CN114859706B - Active fault-tolerant control method of electric scooter system - Google Patents

Active fault-tolerant control method of electric scooter system Download PDF

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CN114859706B
CN114859706B CN202210237197.5A CN202210237197A CN114859706B CN 114859706 B CN114859706 B CN 114859706B CN 202210237197 A CN202210237197 A CN 202210237197A CN 114859706 B CN114859706 B CN 114859706B
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郁明
杨柳
蓝盾
黄云志
平兆武
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Hefei University of Technology
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Abstract

The invention discloses an active fault-tolerant control method of an electric scooter system, which comprises the following steps: establishing a system DBG model, and obtaining BDBG models through separating causal lines; deriving ARRs based on BDBG model, generating residual error by ARRs to obtain a collective fault feature matrix; the sliding mode control law u no under the condition of no fault of the system is designed, so that the speed tracking of the electric scooter is realized; aiming at the sensor fault, designing a neural network observer, reconstructing the sensor fault, and designing a fault-tolerant control law u f1 under the condition of the sensor fault of the system based on the result of the sensor fault reconstruction; aiming at system parameter faults, constructing a fault-tolerant control law u f2 under the system parameter fault condition, and estimating control items for compensating unknown parameter faults in a sliding mode control law u f2 in real time by utilizing an extreme learning machine; and aiming at different fault types, the control law is switched in real time, so that the active fault-tolerant control of the system is realized.

Description

Active fault-tolerant control method of electric scooter system
Technical Field
The invention relates to the technical field of fault diagnosis and fault-tolerant control, in particular to an active fault-tolerant control method of an electric scooter system.
Background
With the rapid development of new energy industry and the trend of population aging becoming increasingly obvious, electric scooter products which provide convenience for the elderly and disabled people to travel are receiving attention in recent years. However, with the rapid increase of the number of electric vehicles, the problem of traffic safety caused by the electric vehicles is increasing. If the system fails during the running process of the electric scooter, the running of the system is unstable, a series of traffic problems can be caused, and even the life safety of a driver is endangered.
At present, researches on electric scooter mainly focus on fault detection, fault isolation, fault parameter estimation and the like, but researches on fault-tolerant control are relatively few. Chinese patent publication No. CN108437798a discloses a fault diagnosis and estimation method for an electric scooter based on a bond map model, and although the fault diagnosis and estimation method is disclosed in the patent, a real-time fault-tolerant control method is not disclosed. Therefore, aiming at the problem of safe operation of the electric scooter, a fault-tolerant control method is urgently needed to be designed, so that acceptable performance of the system can be maintained when the system fails.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides an active fault-tolerant control method of an electric scooter system, which is an active fault-tolerant control method based on a bonding diagram theory, a neural network observer and an extreme learning machine.
In order to achieve the above purpose, the present invention adopts the following technical scheme, including:
an active fault-tolerant control method of an electric scooter system comprises the following steps:
S1, establishing a diagnosis bonding graph model of an electric scooter system, namely a DBG model, and obtaining a double causal relationship diagnosis bonding graph model, namely a BDBG model through separation causal line;
S2, deducing an analytic redundancy relation ARRs of the system based on BDBG model, generating residual errors by ARRs to obtain an aggregate fault feature matrix for fault detection and fault isolation;
s3, designing a sliding mode control law u no under the condition that an electric scooter system has no fault, and realizing speed tracking of the electric scooter;
S4, aiming at the sensor fault of the electric scooter system, designing a neural network observer, reconstructing the sensor fault, and designing a fault-tolerant control law u f1 under the condition of the sensor fault of the electric scooter system based on the result of the sensor fault reconstruction;
S5, aiming at the parameter faults of the electric scooter system, constructing a fault-tolerant control law u f2 under the parameter fault conditions of the electric scooter system, and carrying out real-time estimation on control items for compensating unknown parameter faults in the fault-tolerant control law u f2 by utilizing an extreme learning machine;
S6, designing an active fault-tolerant controller of the electric scooter system, and switching control laws in real time according to fault detection and fault isolation results and aiming at different fault categories, wherein the active fault-tolerant controller is used for realizing active fault-tolerant control of the electric scooter system and specifically comprises the following steps of:
If the system does not have a fault, the control law u no is adopted to realize the speed tracking of the electric scooter under the fault-free condition; if the sensor fault of the system is detected, the system is switched from the control law u no to the fault-tolerant control law u f1 under the condition of no fault; if a parameter failure of the system is detected, the system will switch from the fault-free control law u no to the fault-tolerant control law u f2.
The DBG model and BDBG model of the electric scooter system established in the step S1 are specifically shown as follows:
Modeling an electric scooter system by adopting a bonding diagram theory, wherein bonding diagram elements in a DBG model of the electric scooter system comprise: potential source Se, current source Sf, resistive element { R 1,R2,…,Rj }, capacitive element { C 1,C2,…,Cm }, inertial element { I 1,I2,…,Ii }, flow sensor { Df 1,Df2,…,Dfn }, j, m, I, n represent the number of resistive elements, capacitive elements, inertial elements, and flow sensors, respectively, in the system; the DBG model of the electric scooter system comprises: a direct current motor driver part, a direct current motor part, a rear wheel part, a vehicle body part and a front wheel part;
The BDBG model of the electric scooter system is obtained based on a DBG model, and the specific mode is as follows: converting a source Sf and a source Se of a potential source in the DBG model into a source-source SS, distributing an additional bonding picture element stream amplifier AF and a potential amplifier AE into the DBG model, and distributing double causal relations to all keys connected with the sensor so that the potential and the stream point to nodes connected with the sensor; meanwhile, selecting part of nonlinear elements to be distributed as terminal nodes, and receiving potential and flow information; for type 1 nodes and type 0 nodes, only one of the keys connected to the node has a causal score at the end remote from the node, all other keys have a causal score at the end near the node, or only one of the keys has a causal score at the end near the node, all other keys have a causal score at the end remote from the node.
