CN108983606B - Robust sliding mode self-adaptive control method of mechanical arm system - Google Patents

Robust sliding mode self-adaptive control method of mechanical arm system Download PDF

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CN108983606B
CN108983606B CN201810741941.9A CN201810741941A CN108983606B CN 108983606 B CN108983606 B CN 108983606B CN 201810741941 A CN201810741941 A CN 201810741941A CN 108983606 B CN108983606 B CN 108983606B
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mechanical arm
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bounded
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robust
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CN108983606A (en
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胡健
李向辉
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Nanjing University of Science and Technology
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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
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Abstract

The invention discloses a robust sliding mode self-adaptive control method for a mechanical arm system with progressive tracking performance, and belongs to the field of mechanical arm control. The control method considers the structural uncertainty of parameters and the like of a mechanical arm system and the non-structural uncertainty of external interference and the like, and designs a parameter estimator aiming at the structural uncertainty of parameters and the like; designing a continuous robust controller aiming at the upper bound of non-structural uncertainty such as external interference; the robust adaptive controller designed by the invention has good effect on the mechanical arm system with structural uncertainty such as parameters and the like and non-structural uncertainty such as external interference and the like, and can ensure the position tracking performance of the mechanical arm system; the robust sliding mode self-adaptive controller designed by the invention is simple, has continuous control output and is beneficial to being applied in engineering practice.

Description

Robust sliding mode self-adaptive control method of mechanical arm system
Technical Field
The invention relates to the field of mechanical arm control, in particular to a robust sliding mode self-adaptive control method of a mechanical arm system.
Background
The mechanical arm system is a complex system with multiple inputs and outputs, high nonlinearity and strong coupling. Due to the unique operation flexibility, the device has been widely applied to the fields of industrial assembly, safety explosion prevention and the like, such as paint spraying robots, spot welding robots, bomb disposal robots and the like. Due to the complexity of the mechanical arm system, many modeling uncertainties including structural uncertainties and non-structural uncertainties are encountered in the design process of the controller, and these factors can seriously affect the performance of the controller, which leads to the reduction of the control accuracy of the controller and even the instability of the designed controller, thereby increasing the design difficulty of the controller.
With the continuous progress of the technical level in the industrial field, the control precision of the mechanical arm is also continuously improved. However, the traditional control method obviously cannot meet the high-performance requirement of the system and becomes a factor for limiting the control performance of the mechanical arm. In recent years, with the continuous development of control technology, various control methods based on modern control theory are proposed successively. The sliding mode control is widely used in a mechanical arm, but the sliding mode control cannot estimate structural uncertainty such as parameters in a system, and when the structural uncertainty such as large parameters exists in the system, a designed controller is conservative, so that the performance of the system is deteriorated.
Aiming at the characteristics of uncertain nonlinearity in a mechanical arm system, a system mathematical model is established, and on the basis, sliding mode adaptive robust control of the mechanical arm system is designed to overcome system parameter uncertainty and unmodeled uncertainty.
Disclosure of Invention
The invention provides a robust sliding mode self-adaptive control method for a mechanical arm, aiming at solving the problem of uncertain nonlinearity in mechanical arm system control.
The invention adopts the following steps to solve the problems:
a robust sliding mode self-adaptive control method for a mechanical arm comprises the following steps:
step 1, establishing a dynamic model of a mechanical arm system, wherein according to an Euler-Lagrange method, the dynamic model of the mechanical arm system with n degrees of freedom comprises the following steps:
Figure BDA0001723408380000021
q ∈ R in formula (1)n，
Figure BDA0001723408380000022
Respectively the velocity, angular velocity and angular acceleration of the mechanical arm joint; h (q) epsilon Rn×nIs an inertia matrix of the robotic arm system;
Figure BDA0001723408380000023
G(q)∈Rn，τ∈Rnrepresenting centripetal coriolis force, gravity, and input torque, respectively: fvFor a coefficient of viscous friction, d ∈ RnThe interference vector is an external interference vector, including time-varying interference and constant interference of the system,
the mechanical arm system has the following properties:
properties 1: h (q) is a positive definite symmetric matrix satisfying:
Figure BDA0001723408380000024
wherein m is1And m2E R is a known bounded positive real number;
properties 2: the differential matrix of the mechanical arm inertia matrix and the Coriolis matrix satisfy the following oblique symmetry matrix relationship:
Figure BDA0001723408380000025
properties 3: the dynamic model of the mechanical arm is linear with respect to a set of physical parameters:
Figure BDA0001723408380000026
wherein the content of the first and second substances,
Figure BDA0001723408380000027
is a manipulator joint regression matrix, α ═ κ, ρ, ε, μ]TAre intrinsic parameters in the manipulator model;
the mechanical arm system further meets the following conditions and lemmas:
condition 1: desired position command q for arm systemdFirst order differential of it
Figure BDA0001723408380000028
And second order differential
Figure BDA0001723408380000029
Are all continuously bounded signals;
condition 2: the mechanical arm uncertainty disturbance d is bounded, i.e.
