CN110750050B - Neural network-based mechanical arm system preset performance control method - Google Patents

Neural network-based mechanical arm system preset performance control method Download PDF

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CN110750050B
CN110750050B CN201910961098.XA CN201910961098A CN110750050B CN 110750050 B CN110750050 B CN 110750050B CN 201910961098 A CN201910961098 A CN 201910961098A CN 110750050 B CN110750050 B CN 110750050B
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陈强
丁科新
徐栋
南余荣
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Zhejiang University of Technology ZJUT
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/048—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators using a predictor
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
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    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators

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Abstract

A control method for preset performance of a mechanical arm system based on a neural network comprises the following steps of 1, establishing a mathematical model of a mechanical arm servo system; designing a tangent type obstacle Lyapunov function; step 2, designing a preset performance self-adaptive controller by utilizing a tangent type barrier Lyapunov function in combination with an inversion method; and 3, analyzing the stability. The invention can ensure the steady-state performance and the transient performance of the system by setting the parameter value of the constraint boundary function. In addition, the design of the controller is effectively simplified by using the neural network approximation model uncertain part and the derivative of the virtual control quantity, the robustness of the system is improved to a certain extent, and the mechanical arm servo system can realize accurate and rapid tracking control.

Description

Neural network-based mechanical arm system preset performance control method
Technical Field
The invention relates to a control method for preset performance of a mechanical arm system based on a neural network, in particular to a self-adaptive control method for a mechanical arm servo system of which the system comprises a model uncertainty item and external interference.
Background
The mechanical arm servo system is widely applied to the high-tech fields of robots, medical treatment and the like, has great significance for improving the steady-state performance and the transient performance of mechanical arm movement, and becomes a hot point for studying by scholars at home and abroad. Aiming at improving the motion performance of the system effectively, various control methods have been proposed at home and abroad, including PID control, adaptive control, sliding mode control, neural network control, backstepping control, transient control and the like. The backstepping control has simple algorithm and can decompose a high-order system into a low-order system with the number not more than the system order to design a controller; the neural network has good approximation performance and is often used for approximating uncertain parts such as system external disturbance, parameter perturbation and the like; transient control designs a controller according to specified performance requirements, so that a control system simultaneously meets steady-state performance and transient performance, and the algorithms are more and more widely applied to the control of a mechanical arm servo system.
There are often system uncertainties in the robot arm servo system, which may cause the system motion performance to be poor or even cause the system to operate unstably if the controller is designed by neglecting the influence of the system uncertainties. Algorithms such as PID control and adaptive control are often difficult to ensure both steady-state performance and transient performance of the system, and repeated adjustments of controller parameters are required to improve system performance.
Disclosure of Invention
In order to solve the tracking control problem in a mechanical arm servo system with uncertain items, effectively improve the robustness of the servo system and simultaneously ensure the steady-state performance and the transient performance of the system, the invention provides a preset performance control method based on a neural network. In addition, the neural network is used for estimating the derivative of the uncertain item and the virtual control quantity contained in the servo system, so that the design of the controller is simplified, and the robustness of the system is enhanced.
The technical scheme proposed for solving the technical problems is as follows:
a control method for preset performance of a mechanical arm system based on a neural network comprises the following steps:
step 1, establishing a mechanical arm servo system model;
1.1, the robot arm servo system model is expressed in the following form
Figure BDA0002228929820000021
Wherein the content of the first and second substances,
Figure BDA00022289298200000212
and
Figure BDA0002228929820000022
for system model uncertainty, d 1 ,d 2 The signal is an external interference signal, q is a joint angle position of the mechanical arm, an angle position of a theta motor, K is a joint elastic coefficient, I and J are inertia coefficients of the mechanical arm and the motor, M, g and L are mass, gravitational acceleration and length of the mechanical arm, and tau is a control moment of the mechanical arm;
1.2 design of the obstacle Lyapunov function
Figure BDA0002228929820000023
Wherein tan (·) represents a tangent function, e is a system error, F (t) is a time-varying boundary function decaying exponentially, and the expression is F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies 0 < F ∞ ＜F 0 The initial value of the error needs to satisfy | e (0) | < F 0 (ii) a The requirements of the steady-state performance and the transient performance of the system are ensured by setting the magnitude of the relevant parameter values of F (t); when F (t) approaches infinity, V is converted to a quadratic form, i.e.