In step S2, the derived analytical redundancy relation ARRs and the aggregate fault feature matrix are:
Based on the DBG model and BDBG model of the electric scooter system, the following ARRs is obtained by a causal relation inversion method and analyzing causal paths and constitutive relations of all terminal nodes:
ARRq(Ξ,De,Df,uin)＝0，q＝1，2，...，b (1)
wherein b represents the number of analytical redundancy relations ARRs of the system, xi= [ xi 1...Ξn]T ] represents system parameters, de and Df represent potential sensors and flow sensors in the DBG model, respectively;
u in represents the input of the system, and is selected as a slip form control law u no under the condition that the electric scooter system has no fault, or a fault-tolerant control law u f1 under the condition that the electric scooter system sensor has a fault, or a fault-tolerant control law u f2 under the condition that the electric scooter system parameter has a fault;
Generating residual error r i by carrying out numerical estimation on ARRs, carrying out fault detection according to whether the residual error r i exceeds a corresponding threshold epsilon i, and defining a binary coherence vector CV= [ o 1 o2…ob ] to represent the consistency of all residual errors; wherein the number of elements in the coherence vector CV is b, and the number of elements in the coherence vector CV is consistent with the number of ARRs; the element determination rules in CV are as follows:
Where ρ i represents the threshold for the i-th residual r i;
The aggregate fault signature matrix is constructed according to ARRs of the system as shown in table 1 below:
TABLE 1
In table 1, [ α 1,α2,…,αp ] represents a set of possible faults; q hl represents the relationship between the h possible fault, α h, and the l ARR, ARR l, h=1, 2, … p, l=1, 2, … b, q hl =1 represents ARR l is sensitive to α h, q hl =0 represents ARR l is not sensitive to α h;
In the running process of the system, when the occurrence of the fault is detected, the position where the fault occurs is isolated by comparing the coherence vector CV with the integrated fault characteristic matrix, and the fault type is determined.
In step S3, a slip form control law u no under the condition that the electric scooter system has no fault is designed, which is specifically shown as follows:
S31, the electric scooter is driven by a direct current motor, so that linear proportional relations exist among all measurable variables, and a differential equation model of the electric scooter system is obtained through equivalent substitution and linear transformation and is as follows:
Wherein g and Is a known nonlinear function; /(I)For actually measuring the angular velocity of the rear wheel of the electric scooter,For actually measuring the rear wheel angular acceleration of the electric scooter,Unknown interference for the system and satisfiesD is the upper boundary of system interference;
s32, defining the angular speed of the system relative to the expected rear wheel according to the sliding mode variable structure control method The tracking error of (2) is:
defining a slip-form surface The method comprises the following steps:
Wherein c > 0, deriving the time from both sides of the formula (5) and combining the formula (4) to obtain:
In the method, in the process of the invention, To the desired rear wheel angular velocityIs representative of the desired rear wheel angular acceleration;
The slip form control law u no under the condition of no fault of the electric scooter system is designed as follows:
Where η 1 is a normal number, η 1 is greater than the upper boundary of system interference, and sign (·) is a sign function.
In step S3, for the differential equation model of the electric scooter system shown in formula (3), the sliding mode function defined in formula (5) and the sliding mode control law u no under the condition that the electric scooter system designed in formula (7) has no fault are adopted, so as to prove the stability of the closed loop system, as shown in the following:
selecting a Liapunov function as:
the two sides of the formula (8) are derived from time to obtain:
substitution of formula (5) and formula (6) into formula (9) yields:
Due to Taking eta 1 to be more than D, the following steps are obtained:
from the above evidence can be obtained Is negatively determined, and according to the Liapunov stability principle, the electric scooter system meets the asymptotic stability, namely the system state is from the sliding mode surfaceStarting from any other state, the slide will move to the slide surface.
In step S4, a fault-tolerant control law u f1 under the fault condition of the sensor of the electric scooter system is designed, which is specifically as follows:
The sensor comprises: sensor mounted at rear wheel Sensor mounted at the bodySensor mounted at front wheel
S41, selecting a system state variable of x p＝[x1 x2 x3 x4 x5 x6]T,x2 as the rear wheel angular speed, and x 2 as the derivative of x 1; x 4 is the body line speed, x 4 is the derivative of x 3; x 6 is the front wheel angular velocity, x 6 is the derivative of x 5;
the space state model of the electric scooter system can be established by the formula (1) as follows:
wherein Q u∈R6*2、Ap∈R6*6、Bp∈R6、Cp∈R3*6 is a matrix function, Representing the nonlinear part of the system; y p is a system output vector, f (t) is a sensor fault vector, and t represents time; and satisfies the following:
||f(t)||≤θ (13)
Wherein θ > 0;
S42, converting the sensor fault into fictitious actuator fault, introducing a state z epsilon R 2, and meeting the following conditions:
the construction of a new state space model is represented as follows:
In the method, in the process of the invention, For the augmented state vector, y (t) is the transformed system output, state z is the virtual output of the system described by equation (15), and Q εR 9*2、A∈R9*9、B∈R9、C∈R3*9、F∈R9*3 is the matrix function;
s43, designing a neural network observer aiming at the fault system described by the formula (15), wherein the form is as follows:
In the method, in the process of the invention, The estimated values of x (t), y (t) and f (t) are respectively, L is an observer gain matrix to be designed, and the fault termOn-line estimation using radial basis function neural network:
In the method, in the process of the invention, Is a weight vector having n number of network nodes; /(I) Is a neural network activation function,Is a gaussian function, namely:
Wherein c i is the center of the Gaussian kernel function, and d i is the width of the Gaussian kernel function;
The RBF approximation fault term is obtained by a neural network approximation principle and is as follows:
Wherein epsilon is the approximation error of the neural network and satisfies Is the upper bound of the approximation error of the neural network;
weight vector The following adaptive law is adopted:
Wherein, gamma > 0 is the parameter to be designed, For positive definite number matrix to be designed,To output the estimated error, and there is a positive definite matrix P εR 9*9 satisfying
S44, according to the reconstruction result of the sensor fault vector, considering the compensated rear wheel angular velocity as The differential equation model of the electric scooter system is expressed as:
Wherein, AndIs a known nonlinear function; /(I)Unknown interference for the system and satisfiesD 1 is the current system interference upper bound;
according to the sliding mode variable structure control method, the system is at the moment relative to the expected angular speed of the rear wheel The tracking error e 1 of (a) can be expressed as:
at this time, the sliding surface The method comprises the following steps:
Wherein c 1 > 0;
Because buffeting occurs when the control law is switched, in order to enable the system to quickly recover and stabilize, the sliding mode control adopts an exponential approach law:
Wherein, eta 2 and q 1 are two normal numbers, and the value of eta 2 is required to be larger than the upper boundary of system interference;
substitution of formula (21) into formula (23) yields:
The fault-tolerant control law u f1 under the fault condition of the sensor of the electric scooter is designed as follows:
in step S4, for the electric scooter neural network observer, namely formula (16), the weight adaptive law of formula (20) is adopted to prove the stability of the neural network observer, specifically as follows:
Presence of Lipschitz nonlinear function There is an arbitrary positive definite matrixSo that
In the method, in the process of the invention,
The state estimation error e x and the fault amount estimation error e f are defined as:
deriving the time from both sides of formula (28) and combining formulas (15) and (16) can obtain:
In the method, in the process of the invention,
Selecting Lyapunov function as:
The two sides of the formula (31) are derived for time and the formula (30) is combined to obtain:
In the method, in the process of the invention,
As a result of the fact that,
And because ofW * is an ideal weight vector, soThus, formula (20) andThe method comprises the following steps:
Wherein λ 1 is the minimum eigenvalue of Γ;
as a result of:
So that:
Wherein lambda 2 is the maximum eigenvalue of P;
therefore, when Time,The observer is bounded convergent according to the principle of lisinov stability; therefore, the reconstruction result of the sensor fault vector is
In step S4, for the differential equation model of the electric scooter system shown in formula (21), the slip mode function defined in formula (23) and the fault-tolerant control law u f1 under the fault condition of the electric scooter system sensor designed in formula (26) are adopted to prove the stability of the electric scooter system, as follows:
selecting a Liapunov function as:
the two sides of the formula (38) are derived from time to time:
substitution of formula (23) and formula (24) into formula (38) can be obtained:
Due to Taking eta 2＞D1, the following steps are obtained:
According to the above, V 3 is proved to be negative, and the electric scooter system meets asymptotic stability according to the Liapunov stability principle.
