CN110361967B - Construction method of sliding-mode observer - Google Patents

Construction method of sliding-mode observer Download PDF

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CN110361967B
CN110361967B CN201910420812.4A CN201910420812A CN110361967B CN 110361967 B CN110361967 B CN 110361967B CN 201910420812 A CN201910420812 A CN 201910420812A CN 110361967 B CN110361967 B CN 110361967B
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sliding mode
state
sliding
mode observer
gain
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徐岩
刘辉
韩立金
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a construction method of a sliding-mode observer, which comprises the following steps of 1, simplifying a vehicle driving shaft into a system consisting of a spring and a damper, establishing a state space equation of the sliding-mode observer for the system, and 2, constructing an output estimation error feedback term in the state space equation of the sliding-mode observer by using an exponential approximation law and an exponential function; step 3, constructing a sliding-mode observer equation and a sliding-mode observer structure based on the output estimation error feedback term; the sliding mode observer constructed by the invention can correct the torque estimation result of the vehicle in time, so that the torque estimation value of the driving system approaches to an actual value, the buffeting phenomenon is reduced, the phase delay is reduced, and the observation result is more accurate.

Description

Construction method of sliding-mode observer
Technical Field
The invention belongs to the technical field of sliding mode control, and particularly relates to a construction method of a sliding mode observer.
Background
In the prior art, when the torque of a driving shaft is calculated by using a linear integration method, the torque is influenced by rotational variation errors of a wheel rotational speed sensor and a motor, signal noise of the rotational speed sensor, external interference and the like are gradually accumulated on a torque estimation result due to the integration effect, and finally, a large error is generated between an estimated value and an actual value.
The Luenberger observer has a good observation effect on a system which is mainly represented by linear characteristics, but the Luenberger observer is a typical nonlinear system due to the existence of nonlinear factors such as coulomb damping, air resistance and tooth flank clearance in a vehicle power transmission system, meanwhile, the transmission system modeling process is simplified greatly, unknown interference can also exist in an actual system, the linear observer is adopted, so that the estimation error of the output of the driving system cannot approach zero, and the estimated system state is difficult to converge to the actual state of the system.
Therefore, the actual measurable quantity is needed to be used for feedback, the torque estimation result is corrected, the estimation value can follow the actual value, and the influence of external interference is resisted, namely, a sliding mode observer needs to be designed to observe the torque of the driving shaft.
Disclosure of Invention
The invention aims to provide a construction method of a sliding mode observer, which is simple in construction process, realizes that the deviation between the torque estimated value and the actual value of a driving system is reduced in a nonlinear state by using an exponential function, eliminates buffeting when the state of the driving system approaches a sliding mode surface, reduces phase delay and enables the observation result of the constructed sliding mode observer to be higher in accuracy.
The invention adopts the technical scheme that the construction method of the sliding-mode observer specifically comprises the following steps:
step 1, simplifying a vehicle driving shaft into a system consisting of a spring and a damper, and establishing a state space equation of the sliding mode observer, wherein the general form of the sliding mode observer is as formula (1):
Figure BDA0002065953820000011
let x be the state variable and,
Figure BDA0002065953820000012
y is an output quantity,
Figure BDA0002065953820000013
in order to estimate the error in the state,
Figure BDA0002065953820000014
for output estimation error, v is feedback term of output estimation error, GnIn order to be a matrix of gains, the gain matrix,
Figure BDA0002065953820000021
the number of revolutions of the motor is,
Figure BDA0002065953820000022
to output rotational speed, TsAs output shaft torque, CtFor retarder damping, CvFor wheel damping, CsFor drive shaft damping, JvEquivalent of the vehicle body to the sum of the equivalent inertia of the wheel and the inertia of the wheel, JpIs the rotor inertia of motor B, irIs the main reducer transmission ratio, ksFor drive shaft stiffness, i is the final drive ratio, TpFor driving torque, TvIs the vehicle load torque;
step 2, setting an output estimation error feedback term of the sliding mode observer by using an exponential approaching law and an exponential function;
and 3, establishing an equation and a structure of the sliding-mode observer.