Figure BDA0002228929820000024
Thus V applies to both constrained and unconstrained cases;
1.3, the neural network has good approximation characteristics, is used for approximating a nonlinear function, and can approximate any continuously unknown nonlinear function H (X) into
Figure BDA0002228929820000025
Wherein W *T Is an ideal weight, X is the neural network input, ε is the approximation error and satisfies
Figure BDA0002228929820000026
Figure BDA0002228929820000027
Is a constant number greater than zero and is,
Figure BDA0002228929820000028
is a neuron excitation function expressed as
Figure BDA0002228929820000029
Wherein a, b, c and d are given parameters;
1.4, define the State variable x 1 ＝q，
Figure BDA00022289298200000210
x 3 ＝θ，
Figure BDA00022289298200000211
The formula (1) is rewritten into the following state space form
Figure BDA0002228929820000031
Wherein y is the system output;
step 2, designing an inversion controller;
2.1, defining a tracking error e 1 Is composed of
e 1 ＝x 1 -y d (6)
Wherein, y d Is a reference track; defining Lyapunov functions
Figure BDA0002228929820000032
Wherein F (t) is a boundary function which is greater than zero and decays exponentially, denoted as F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies the condition 0 < F ∞ ＜F 0 And satisfy | e 1 (0)|＜F 0 (ii) a Derived from formula (7)
Figure BDA0002228929820000033
Will be in the formula (5)
Figure BDA0002228929820000034
Partial substitution into formula (8) to obtain
Figure BDA0002228929820000035
Wherein e is 2 ＝x 2 -α 1 ，α 1 For the virtual control amount, the virtual control law is designed according to the formula (9)
Figure BDA0002228929820000036
Wherein k is 1 Is a constant greater than zero;
note book
Figure BDA0002228929820000041
Wherein, in e 1 Is limited to 0
Figure BDA0002228929820000042
To S (e) 1 ) Derived by derivation
Figure BDA0002228929820000043
At e 1 Is limited at position of =0
Figure BDA0002228929820000044
Thereby obtaining alpha 1 And the derivative has no singular value problem, and the formula (10) is substituted into the formula (9) to obtain
Figure BDA0002228929820000045
2.2 defining the Lyapunov function
Figure BDA0002228929820000046
Wherein eta 1 Is a constant number greater than zero and is,
Figure BDA0002228929820000047
W 1 * the weight value is an ideal weight value of the neural network,
Figure BDA0002228929820000048
is W 1 * An estimated value of (d); derived from the formula (12)
Figure BDA0002228929820000049
Wherein e is 3 ＝x 3 -α 2 ，α 2 The indeterminate part Δ existing in equation (13) for the virtual control amount 1 And
Figure BDA00022289298200000410
approximating an uncertainty portion delta using a neural network 1 And
Figure BDA00022289298200000411
is shown as
Figure BDA00022289298200000412
Wherein epsilon 1 Is an approximation error, and has
Figure BDA00022289298200000413
Figure BDA00022289298200000414
Substituting formula (14) into formula (13) for neural network input
Figure BDA00022289298200000415
Design of virtual control law α 2 Is composed of
Figure BDA00022289298200000416
Wherein k is 2 Is a constant greater than zero, and is obtained by substituting equation (11) and equation (15) into equation (14)
Figure BDA00022289298200000417
The design update law according to the formula (16) is
Figure BDA0002228929820000051
Wherein σ 1 Is a constant greater than zero, and is obtained by substituting formula (17) into formula (16)
Figure BDA0002228929820000052
Wherein, delta 1 ＝ε 1 +d 1 There is a positive constant
Figure BDA0002228929820000053
Satisfy the requirements of
Figure BDA0002228929820000054
According to the Young inequality
Figure BDA0002228929820000055
Figure BDA0002228929820000056
Substituting the formula (19) and the formula (20) into the formula (18) to obtain
Figure BDA0002228929820000057
2.3 defining the Lyapunov function
Figure BDA0002228929820000058
Wherein eta 2 Is a constant number greater than zero and is,
Figure BDA0002228929820000059
the weight value is an ideal weight value,
Figure BDA00022289298200000510
is composed of
Figure BDA00022289298200000511
An estimated value of (d); derived from the formula (22)
Figure BDA00022289298200000512
Wherein e is 4 ＝x 4 -α 3 ，α 3 For virtual control of quantities, in order to avoid the need for
Figure BDA00022289298200000513
It is approximated by a neural network, denoted as
Figure BDA00022289298200000514
Wherein epsilon 2 Is an approximation error and has