In step S5, a fault-tolerant control law u f2 under the fault condition of the electric scooter system parameter is designed, which is specifically as follows:
In the running process of the electric scooter, considering that the system possibly has parameter faults, the differential equation model of the electric scooter under the parameter fault condition is defined as follows:
In the method, in the process of the invention, Is an unknown fault function; when the system is normal,When there is a parameter failure in the system,
Slip form surface using (5)The method comprises the following steps:
Wherein η 3 and q 2 are two normal numbers, and the value of η 3 needs to be larger than the upper boundary of system interference; substituting formula (41) into formula (42) yields:
In the method, in the process of the invention, ForIs a function of the estimated value of (2);
The fault-tolerant control law u f2 under the condition of the parameter fault of the electric scooter is designed as follows:
Utilizing extreme learning machine pairs And (3) performing real-time estimation:
In the method, in the process of the invention, The output weight vector estimated value of the output node is beta *, and according to the universal approximation theorem of the extreme learning machine, the optimal beta * exists, so that:
Where ε 2 is the approximation error, satisfy:
|ε2|≤εN (47)
Where ε N is the upper approximation error bound; h (z, ω, b) is L inputs in ELM and has Hidden layer output matrix of individual neurons:
wherein, z= [ z 1,z2,…,zL ] is an input vector, ω i＝[ω1,ω2,…,ωL]T is an input weight vector, and b i is the bias of hidden layer nodes; g (·) is an activation function, in the following specific form:
In step S5, for the differential equation model of the electric scooter system shown in formula (41), the fault-tolerant control law u f2 under the condition of the fault of the electric scooter system parameter designed in formula (44) is adopted to prove the stability of the electric scooter system, as follows:
output weight of extreme learning machine Updating is performed by the following adaptive law:
wherein, the adaptive gain sigma is more than 0;
substitution of formula (44) into formula (43) yields:
In the method, in the process of the invention,
Definition:
selecting a Liapunov function as:
The two sides of the formula (54) are derived from time to time:
Due to Substituting formula (50) into formula (55) yields:
Due to Taking η 3＞D+εN, we can get ∈ 2|≤εN:
according to the above, V 4 is proved to be negative, and the electric scooter system meets asymptotic stability according to the Liapunov stability principle.
The invention has the advantages that:
(1) The invention firstly uses the bonding diagram theory to model the electric scooter system, and uses the separation causal line method to construct BDBG model of the system, deduce ARRs and construct the aggregate fault feature matrix, which can improve the isolation of faults compared with the traditional fault diagnosis method based on the bonding diagram.
(2) The active fault-tolerant control method is designed based on a sliding mode control theory, and the sliding mode variable structure control system has a series of advantages of quick response, strong robustness and the like.
(3) The active fault-tolerant controller designed by the invention considers possible sensor faults and parameter faults of the electric scooter system, performs fault diagnosis by utilizing ARRs and a fault characteristic matrix, reconstructs the sensor faults by combining a neural network observer, and designs a fault-tolerant control law under the condition of the sensor faults based on a fault reconstruction result. In the running process of the system, no matter what faults occur, the control law can be switched according to different types of faults, so that the active fault-tolerant control of the electric scooter is realized.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
fig. 2 is a DBG model of the electric walker system.
Fig. 3 shows that the terminal nodes are C 1、R4, I: and BDBG model of an electric scooter system.
Fig. 4 is a BDBG model of an electric scooter system with a terminal node R 3、R4、C2.
Fig. 5 is a schematic diagram of the active fault-tolerant control of the present invention.
Fig. 6 to 10 are schematic diagrams of residuals obtained by ARRs when the rear wheel angular velocity sensor fails.
Fig. 11 is an output of the electric scooter system when the rear wheel angular velocity sensor fails.
Fig. 12 is a failure reconstruction result of the neural network observer when the rear wheel angular velocity sensor fails.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
As shown in fig. 1, an active fault-tolerant control method of an electric scooter system includes the following steps:
S1, establishing a diagnostic bonding diagram model (DBG model diagnosis bond graph) of an electric scooter system, and obtaining a dual causal relationship diagnostic bonding diagram model (BDBG model (bicausal diagnosis bond graph) through separation causal line;
S2, based on BDBG models, deducing corresponding analytic redundancy relation ARRs (analytical redundancy relations), generating residual errors by ARRs to obtain an aggregate fault feature matrix, and performing fault diagnosis by using ARRs and the fault feature matrix;
S3, designing a control law under the condition that the electric scooter system has no fault so as to realize speed tracking of the electric scooter;
s4, aiming at the sensor fault of the electric scooter system, designing a neural network observer, reconstructing the sensor fault, and designing a fault-tolerant control law under the condition of the sensor fault of the electric scooter system based on the result of the sensor fault reconstruction;
S5, aiming at the parameter faults of the electric scooter system, constructing a fault-tolerant control law under the parameter fault condition, wherein a limit learning machine is utilized to estimate control items for compensating unknown parameter faults in real time in the control law;
S6, designing an active fault-tolerant controller of the electric scooter, and switching control laws in real time according to fault detection and fault isolation results and different fault categories, so that active fault-tolerant control of the electric scooter is realized.
