters based on the fuzzy control model is corrected by the following formula to obtain the large accelerator opening corrected slip speed e f_max :
wherein ,
in the formula ,ef E is the actual slip speed f ' is the actual slip speed after adjustment, e t_max The target slip rotation speed is set when the accelerator opening is at a large accelerator opening.
Preferably, in the second step, when the accelerator opening is at the small accelerator opening, the actual slip speed is controlled by the following formula to obtain a small accelerator opening slip speed e f_min :
Wherein F (β) = -0.0013 β 2 +0.0284β+A 1 ;
in the formula ,ef_minA E is the maximum actual slip rotation speed which can be achieved when the accelerator opening is at a small accelerator opening f_minB E is the minimum actual slip rotating speed which can be achieved when the accelerator opening is at the small accelerator opening f E is the actual slip speed t_min Target slip rotation speed beta set when the accelerator opening is at a small accelerator opening minA To control the maximum accelerator opening that can be reached when the accelerator opening is in the small accelerator opening range, β is the accelerator opening, a 1 The value range is 0.97 to 1.05 for the first tested correction constant, A 2 For the second empirical correction constant, the value range is 4.24-4.38, delta is the first correction coefficient, the value range is 1.97-2.13, phi is the second correction coefficient, and the value range is 5.67-5.83.
Preferably, in the second step, when the accelerator opening is at the full accelerator opening, the actual slip speed is controlled by the following formula to obtain the full accelerator opening slip speed e f_all :
wherein ,
in the formula ,ef E is the actual slip speed t_max A target slip rotation speed set when the accelerator opening is at a large accelerator opening,A 3 the third empirical correction constant is in the range of 0.083 to 0.096.
Preferably, A 1 The value is 1.01, A 2 The value is 4.31, A 3 The value is 0.092, delta is 2.06, and psi is 5.75; and
e t_min 800r/min, e t_max 3000r/min.
Preferably, the small accelerator opening degree beta min The value range of (2) is 0 < beta min Less than or equal to 20 percent, the opening beta of the middle throttle mid The range of the value of (2) is 20% < beta mid Less than or equal to 70 percent, the large accelerator opening degree beta max The range of the value of (2) is 70% < beta max <100%。
Preferably, the difference e between the target slip speed and the actual slip speed in the fuzzy control model is divided into 9 quantization levels, the slip speed change rateDivided into 9 quantization levels, the difference e between the target slip speed and the actual slip speed and the slip speed change rate +.>Is divided into { NB, NM, NS, ZO, PS, PM, PB };
ΔK of PID controller parameters p 、ΔK i 、ΔK d The average is divided into 9 quantization levels, and the fuzzy set is divided into { NB, NM, NS, ZO, PS, PM, PB }.
Preferably, the difference e between the target slip speed and the actual slip speed and the slip speed change rateIs of the actual quantization level [ -9,9]Parameter delta K regulated by PID controller i 、ΔK d 、ΔK p Is of the actual quantization level [ -9,9]。
Preferably, the difference e between the target slip rotational speed and the actual slip rotational speed, and the slip rotational speed change rateParameter delta K regulated by PID controller i 、ΔK d The actual discrete domains of (a) are [ -1,1]Parameter delta K regulated by PID controller p The actual discrete domain of (2) is [ -2,2]。
Compared with the prior art, the invention has the following beneficial effects:
1. the invention adopts fuzzy PID control, effectively solves the problem that the actual slip rotating speed cannot be consistent with the ideal slip rotating speed in real time;
2. the existing slip rotating speed PID control technology has poor parameter setting, poor adaptability to operation conditions and larger hysteresis, cannot meet the requirement of real-time adjustment to the target slip rotating speed, is easy to cause the phenomena of overlarge deviation from the target slip rotating speed, overheat of a friction plate and the like when in slip friction, and effectively solves the problems;
3. compared with the traditional PID control, the proportional, integral and differential coefficients of the PID controller are adjusted more flexibly, rapidly and accurately, and the real-time requirement can be better met;
4. according to the invention, when the accelerator opening is in the small accelerator opening, the large accelerator opening or the full accelerator opening, different corrections are carried out on the actual slip rotating speed, so that the slip rotating speed is better controlled.