Figure BDA00022289298200000515
Figure BDA00022289298200000516
Inputting a neural network; design of virtual control law α 3 Is composed of
Figure BDA00022289298200000517
Wherein k is 3 Substituting the equation (24) and the equation (25) into the equation (23) to obtain a constant greater than zero
Figure BDA00022289298200000518
The design update law is
Figure BDA0002228929820000061
Wherein σ 2 Is a constant greater than zero; substituting formula (27) into formula (26) to obtain
Figure BDA0002228929820000062
Wherein, delta 2 ＝ε 2 There is a positive constant
Figure BDA0002228929820000063
Satisfy the requirements of
Figure BDA0002228929820000064
According to the Young's inequality
Figure BDA0002228929820000065
Figure BDA0002228929820000066
Substituting the formulas (21), (29) and (30) into the formula (28) to obtain
Figure BDA0002228929820000067
2.4 defining the Lyapunov function
Figure BDA0002228929820000068
Wherein eta 3 Is a constant greater than zeroDerived from (32)
Figure BDA0002228929820000069
Using neural network approximation
Figure BDA00022289298200000610
Is shown as
Figure BDA00022289298200000611
Wherein epsilon 3 Is an approximation error and has
Figure BDA00022289298200000612
Figure BDA00022289298200000613
Inputting a neural network; design the controller τ to
Figure BDA00022289298200000614
Wherein k is 4 Substituting the equations (34) and (35) into the equation (33) to obtain a constant greater than zero
Figure BDA00022289298200000615
The design update law according to equation (36) is
Figure BDA00022289298200000616
Wherein σ 3 Is a constant greater than zero.
The control method further comprises the following steps:
step 3, stability analysis;
substituting formula (37) into formula (36) to obtain
Figure BDA0002228929820000071
Wherein, delta 3 ＝ε 2 +d 2 According to the Young's inequality
Figure BDA0002228929820000072
Figure BDA0002228929820000073
Substituting the formulas (31), (39) and (40) into the formula (38) to obtain
Figure BDA0002228929820000074
Wherein the controller gain k i The value is required to satisfy
Figure BDA0002228929820000075
Formula (41) is represented as
Figure BDA0002228929820000076
Wherein ρ, μ is
Figure BDA0002228929820000077
Integrating the formula (42) to
Figure BDA0002228929820000078
V 4 Satisfy inequality
0≤V 4 (t)≤C(t) (44)
Wherein the content of the first and second substances,
Figure BDA0002228929820000079
V 4 (0) Is a V 4 Thereby proving that all signals of the closed loop system are consistent and ultimately bounded;
according to formula (32) and formula (44)
Figure BDA00022289298200000710
Solve inequality (45) to obtain
Figure BDA00022289298200000711
Thus demonstrating that the tracking error of the system is always constrained to the time-varying boundaries (-F (t), F (t)).
The invention provides a preset performance control method of a mechanical arm system based on a neural network, which can simultaneously ensure the steady-state performance and the transient performance of the system, effectively solve the influence of uncertain items in the system on the control effect, simplify the design of a controller, improve the robustness of the system and realize the accurate tracking control of the mechanical arm system.
The technical conception of the invention is as follows: aiming at a mechanical arm servo system with model uncertainty, the method constructs a tangent type barrier Lyapunov function, designs an exponential attenuation type time-varying constraint boundary, and can simultaneously ensure the steady-state performance and the transient performance of the system by setting the parameter value of the constraint boundary function. In addition, the neural network is adopted to estimate the derivative of the uncertain item and the virtual control quantity of the system, so that the reality of the controller is simplified, the robustness of the system is improved, and the mechanical arm servo system can realize accurate and quick tracking control.
The invention has the beneficial effects that: the output limitation processing is carried out on the mechanical arm servo system, the steady-state performance and the transient performance of the mechanical arm servo system are guaranteed, the uncertain part of the model and the derivative of the virtual control quantity are approximated by the neural network, the design of the controller is simplified, and the robustness of the system is improved.