The DBG model established in step S1 is shown in fig. 2, and the BDBG model established is shown in fig. 3 and 4.
Specifically, the electric scooter system is modeled by adopting a bonding diagram theory, and fig. 2 is a DBG model of the electric scooter, wherein the DBG model comprises a current source Sf, a resistive element { R 1,R2,R3,R4 }, a capacitive element { C 1,C2 }, an inertial element { I 1,I2,I3,I4 }, and a flow sensorThe built DBG model of the electric scooter is divided into a direct current motor driver part, a direct current motor part, a rear wheel part, a vehicle body part and a front wheel part.
Modeling of a driving part: u in represents an electric scooter voltage source, consisting of a 24V lead-acid battery, TF: the k 1 converter represents the conversion of voltage to current through the ratio k 1.
Modeling a direct current motor part: the internal resistance of the direct current motor is R: r 1 describes. I: j 1 denotes the moment of inertia of the dc motor. Gyrator GY: k 2 denotes that the direct current motor converts electric energy into mechanical energy, and the conversion ratio is k 2. The converter TF: k 3 denotes that the rotational speed of the motor is converted into the electric scooter rear wheel angular velocity in the proportion k 3. R: r 2 is the mechanical friction of the motor, and the specific form is as follows:
where k f is the motor viscous friction coefficient, Is the rear wheel angular velocity. F u is the motor coulomb friction torque, sign (·) is a sign function.
Rear wheel partial modeling: i: j 2 is the rear wheel rotational inertia of the electric scooter.Is a sensor mounted on the rear wheel. R: r 3 is rear wheel friction force, and the specific form is as follows:
where k f1 is the rear wheel viscous friction coefficient and F u1 is the rear wheel coulomb friction torque.
Modeling a vehicle body part: c 1：1/N1 is the transmission shaft of the rear wheel to the body, and N 1 is the transmission shaft rigidity. The converter TF: k 4 denotes a conversion of the rear wheel angular velocity into the vehicle body linear velocity, and the conversion ratio is the wheel radius k 4. I: m is an energy storage element representing the weight of the vehicle body, and m is the weight of the vehicle body.Is a sensor mounted on a vehicle body, whereinIs the vehicle body line speed. C 2：1/N2 is the drive shaft from the body to the front wheels, and N 2 is the drive shaft stiffness.
Front wheel partial modeling: i: j 3 is the moment of inertia of the front wheel.Is a front-wheel mounted sensor, whereinIs the front wheel angular velocity. R: r 4 is the friction force of the front wheel, and the specific form is as follows:
Where k f2 is the front wheel viscous friction coefficient and F u2 is the front wheel coulomb friction torque.
The electric scooter BDBG model shown in fig. 3 and 4 selects different terminal nodes based on the DBG model, and performs causal division on part of keys to obtain a BDBG model.
Specifically, the and flow sensor in the DBG model of FIG. 2Alternative to potential-flow source Representing the potential source at the rear wheel-the source of the current,Representing the potential source-the source of the current at the body,Representing the potential source at the front wheel-the source of the current. Using a flow amplifier AF: k 5 performs the conversion of the potential (flow) relationship. Meanwhile, nonlinear elements C 1、R4, I are selected: m is terminal node receiving potential and flow information, separating the keys on the path from the potential source-flow source to the terminal node, and obtaining terminal nodes shown in fig. 3 as C 1、R4 and I: and BDBG model of m. Similarly, the nonlinear element R 3、R4、C2 is selected to receive the potential and flow information for the end node, and a BDBG model for the end node R 3、R4、C2 shown in fig. 4 can be obtained.
In step S2, based on the DBG model and BDBG model, an analytical redundancy relation ARRs is obtained, and a residual error is generated according to the analytical redundancy relation ARRs, so as to obtain a collective fault feature matrix, and the specific deduction process is as follows:
Considering the DBG model of fig. 2, according to the sensor causal relationship inversion method, considering the constitutive relationship of nodes 1 3、14 and 1 5 of the sensor connection, eliminating the unknown variable in the constitutive relationship with the known variable can obtain the following ARRs:
Wherein, Is the rear wheel angular acceleration,Is the acceleration of the vehicle body,Is the front wheel angular acceleration.
According to the electric scooter BDBG model of fig. 3, terminal node I: m has the following causal path:
uin→f1→f2→f4→e5→e8→e9→e15→e17→e18→e19
meanwhile, I: the constitutive relation of m is as follows:
From the constitutive relationship of node 1 5, we can obtain:
e19＝e18-e23 (1.10)
Wherein,
Simultaneous (1.7) -formula (1.11), can be obtained:
similarly, analysis terminal node And R: the causal path for R 4 is available:
According to the electric scooter BDBG model of fig. 3, the terminal node R is analyzed: r 4, And R: the causal path of R 3 can be obtained:
From the above ARRs deductions, considering that the expression of ARR 1 and ARR 5、ARR9 are consistent and the expression of ARR 3 and ARR 6、ARR7 are consistent, the ARRs relational expression of the electric scooter system is shown in formula (1.18).
By performing numerical estimation on ARRs, a residual r i can be generated, and fault detection can be performed according to whether the residual exceeds a threshold. Defining a binary coherence vector CV= [ o 1 o2 o3 o4 o5 ] according to the residuals, wherein the element determination rule in CV is as follows:
where ρ i represents the threshold for the ith residual. Because each ARR has sensitivity to different faults, the occurrence position of the fault can be determined according to the CV value, when the CV is not equal to zero vector, the fault of the system is indicated, and after the fault is detected, fault isolation is carried out.