Drawings
Fig. 1 is a schematic diagram of the operating characteristics of a torque converter according to the present invention.
FIG. 2 is a schematic diagram of the fuzzy PID operating principle of the present invention.
FIG. 3 is a fuzzy PID workflow diagram in accordance with the present invention.
FIG. 4 is a graph of membership functions according to the present invention.
Detailed Description
The present invention is described in further detail below with reference to the drawings to enable those skilled in the art to practice the invention by referring to the description.
The invention provides a slip rotating speed control method of a locking clutch under different starting working conditions of a vehicle, which comprises the following steps:
step one, collecting the opening degree of an accelerator and the actual slip rotating speed, and determining the starting working condition of a vehicle and the target slip rotating speed of the vehicle;
the accelerator opening is divided into a small accelerator opening, a medium accelerator opening, a large accelerator opening and a full accelerator opening;
step two, controlling the slip rotating speed of the closed clutch according to the opening degree of the accelerator;
when the accelerator opening is at a middle accelerator opening or a large accelerator opening, parameters of a PID controller are adjusted through a fuzzy control model, and three parameters delta K are adjusted through the PID controller p 、ΔK i 、ΔK d The actual slip speed is adjusted.
In another embodiment, as shown in FIG. 1, in step two, slip, i.e., incomplete lock-up, i.e., friction between the impeller and stator of the torque converter, but not complete binding, the slip region is primarily determined by the relationship between torque converter efficiency and speed ratio, i represents the speed ratio, i.e., the ratio of torque converter turbine speed to impeller speed, i 0 The speed ratio is the highest efficiency point of the hydraulic torque converter, i 1 The speed ratio is the torque converter coupling operating point at which the torque converter is normally fully locked, and therefore the region between the highest efficiency point and the fully locked point is the most suitable slip region.
As shown in fig. 2, in the whole process of starting the automobile, the engine is subjected to a complex change process from stopping to idling, to medium-speed operation and finally to higher-speed operation, when a driver faces different situations, the requirements on starting time are different, and the specific differences are reflected on different accelerator opening degrees and can be divided into starting under a small accelerator opening degree, starting under a medium accelerator opening degree, starting under a large accelerator opening degree and starting under a full accelerator opening degree; meanwhile, it is particularly pointed out that when the small accelerator opening and the full accelerator opening are used, the engine torsional vibration is particularly severe and is not suitable for the fuzzy PID to carry out slip control, so that the adopted fuzzy PID working principle is suitable for starting working conditions under the middle accelerator and the large accelerator opening.
As shown in fig. 3 and 4, first, the target slip rotation speed is input, and then, the difference between the target slip rotation speed and the actual slip rotation speed, and the slip change rate are calculated. The rotational speed difference and the slip change rate are used as input quantity of the fuzzy controller, and three coefficients K of the PID controller are used P ,K i ,K d Adjusting, and outputting K after fuzzy control p ,K i ,K d The variation ΔK of (a) p ,ΔK i ,ΔK d So that the three coefficients of the PID controller can be adjusted. Then the K after adjustment P ,K i ,K d The actual slip speed will be corrected to be as close as possible to the target slip speed. When the difference value between the target slip rotating speed and the actual slip rotating speed is within the error allowable range, the PID controller does not work any more; and when the difference value is too large, the PID controller can adjust the actual slip rotating speed to generate a new slip rotating speed, so that the new slip rotating speed is compared with the target slip rotating speed again. The adjustments are thus made continuously in an effort to ensure that the actual slip speed is at an optimal condition.