Drawings
FIG. 1 is a control flow diagram of the present invention;
FIG. 2 shows a reference trajectory y d A schematic diagram of the position tracking trajectory of the present invention at =0.5 (sint + sin0.5t);
FIG. 3 shows a reference trajectory y d Schematic diagram of position tracking error of the present invention at =0.5 (sint + sin0.5t);
FIG. 4 shows a reference trajectory y d Schematic diagram of the control signal of the invention at 0.5 (sint + sin0.5t);
FIG. 5 is a schematic diagram of a position tracking trace of the present invention with a unit step signal as a reference trace;
FIG. 6 is a schematic diagram of the position tracking error of the present invention with the reference track as a unit step signal;
FIG. 7 is a diagram illustrating control signals according to the present invention when the reference trace is a unit step signal.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 7, a method for controlling preset performance of a mechanical arm system based on a neural network includes the following steps:
step 1, establishing a mechanical arm servo system model;
1.1, the robot arm servo system model is expressed in the following form
Figure BDA0002228929820000091
Wherein the content of the first and second substances,
Figure BDA0002228929820000092
and
Figure BDA0002228929820000093
for system model uncertainty, d 1 ,d 2 Q is an external interference signal, q is the angular position of a joint of the mechanical arm, the angular position of a theta motor, K is the elastic coefficient of the joint, I and J are the inertia coefficients of the mechanical arm and the motor, M, g and L are the mass, the gravity acceleration and the length of the mechanical arm, and tauControlling the moment for the mechanical arm;
1.2 design obstacle Lyapunov function
Figure BDA0002228929820000094
Where tan (·) represents a tangent function, e is a system error, F (t) is a time-varying boundary function decaying exponentially, and the expression is F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies 0 < F ∞ ＜F 0 The initial value of the error needs to satisfy | e (0) | < F 0 (ii) a The steady-state and transient performance requirements of the system are ensured by setting the magnitude of the relevant parameter values of F (t); when F (t) approaches infinity, V is converted to a quadratic form, i.e.
Figure BDA0002228929820000095
Thus V applies to both constrained and unconstrained cases;
1.3, the neural network has good approximation characteristics, is used for approximating a nonlinear function, and can approximate any continuously unknown nonlinear function H (X) into
Figure BDA0002228929820000096
Wherein W *T Is an ideal weight, X is the neural network input, ε is the approximation error and satisfies
Figure BDA0002228929820000097
Figure BDA0002228929820000098
Is a constant number greater than zero and is,
Figure BDA0002228929820000099
is a neuron excitation function expressed by
Figure BDA00022289298200000910
Wherein a, b, c and d are given parameters;
1.4, define the State variable x 1 ＝q，
Figure BDA00022289298200000911
x 3 ＝θ，
Figure BDA00022289298200000912
The formula (1) is rewritten into the following state space form
Figure BDA0002228929820000101
Wherein y is the system output;
step 2, designing an inversion controller;
2.1, defining a tracking error e 1 Is composed of
e 1 ＝x 1 -y d (6)
Wherein, y d Is a reference track; defining Lyapunov functions
Figure BDA0002228929820000102
Wherein F (t) is a boundary function which is greater than zero and decays exponentially, expressed as F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies 0 < F ∞ ＜F 0 And satisfy | e 1 (0)|＜F 0 (ii) a Derived from formula (7)
Figure BDA0002228929820000103
Will be in the formula (5)
Figure BDA0002228929820000104
Part is substituted into formula (8) to obtain
Figure BDA0002228929820000105
Wherein e is 2 ＝x 2 -α 1 ，α 1 For the virtual control amount, a virtual control law is designed according to the formula (9)
Figure BDA0002228929820000106
Wherein k is 1 Is a constant greater than zero;
note book
Figure BDA0002228929820000111
Wherein, in e 1 Is limited at position of =0
Figure BDA0002228929820000112
To S (e) 1 ) Derived by derivation
Figure BDA0002228929820000113
At e 1 Is limited at position of =0
Figure BDA0002228929820000114
Thereby obtaining alpha 1 And the derivative has no singular value problem, and the formula (10) is substituted into the formula (9) to obtain
Figure BDA0002228929820000115
2.2 defining the Lyapunov function
Figure BDA0002228929820000116
Wherein eta is 1 Is a constant number greater than zero and is,
Figure BDA0002228929820000117
W 1 * is an ideal weight value of the neural network,
Figure BDA0002228929820000118
is W 1 * An estimated value of (d); derived from the formula (12)