||d||≤d0 (5)
Wherein d is0Is a known bounded normal number;
lesion 1. consider a first order system
Figure BDA00017234083800000210
The following control laws are achieved in the above system for a finite time to stabilize:
θ＝-θ-λθ-μθq/p (7)
wherein theta is a system state variable, lambda and mu are normal numbers, q is more than 0, p is more than 0, and q and p are odd integers, and q/p is less than 1; thus, the convergence time tsComprises the following steps:
Figure BDA0001723408380000031
2 in the introduction.
Figure BDA0001723408380000032
Is provided with
0≤xtanh(x/a)≤|x| (8)
And 3, leading.
Figure BDA0001723408380000033
Or x > and y ≧ 0
x/x+y≤1 (9)；
Step 2, designing the self-adaptive robust control steps of the mechanical arm sliding mode as follows:
step 2.1, definition
Figure BDA0001723408380000034
Therefore, it is not only easy to use
Figure BDA0001723408380000035
qdIs a position instruction expected to be tracked by the system, the second order of the instruction can be minimized, a sliding mode surface is designed, the tracking error is limited, the error is ensured to be converged to 0,
Figure BDA0001723408380000036
wherein Λ is a constant matrix and its eigenvalues are strictly located in the right complex half plane, and designing a virtual reference trajectory qrInstead of the desired trajectory qd，
Figure BDA0001723408380000037
Therefore, the temperature of the molten metal is controlled,
Figure BDA0001723408380000038
and
Figure BDA0001723408380000039
is replaced by
Figure BDA00017234083800000310
Figure BDA00017234083800000311
Definition of
Figure BDA00017234083800000312
Wherein s ═ s1,s2,…sn]T，
In order to avoid the jitter of sliding mode control, a control law is designed:
Figure BDA00017234083800000313
wherein λ1And mu1Is a normal number, q1> 0 and p1Is more than 0 and is an integer odd number, and satisfies q1/p1＜1，
Obtained according to equations (1), (14) and (15):
Figure BDA0001723408380000041
according to formula (16):
Figure BDA0001723408380000042
step 2.2, designing a control law, and combining a formula (17) and the property 1, wherein the control law based on the mechanical arm system dynamic model is designed as follows:
τ＝τa+τs (18)
Figure BDA0001723408380000043
τs＝τs1+τs2 (20)
τs1＝-KDs (21)
wherein, tauaFeeding forward a compensation term for the model; tau iss1The linear feedback item ensures the stability of the system; tau iss2The method is a continuous robust item and is used for overcoming external bounded interference in the system, and a specific design form is given in subsequent design; kDIs a symmetrical positive definite matrix which is a diagonal matrix;
Figure BDA0001723408380000044
and
Figure BDA0001723408380000045
is a mixture of H (q),
Figure BDA0001723408380000046
g (q) and FvAn estimated value of (d);
step 2.3, designing a parameter regressor and a parameter estimator,
the parameter adaptation law using a parameter regressor is:
Figure BDA0001723408380000047
Figure BDA0001723408380000048
wherein gamma is a diagonal adaptive law matrix, gamma is more than 0, gamma is a normal number, and gamma influence the adaptive rate of the parameters;
step 2.4, designing a continuous robust item according to the condition 2, and overcoming system interference
τs2＝[τs21,τs22,…τs2n]T (24)
Wherein tau iss2iIn the form of
Figure BDA0001723408380000051
In the formula (25), kiFor a known normal number, ξ (t) satisfies the following condition
|ξi(t)|≤δi *
Figure BDA0001723408380000052
Wherein
Figure BDA0001723408380000053
And deltaiAre all normal numbers;
step 3, analyzing the stability of the mechanical arm system, carrying out stability verification on the system by utilizing the Lyapunov stability theory according to the sliding mode self-adaptive robust control method designed in the step 2 to obtain a gradual stable result of the system,
the lyapunov function is defined as follows:
Figure BDA0001723408380000054
wherein
Figure BDA0001723408380000055
