CN107662208B - Flexible joint mechanical arm finite time self-adaptive backstepping control method based on neural network - Google Patents

Flexible joint mechanical arm finite time self-adaptive backstepping control method based on neural network Download PDF

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CN107662208B
CN107662208B CN201710732672.5A CN201710732672A CN107662208B CN 107662208 B CN107662208 B CN 107662208B CN 201710732672 A CN201710732672 A CN 201710732672A CN 107662208 B CN107662208 B CN 107662208B
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neural network
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CN107662208A (en
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陈强
施卉辉
孙明轩
何熊熊
庄华亮
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Zhejiang University of Technology ZJUT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J17/00—Joints
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/161—Hardware, e.g. neural networks, fuzzy logic, interfaces, processor
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1638—Programme controls characterised by the control loop compensation for arm bending/inertia, pay load weight/inertia

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Abstract

A flexible joint mechanical arm finite time self-adaptive backstepping control method based on a neural network is designed by aiming at a flexible joint mechanical arm containing unknown uncertain items and utilizing the neural network and the finite time control method. In each step of the backstepping control, the self-adaptive finite time virtual controller is provided to realize that the system tracking error converges to the area near the balance point in a finite time. Two simple neural networks are applied to approximate and compensate the uncertain items of the system, and a large amount of calculation in the traditional backstepping control is reduced. The invention provides a control method which can compensate unknown uncertainty of a system, solve the problem of large calculation amount of the traditional backstepping control and realize the convergence of the tracking error of the system in limited time and the tracking of the system in limited time.

Description

Flexible joint mechanical arm finite time self-adaptive backstepping control method based on neural network
Technical Field
The invention relates to a finite time self-adaptive backstepping control method of a flexible joint mechanical arm based on a neural network, in particular to a flexible joint mechanical arm control method with unknown uncertainty.
Background
The mechanical arm has the advantages of flexible action, small movement inertia, high working efficiency, stability, reliability and the like, has prominent effect in actual life, and is particularly widely applied in the field of high precision, such as industrial design, aerospace, medical appliances and the like. With the development of science and technology, people have higher and higher precision requirements on the mechanical arm, but the control performance and the technical level limitation of the mechanical arm are seriously influenced by the actually existing complex uncertain factors in the mechanical arm. In order to meet the requirements of higher precision and performance, the flexibility of the mechanical arm joint is considered, and a flexible joint mechanical arm is adopted in the modeling and design control method. For flexible joint mechanical arms, researchers have proposed many control methods, such as adaptive control, fuzzy control, sliding mode control, and backstepping control.
The backstepping control method is a nonlinear system design method, and the basic idea is to decompose a complex nonlinear system into subsystems with the order not exceeding the system order, then design a virtual controller in each subsystem respectively, and back to the whole system until the design of the whole control law is completed. A controller of the flexible joint mechanical arm is designed by utilizing a backstepping control technology, so that the problem of non-matching uncertainty in a system can be solved.
Although many control strategies can effectively solve the tracking control of the flexible joint mechanical arm, most of the control strategies can only ensure that the system state error is finally and consistently bounded. In order to ensure that the system is stable within a limited time, limited time control is adopted. The finite time control is successfully applied to a plurality of control fields such as a mechanical arm system, a spacecraft system, a multi-agent system, a permanent magnet synchronous motor system and the like, is a control technology based on a finite time stability theory, can improve the robust performance of the system, and ensures that the system reaches a stable state in a finite time. Neural networks can approximate a position function within arbitrary precision, and therefore, are widely used to solve the problem of system uncertainty. The control method application of the above control strategy has certain limitations, or each state error variable cannot be guaranteed to be converged in a limited time, or a system model must be known.
Disclosure of Invention
In order to overcome the problem of unknown uncertainty of the flexible joint mechanical arm, the invention provides a flexible joint mechanical arm finite time self-adaptive backstepping control method based on a neural network.