Figure BDA0002228929820000119
Wherein e is 3 ＝x 3 -α 2 ，α 2 The indeterminate part Δ existing in equation (13) for the virtual control amount 1 And
Figure BDA00022289298200001110
approximating an uncertainty portion delta using a neural network 1 And
Figure BDA00022289298200001111
is shown as
Figure BDA00022289298200001112
Wherein epsilon 1 Is an approximation error and has
Figure BDA00022289298200001113
Figure BDA00022289298200001114
Substituting formula (14) into formula (13) for neural network input
Figure BDA00022289298200001115
Design of virtual control law α 2 Is composed of
Figure BDA00022289298200001116
Wherein k is 2 The constant is greater than zero, and formula (11) and formula (15) are substituted into formula (14) to obtain
Figure BDA00022289298200001117
The design update law according to the formula (16) is
Figure BDA0002228929820000121
Wherein σ 1 Is a constant greater than zero, and is obtained by substituting formula (17) into formula (16)
Figure BDA0002228929820000122
Wherein, delta 1 ＝ε 1 +d 1 There is a positive constant
Figure BDA0002228929820000123
Satisfy the requirements of
Figure BDA0002228929820000124
According to the Young inequality
Figure BDA0002228929820000125
Figure BDA0002228929820000126
Substituting the formula (19) and the formula (20) into the formula (18) to obtain
Figure BDA0002228929820000127
2.3 defining the Lyapunov function
Figure BDA0002228929820000128
Wherein eta is 2 Is a constant number greater than zero and is,
Figure BDA0002228929820000129
in order to be the ideal weight value,
Figure BDA00022289298200001210
is composed of
Figure BDA00022289298200001211
An estimated value of (d); derived from the formula (22)
Figure BDA00022289298200001212
Wherein e is 4 ＝x 4 -α 3 ，α 3 For virtually controlling the quantity, in order to avoid seeking
Figure BDA00022289298200001213
It is approximated by a neural network, denoted as
Figure BDA00022289298200001214
Wherein epsilon 2 Is an approximation error and has
Figure BDA00022289298200001215
Figure BDA00022289298200001216
Inputting a neural network; design of virtual control law α 3 Is composed of
Figure BDA00022289298200001217
Wherein，k 3 Is a constant greater than zero, and is obtained by substituting the formula (24) and the formula (25) into the formula (23)
Figure BDA00022289298200001218
The design update law is
Figure BDA0002228929820000131
Wherein σ 2 Is a constant greater than zero; substituting formula (27) into formula (26) to obtain
Figure BDA0002228929820000132
Wherein, delta 2 ＝ε 2 There is a positive constant
Figure BDA0002228929820000133
Satisfy the requirement of
Figure BDA0002228929820000134
According to the Young's inequality
Figure BDA0002228929820000135
Figure BDA0002228929820000136
Substituting the formulas (21), (29) and (30) into the formula (28) to obtain
Figure BDA0002228929820000137
2.4, defining Lyapunov function
Figure BDA0002228929820000138
Wherein eta 3 Is a constant greater than zero, derived by the formula (32)
Figure BDA0002228929820000139
Using neural network approximation
Figure BDA00022289298200001310
Is shown as
Figure BDA00022289298200001311
Wherein epsilon 3 Is an approximation error and has
Figure BDA00022289298200001312
Figure BDA00022289298200001313
Inputting a neural network; design the controller τ to
Figure BDA00022289298200001314
Wherein k is 4 Is a constant greater than zero, and is obtained by substituting equations (34) and (35) into equation (33)
Figure BDA00022289298200001315
The design update law according to equation (36) is
Figure BDA00022289298200001316
Wherein σ 3 Is a constant greater than zero.
The control method further comprises the following steps:
step 3, stability analysis;
substituting the formula (37) into the formula (36) to obtain
Figure BDA0002228929820000141
Wherein, delta 3 ＝ε 2 +d 2 According to the Young's inequality
Figure BDA0002228929820000142
Figure BDA0002228929820000143
Substituting the formulas (31), (39) and (40) into the formula (38) to obtain
Figure BDA0002228929820000144
Wherein the controller gain k i The value is required to satisfy
Figure BDA0002228929820000145
Formula (41) is represented as
Figure BDA0002228929820000146
Wherein ρ and μ are
Figure BDA0002228929820000147
Integrating the formula (42) to
Figure BDA0002228929820000148
V 4 Satisfy the inequality
0≤V 4 (t)≤C(t) (44)
Wherein the content of the first and second substances,
Figure BDA0002228929820000149
V 4 (0) Is a V 4 Thereby proving that all signals of the closed loop system are consistent and ultimately bounded;
according to formula (32) and formula (44)
Figure BDA00022289298200001410
Solve inequality (45) to obtain
Figure BDA00022289298200001411
Thus demonstrating that the tracking error of the system is always constrained to the time-varying boundaries (-F (t), F (t)).