In order to achieve fault isolation, a collective fault feature matrix is constructed according to the relationship between the plurality of ARRs derived in step S2 and the system parameters, as shown in the following table 1:
TABLE 1
Wherein D b represents detectability, I b represents isolability, N 1 represents propeller shaft rigidity of the rear wheel to the vehicle body, N 2 represents propeller shaft rigidity of the vehicle body to the front wheel,Representing a sensor mounted at the rear wheel,Representing a sensor mounted at the body of a vehicle,Representing the sensor mounted at the front wheel.
In addition, a fault feature matrix that can be obtained according to the related art, i.e., based on the bond map fault diagnosis method, is shown in the following table 2:
TABLE 2
As can be seen by comparing table 1 and table 2, the isolation of the aggregate fault feature matrix from faults is improved compared with the fault feature matrix. For example, whenWhen faults occur, residual errors are obtained through ARRs under the DBG model, the occurrence of faults can be detected through the obtained coherence vector CV= [ 1] 0, and the isolated possible fault parameter set is/>, by comparing fault feature matrixesUnder BDBG model, residual error is obtained according to ARRs to obtain coherence vector CV= [ 110 11 ], system failure is detected, and/> can be isolated by comparing the set failure feature matrixA fault occurred.
In step S3, according to the sliding mode variable structure control principle, a control law of the electric scooter system under the condition of no fault is designed, namely, a control law when a coherence vector CV is equal to a zero vector is designed to realize speed tracking of the electric scooter, and the specific steps are as follows:
S31, the electric scooter is driven by a direct current motor, so that linear proportional relations exist among all measurable variables, and a differential equation model of the electric scooter system can be obtained through equivalent substitution and linear transformation, wherein the differential equation model is as follows:
Wherein, For the actually measured angular velocity of the electric scooter rear wheel,Is the angular acceleration of the rear wheel of the electric scooter,Unknown interference for the system and satisfiesD is the upper boundary of system interference, g speciesIs a known function, and the specific form is as follows:
s32, defining the angular speed of the system relative to the expected rear wheel according to the sliding mode variable structure control method The tracking error of (2) is:
defining a slip-form surface The method comprises the following steps:
Wherein c > 0, deriving the time from both sides of the formula (1.23) and combining the formula (1.20) to obtain:
In the method, in the process of the invention, U no is the control law of the system under the condition of no fault for the expected rear wheel angular acceleration of the electric scooter.
The control law u no is:
where η 1 is a normal number, and the value of η 1 needs to be greater than the upper boundary of system interference.
S33, for the electric scooter system (1.20) to be considered, adopting a sliding mode function defined by a formula (1.23) and a control law designed by a formula (1.25), the system stability is proved as follows:
selecting a Liapunov function as:
The two sides of the formula (1.26) are derived from time to time:
substitution of formula (1.23) and formula (1.24) into formula (1.27) can be obtained:
Due to Taking eta 1 to be more than D, the following steps are obtained:
from the above analysis it can be seen that V 1 is negative and that the system satisfies the asymptotic stability, i.e. the system state is determined from the slip form surface, according to the Liapunov stability principle Starting from any other state, the slide will move to the slide surface.
In step S4, for a sensor failure that may occur in the electric scooter system, the sensor failure needs to be reconstructed, where the sensor includes: sensor mounted at rear wheelSensor mounted at the bodySensor mounted at front wheel
The sensor fault reconstruction method of the electric scooter system mainly comprises the following steps of:
s41, selecting the system state variable as The space state model of the electric scooter system can be established by the formula (1.18) as follows: /(I)
Wherein,As a state variable of the system,
Bp＝[0 b 0 0 0 0]T，
Wherein,
Nonlinear partThe method comprises the following steps:
for system output variables,/> Representing an unknown sensor fault vector. And satisfies the following:
||f(t)||≤θ (1.32)
wherein θ > 0.
S42, in order to convert the sensor fault into the fictitious actuator fault, a state z (t) E R 2 is introduced, which satisfies:
the construction of a new state space model is represented as follows:
In the method, in the process of the invention, For the augmented state vector, state z is the virtual output of the system described by equation (1.34)/>
B＝[0 0 0 0 b0 0 0 0]T，
S43, designing a neural network observer aiming at the system described by the formula (1.34), wherein the form is as follows:
In the method, in the process of the invention, The estimated values of x (t), y (t) and f (t) are respectively, L is an observer gain matrix to be designed, and the fault termOn-line estimation using radial basis function (Radial basis function, RBF) neural networks:
In the method, in the process of the invention, Is a weight vector having n numbers of network nodes. /(I) Is a neural network activation function,The method meets the following conditions:
Where c i is the center of the gaussian kernel and d i is the width of the gaussian kernel.
The RBF approximation fault term is formed by a neural network approximation principle as follows:
Wherein epsilon is the approximation error of the neural network and satisfies
Weight vectorThe following adaptive law is adopted: /(I)
Wherein, gamma > 0 is the parameter to be designed,For positive definite number matrix to be designed,To output the estimated error, and there is a positive definite matrix P εR 9*9 satisfying
S44, considering the electric scooter neural network observer (1.35), adopting a weight self-adaptive law of the formula (1.39), wherein the observer stability is proved as follows:
Presence of Lipschitz nonlinear function There is an arbitrary positive definite matrix P εR n*n, so that
In the method, in the process of the invention,
The state estimation error e x and the fault amount estimation error e f are defined as:
Deriving the time from both sides of formula (1.42) and combining formula (1.34) and formula (1.35) yields:
In the method, in the process of the invention,
Selecting Lyapunov function as:
deriving both sides of formula (1.44) over time and combining formula (38) yields:
In the method, in the process of the invention,
As a result of the fact that,
And because ofW * is an ideal weight vector, soThus, formula (44) andThe method can obtain the following steps:
Where λ 1 is the minimum eigenvalue of Γ.
As a result of:
So that:
Where lambda 2 is the maximum eigenvalue of P.
Therefore, whenTime,The observer described above is bounded convergent according to the principle of lispro stability. Thus, the sensor fault vector may be reconstructed as:
S45, according to the fault reconstruction result described by the formula (1.50), considering that the compensated rear wheel angular velocity is The differential equation model of the electric scooter system can be expressed as:
Wherein, Unknown interference for the system and satisfiesD 1 is the current upper system disturbance bound. g andHas the following form:
According to the sliding mode variable structure control method, the angular speed of the system relative to the expected rear wheel is defined The tracking error of (2) is expressed as:
Here, the slip-form surface is denoted as:
Wherein c 1 > 0.