The technical scheme proposed for solving the technical problems is as follows:
a flexible joint mechanical arm finite time self-adaptive backstepping control method based on a neural network comprises the following steps:
step 1, establishing a dynamic model of the flexible joint mechanical arm, and initializing a system state, sampling time and control parameters, wherein the process comprises the following steps:
1.1 the dynamic model expression form of an n-order flexible joint mechanical arm is as follows:
Figure GDA0002473393710000021
wherein q ∈ Rn,θ∈RnRespectively are a joint position vector and a motor position vector, and n is the order of the system;
Figure GDA0002473393710000022
is the joint acceleration vector;
Figure GDA0002473393710000023
is motor acceleration vector, M (q) ∈ Rn×nJ ∈ R as an unknown nonsingular symmetric positive definite matrix representing the inertia of the jointn×nK ∈ R as an unknown nonsingular symmetrical positive definite matrix representing motor inertian×nIs an unknown diagonal positive definite matrix representing the joint spring stiffness h (q, theta) ∈ Rn×nU ∈ R as a function of centripetal, Coriolis, and gravitational accelerationnRepresenting a control torque vector;
1.2 redefine the variables, writing equation (1) in the form of a state space equation:
definition of x1＝q,
Figure GDA0002473393710000024
x3＝θ,
Figure GDA0002473393710000025
Equation (1) is written in the following state space form:
Figure GDA0002473393710000026
wherein xiI is measurable at 1,2,3,4Of M (x)1)，h(x1,x2) K and J are both unknown terms;
step 2, calculating the tracking error of the system, wherein the process is as follows:
the system tracking error is defined as follows:
z1＝x1-xd(3)
wherein xdIs a given smoothly bounded reference trajectory;
the derivation of equation (3) yields:
Figure GDA0002473393710000031
wherein
Figure GDA0002473393710000032
Is the first derivative of the error and is,
Figure GDA0002473393710000033
is the first derivative of the reference trajectory;
step 3, defining an error variable, and designing a virtual controller, wherein the process is as follows:
3.1 define error variables as:
zj＝xj-aj-1,j＝2,3,4 (5)
wherein, aj-1J is 2,3,4 is a virtual controller in the design controller process;
derivation of equation (5) yields:
Figure GDA0002473393710000034
wherein
Figure GDA0002473393710000035
Is the first derivative of the error and is,
Figure GDA0002473393710000036
j is 2,3,4 is the first derivative of the virtual controller in the design controller process;
substituting formulae (2) and (5) into formulae (4) and (6) to obtain:
Figure GDA0002473393710000037
3.2 definition of
Figure GDA0002473393710000038
Wherein
Figure GDA0002473393710000039
Is to approach
Figure GDA00024733937100000310
And
Figure GDA00024733937100000311
the following two neural networks were designed:
definition of
Figure GDA00024733937100000312
Is an ideal weight matrix of the neural network, and m is the number of the neurons; then
Figure GDA00024733937100000313
Is approximated as follows:
Figure GDA00024733937100000314
Figure GDA00024733937100000315
wherein
Figure GDA00024733937100000316
Is a basis function of the neural network;1,2representing approximation error of neural network and satisfying | non-calculation1||≤1N,||2||≤2N；1N，2NIs a positive constant, | · | | | represents a two-norm of the value;
Figure GDA00024733937100000317
and
Figure GDA00024733937100000318
the form of (A) is as follows:
Figure GDA0002473393710000041
Figure GDA0002473393710000042
wherein, al,bl,cl,dlAre all constant parameters, l ═ 1, 2;
3.3 designing neural network weight and estimation error updating law:
Figure GDA0002473393710000043
Figure GDA0002473393710000044
Figure GDA0002473393710000045
Figure GDA0002473393710000046
wherein
Figure GDA0002473393710000047
Is a positive definite diagonal matrix; sigma1,σ2,ρ1,ρ2Are suitable parameters;
Figure GDA0002473393710000048
and
Figure GDA0002473393710000049
are respectively
Figure GDA00024733937100000410
And
Figure GDA00024733937100000411
an estimated value of (d);
Figure GDA00024733937100000412
and
Figure GDA00024733937100000413
are respectively1NAnd2Nan estimated value of (d);
3.4 design virtual controller, as follows:
Figure GDA00024733937100000414
wherein h is1,h2,h3,k1,k2,k3Is a normal number;
3.5 design the actual controller as follows:
Figure GDA00024733937100000415
wherein h is4，k4Is a normal number;
3.6 substituting formula (8), formula (9), formula (16) and formula (17) into formula (7) yields:
Figure GDA0002473393710000051