In order to verify the effectiveness and superiority of the proposed method, the following control methods are simulated and compared
M1: the invention provides a self-adaptive control method for the preset performance of a mechanical arm servo system based on a neural network. The expressions of the virtual control law are shown as (10), (15) and (25), the expressions of the weight updating law are shown as (17), (27) and (37), and the expression of the controller is shown as (35).
M2: the neural network self-adaptive control method based on the constant value constraint obstacle Lyapunov function design is characterized in that the neural network parameters and the weight value updating law are the same as those of the M1 method, and the virtual control law and the controller are respectively designed as follows:
Figure BDA0002228929820000151
Figure BDA0002228929820000152
Figure BDA0002228929820000153
Figure BDA0002228929820000154
m3: the neural network self-adaptive control method based on the backstepping method design is characterized in that neural network parameters and a weight value updating law are the same as those of the M1 method, and a virtual control law and a controller are respectively designed as follows:
Figure BDA0002228929820000155
Figure BDA0002228929820000156
Figure BDA0002228929820000158
Figure BDA0002228929820000157
initial conditions and control parameters in the simulation experiment were set as:
system parameters:
mgl＝5,I＝1,J＝1,K＝40
initial state:
x 1 (0)＝0.4,x 2 (0)＝0,x 3 (0)＝0,x 4 (0)＝0
expected trajectory:
y d ＝0.5(sint+sin0.5t)
constraint boundary parameters:
F(t)＝(1-0.02)exp -5t +0.02
k b ＝0.5
neural network parameters:
a＝2,b＝10,c＝1,d＝-1
Figure BDA0002228929820000161
controller gain parameters:
K 1 ＝6,K 2 ＝6,K 3 ＝6,K 4 ＝6,
FIG. 2 is a diagram when the reference trajectory is y d Simulation effect diagram when =0.5 (sint + sin0.5t), fig. 3 is a schematic diagram of angular position tracking error, and fig. 4 is a schematic diagram of control signal. It can be seen from fig. 2 and 3 that the three control methods can track the desired trajectory, but the M1 method proposed herein has a faster tracking speed than the other two methods. It is important to note that the tracking errors of the M2 and M3 methods cross the time-varying boundary (-F (t), F (t)).
To further compare the transient performance of the three methods, the unit step signal was chosen as the desired trace. The initial state of the system is: x is a radical of a fluorine atom 1 (0)＝0.6,x 2 (0)＝0,x 3 (0)＝0,x 4 (0) =0; controller gain set to K i =5,i =1,2,3,4. Constraint boundary parameter set to
F(t)＝(1-0.02)exp -4t +0.02
k b ＝0.5
Fig. 5 is a diagram showing the effect of tracking the joint angle position of the robot arm. As can be seen from fig. 5, compared with the M2 and M3 methods, the M1 method proposed by the present invention has a smaller overshoot and a faster tracking speed. Fig. 6 is a graph of the effect of angular position tracking error. As shown in fig. 6, the tracking errors of the M2 and M3 methods cross the time-varying boundary (-F (t), F (t)), while the tracking error under the M1 method always remains within the boundary (-F (t), F (t)). The good transient performance and steady-state performance of the system can be ensured by presetting the magnitude of the relevant parameters of the F (t). Fig. 7 is a diagram of the effect of the controller output.
In summary, it can be seen from the two sets of example simulation results that the neural network-based predetermined performance control method provided herein can effectively eliminate the influence of system uncertainty and external interference on the performance of the mechanical arm servo system in the control of the mechanical arm servo system, enhance the robustness of the system, and can simultaneously ensure good steady-state performance and transient performance of the mechanical arm servo system by setting relevant parameters of the time-varying constraint boundary F (t), so that the system has a good tracking control effect.
While two comparative simulations have been set forth above to demonstrate the advantages of the designed method, it will be understood that the invention is not limited to the examples described herein, but is capable of numerous modifications without departing from the spirit and scope of the invention. The control scheme designed by the invention has a good control effect on the mechanical arm servo system containing output constraint and uncertainty items, enhances the robustness of the system, and simultaneously ensures the steady-state performance and the transient performance of the mechanical arm servo system, so that the system has a good tracking control effect.