Further, the feedback term of the estimation error output in step 2 is:
v＝-εsigmoid(s)-qs ε＞0,q＞0
Figure BDA0002065953820000023
wherein epsilon is the sliding mode gain, q is the exponential approach gain, s is the distance between the driving system state and the sliding mode surface,
Figure BDA0002065953820000029
is any positive integer.
Further, the equation of the sliding-mode observer established in step 3 is:
Figure BDA0002065953820000024
wherein
Figure BDA0002065953820000025
Is a state variable x1Is measured by the measurement of (a) a,
Figure BDA0002065953820000026
is a state variable x2Is measured by the measurement of (a) a,
Figure BDA0002065953820000027
is a state variable x3Measured value of s1Is a state variable of x1Distance between the state of the time-driven system and the slip form surface, s2Is a state variable of x2Distance between the state of the time-driven system and the slip form surface, q1Is a state variable x1Is close to the gain, q2Is a state variable x2Is close to the gain, epsilon1Is a state variable x1Gain of sliding mode of epsilon2Is a state variable x2Gain of sliding mode, L1、L2Is a feedback gain factor;
state error of sliding mode observer
Figure BDA0002065953820000028
The equation is:
Figure BDA0002065953820000031
further, the sliding mode gain selection of the sliding mode observer meets the following conditions:
Figure BDA0002065953820000032
wherein e3Is a state variable x3The state estimation error of (2).
The invention has the beneficial effects that: the process for constructing the sliding mode observer is simple, and buffeting generated when the state of the driving system approaches to the sliding mode surface is eliminated by using the exponential function in construction, so that the constructed sliding mode observer has high detection accuracy when used for detection.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a simplified model diagram of a powertrain system.
Fig. 2 is a structure diagram of a torque sliding mode observer.
FIG. 3 is a graph of the torsional damping results of the present invention on a linear model.
FIG. 4 is a graph of the torsional vibration reduction results of the present invention on a nonlinear model.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The construction method of the sliding-mode observer specifically comprises the following steps:
step 1: simplifying a vehicle driving shaft into a system consisting of a spring and a damper, wherein the structure is shown in FIG. 1, establishing a state space equation of a sliding mode observer, and the general form of the sliding mode observer is shown in formula (1):
Figure BDA0002065953820000033
in the formula (1), x is a state variable,
Figure BDA0002065953820000041
y is an output quantity, provided
Figure BDA0002065953820000042
In order to estimate the error in the state,
Figure BDA0002065953820000043
for output estimation error, v is feedback term of output estimation error, GnIn order to be a matrix of gains, the gain matrix,
Figure BDA0002065953820000044
wherein
Figure BDA0002065953820000045
The number of revolutions of the motor is,
Figure BDA0002065953820000046
to output rotational speed, TsAs output shaft torque, CtFor retarder damping, CvFor wheel damping, CsFor drive shaft damping, JvEquivalent of the vehicle body to the sum of the equivalent inertia of the wheel and the inertia of the wheel, JpIs the rotor inertia of motor B, irIs the main reducer transmission ratio, ksFor drive shaft stiffness, i is the final drive ratio, TpFor driving torque, TvIs the vehicle load torque;
step 2: setting an output estimation error feedback term v of the sliding mode observer by using an exponential approaching law and an exponential function;
and (3) utilizing an exponential approximation law as a feedback term of the output estimation error to enable the torque estimation value to quickly converge to an actual value, wherein the exponential approximation law is shown as a formula (2):
v＝-εsgn(s)-qs ε＞0,q＞0 (2)
wherein epsilon is the sliding mode gain, q is the exponential approach gain, and s is the distance between the state of the driving system and the sliding mode surface; the index approach law enables any point in the state space to converge on the sliding mode surface in an index form, the convergence speed is accelerated along with the increase of the index approach gain q, but the state variable has larger vibration on the sliding mode surface due to the overlarge index approach gain; the state of the driving system is far away from the sliding mode surface, namely when the absolute value of s is large, the state of the driving system can rapidly approach the sliding mode surface, so that s is 0; the state of the driving system is close to the sliding mode surface, namely when the absolute value of s is small, the speed of the state of the driving system approaching the sliding mode surface is gradually reduced, and the switching term-epsilon sgn(s) can ensure that the driving system reaches the sliding mode surface within limited time;