And Fv＞0，
Derivation of formula (27) to obtain
Figure BDA0001723408380000056
Substituting (28) the equations (16), (18) - (21) into the equation (28) results from the mechanical arm properties 2, 3
Figure BDA0001723408380000057
The adaptive laws (22) and (23) are brought into (29)
Figure BDA0001723408380000058
Bringing the formulae (24), (25) into (30)
Figure BDA0001723408380000061
According to 2 can be obtained
Figure BDA0001723408380000062
According to the introduction 3
Figure BDA0001723408380000063
Wherein W is a positive function, and integrating the two sides of equation (33) to obtain
Figure BDA0001723408380000064
V (t) e L is obtained from the formula (34)∞，W∈L2So that the position of s,
Figure BDA0001723408380000065
and
Figure BDA0001723408380000066
is bounded, according to the condition 1, the system state q is bounded, according to the equation (26), the system output tau is bounded, therefore, the closed-loop signals of the system are bounded, and W can be obtained bounded, therefore, W is consistent and continuous, and the system is gradually stable according to the Barbalt theorem,
according to the introduction of 1 to obtain siE s at a finite time tiConverge to 0, tiThe moments are as follows
Figure BDA0001723408380000071
And (3) proving that: designing Lyapunov functions
Figure BDA0001723408380000072
Derived from (36)
Figure BDA0001723408380000073
Substituting equation (15) into (37)
Figure BDA0001723408380000074
Wherein λ1And mu1Is a normal number, q1＞0，q1Is more than 0 and is an odd integer,
Figure BDA0001723408380000075
so at a limited time tiWhen s isiConverge to 0.
The invention has the beneficial effects that: the invention takes a mechanical arm system as a research object, establishes a dynamic model of the mechanical arm system, takes an instruction which is output by the joint position and can accurately track an expected position as a control target, simultaneously considers structural uncertainty of parameters of the system and the like and non-structural uncertainty of external interference and the like, designs a parameter estimator aiming at the structural uncertainty of the parameters, designs a continuous robust control item aiming at the external disturbance uncertainty, and simultaneously combines finite time control to ensure that the position output of the mechanical arm system can accurately track the expected position instruction; the robust sliding mode self-adaptive control method of the mechanical arm system has smooth and continuous control output and is more beneficial to application in engineering practice. The simulation result verifies the effectiveness of the test paper.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the accompanying drawings.
Drawings
FIG. 1 is a diagram of a simulated two-degree-of-freedom robot arm according to the present invention.
FIG. 2 is a schematic diagram and a flow chart of a robust sliding mode adaptive control principle of a mechanical arm system.
FIG. 3 is a diagram of system parameter adaptation under the action of a controller designed by the present invention.
Fig. 4 is a graph of the tracking of the joints of the robot arm of the controller designed by the invention.
Fig. 5 is a diagram of the tracking error of the robot arm joint of the controller designed by the invention.
FIG. 6 is a graph of PID controller robot arm joint tracking error.
Fig. 7 is a diagram of the control output of the joint of the controller designed by the invention.
Detailed Description
The invention is further described with reference to the accompanying drawings.
The embodiment is implemented by combining a two-degree-of-freedom mechanical arm (as shown in fig. 1), and the lengths of the mechanical arm connecting rods 1 and 2 are l respectively1And l2Mass is m respectively1And m2And the center of mass is located at one half of the connecting rod, q1And q is2The joint angle of the mechanical arm joint space.