Because buffeting occurs when the control law is switched, in order to enable the system to quickly recover and stabilize, the sliding mode control adopts an exponential approach law:
Where η 2 and q are two normal numbers, and the value of η 2 needs to be greater than the upper boundary of system interference. Substitution of formula (1.54) into formula (1.55) yields:
the fault-tolerant control law under the condition of system fault is designed as follows:
s46, regarding the electric scooter system (1.51) considered, the system stability is demonstrated as follows, using the sliding mode function defined by equation (1.54) and the control law designed by equation (1.57):
selecting a Liapunov function as:
the two sides of the formula (1.58) are derived from time to time:
substitution of formula (1.56) into formula (1.59) yields:
Due to Taking eta 2＞D1, the following steps are obtained:
From the above analysis it can be seen that V 3 is negative and that the system satisfies the asymptotic stability principle according to the li-apunov stability principle.
Aiming at possible parameter faults of the system, the main steps of the fault-tolerant control law under the condition of the parameter faults of the electric scooter system designed in the step S5 are as follows:
s51, in the running process of the electric scooter, considering that a system possibly has parameter faults, a differential equation model of the electric scooter under the parameter fault condition can be defined as:
In the method, in the process of the invention, Is an unknown fault function. When the system is normal,When there is a parameter failure in the system,
Here, the slip-form surface of the formula (1.23) can be used to obtain:
Where η 3 and q 2 are two normal numbers, and the value of η 3 needs to be greater than the upper boundary of system interference. Substitution of formula (1.62) into formula (1.63) yields:
the fault-tolerant control law under the condition of the design parameter fault is as follows:
In the method, in the process of the invention, ForIs a normal number, η 3 is a value of η 3, and is required to be greater than the upper boundary of system interference.
Utilizing extreme learning machine pairsAnd (3) performing real-time estimation:
In the method, in the process of the invention, The output weight vector estimated value of the output node is beta *, and according to the universal approximation theorem of the extreme learning machine, the optimal beta * exists, so that:
Where ε 2 is the approximation error, satisfy:
|ε2|≤εN (1.68)
Where ε N is the upper approximation error bound. H (z, ω, b) is L inputs in ELM and has Hidden layer output matrix of individual neurons:
Where z= [ z 1,z2,…,zL ] is an input vector, ω i＝[ω1,ω2,…,ωL]T is an input weight vector, and b i is the bias of the hidden layer node. G (·) is an activation function, in the following specific form:
S52, regarding the control law designed taking into consideration the electric scooter system (1.62), the system stability is proved as follows, using the formula (1.65):
output weight of extreme learning machine Updating is performed by the following adaptive law:
Where the adaptive gain σ > 0.
Substitution of formula (1.65) into formula (1.64) yields:
In the method, in the process of the invention,
Definition:
selecting a Liapunov function as
The two sides of the formula (1.75) are derived from time to time:
Due to Substitution of formula (1.71) into formula (1.76) can be obtained
Due toTaking η 3＞D+εN, we can get ∈ 2|≤εN:
From the above analysis it can be seen that V 4 is negative and that the system satisfies the asymptotic stability principle according to the li-apunov stability principle.
In step S6, according to the above designed fault diagnosis module and control law, the electric scooter active fault-tolerant controller operates as follows:
When the defined binary coherence vector CV is a non-zero vector, the occurrence of a fault is detected. By comparing the non-zero coherence vector with the integrated fault signature matrix, a possible fault is isolated.
If the sensor fault of the system is detected, the system is switched from the control law u no to the fault-tolerant control law u f1 under the fault-free condition; if the system is detected to have a parameter fault, the system is switched from the sliding mode control law u no to the fault-tolerant control law u f2 under the fault-free condition, so that the active fault-tolerant control of the electric scooter system is realized.
In this embodiment, in order to verify the effectiveness of the proposed electric scooter active error-tolerant control method, simulation analysis is performed, and simulation experiment results are as follows:
Taking the fault of the rear wheel angular velocity sensor as an example, the simulation time is 12s, and the sudden change fault of the rear wheel sensor is injected at the 6 th s. According to table 1, the corresponding o 1＝1,o2＝1,o3＝0,o4＝1,o5 =1 corresponds to the detection results of fig. 6 to 10, and the broken line in fig. 6 to 10 indicates a given threshold value. Fig. 11 shows the angular velocity response of the rear wheels of the electric scooter, and fig. 12 shows the sensor failure reconstruction result.
The above embodiments are merely preferred embodiments of the present invention and are not intended to limit the present invention, and any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of the present invention.

Claims (10)

1. An active fault-tolerant control method of an electric scooter system is characterized by comprising the following steps:
S1, establishing a diagnosis bonding graph model of an electric scooter system, namely a DBG model, and obtaining a double causal relationship diagnosis bonding graph model, namely a BDBG model through separation causal line;
S2, deducing an analytic redundancy relation ARRs of the system based on BDBG model, generating residual errors by ARRs to obtain an aggregate fault feature matrix for fault detection and fault isolation;
s3, designing a sliding mode control law u no under the condition that an electric scooter system has no fault, and realizing speed tracking of the electric scooter;
S4, aiming at the sensor fault of the electric scooter system, designing a neural network observer, reconstructing the sensor fault, and designing a fault-tolerant control law u f1 under the condition of the sensor fault of the electric scooter system based on the result of the sensor fault reconstruction;
S5, aiming at the parameter faults of the electric scooter system, constructing a fault-tolerant control law u f2 under the parameter fault conditions of the electric scooter system, and carrying out real-time estimation on control items for compensating unknown parameter faults in the fault-tolerant control law u f2 by utilizing an extreme learning machine;
S6, designing an active fault-tolerant controller of the electric scooter system, and switching control laws in real time according to fault detection and fault isolation results and aiming at different fault categories, wherein the active fault-tolerant controller is used for realizing active fault-tolerant control of the electric scooter system and specifically comprises the following steps of:
If the system does not have a fault, the control law u no is adopted to realize the speed tracking of the electric scooter under the fault-free condition; if the sensor fault of the system is detected, the system is switched from the control law u no to the fault-tolerant control law u f1 under the condition of no fault; if a parameter failure of the system is detected, the system will switch from the fault-free control law u no to the fault-tolerant control law u f2.
