CN106844967A - A kind of Harmonic Drive System quadratic form integral sliding mode control device method for designing - Google Patents

Abstract

The invention discloses a kind of Harmonic Drive System quadratic form integral sliding mode control device method for designing, using a kind of novel secondary type Integral Sliding Mode face, it is proposed that a kind of method for designing of novel Harmonic Drive System sliding mode controller.Mathematical modeling is carried out to harmonic gear system first, modelling quadratic form Integral Sliding Mode face is directed to again, and based on this quadratic form Integral Sliding Mode face, non-matching uncertain harmonic gear quadratic form integral sliding mode control device is designed, also by being suppressed to shake using hyperbolic tangent function.In view of uncertain and non-linear present in harmonic gear system, controller proposed by the present invention can effectively reduce the steady-state error of system, the dynamic property of improvement system, and there is strong robustness to uncertainty, realize the robust control to Harmonic Drive System, reach driving error is compensated by control device, improve the purpose of systematic function.

Description

Design method of quadratic integral sliding mode controller of harmonic gear transmission system

Technical Field

The invention belongs to the technical field of power and transmission, and particularly relates to a design method of a quadratic integral sliding mode controller of a harmonic gear transmission system. The invention relates to a harmonic gear transmission system controller design technology, which adopts a novel quadratic integral sliding mode surface, and can perform robust control on a harmonic gear transmission system and realize error compensation through quadratic integral sliding mode controller design, nonlinear torque modeling, unmatched uncertain item modeling and harmonic gear transmission system modeling.

Background

The harmonic gear is a novel mechanism which realizes transmission by means of elastic deformation movement, breaks through the mode that mechanical transmission adopts a rigid member mechanism, and uses a flexible member to realize mechanical transmission, thereby obtaining a series of special functions which are difficult to achieve by other transmissions. The harmonic gear is composed of three basic components including a rigid gear, a flexible gear and a wave generator. A moving deformation wave is generated in the flexible gear under the action of the wave generator and is meshed with the rigid wheel, so that the transmission purpose is achieved. The harmonic gear has the advantages of large transmission ratio range, more meshing teeth, large bearing capacity, high precision, stable operation, no impact and the like. With the development of science and technology, the fields of aerospace aircraft control systems, instrumentation, robots and the like have new requirements on mechanical transmission, such as: the harmonic gear has the advantages of large transmission ratio, small volume, light weight, high transmission precision, small return difference and the like, and is widely applied to the fields due to the characteristics of the harmonic gear. Due to the characteristics of the harmonic gear, assembly errors, abrasion, deterioration of working environment and the like, the control of the transmission errors of the harmonic gear is particularly important.

Scholars at home and abroad carry out deep research on harmonic gear transmission, and the main research directions are a harmonic gear mechanical simulation model, transmission error analysis, nonlinear friction model modeling and the like. Such as the literature^[1]In the method, a mechanical transmission system Lugre friction model is established, and documents are provided^[2]The modeling of the harmonic gear transmission system is realized by applying system kinetic equation analysis and a least square method, and documents^[3]The elastic theory, the nonlinear finite element analysis theory and the modern CAD and CAE technology are applied to establish a three-dimensional entity finite element analysis model of the flexible gear.

The research on the aspect of compensating errors by controlling the motor is less due to the combination of the motor system and the transmission system, and the research on the aspect is slow due to the fact that the harmonic gear transmission system is a nonlinear system and unmatched uncertainty is added to the system due to nonlinear torsional rigidity and tooth side clearance in work. Literature reference^[4]The method applies the traditional PID controller to carry out error compensation control on a harmonic gear transmission system containing a friction model, but does not consider mismatching terms of the system caused by nonlinear elastic deformation.

Sliding mode control is a new variable structure robust control method, and through the design of a sliding mode surface and the selection of a approximation rule, a system can have the advantages of good response speed, insensitivity to external interference, parameter change, strong robustness and the like, and the method is widely and deeply researched as an important method of nonlinear control in recent years. Literature reference^[5]The sliding mode controller is designed for a hydraulic pump control motor system by Zhongxie construction et al, and simulation results show that compared with a common PID control method, the sliding mode control method has stronger anti-interference capability and good tracking performance, and improves the control precision and stability of the hydraulic pump control motor system. Literature reference^[6]In the prior art, NE Sadr et al designs a sliding mode controller for an automatic guided missile navigation system, and overcomes buffeting by adopting a boundary layer method, so that the system has strong robustness to external interference and uncertainty, and documents^[7]In the Murat fura et al, a second-order integral sliding mode controller is designed for a single-input single-output uncertain system, so that the system overcomes parameter fluctuation caused by external load and uncertainty. Literature reference^[8]In the middle, Azar AT and the like design a self-adaptive sliding mode controller for a Gutian pendulum system, and simulation results show that compared with other control methods, the sliding mode control method has obvious advantages. Literature reference^[9]In Ginoya D et al, aiming at a system containing mismatch uncertainty, a sliding mode controller containing an augmented disturbance observer is designed, and a simulation result shows that the system can overcome the mismatch uncertainty in the system. Literature reference^[10]Hess RA et al for a class of non-linear unmanned aerial vehicle systemsThe sliding mode controller is designed, and the sliding mode controller has good control effect on the nonlinear system