The embodiment is described with reference to fig. 1 to fig. 2, and the specific steps of the robust sliding mode adaptive control method of the robot arm system according to the embodiment are as follows:
step 1, establishing a dynamic model of the mechanical arm system, wherein according to an Euler-Lagrange method, the dynamic model of the mechanical arm system is as follows:
Figure BDA0001723408380000081
q ∈ R in formula (1)n，
Figure BDA0001723408380000082
Respectively the velocity, angular velocity and angular acceleration of the mechanical arm joint; h (q) epsilon Rn×nIs an inertia matrix of the robotic arm system;
Figure BDA0001723408380000083
G(q)∈Rn，τ∈Rnrepresenting centripetal coriolis force, gravity, and input torque: fvFor a coefficient of viscous friction, d ∈ RnThe external interference vector comprises time-varying interference and constant interference of the system.
There are several properties of robotic arm systems:
properties 1: h (q) is a positive definite symmetric matrix satisfying:
Figure BDA0001723408380000084
wherein m is1And m2E R is a known bounded positive real number.
Properties 2: the differential matrix of the mechanical arm inertia matrix and the Coriolis matrix satisfy the following oblique symmetry matrix relationship:
Figure BDA0001723408380000085
properties 3: the dynamic model of the mechanical arm is linear with respect to a set of physical parameters:
Figure BDA0001723408380000086
wherein the content of the first and second substances,
Figure BDA0001723408380000091
is a manipulator joint regression matrix, α ═ κ, ρ, ε, μ]TAre intrinsic parameters in the robot arm model.
For subsequent controller design and analysis, the robotic arm system needs to satisfy the following conditions and lemma:
condition 1: desired position command q for arm systemdFirst order differential of it
Figure BDA0001723408380000092
And second order differential
Figure BDA0001723408380000093
Are continuously bounded signals.
Condition 2: the mechanical arm uncertainty disturbance d is bounded, i.e.
||d||≤d0 (5)
Wherein d is0Is a known bounded normal number.
Lesion 1. consider a first order system
Figure BDA0001723408380000094
The following control laws are achieved in the above system for a finite time to stabilize:
θ＝-θ-λθ-μθq/p (7)
wherein theta is a system state variable, lambda and mu are normal numbers, q is more than 0 and p is more than 0, and q and p are odd integers, and q/p is less than 1. Thus, the convergence time tsComprises the following steps:
Figure BDA0001723408380000095
2 in the introduction.
Figure BDA0001723408380000096
Is provided with
0≤xtanh(x/a)≤|x| (8)
And 3, leading.
Figure BDA0001723408380000097
Or x > and y ≧ 0
x/x+y≤1 (9)
Step 2, designing the self-adaptive robust control steps of the mechanical arm sliding mode by combining the step 2 as follows:
step 2.1, definition
Figure BDA0001723408380000098
Therefore, it is not only easy to use
Figure BDA0001723408380000099
qdThe method is a position instruction expected to be tracked by a system, the second order of the instruction can be minimized, a sliding mode surface is designed, tracking errors are limited, and the errors are guaranteed to be converged to 0.
Figure BDA00017234083800000910
Where Λ is a constant matrix and its eigenvalues lie strictly in the right complex half plane. Then we design a virtual reference trajectory qrInstead of the desired trajectory qd。
Figure BDA0001723408380000101
Therefore, the temperature of the molten metal is controlled,
Figure BDA0001723408380000102
and
Figure BDA0001723408380000103
is replaced by
Figure BDA0001723408380000104
Figure BDA0001723408380000105
Definition of
Figure BDA0001723408380000106
Wherein s ═ s1,s2,…sn]T。
In order to avoid the jitter of the sliding mode control, a control law is designed:
Figure BDA0001723408380000107
wherein λ1And mu1Is a normal number, q1> 0 and p1Is more than 0 and is an integer odd number, and satisfies q1/p1＜1。
From equations (1), (14) and (15) we can obtain:
Figure BDA0001723408380000108
according to formula (16):
Figure BDA0001723408380000109
step 2.2, designing a control law, and combining a formula (17) and the property 1, wherein the control law based on the mechanical arm dynamic model is designed as follows:
τ＝τa+τs (18)
Figure BDA00017234083800001010
τs＝τs1+τs2 (20)
τs1＝-KDs (21)
wherein, tauaFeeding forward a compensation term for the model; tau iss1The linear feedback item ensures the stability of the system; tau iss2The method is a continuous robust item and is used for overcoming external bounded interference in the system, and a specific design form is given in subsequent design; kDIs a symmetric positive definite matrix, generally a diagonal matrix;
Figure BDA00017234083800001011
and
Figure BDA00017234083800001012
is a mixture of H (q),
Figure BDA0001723408380000111
g (q) and FvAn estimate of (d).