step 5, designing the Lyapunov function into the following form:
Figure GDA0002473393710000052
wherein
Figure GDA0002473393710000053
Derivation of equation (19) and substitution of (18) yields:
Figure GDA0002473393710000054
if equation (20) is written as
Figure GDA0002473393710000055
η therein1＝Min{2h1,2h2λmin{M-1(x1)K},2h3,2h4λmin{J-1}}，
η2＝Min{2αk1,(2M-1(x1)K)αk2,2αk3,(2J-1)αk4}，
＝Z2NW1NΦ1N+Z4NW2NΦ2N+Z2NE1μ+Z4NE2μ,
Where Min {. cndot.) represents a minimum value, λmin{. denotes the minimum eigenvalue; z2NAnd Z4NRespectively represent | | z2And z4The maximum value of | l; w1NAnd W2NRespectively represent
Figure GDA0002473393710000056
And
Figure GDA0002473393710000057
maximum value of (d); phi1NAnd phi2NRespectively represent
Figure GDA0002473393710000058
And
Figure GDA0002473393710000059
maximum value of (d); e1μAnd E2μRespectively represent
Figure GDA00024733937100000510
And
Figure GDA00024733937100000511
maximum value of (d);
it is determined that the system tracking error has a finite time to converge into a domain near the equilibrium point.
The invention designs the finite-time self-adaptive backstepping control method of the flexible joint mechanical arm based on the neural network based on the flexible joint mechanical arm containing unknown uncertainty items and combining the self-adaptive backstepping control, the neural network and the finite-time control method, solves the problem of uncertainty in a system, reduces the calculated amount in the traditional backstepping control, and realizes the finite-time convergence of the tracking error of the system.
The technical conception of the invention is as follows: the method is characterized in that self-adaptive backstepping control is designed aiming at the flexible joint mechanical arm containing unknown uncertainty items, and a flexible joint mechanical arm finite time self-adaptive backstepping control method based on a neural network is designed by utilizing the neural network and a finite time control method. In each step of the backstepping control, the self-adaptive finite time virtual controller is provided to realize that the system tracking error converges to the area near the balance point in a finite time. Two simple neural networks are applied to approximate and compensate the uncertain items of the system, and a large amount of calculation in the traditional backstepping control is reduced. The invention provides a control method which can compensate unknown uncertainty of a system, solve the problem of large calculation amount of the traditional backstepping control and realize the convergence of the tracking error of the system in limited time and the tracking of the system in limited time.
The invention has the beneficial effects that: the unknown uncertainty of the system is compensated, a large amount of calculation amount of the traditional backstepping control is reduced, the convergence of the tracking error of the system in limited time is realized, and the limited time tracking of the system is realized.
Drawings
FIG. 1 is a graph of the tracking effect of the present invention;
FIG. 2 is a tracking error map of the present invention;
FIG. 3 is a diagram of the neural network approximation unknowns of the present invention;
FIG. 4 is a graph of the weight norm of the neural network approximation of the present invention;
FIG. 5 is a state variable diagram of the present invention;
FIG. 6 is a control input diagram of the present invention;
FIG. 7 is a control flow diagram of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1-7, a finite-time adaptive backstepping control method of a flexible joint mechanical arm based on a neural network comprises the following steps:
step 1, establishing a dynamic model of the flexible joint mechanical arm, and initializing a system state, sampling time and control parameters, wherein the process comprises the following steps:
1.1 the dynamic model expression form of an n-order flexible joint mechanical arm is as follows:
Figure GDA0002473393710000061
wherein q ∈ Rn,θ∈RnRespectively are a joint position vector and a motor position vector, and n is the order of the system;
Figure GDA0002473393710000071
is the joint acceleration vector;
Figure GDA0002473393710000072
is motor acceleration vector, M (q) ∈ Rn×nJ ∈ R as an unknown nonsingular symmetric positive definite matrix representing the inertia of the jointn×nK ∈ R as an unknown nonsingular symmetrical positive definite matrix representing motor inertian×nIs an unknown diagonal positive definite matrix representing the joint spring stiffness h (q, theta) ∈ Rn×nU ∈ R as a function of centripetal, Coriolis, and gravitational accelerationnRepresenting a control torque vector;