Claims (2)

1. A control method for preset performance of a mechanical arm system based on a neural network is characterized by comprising the following steps:
step 1, establishing a mechanical arm servo system model;
1.1, the robot arm servo system model is expressed in the form
Figure FDA0003792152980000011
Wherein the content of the first and second substances,
Figure FDA0003792152980000018
and
Figure FDA0003792152980000012
for system model uncertainty, d 1 ,d 2 The method comprises the steps of obtaining external interference signals, obtaining q of a joint angle position of a mechanical arm, obtaining an angle position of a theta motor, obtaining K of a joint elastic coefficient, obtaining I and J of inertia coefficients of the mechanical arm and the motor respectively, obtaining M, g and L of the mechanical arm, obtaining gravity acceleration and mechanical arm length respectively, and obtaining tau of the mechanical armControlling the moment;
1.2 design obstacle Lyapunov function
Figure FDA0003792152980000013
Wherein tan (·) represents a tangent function, e is a system error, F (t) is a boundary function that is greater than zero and exponentially decays, and the expression is F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies 0 < F ∞ ＜F 0 The initial value of the error needs to satisfy | e (0) | < F 0 (ii) a The requirements of the steady-state performance and the transient performance of the system are ensured by setting the magnitude of the relevant parameter values of F (t); when F (t) approaches infinity, V is converted to a quadratic form, i.e.
Figure FDA0003792152980000014
Thus V applies to both constrained and unconstrained cases;
1.3, the neural network has good approximation characteristics and is used for approximating a nonlinear function to approximate any continuously unknown nonlinear function H (X) into
Figure FDA00037921529800000111
Wherein W *T Is an ideal weight, X is the neural network input, ε is the approximation error and satisfies
Figure FDA0003792152980000019
Figure FDA00037921529800000110
Is a constant number greater than zero and is,
Figure FDA00037921529800000112
is a neuron excitation function expressed by
Figure FDA0003792152980000015
Wherein a, b, c and d are given parameters;
1.4, define the State variable x 1 ＝q，
Figure FDA0003792152980000016
x 3 ＝θ，
Figure FDA0003792152980000017
The formula (1) is rewritten into the following state space form
Figure FDA0003792152980000021
Wherein y is the system output;
step 2, designing an inversion controller;
2.1, defining a tracking error e 1 Is composed of
e 1 ＝x 1 -y d (6)
Wherein, y d Is a reference track; defining Lyapunov functions
Figure FDA0003792152980000022
Wherein F (t) is a boundary function which is greater than zero and decays exponentially, denoted as F (t) = (F) 0 -F ∞ )exp -nt +F ∞ ，F 0 ,F ∞ N is a constant greater than zero and satisfies 0 < F ∞ ＜F 0 And satisfy | e 1 (0)|＜F 0 (ii) a Derived from formula (7)
Figure FDA0003792152980000023
Will be in the formula (5)
Figure FDA0003792152980000024
Part is substituted into formula (8) to obtain
Figure FDA0003792152980000025
Wherein e is 2 ＝x 2 -α 1 ，α 1 For the virtual control amount, the virtual control law is designed according to the formula (9)
Figure FDA0003792152980000026
Wherein k is 1 Is a constant greater than zero;
note the book
Figure FDA0003792152980000031
Wherein, in e 1 Is limited at position of =0
Figure FDA0003792152980000032
To S (e) 1 ) Is derived by
Figure FDA0003792152980000033
At e 1 Is limited to 0
Figure FDA0003792152980000034
Thereby obtaining alpha 1 And the derivative thereof has no singular value problem, and the formula (10) is substituted into the formula (9) to obtain
Figure FDA0003792152980000035
2.2 defining the Lyapunov function
Figure FDA0003792152980000036
Wherein eta is 1 Is a constant number greater than zero and is,
Figure FDA0003792152980000037
W 1 * is an ideal weight value of the neural network,
Figure FDA0003792152980000038
is W 1 * An estimated value of (d); derived from the formula (12)
Figure FDA0003792152980000039
Wherein e is 3 ＝x 3 -α 2 ，α 2 The indeterminate part Delta existing in the formula (13) for the virtual control quantity 1 And
Figure FDA00037921529800000310
approximating an uncertainty portion delta using a neural network 1 And
Figure FDA00037921529800000311
is shown as
Figure FDA00037921529800000312
Wherein epsilon 1 Is an approximation error, and has
Figure FDA00037921529800000313
Figure FDA00037921529800000314