in the feedback item for outputting the estimation error, a sign function sgn(s) is adopted, so that the state of the driving system has a certain speed when reaching the sliding mode surface, the driving system can cross the sliding mode surface under the action of inertia, then the control action is reversed, the state of the driving system is decelerated and reversely approaches the sliding mode surface, and the chattering is caused by repeating the steps, the chattering generated when the state of the driving system approaches the sliding mode surface is eliminated by using an exponential function sigmoid(s), and the calculation of the exponential function is shown as a formula (3):
Figure BDA0002065953820000047
wherein
Figure BDA0002065953820000051
Is any positive integer;
replacing sgn(s) in formula (2) with formula (3), and the feedback term of the output error in the sliding-mode observer is shown in formula (4):
v＝-εsigmoid(s)-qs ε＞0,q＞0 (4)
step 3, establishing an equation and a structure of the sliding-mode observer;
according to a state space equation of the sliding mode observer, setting a sliding mode surface function S as:
Figure BDA0002065953820000052
gain matrix GnIn the form of
Figure BDA0002065953820000053
Wherein L ∈ R1 ×2I is a unit matrix, L is a coefficient matrix of 1 x 2, and R represents that L is a real matrix;
the expression of the sliding-mode observer is shown in equation (5):
Figure BDA0002065953820000054
wherein JpIs the rotor inertia of motor B, JvEquivalent of the vehicle body to the sum of the equivalent inertia of the wheel and the inertia of the wheel, TvFor loading vehicle torque，CtFor retarder damping, CvFor wheel damping, CsFor drive shaft damping, ksFor drive shaft stiffness, TPIn order for the power system to drive the torque,
Figure BDA0002065953820000055
is a state variable x1Is measured by the measurement of (a) a,
Figure BDA0002065953820000056
is a state variable x2Is measured by the measurement of (a) a,
Figure BDA0002065953820000057
is a state variable x3Measured value of s1Is a state variable of x1Distance between the state of the time-driven system and the slip form surface, s2Is a state variable of x2Distance between the state of the time-driven system and the slip form surface, q1Is a state variable x1Is close to the gain, q2Is a state variable x2Is close to the gain, epsilon1Is a state variable x1Gain of sliding mode of epsilon2Is a state variable x2Gain of sliding mode, L1、L2Is a feedback gain factor;
error of sliding-mode observer
Figure BDA0002065953820000058
The equation is:
Figure BDA0002065953820000059
the error of the state variable measured by the sliding-mode observer from the actual state is shown in equation (7):
Figure BDA0002065953820000061
e1is a state variable x1State estimation error of e2Is a state variable x2The state estimation error of (a) is,e3is a state variable x3State estimation error of (2);
the stability of the sliding mode observer is verified by utilizing a Lyapunov stability theory, and a Lyapunov function of the sliding mode observer is shown as a formula (8):
Figure BDA0002065953820000062
wherein S is a matrix formed by the state of the driving system and the distance S of the sliding mode surface;
the derivation is obtained by applying the derivation to the formula (8) and substituting the derivation result into the formula (7):
Figure BDA0002065953820000063
as can be seen from equation (9), in order to ensure the stability of the sliding mode observer, the sliding mode gain of the sliding mode observer should satisfy the following condition when selecting:
1)e1>0 and e2>0，
Figure BDA0002065953820000064
2)e1>0 and e2<0，
Figure BDA0002065953820000065
3)e1<0 and e2>0，
Figure BDA0002065953820000066
4)e1<0 and e2<0，
Figure BDA0002065953820000067
JPThe rotor inertia of the motor B is represented by epsilon, the sliding mode gain is represented by i, the transmission ratio is represented by i, and the state estimation error is represented by e;
calculating and simplifying the sliding mode gain selection condition of the sliding mode observer to obtain the condition that the sliding mode gain of the sliding mode observer meets the following condition for ensuring the progressive stability of the sliding mode observer:
Figure BDA0002065953820000071
according to the sliding mode observer constructed by the invention, the feedback item for outputting the estimation error is constructed by utilizing the index approaching law and the index function, so that the state of the driving system can be decelerated in time when approaching the sliding mode surface, the buffeting phenomenon of the driving system near the sliding mode surface is reduced, the speed of the estimated value approaching the actual value is higher, the phase delay is reduced, and the observation result of the sliding mode observer is more accurate.