1.2 redefine the variables, writing equation (1) in the form of a state space equation:
definition of x1＝q,
Figure GDA0002473393710000073
x3＝θ,
Figure GDA0002473393710000074
Equation (1) is written in the following state space form:
Figure GDA0002473393710000075
wherein xiI-1, 2,3,4 are all measurable, M (x)1)，h(x1,x2) K and J are both unknown terms;
step 2, calculating the tracking error of the system, wherein the process is as follows:
the system tracking error is defined as follows:
z1＝x1-xd(3)
wherein xdIs a given smoothly bounded reference trajectory;
the derivation of equation (3) yields:
Figure GDA0002473393710000076
wherein
Figure GDA0002473393710000077
Is the first derivative of the error and is,
Figure GDA0002473393710000078
is the first derivative of the reference trajectory;
step 3, defining an error variable, and designing a virtual controller, wherein the process is as follows:
3.1 define error variables as:
zj＝xj-aj-1,j＝2,3,4 (5)
wherein, aj-1J is 2,3,4 is a virtual controller in the design controller process;
derivation of equation (5) yields:
Figure GDA0002473393710000079
wherein
Figure GDA00024733937100000710
Is the first derivative of the error and is,
Figure GDA00024733937100000711
j is 2,3,4 is the first derivative of the virtual controller in the design controller process;
substituting formulae (2) and (5) into formulae (4) and (6) to obtain:
Figure GDA0002473393710000081
3.2 definition of
Figure GDA0002473393710000082
Wherein
Figure GDA0002473393710000083
Is to approach
Figure GDA0002473393710000084
And
Figure GDA0002473393710000085
the following two neural networks were designed:
definition of
Figure GDA0002473393710000086
Is an ideal weight matrix of the neural network, and m is the number of the neurons; then
Figure GDA0002473393710000087
Is approximated as follows:
Figure GDA0002473393710000088
Figure GDA0002473393710000089
wherein
Figure GDA00024733937100000810
Is a basis function of the neural network;1,2representing approximation error of neural network and satisfying | non-calculation1||≤1N,||2||≤2N；1N，2NIs a positive constant, | · | | | represents a two-norm of the value;
Figure GDA00024733937100000811
and
Figure GDA00024733937100000812
the form of (A) is as follows:
Figure GDA00024733937100000813
Figure GDA00024733937100000814
wherein, al,bl,cl,dlAre all constant parameters, l ═ 1, 2;
3.3 designing neural network weight and estimation error updating law:
Figure GDA00024733937100000815
Figure GDA00024733937100000816
Figure GDA00024733937100000817
Figure GDA0002473393710000091
wherein
Figure GDA0002473393710000092
Is a positive definite diagonal matrix; sigma1,σ2,ρ1,ρ2Are suitable parameters;
Figure GDA0002473393710000093
and
Figure GDA0002473393710000094
are respectively
Figure GDA0002473393710000095
And
Figure GDA0002473393710000096
an estimated value of (d);
Figure GDA0002473393710000097
and
Figure GDA0002473393710000098
are respectively1NAnd2Nan estimated value of (d);
3.4 design virtual controller, as follows:
Figure GDA0002473393710000099
wherein h is1,h2,h3,k1,k2,k3Is a normal number;
3.5 design the actual controller as follows:
Figure GDA00024733937100000910
wherein h is4，k4Is a normal number;
3.6 substituting formula (8), formula (9), formula (16) and formula (17) into formula (7) yields:
Figure GDA00024733937100000911
step 5, designing the Lyapunov function into the following form:
Figure GDA00024733937100000912
wherein
Figure GDA00024733937100000913
Derivation of equation (19) and substitution of (18) yields:
Figure GDA0002473393710000101
if equation (20) is written as
Figure GDA0002473393710000102
η therein1＝Min{2h1,2h2λmin{M-1(x1)K},2h3,2h4λmin{J-1}}，
η2＝Min{2αk1,(2M-1(x1)K)αk2,2αk3,(2J-1)αk4}，
＝Z2NW1NΦ1N+Z4NW2NΦ2N+Z2NE1μ+Z4NE2μ,
Where Min {. cndot.) represents a minimum value, λmin{. denotes the minimum eigenvalue; z2NAnd Z4NRespectively represent | | z2And z4The maximum value of | l; w1NAnd W2NRespectively represent
Figure GDA0002473393710000103
And
Figure GDA0002473393710000104
maximum value of (d); phi1NAnd phi2NRespectively represent
Figure GDA0002473393710000105
And
Figure GDA0002473393710000106
maximum value of (d); e1μAnd E2μRespectively represent
Figure GDA0002473393710000107
And
Figure GDA0002473393710000108
maximum value of (d);
it is determined that the system tracking error has a finite time to converge into a domain near the equilibrium point.