For a common PID controller, the mathematical model formula is that,
wherein u (t) is the system output value, K P For proportional control coefficient, K i For integrating the control coefficient, K d E (t) is the difference between the target slip speed and the actual slip speed for the differential control coefficient,is the slip rotation speed change rate;
for the fuzzy controller, a two-input and three-output structure is adopted, and the difference e between the target slip rotating speed and the actual slip rotating speed and the slip rotating speed change rate are adoptedAs input, three coefficients (proportional coefficient K) P Integral coefficient K i Differential coefficient K d ) As output, i.e. delta K P ,ΔK i ,ΔK d 。
Wherein e= |e f -e t |;
Wherein e is the difference between the actual slip speed and the target slip speed, e f E is the actual slip speed t For the target slip speed, after the input and output variables are determined, three output variables ΔK are then determined P ,ΔK i ,ΔK d Two input variables e andand (5) blurring processing is performed.
Firstly, according to the data of the whole vehicle test, determining a variable e,ΔK i 、ΔK d The actual discrete domain of (1) is [ -1,1]While DeltaK p The actual discrete domain of (2) is [ -2,2]。
Then, according to the actual situation, in order to make the control accuracy sufficient and avoid excessively complicated control rules, the actual quantization levels of the two input variables are determined as { -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and the actual quantization levels of the three output variables are determined as { -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, according to the same principle.
As shown in FIG. 4, empirically, e is summedIs divided into 7 fuzzy states of { Positive Big (PB), median (PN), positive Small (PS), zero (ZO), negative Small (NS), negative Medium (NM), negative Big (NB) }, and then ΔK is also calculated P ,ΔK i ,ΔK d Is divided into { positive large (PB)7 fuzzy states such as median (PN), small Positive (PS), zero (ZO), small Negative (NS), medium Negative (NM), large Negative (NB); the PB and NB subsets select Gaussian membership functions, and the rest subsets select triangular membership functions.
After determining the membership function, the fuzzy control rule needs to be formulated. When the deviation is large, the K of the PID controller should be made P Take a larger value, K i Taking 0, so that the deviation can be eliminated as soon as possible; when the deviation is moderate, the deviation is eliminated and the overshoot is prevented, so K should be made i Take on smaller values, K P Reduce somewhat, and K d The values are appropriate; when the deviation is small, the static error is mainly eliminated, and K is needed to be calculated i Increase and let K P Smaller ones; meanwhile, in order to better adapt to various road conditions, experience of a skilled driver is widely collected, and the actual situation of locking slip control is combined, as shown in tables 1-3, the fuzzy control rule is as follows:
TABLE 1 DeltaK P Fuzzy rule table
TABLE 2 DeltaK i Fuzzy rule table
TABLE 3 DeltaK d Fuzzy rule table
In the slip control process, after the fuzzy language of three output variables is obtained through fuzzy control rules, the intelligent control theory is adoptedThe common area gravity center method carries out anti-blurring treatment on three output variables so as to obtain delta K P ,ΔK i ,ΔK d Is a precise numerical value of (a).
After obtaining the accurate values of the three output variables, the dynamic setting of the PID control parameters can be performed finally, i.e. according to the formula,
K P =K P0 +ΔK P
K i =K i0 +ΔK i
K d =K d0 +ΔK d
wherein ,Ki0 ,K P0 ,K d0 Is the parameter set value of the PID controller; ΔK P ,ΔK i ,ΔK d The three output quantities of the fuzzy controller are three, so that the three parameters of the PID controller can be dynamically adjusted in real time, and the control precision is improved.
In another embodiment, when the accelerator opening is at a large accelerator opening, the actual slip speed adjusted by adjusting the PID controller parameters based on the fuzzy control model is corrected by the following formula to obtain a large accelerator opening corrected slip speed e f_max :
wherein ,
in the formula ,ef E is the actual slip speed f ' is the actual slip speed after adjustment, e t_max The target slip rotation speed is set when the accelerator opening is at a large accelerator opening.