Step 2.3, design parameter regressor and parameter estimator
The parameter adaptation law using a parameter regressor is:
Figure BDA0001723408380000112
Figure BDA0001723408380000113
in the formula, gamma is a diagonal adaptive law matrix, gamma is greater than 0, gamma is a normal number, and gamma influence the adaptive rate of the parameters.
Step 2.4, designing a continuous robust item according to the condition 2 to overcome system interference
τs2＝[τs21，τs22，…τs2n]T (24)
Wherein tau iss2iIn the form of
Figure BDA0001723408380000114
In the formula (25), kiFor a known normal number, ξ (t) satisfies the following condition
|ξi(t)|≤δi *
Figure BDA0001723408380000115
Wherein deltaiSum deltaiAre all normal numbers.
And 3, analyzing the stability of the mechanical arm system, and performing stability verification on the system by using the Lyapunov stability theory according to the sliding mode adaptive robust control method designed in the step two to obtain a gradual stable result of the system.
The lyapunov function is defined as follows:
Figure BDA0001723408380000116
wherein
Figure BDA0001723408380000117
And Fv＞0。
Derivation of formula (27) to obtain
Figure BDA0001723408380000118
Substituting (28) the equations (16), (18) - (21) into the equation (28) results from the mechanical arm properties 2, 3
Figure BDA0001723408380000121
The adaptive laws (22) and (23) are brought into (29)
Figure BDA0001723408380000122
Bringing the formulae (24), (25) into (30)
Figure BDA0001723408380000123
According to 2 can be obtained
Figure BDA0001723408380000124
According to the introduction 3
Figure BDA0001723408380000125
Wherein W is a positive function, and integrating the two sides of equation (33) to obtain
Figure BDA0001723408380000131
V (t) e L is obtained from the formula (34)∞，W∈L2So that the position of s,
Figure BDA0001723408380000132
and
Figure BDA0001723408380000133
are bounded. From condition 1, the system state q is known to be bounded. According to equation (26), the system output τ is bounded. Therefore, the closed loop signals of the system are all bounded and W can be obtained bounded, so that W is consistently continuous. The system becomes progressively more stable according to the barbalt theorem.
According to the introduction of 1 to obtain siE s at a finite time tiConverge to 0, tiThe moments are as follows
Figure BDA0001723408380000134
And (3) proving that: designing Lyapunov functions
Figure BDA0001723408380000135
Derived from (36)
Figure BDA0001723408380000136
Substituting equation (15) into (37)
Figure BDA0001723408380000137
Wherein λ1And mu1Is a normal number, q1＞0，q1Is more than 0 and is an odd integer.
Figure BDA0001723408380000138
So at a limited time tiWhen s isiConverge to 0.
Example (b):
the two-degree-of-freedom mechanical arm is used as a simulation model, wherein the system parameters of the mechanical arm are as follows: kappa is 6.7kg m2、ρ＝3.4kg·m2、ε＝3.0kg·m2、μ＝0、Fv5N · m · s/rad, interference added
Figure BDA0001723408380000139
The position command that the system expects to track is a curve q1dSin (3.14 × t) (rad) and q2dSin (3.14 × t) (rad). The manipulator model matrix can be obtained according to the simulation parameters as follows:
Figure BDA00017234083800001310
Figure BDA0001723408380000141
Figure BDA0001723408380000142
comparing simulation results: the parameter selection of the robust sliding mode self-adaptive control method of the mechanical arm system is that Lambda is diag [10,10]，Γ＝diag[5,5,5,5]，KD＝diag[20,20]，γ＝80，λ1＝5，q1＝3，p1＝5，k1＝k2＝1，ξ1(t)＝ξ2(t) 5000/(1+ t2), which satisfies equation (26), and the initial value of the arm parameter is α ═ 0,0,0,0]，F v0. The PID controller parameters are selected as follows: kp＝diag[1000,3000],Ki＝0,Kd＝diag[500,1000]。
FIG. 3 shows system parameters α and F under the action of a robust sliding mode adaptive control method of a mechanical arm system designed by the inventionvThe curve of the estimated value of (2) changing with time can be seen from the figure that the estimated value of the curve gradually approaches to the nominal value of the system parameter and fluctuates in a certain range near the nominal value, so that the parameter of the system can be accurately estimated.