In order to verify the effectiveness of the method, the method carries out simulation verification on the flexible joint mechanical arm of one joint. The system initialization parameters are set as follows:
the parameters of the neural network basis functions are as follows: a is1＝10,b1＝2,c1＝1,d1＝-1.8，a2＝5,b2＝0.2,c2＝1,d2The update law of the neural network weight and the error estimation is as follows:1＝0.05,2＝0.1,σ1＝0.1,σ2＝0.1,γ1＝1,γ2＝0.5,ρ1＝5,ρ2the coefficients of the virtual controller are as follows: h is1＝4,h2＝0.8,h3＝2,h4＝5,k1＝2.2,k2＝0.3,k3＝3,k 45, α, 13/15, and xdSin (t); initial value given x of systemd(0)＝0,x1(0)＝x2(0)＝x3(0)＝x4(0) The weight and the initial value of the estimation error are selected to be distributed in [ -1,1 [ -0 ]]An arbitrary vector of (a).
FIG. 1 and FIG. 2 show system traceabilityCan be matched with the corresponding tracking error, and the output x can be seen1Can track ideal track x welldAnd the tracking error converges in a zero domain; fig. 3 and 4 show the approximation effect of the neural network, and it can be seen that the neural network can well approximate the unknown function under the bounded weight norm. In fig. 5 and 6, several other state variables and control inputs are shown.
Therefore, the invention can provide a control method which can compensate the uncertain unknown items of the system, reduce a large amount of calculation amount of the traditional backstepping control and realize the convergence of the tracking error of the system in limited time and the limited time tracking of the system.
While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.

Claims (1)

1. A flexible joint mechanical arm finite time self-adaptive backstepping control method based on a neural network is characterized by comprising the following steps: the control method comprises the following steps:
step 1, establishing a dynamic model of the flexible joint mechanical arm, and initializing a system state, sampling time and control parameters, wherein the process comprises the following steps:
1.1 the dynamic model expression form of an n-order flexible joint mechanical arm is as follows:
Figure FDA0002473393700000011
wherein q ∈ Rn,θ∈RnRespectively are a joint position vector and a motor position vector, and n is the order of the system;
Figure FDA0002473393700000012
is the joint acceleration vector;
Figure FDA0002473393700000013
is motor acceleration vector, M (q) ∈ Rn×nJ ∈ R as an unknown nonsingular symmetric positive definite matrix representing the inertia of the jointn×nK ∈ R as an unknown nonsingular symmetrical positive definite matrix representing motor inertian×nIs an unknown diagonal positive definite matrix representing the joint spring stiffness h (q, theta) ∈ Rn×nU ∈ R as a function of centripetal, Coriolis, and gravitational accelerationnRepresenting a control torque vector;
1.2 redefine the variables, writing equation (1) in the form of a state space equation:
definition of x1＝q,
Figure FDA0002473393700000014
x3＝θ,
Figure FDA0002473393700000015
Equation (1) is written in the following state space form:
Figure FDA0002473393700000016
wherein xiI-1, 2,3,4 are all measurable, M (x)1)，h(x1,x2) K and J are both unknown terms;
step 2, calculating the tracking error of the system, wherein the process is as follows:
the system tracking error is defined as follows:
z1＝x1-xd(3)
wherein xdIs a given smoothly bounded reference trajectory;
the derivation of equation (3) yields:
Figure FDA0002473393700000017
wherein
Figure FDA0002473393700000021
Is the first derivative of the error and is,
Figure FDA0002473393700000022
is the first derivative of the reference trajectory;