2. The method for active fault-tolerant control of an electric scooter system according to claim 1, wherein the DBG model and BDBG model of the electric scooter system established in step S1 are specifically as follows:
Modeling an electric scooter system by adopting a bonding diagram theory, wherein bonding diagram elements in a DBG model of the electric scooter system comprise: potential source Se, current source Sf, resistive element { R 1,R2,…,Rj }, capacitive element { C 1,C2,…,Cm }, inertial element { I 1,I2,…,Ii }, flow sensor { Df 1,Df2,…,Dfn }, j, m, I, n represent the number of resistive elements, capacitive elements, inertial elements, and flow sensors, respectively, in the system; the DBG model of the electric scooter system comprises: a direct current motor driver part, a direct current motor part, a rear wheel part, a vehicle body part and a front wheel part;
The BDBG model of the electric scooter system is obtained based on a DBG model, and the specific mode is as follows: converting a source Sf and a source Se of a potential source in the DBG model into a source-source SS, distributing an additional bonding picture element stream amplifier AF and a potential amplifier AE into the DBG model, and distributing double causal relations to all keys connected with the sensor so that the potential and the stream point to nodes connected with the sensor; meanwhile, selecting part of nonlinear elements to be distributed as terminal nodes, and receiving potential and flow information; for type 1 nodes and type 0 nodes, only one of the keys connected to the node has a causal score at the end remote from the node, all other keys have a causal score at the end near the node, or only one of the keys has a causal score at the end near the node, all other keys have a causal score at the end remote from the node.
3. The method for active fault-tolerant control of an electric scooter system according to claim 2, wherein in step S2, the derived analytical redundancy relation ARRs and the integrated fault feature matrix are:
Based on the DBG model and BDBG model of the electric scooter system, the following ARRs is obtained by a causal relation inversion method and analyzing causal paths and constitutive relations of all terminal nodes:
ARRq(Ξ,De,Df,uin)＝0，q＝1，2，…，b (1)
wherein b represents the number of analytical redundancy relations ARRs of the system, xi= [ xi 1...Ξn]T ] represents system parameters, de and Df represent potential sensors and flow sensors in the DBG model, respectively;
u in represents the input of the system, and is selected as a slip form control law u no under the condition that the electric scooter system has no fault, or a fault-tolerant control law u f1 under the condition that the electric scooter system sensor has a fault, or a fault-tolerant control law u f2 under the condition that the electric scooter system parameter has a fault;
Generating residual error r i by carrying out numerical estimation on ARRs, carrying out fault detection according to whether the residual error r i exceeds a corresponding threshold epsilon i, and defining a binary coherence vector CV= [ o 1 o2 … ob ] to represent the consistency of all residual errors; wherein the number of elements in the coherence vector CV is b, and the number of elements in the coherence vector CV is consistent with the number of ARRs; the element determination rules in CV are as follows:
Where ρ i represents the threshold for the i-th residual r i;
The aggregate fault signature matrix is constructed according to ARRs of the system as shown in table 1 below:
TABLE 1
In table 1, [ α 1,α2,…,αp ] represents a set of possible faults; q hl represents the relationship between the h possible fault, α h, and the l ARR, ARR l, h=1, 2, … p, l=1, 2, … b, q hl =1 represents ARR l is sensitive to α h, q hl =0 represents ARR l is not sensitive to α h;
In the running process of the system, when the occurrence of the fault is detected, the position where the fault occurs is isolated by comparing the coherence vector CV with the integrated fault characteristic matrix, and the fault type is determined.
4. The method for active fault-tolerant control of electric scooter system according to claim 3, wherein in step S3, a slip form control law u no under the condition of no fault of the electric scooter system is designed, specifically as follows:
S31, the electric scooter is driven by a direct current motor, so that linear proportional relations exist among all measurable variables, and a differential equation model of the electric scooter system is obtained through equivalent substitution and linear transformation and is as follows:
Wherein g and Is a known nonlinear function; /(I)For actually measuring the angular velocity of the rear wheel of the electric scooter,For actually measuring the rear wheel angular acceleration of the electric scooter,Unknown interference for the system and satisfiesD is the upper boundary of system interference;
s32, defining the angular speed of the system relative to the expected rear wheel according to the sliding mode variable structure control method The tracking error of (2) is:
defining a slip-form surface The method comprises the following steps:
Wherein c >0, deriving the time from both sides of formula (5) and combining formula (4) to obtain:
In the method, in the process of the invention, To the desired rear wheel angular velocityIs representative of the desired rear wheel angular acceleration;
The slip form control law u no under the condition of no fault of the electric scooter system is designed as follows:
Where η 1 is a normal number, η 1 is greater than the upper boundary of system interference, and sign (·) is a sign function.
5. The method for active fault-tolerant control of electric scooter system according to claim 4, wherein in step S3, for the differential equation model of the electric scooter system shown in formula (3), a sliding mode function defined in formula (5) and a sliding mode control law u no of the electric scooter system designed in formula (7) under no fault condition are adopted to prove the stability of the closed loop system, as follows:
selecting a Liapunov function as:
the two sides of the formula (8) are derived from time to obtain:
substitution of formula (5) and formula (6) into formula (9) yields:
Due to Taking η 1 > D, we can get:
from the above evidence can be obtained Is negatively determined, and according to the Liapunov stability principle, the electric scooter system meets the asymptotic stability, namely the system state is from the sliding mode surfaceStarting from any other state, the slide will move to the slide surface.