The controller has the following effects: fig. 4 is a graph showing the tracking curve of the double joints of the mechanical arm of the controller designed by the invention, and it can be seen that the controller designed by the invention accurately tracks the expected curve with good tracking capability.
Fig. 5 and 6 are comparison curves of the tracking error of the system with time under the respective actions of the controller designed by the present invention and the conventional PID controller, and it can be seen from the comparison curves that the tracking error of the double joints of the controller designed by the present invention is much smaller than that of the conventional PID controller, which shows the good performance of the controller designed by the present invention.
Fig. 7 is a curve of the control input of the robust sliding mode adaptive control method for the manipulator system, which is designed by the present invention, changing with time, and it can be seen from the figure that the control input signal obtained by the present invention is smooth and continuous, which is beneficial to the application in engineering practice.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (1)

1. A robust sliding mode self-adaptive control method of a mechanical arm system is characterized by comprising the following steps:
step 1, establishing a dynamic model of a mechanical arm system, wherein according to an Euler-Lagrange method, the dynamic model of the mechanical arm system with n degrees of freedom comprises the following steps:
Figure FDA0002920318530000011
q ∈ R in formula (1)n，
Figure FDA0002920318530000012
Respectively the velocity, angular velocity and angular acceleration of the mechanical arm joint; h (q) epsilon Rn×nIs an inertia matrix of the robotic arm system;
Figure FDA0002920318530000013
G(q)∈Rn，τ∈Rnrepresenting centripetal coriolis force, gravity, and input torque, respectively: fvFor a coefficient of viscous friction, d ∈ RnThe interference vector is an external interference vector, including time-varying interference and constant interference of the system,
the mechanical arm system has the following properties:
properties 1: h (q) is a positive definite symmetric matrix satisfying:
Figure FDA0002920318530000014
wherein m is1And m2E R is a known bounded positive real number;
properties 2: the differential matrix of the mechanical arm inertia matrix and the Coriolis matrix satisfy the following oblique symmetry matrix relationship:
Figure FDA0002920318530000015
properties 3: the dynamic model of the mechanical arm is linear with respect to a set of physical parameters:
Figure FDA0002920318530000016
wherein the content of the first and second substances,
Figure FDA0002920318530000017
is a regression matrix of the mechanical arm joint, alpha ═ k, rho, epsilon, upsilon]TAre intrinsic parameters in the manipulator model;
the mechanical arm system further meets the following conditions and lemmas:
condition 1: position command q that the arm system expects to trackdFirst order differential of it
Figure FDA0002920318530000021
And second order differential
Figure FDA0002920318530000022
Are all continuously bounded signals;
condition 2: the mechanical arm uncertainty disturbance d is bounded, i.e.
||d||≤d0 (5)
Wherein d is0Is a known bounded normal number;
lesion 1. consider a first order system
Figure FDA0002920318530000023
The following control laws are achieved in the above system for a finite time to stabilize:
θ＝-θ-λθ-μθq/p (7)
wherein theta is a system state variable, lambda and mu are normal numbers, q is more than 0, p is more than 0, and q and p are odd integers, and q/p is less than 1; thus, the convergence time tsComprises the following steps:
Figure FDA0002920318530000024
2 in the introduction.
Figure FDA0002920318530000025
Is provided with
0≤xtanh(x/a)≤|x| (8)
And 3, leading.