Substituting the formula (14-1) into the formula (13) for neural network input
Figure FDA00037921529800000315
Design of virtual control law α 2 Is composed of
Figure FDA00037921529800000316
Wherein k is 2 Is a constant greater than zero, and is obtained by substituting formula (11) and formula (15) into formula (14-2)
Figure FDA00037921529800000317
The design update law according to equation (16) is
Figure FDA0003792152980000041
Wherein σ 1 Is a constant greater than zero, and is obtained by substituting formula (17) into formula (16)
Figure FDA0003792152980000042
Wherein, delta 1 ＝ε 1 +d 1 There is a positive constant
Figure FDA0003792152980000043
Satisfy the requirements of
Figure FDA0003792152980000044
According to the Young inequality
Figure FDA0003792152980000045
Figure FDA0003792152980000046
Substituting the formula (19) and the formula (20) into the formula (18) to obtain
Figure FDA0003792152980000047
2.3 defining the Lyapunov function
Figure FDA0003792152980000048
Wherein eta 2 Is a constant number greater than zero and is,
Figure FDA0003792152980000049
Figure FDA00037921529800000410
in order to be the ideal weight value,
Figure FDA00037921529800000411
is composed of
Figure FDA00037921529800000412
An estimated value of (d); derived from the formula (22)
Figure FDA00037921529800000413
Wherein e is 4 ＝x 4 -α 3 ，α 3 For virtually controlling the quantity, in order to avoid seeking
Figure FDA00037921529800000414
It is approximated by a neural network, denoted as
Figure FDA00037921529800000415
Wherein epsilon 2 Is an approximation error, and has
Figure FDA00037921529800000416
Figure FDA00037921529800000417
Inputting a neural network; design of virtual control law α 3 Is composed of
Figure FDA00037921529800000418
Wherein k is 3 Is a constant greater than zero, and is obtained by substituting the formula (24) and the formula (25) into the formula (23)
Figure FDA00037921529800000419
The design update law is
Figure FDA0003792152980000051
Wherein σ 2 Is a constant greater than zero; substituting formula (27) into formula (26) to obtain
Figure FDA0003792152980000052
Wherein, delta 2 ＝ε 2 There is a positive constant
Figure FDA0003792152980000053
Satisfy the requirements of
Figure FDA0003792152980000054
According to the Young's inequality
Figure FDA0003792152980000055
Figure FDA0003792152980000056
Substituting the formulas (21), (29) and (30) into the formula (28) to obtain
Figure FDA0003792152980000057
2.4 defining the Lyapunov function
Figure FDA0003792152980000058
Wherein eta is 3 Is a constant greater than zero, derived by the formula (32)
Figure FDA0003792152980000059
Using neural network approximation
Figure FDA00037921529800000510
Is shown as
Figure FDA00037921529800000511
Wherein epsilon 3 Is an approximation error and has
Figure FDA00037921529800000512
Figure FDA00037921529800000513
Inputting a neural network; design the controller w to
Figure FDA00037921529800000514
Wherein k is 4 Is a constant greater than zero, and is obtained by substituting equations (34) and (35) into equation (33)
Figure FDA00037921529800000515
The design update law according to equation (36) is
Figure FDA00037921529800000516
Wherein σ 3 A constant greater than zero.
2. The neural network-based preset performance control method for the mechanical arm system, as claimed in claim 1, wherein the control method further comprises the steps of:
step 3, stability analysis;
substituting the formula (37) into the formula (36) to obtain
Figure FDA0003792152980000061
Wherein, delta 3 ＝ε 2 +d 2 According to the Young's inequality
Figure FDA0003792152980000062
Figure FDA0003792152980000063
Substituting the formulas (31), (39) and (40) into the formula (38) to obtain
Figure FDA0003792152980000064
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003792152980000065
formula (41) is represented as
Figure FDA0003792152980000066
Wherein ρ, μ is
Figure FDA0003792152980000067
Integral of formula (42) to
Figure FDA0003792152980000068
V 4 Satisfy inequality
0≤V 4 (t)≤C(t) (44)
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003792152980000069
V 4 (0) Is a V 4 Thereby proving that all signals of the closed loop system are consistent and ultimately bounded;
according to formula (32) and formula (44)
Figure FDA00037921529800000610
Solve inequality (45) to obtain
Figure FDA00037921529800000611
Thus demonstrating that the tracking error of the system is always constrained to the time-varying boundaries (-F (t), F (t)).
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