6. The method for active fault-tolerant control of electric scooter system according to claim 4, wherein in step S4, a fault-tolerant control law u f1 under the fault condition of the sensor of the electric scooter system is designed, specifically as follows:
The sensor comprises: sensor mounted at rear wheel Sensor mounted at the bodySensor mounted at front wheel
S41, selecting a system state variable of x p＝[x1 x2 x3 x4 x5 x6]T,x2 as the rear wheel angular speed, and x 2 as the derivative of x 1; x 4 is the body line speed, x 4 is the derivative of x 3; x 6 is the front wheel angular velocity, x 6 is the derivative of x 5;
the space state model of the electric scooter system can be established by the formula (1) as follows:
wherein Q u∈R6*2、Ap∈R6*6、Bp∈R6、Cp∈R3*6 is a matrix function, Representing the nonlinear part of the system; y p is a system output vector, f (t) is a sensor fault vector, and t represents time; and satisfies the following:
Wherein,
S42, converting the sensor fault into fictitious actuator fault, introducing a state z epsilon R 2, and meeting the following conditions:
the construction of a new state space model is represented as follows:
In the method, in the process of the invention, For the augmented state vector, y (t) is the transformed system output, state z is the virtual output of the system described by equation (15), and Q εR 9*2、A∈R9*9、B∈R9、C∈R3*9、F∈R9*3 is the matrix function;
s43, designing a neural network observer aiming at the fault system described by the formula (15), wherein the form is as follows:
In the method, in the process of the invention, The estimated values of x (t), y (t) and f (t) are respectively, L is an observer gain matrix to be designed, and the fault termOn-line estimation using radial basis function neural network:
In the method, in the process of the invention, Is a weight vector having n number of network nodes; /(I) ' Is a neural network activation function,Is a gaussian function, namely:
Wherein c i is the center of the Gaussian kernel function, and d i is the width of the Gaussian kernel function;
The RBF approximation fault term is obtained by a neural network approximation principle and is as follows:
Wherein epsilon is the approximation error of the neural network and satisfies Is the upper bound of the approximation error of the neural network;
weight vector The following adaptive law is adopted:
Wherein, gamma > 0 is the parameter to be designed, For positive definite number matrix to be designed,To output the estimated error, and there is a positive definite matrix P εR 9*9 satisfying
S44, according to the reconstruction result of the sensor fault vector, considering the compensated rear wheel angular velocity as The differential equation model of the electric scooter system is expressed as:
Wherein, AndIs a known nonlinear function; /(I)Unknown interference for the system and satisfiesD 1 is the current system interference upper bound;
according to the sliding mode variable structure control method, the system is at the moment relative to the expected angular speed of the rear wheel The tracking error e 1 of (a) can be expressed as:
at this time, the sliding surface The method comprises the following steps:
Wherein c 1 > 0;
Because buffeting occurs when the control law is switched, in order to enable the system to quickly recover and stabilize, the sliding mode control adopts an exponential approach law:
Wherein, eta 2 and q 1 are two normal numbers, and the value of eta 2 is required to be larger than the upper boundary of system interference;
substitution of formula (21) into formula (23) yields:
The fault-tolerant control law u f1 under the fault condition of the sensor of the electric scooter is designed as follows:
7. The method for active fault-tolerant control of electric scooter system according to claim 6, wherein in step S4, for the electric scooter neural network observer, formula (16), a weight adaptive law of formula (20) is adopted to prove the stability of the neural network observer, specifically as follows:
Presence of Lipschitz nonlinear function There is an arbitrary positive definite matrixSo that
In the method, in the process of the invention,
The state estimation error e x and the fault amount estimation error e f are defined as:
deriving the time from both sides of formula (28) and combining formulas (15) and (16) can obtain:
In the method, in the process of the invention,
Selecting Lyapunov function as:
The two sides of the formula (31) are derived for time and the formula (30) is combined to obtain:
In the method, in the process of the invention,
As a result of the fact that,
And because ofIs an ideal weight vector, so
Thus, the combination of (20) andThe method comprises the following steps:
Wherein λ 1 is the minimum eigenvalue of Γ;
as a result of:
So that:
Wherein lambda 2 is the maximum eigenvalue of P;
therefore, when Time,The observer is bounded convergent according to the principle of lisinov stability; therefore, the reconstruction result of the sensor fault vector is
8. The method for active fault-tolerant control of electric scooter system according to claim 6, wherein in step S4, for the differential equation model of the electric scooter system shown in formula (21), the stability of the electric scooter system is proved by adopting a sliding mode function defined in formula (23) and a fault-tolerant control law u f1 under the fault condition of the sensor of the electric scooter system designed in formula (26), as follows:
selecting a Liapunov function as:
the two sides of the formula (38) are derived from time to time:
substitution of formula (23) and formula (24) into formula (38) can be obtained:
Due to Taking eta 2＞D1, the following steps are obtained:
According to the above, V 3 is proved to be negative, and the electric scooter system meets asymptotic stability according to the Liapunov stability principle.
9. The method for active fault-tolerant control of electric scooter system according to claim 4, wherein in step S5, a fault-tolerant control law u f2 under the fault condition of the electric scooter system parameter is designed, specifically as follows:
In the running process of the electric scooter, considering that the system possibly has parameter faults, the differential equation model of the electric scooter under the parameter fault condition is defined as follows:
In the method, in the process of the invention, Is an unknown fault function; when the system is normal,When there is a parameter failure in the system,
Slip form surface using (5)The method comprises the following steps:
Wherein η 3 and q 2 are two normal numbers, and the value of η 3 needs to be larger than the upper boundary of system interference; substituting formula (41) into formula (42) yields:
In the method, in the process of the invention, ForIs a function of the estimated value of (2);
The fault-tolerant control law u f2 under the condition of the parameter fault of the electric scooter is designed as follows:
Utilizing extreme learning machine pairs And (3) performing real-time estimation:
In the method, in the process of the invention, The output weight vector estimated value of the output node is beta *, and according to the universal approximation theorem of the extreme learning machine, the optimal beta * exists, so that:
Where ε 2 is the approximation error, satisfy:
|ε2|≤εN (47)
Where ε N is the upper approximation error bound; h (z, ω, b) is L inputs in ELM and has Hidden layer output matrix of individual neurons:
wherein, z= [ z 1,z2,…,zL ] is an input vector, ω i＝[ω1,ω2,…,ωL]T is an input weight vector, and b i is the bias of hidden layer nodes; g (·) is an activation function, in the following specific form:
。
10. The method for active fault-tolerant control of electric scooter system according to claim 9, wherein in step S5, for the differential equation model of the electric scooter system shown in formula (41), fault-tolerant control law u f2 under the condition of failure of the electric scooter system parameters designed in formula (44) is adopted to prove the stability of the electric scooter system, as follows:
output weight of extreme learning machine Updating is performed by the following adaptive law:
wherein, the adaptive gain sigma is more than 0;
substitution of formula (44) into formula (43) yields:
In the method, in the process of the invention,
Definition:
selecting a Liapunov function as:
The two sides of the formula (54) are derived from time to time:
Due to Substituting formula (50) into formula (55) yields:
Due to Taking η 3＞D+εN, we can get ∈ 2|≤εN:
according to the above, V 4 is proved to be negative, and the electric scooter system meets asymptotic stability according to the Liapunov stability principle.
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