Figure FDA0002920318530000026
Or x > and y ≧ 0
x/x+y≤1 (9)；
Step 2, designing the self-adaptive robust control steps of the mechanical arm sliding mode as follows:
step 2.1, definition
Figure FDA0002920318530000027
Therefore, it is not only easy to use
Figure FDA0002920318530000028
qdIs a position instruction expected to be tracked by the system, the second order of the instruction can be minimized, a sliding mode surface is designed, the tracking error is limited, the error is ensured to be converged to 0,
Figure FDA0002920318530000029
where Λ is a constant matrix and its eigenvalues lie strictly in the right complex half plane,
designing a virtual reference trajectory qrPlace instruction q instead of the system's expected trackingd，
Figure FDA0002920318530000031
Therefore, the temperature of the molten metal is controlled,
Figure FDA0002920318530000032
and
Figure FDA0002920318530000033
is replaced by
Figure FDA0002920318530000034
Figure FDA0002920318530000035
Definition of
Figure FDA0002920318530000036
Wherein s ═ s1,s2,…sn]T，
In order to avoid the jitter of sliding mode control, a control law is designed:
Figure FDA0002920318530000037
wherein λ1And mu1Is a normal number, q1> 0 and p1Is more than 0 and is an integer odd number, and satisfies q1/p1< 1, obtained according to the formulae (1), (14) and (15):
Figure FDA0002920318530000038
according to formula (16):
Figure FDA0002920318530000039
step 2.2, designing a control law, and combining a formula (17) and the property 1, wherein the control law based on the mechanical arm system dynamic model is designed as follows:
τ＝τa+τs (18)
Figure FDA00029203185300000310
τs＝τs1+τs2 (20)
τs1＝-KDs (21)
wherein, tauaFeeding forward a compensation term for the model; tau iss1Is a linear feedback term; tau iss2Is a continuous robust term; kDIs a symmetrical positive definite matrix which is a diagonal matrix;
Figure FDA00029203185300000311
and
Figure FDA00029203185300000312
is a mixture of H (q),
Figure FDA0002920318530000041
g (q) and FvAn estimated value of (d);
step 2.3, designing a parameter regressor and a parameter estimator,
the parameter adaptation law using a parameter regressor is:
Figure FDA0002920318530000042
Figure FDA0002920318530000043
wherein gamma is a diagonal adaptive law matrix, gamma is more than 0, gamma is a normal number, and gamma influence the adaptive rate of the parameters;
step 2.4, designing a continuous robust item according to the condition 2, and overcoming system interference
τs2＝[τs21,τs22,…τs2n]T (24)
Wherein tau iss2iIn the form of
Figure FDA0002920318530000044
In the formula (25), kiFor a known normal number, ξ (t) satisfies the following condition
Figure FDA0002920318530000045
Wherein deltai *And deltaiAre all normal numbers;
and 3, analyzing the stability of the mechanical arm system, and according to the sliding mode self-adaptive robust control method designed in the step 2, carrying out stability verification on the system by using the Lyapunov stability theory to obtain a gradual system stability result:
the lyapunov function is defined as follows:
Figure FDA0002920318530000046
wherein
Figure FDA0002920318530000047
And Fv＞0，
Derivation of formula (27) to obtain
Figure FDA0002920318530000051
Substituting (28) the equations (16), (18) - (21) into the equation (28) results from the mechanical arm properties 2, 3
Figure FDA0002920318530000052
The adaptive laws (22) and (23) are brought into (29)
Figure FDA0002920318530000053
Bringing the formulae (24), (25) into (30)
Figure FDA0002920318530000054
According to 2 can be obtained
Figure FDA0002920318530000061
According to the introduction 3
Figure FDA0002920318530000062
Wherein W is a positive function, and integrating the two sides of equation (33) to obtain
Figure FDA0002920318530000063
V (t) e L is obtained from the formula (34)∞，W∈L2So that the position of s,
Figure FDA0002920318530000064
and
Figure FDA0002920318530000065
is bounded, according to the condition 1, the system state q is bounded, according to the equation (26), the system output tau is bounded, and W can be obtained bounded, so that W is consistent and continuous, and the system is gradually stable according to the Barbalt theorem,
according to the introduction of 1 to obtain siE s at a finite time tiConverge to 0, tiThe moments are as follows
Figure FDA0002920318530000066
And (3) proving that: designing Lyapunov functions
Figure FDA0002920318530000071
Derived from (36)
Figure FDA0002920318530000072
Substituting equation (15) into (37)
Figure FDA0002920318530000073
Wherein λ1And mu1Is a normal number, q1＞0，q1Is more than 0 and is an odd integer,
Figure FDA0002920318530000074
so at a limited time tiWhen s isiConverge to 0.
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