In another embodiment, in the second step, when the accelerator opening is at the small accelerator opening, the actual slip speed is controlled to obtain the small accelerator opening slip speed e by the following formula f_min :
Wherein F (β) = -0.0013 β 2 +0.0284β+A 1 ;
In another embodiment, when the accelerator opening is at full accelerator opening, the actual slip speed is controlled to obtain full accelerator opening slip speed e by the following formula f_all :
wherein ,
in the formula ,ef_minA E is the maximum actual slip rotation speed which can be achieved when the accelerator opening is at a small accelerator opening f_minB E is the minimum actual slip rotating speed which can be achieved when the accelerator opening is at the small accelerator opening f E is the actual slip speed t_min E is a target slip rotation speed set when the accelerator opening is at a small accelerator opening t_max Target slip rotation speed beta set when accelerator opening is at large accelerator opening minA To control the maximum accelerator opening that can be reached when the accelerator opening is in the small accelerator opening range, β is the accelerator opening, a 1 The value range is 0.97 to 1.05 for the first tested correction constant, A 2 For the second empirical correction constant, the value range is 4.24-4.38, A 3 For the third empirical correction constant, the value range is 0.083-0.096, delta is the first correction coefficient, and the value range is 1.97 to 2.13, wherein psi is a second correction coefficient, and the value range is 5.67 to 5.83; as a preference, A 1 The value is 1.01, A 2 The value is 4.31, A 3 The value of delta is 2.06, the value of phi is 5.75 and the value of e is 0.092 t_min 800r/min, e t_max 3000r/min.
In another embodiment, the small throttle opening beta min The value range of (2) is 0 < beta min Less than or equal to 20 percent, and the opening degree beta of the middle throttle mid The range of the value of (2) is 20% < beta mid Less than or equal to 70 percent, and the large accelerator opening degree beta max The range of the value of (2) is 70% < beta max <100%。
Although embodiments of the present invention have been disclosed above, it is not limited to the details and embodiments shown and described, it is well suited to various fields of use for which the invention would be readily apparent to those skilled in the art, and accordingly, the invention is not limited to the specific details and illustrations shown and described herein, without departing from the general concepts defined in the claims and their equivalents.
Claims (9)
1. The lockup clutch slip rotating speed control method for the vehicle under different starting conditions is characterized by comprising the following steps of:
step one, collecting the opening degree of an accelerator and the actual slip rotating speed, and determining the starting working condition of a vehicle and the target slip rotating speed of the vehicle;
the accelerator opening is divided into a small accelerator opening, a medium accelerator opening, a large accelerator opening and a full accelerator opening;
step two, controlling the slip rotating speed of the locking clutch according to the opening degree of the accelerator;
when the accelerator opening is at a middle accelerator opening or a large accelerator opening, parameters of a PID controller are adjusted through a fuzzy control model, and three parameters delta K are adjusted through the PID controller p 、ΔK i 、ΔK d The actual slip rotating speed is adjusted;
in the second step, the process of adjusting the parameters of the PID controller through the fuzzy control model comprises the following steps:
setting an initial parameter K regulated by a PID controller p 、K i 、K d Then, determining a sampling period;
the difference e between the target slip rotation speed and the actual slip rotation speed and the slip rotation speed change rate are respectively calculatedΔK for adjusting PID controller parameters p 、ΔK i 、ΔK d Converting into quantization levels in the ambiguity domain; inputting the difference e between the target slip rotating speed and the actual slip rotating speed and the slip rotating speed change rate ec into a fuzzy controller, wherein the output of the fuzzy controller is delta K for adjusting the parameters of the PID controller p 、ΔK i 、ΔK d ;
Three parameters delta K are regulated by PID controller p 、ΔK i 、ΔK d And obtaining an adjusted actual slip speed, calculating the difference between the target slip speed and the adjusted actual slip speed, and if the difference is within the error allowable range, the adjusted actual slip speed is the controlled slip speed.