step 3, defining an error variable, and designing a virtual controller, wherein the process is as follows:
3.1 define error variables as:
zj＝xj-aj-1,j＝2,3,4 (5)
wherein, aj-1J is 2,3,4 is a virtual controller in the design controller process;
derivation of equation (5) yields:
Figure FDA0002473393700000023
wherein
Figure FDA0002473393700000024
Is the first derivative of the error and is,
Figure FDA0002473393700000025
j is 2,3,4 is the first derivative of the virtual controller in the design controller process;
substituting formulae (2) and (5) into formulae (4) and (6) to obtain:
Figure FDA0002473393700000026
3.2 definition of
Figure FDA0002473393700000027
Wherein
Figure FDA0002473393700000028
Is to approach
Figure FDA0002473393700000029
And
Figure FDA00024733937000000210
the following two neural networks were designed:
definition of
Figure FDA00024733937000000211
Is an ideal weight matrix of the neural network, and m is the number of the neurons; then
Figure FDA00024733937000000212
Is approximated as follows:
Figure FDA00024733937000000213
Figure FDA00024733937000000214
wherein
Figure FDA00024733937000000215
Is a basis function of the neural network;1,2representing approximation error of neural network and satisfying | non-calculation1||≤1N,||2||≤2N；1N，2NIs a positive constant, | · | | | represents a two-norm of the value;
Figure FDA00024733937000000216
and
Figure FDA00024733937000000217
the form of (A) is as follows:
Figure FDA00024733937000000218
Figure FDA0002473393700000031
wherein, al,bl,cl,dlAre all constant parameters, l ═ 1, 2;
3.3 designing neural network weight and estimation error updating law:
Figure FDA0002473393700000032
Figure FDA0002473393700000033
Figure FDA0002473393700000034
Figure FDA0002473393700000035
wherein
Figure FDA0002473393700000036
Is a positive definite diagonal matrix; sigma1,σ2,ρ1,ρ2Are suitable parameters;
Figure FDA0002473393700000037
and
Figure FDA0002473393700000038
are respectively
Figure FDA0002473393700000039
And
Figure FDA00024733937000000310
an estimated value of (d);
Figure FDA00024733937000000311
and
Figure FDA00024733937000000312
are respectively1NAnd2Nan estimated value of (d);
3.4 design virtual controller, as follows:
Figure FDA00024733937000000313
wherein h is1,h2,h3,k1,k2,k3Is a normal number;
3.5 design the actual controller as follows:
Figure FDA00024733937000000314
wherein h is4，k4Is a normal number;
3.6 substituting formula (8), formula (9), formula (16) and formula (17) into formula (7) yields:
Figure FDA00024733937000000315
step 5, designing the Lyapunov function into the following form:
Figure FDA0002473393700000041
wherein
Figure FDA0002473393700000042
Derivation of equation (19) and substitution of (18) yields:
Figure FDA0002473393700000043
if equation (20) is written as
Figure FDA0002473393700000044
η therein1＝Min{2h1,2h2λmin{M-1(x1)K},2h3,2h4λmin{J-1}}，
η2＝Min{2αk1,(2M-1(x1)K)αk2,2αk3,(2J-1)αk4}，
＝Z2NW1NΦ1N+Z4NW2NΦ2N+Z2NE1μ+Z4NE2μ,
Where Min {. cndot.) represents a minimum value, λmin{. denotes the minimum eigenvalue; z2NAnd Z4NRespectively represent | | z2And z4The maximum value of | l; w1NAnd W2NRespectively represent
Figure FDA0002473393700000045
And
Figure FDA0002473393700000046
maximum value of (d); phi1NAnd phi2NRespectively represent
Figure FDA0002473393700000047
And
Figure FDA0002473393700000048
maximum value of (d); e1μAnd E2μRespectively represent
Figure FDA0002473393700000049
And
Figure FDA00024733937000000410
maximum value of (d);
it is determined that the system tracking error has a finite time to converge into a domain near the equilibrium point.
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