Examples
A simplified model of a transmission system is built in Matlab/Simulink, the delay characteristic of a power source, the friction resistance of the power source and a transmission element and the like are considered in the modeling process, the observation effect of a sliding mode observer is simulated and verified based on the simplified model, the result is shown in figures 3 and 4, as can be seen from figure 3, the sliding mode observer in a linear system has good observation capability, the observation result of the sliding mode observer can well track the actual state of a vehicle, and a good observation effect is achieved, the sliding mode observer in figure 4 can still keep a good observation result in a nonlinear model, a stable error exists between the observation result and the actual torque, when the sliding mode observer observes the torque of a driving shaft, the torque command of the power transmission system is directly taken as the torque of the system to be output and input into the sliding mode observer to be calculated, and due to the existence of the friction resistance torque, therefore, a steady-state difference exists between the actual output torque of the power source and the torque command, and the steady-state difference is reflected on an observation result of the sliding-mode observer and is a steady-state observation error, but the error can be basically ensured to be within 5%.
All the embodiments in the present specification are described in a related manner, and the same and similar parts among the embodiments may be referred to each other, and each embodiment focuses on the differences from the other embodiments. In particular, for the system embodiment, since it is substantially similar to the method embodiment, the description is simple, and for the relevant points, reference may be made to the partial description of the method embodiment.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (1)

1. The construction method of the sliding-mode observer is characterized by comprising the following steps:
step 1, simplifying a vehicle driving shaft into a system consisting of a spring and a damper, and establishing a state space equation of the sliding mode observer, wherein the general form of the sliding mode observer is as formula (1):
Figure FDA0003188404940000011
let x be the state variable and,
Figure FDA0003188404940000012
y is an output quantity,
Figure FDA0003188404940000013
in order to estimate the error in the state,
Figure FDA0003188404940000014
for output estimation error, v is feedback term of output estimation error, GnIn order to be a matrix of gains, the gain matrix,
Figure FDA0003188404940000015
Figure FDA0003188404940000016
the number of revolutions of the motor is,
Figure FDA0003188404940000017
to output rotational speed, TsAs output shaft torque, CtFor retarder damping, CvFor wheel damping, CsFor drive shaft damping, JvEquivalent of the vehicle body to the sum of the equivalent inertia of the wheel and the inertia of the wheel, JpIs the rotor inertia of motor B, irIs the main reducer transmission ratio, ksFor drive shaft stiffness, i is the final drive ratio, TpFor driving torque, TvIs the vehicle load torque;
step 2, setting an output estimation error feedback term v of the sliding mode observer by using an exponential approaching law and an exponential function:
v＝-εsigmoid(s)-qsε＞0,q＞0
Figure FDA0003188404940000018
wherein epsilon is the sliding mode gain, q is the exponential approach gain, s is the distance between the driving system state and the sliding mode surface,
Figure FDA0003188404940000019
is any positive integer;
step 3, establishing a sliding-mode observer equation and a sliding-mode observer structure:
Figure FDA0003188404940000021
wherein
Figure FDA0003188404940000022
Is a state variable x1Is measured by the measurement of (a) a,
Figure FDA0003188404940000027
is a state variable x2Is measured by the measurement of (a) a,
Figure FDA0003188404940000023
is a state variable x3Measured value of s1Is a state variable of x1Distance between the state of the time-driven system and the slip form surface, s2Is a state variable of x2Distance between the state of the time-driven system and the slip form surface, q1Is a state variable x1Is close to the gain, q2Is a state variable x2Is close to the gain, epsilon1Is a state variable x1Gain of sliding mode of epsilon2Is a state variable x2Gain of sliding mode, L1、L2Is a feedback gain factor;
the sliding mode gain selection should meet the following conditions:
Figure FDA0003188404940000024
wherein e3Is a state variable x3State estimation error of (2);
state error of sliding mode observer
Figure FDA0003188404940000025
The equation is:
Figure FDA0003188404940000026
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