2. The lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 1, characterized in that when the accelerator opening is at a large accelerator opening, the actual slip speed adjusted by adjusting the PID controller parameters based on the fuzzy control model is corrected by the following formula, resulting in a large accelerator opening corrected slip speed e f_max :
wherein ,
in the formula ,ef E is the actual slip speed f ' as actual slip rotation after adjustmentSpeed e t_max The target slip rotation speed is set when the accelerator opening is at a large accelerator opening.
3. The lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 1, characterized in that in the second step, when the accelerator opening is at a small accelerator opening, the actual slip speed is controlled to obtain a small accelerator opening slip speed e by the following formula f_min :
Wherein F (β) = -0.0013 β 2 +0.0284β+A 1 ;
in the formula ,ef_minA E is the maximum actual slip rotation speed which can be achieved when the accelerator opening is at a small accelerator opening f_minB E is the minimum actual slip rotating speed which can be achieved when the accelerator opening is at the small accelerator opening f E is the actual slip speed t_min Target slip rotation speed beta set when the accelerator opening is at a small accelerator opening minA To control the maximum accelerator opening that can be reached when the accelerator opening is in the small accelerator opening range, β is the accelerator opening, a 1 The value range is 0.97 to 1.05 for the first tested correction constant, A 2 For the second empirical correction constant, the value range is 4.24-4.38, delta is the first correction coefficient, the value range is 1.97-2.13, phi is the second correction coefficient, and the value range is 5.67-5.83.
4. As claimed inThe lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 3, characterized in that in the second step, when the accelerator opening is at a full accelerator opening, the actual slip speed is controlled by the following formula to obtain a full accelerator opening slip speed e f_all :
wherein ,
in the formula ,ef E is the actual slip speed t_max A is a target slip rotation speed set when the accelerator opening is at a large accelerator opening 3 The third empirical correction constant is in the range of 0.083 to 0.096.
5. The lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 4, wherein a 1 The value is 1.01, A 2 The value is 4.31, A 3 The value is 0.092, delta is 2.06, and psi is 5.75; and
e t_min 800r/min, e t_max 3000r/min.
6. The lockup clutch slip speed control method for different starting conditions of a vehicle according to any one of claims 1 to 4, characterized in that the small accelerator opening degree β min The value range of (2) is 0 < beta min Less than or equal to 20 percent, the opening beta of the middle throttle mid The range of the value of (2) is 20% < beta mid Less than or equal to 70 percent, the large accelerator opening degree beta max The range of the value of (2) is 70% < beta max <100%。
7. The lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 2, characterized in thatCharacterized in that the difference e between the target slip speed and the actual slip speed in the fuzzy control model is divided into 9 quantization levels, and the slip speed change rateDivided into 9 quantization levels, the difference e between the target slip speed and the actual slip speed and the slip speed change rate +.>Is divided into { NB, NM, NS, ZO, PS, PM, PB };
ΔK of PID controller parameters p 、ΔK i 、ΔK d The average is divided into 9 quantization levels, and the fuzzy set is divided into { NB, NM, NS, ZO, PS, PM, PB }.
8. The lockup clutch slip speed control method for different starting conditions of a vehicle as claimed in claim 7, characterized in that a difference e between a target slip speed and an actual slip speed and a slip speed change rateIs of the actual quantization level [ -9,9]Parameter delta K regulated by PID controller i 、ΔK d 、ΔK p Is of the actual quantization level [ -9,9]。
9. The lockup clutch slip speed control method for different starting conditions of a vehicle according to claim 8, characterized in that a difference e between a target slip speed and an actual slip speed, a slip speed change rateParameter delta K regulated by PID controller i 、ΔK d The actual discrete domains of (a) are [ -1,1]Parameter delta K regulated by PID controller p The actual discrete domain of (2) is [ -2,2]。
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