The invention provides a novel quadratic integral sliding mode controller design method, which aims at a non-linear harmonic gear transmission system containing mismatching uncertainty, considers time-varying non-linear torque caused by tooth side clearance and parameter perturbation and non-linear friction of the system, strengthens the effect of a non-linear term on system control and provides the novel quadratic integral sliding mode controller design method. According to the Lyapunov stability theory, the quadratic integral sliding mode surface is proved to reach in effective time, and the conclusion that the closed loop system robustness is gradually stable is given. Incorporation of documentation in modeling systems^[10]The friction model of (1) taking into account the nonlinear elastic deformation caused by backlash^[11]So that the model is more fit to the actual system. Finally, compared with the traditional linear sliding mode control and integral sliding mode control, the simulation result shows that the system adopting the quadratic integral sliding mode controller can respond quickly, has smaller error and has strong robustness to the non-linear terms which are not matched and uncertain.

Reference documents:

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[3] Study on finite element analysis of harmonic gear drive flexible gear in the country [ D ]. university of Sichuan, 2005.

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[5] Xie Jian, tension, Xie Zheng, and the like.

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[8]Azar A T,Serrano F E.Adaptive Sliding Mode Control of the FurutaPendulum[M]//Advances and Applications in Sliding Mode Controlsystems.Springer International Publishing,2015:1-Shtessel Y,Edwards C,FridmanL,et al.Sliding Mode Control and Observation[J].2014:213-249.

[9]Ginoya D,Shendge P D,Phadke S B.Sliding Mode Control forMismatched Uncertain Systems Using an Extended Disturbance Observer[J].IEEETransactions on Industrial Electronics,2014,61(4):1983-1992.

[10]Hess R A,Bakhtiarinej ad M.Sliding-Mode Control Applied of aNonlinear Model of an Unmanned Aerial Vehicle[J].Journal of Guidance Control&Dynamics,2015,31(4):1163-1166.

[11] Modeling and motion control of [ D ] southern hua university, 2014 of takawa.

[12] Li cong jun a harmonic gear mathematical model analysis for precise position control [ J ] mechanical transmission, 2010,34(1):26-29.

[13]Saito Y.Harmonic gear drive:US,US 8020470B2[P].2011.

[14]Tuttle T D,Seering W.Modeling a harmonic drive gear transmission[C]//IEEE International Conference on Robotics and Automation,1993.Proceedings.IEEE,1993:624-629vol.2.

[15] Analysis and control of gear transmission accuracy in the Shenhui orientation controller [ D ]. Chongqing university, 2006.

[16]Zhang X.Integral sliding mode control for non-linear systems withmismatched uncertainty based on quadratic sliding mode[J].Journal ofEngineering,2015,1.

[17]Liao H,Fan S,Fan D.Friction compensation of harmonic gear basedon location relationship[J].Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems&Control Engineering,2016,230.

[18] Royal gorgeous, parametric modeling of aircraft engine gear drive systems and dynamic performance studies [ D ]. shenyang aerospace university, 2013.

[19]Tuttle T D,Seering W P.A Nonlinear Model of a Harmonic Drive GearTransmission[J].IEEE Transactions on Robotics&Automation,1996,12(3)(3):368-374.

[20] The method comprises the following steps of (1) carrying out fine analysis on transmission errors of a harmonic gear transmission system by the Xinhong soldier [ J ]. modern manufacturing engineering, 2005(2): 109-.

[21]Yang J,Liang D,Yu D,et al.System identification and sliding modecontrol design for electromechanical actuator with harmonic gear drive[C]//Chinese Control and Decision Conference.IEEE,2016.

[22]Zhang H B,Pan B,Wang L,et al.Non-Linear Dynamic Modeling andExperiment of Harmonic Gear Drive[J].Applied Mechanics&Materials,2014,668:217-220.

[23] Ten thousand celebrations, Junggui, Wanke, etc. precision harmonic gear reducer transmission error analysis [ J ] instrument technology and sensors, 2013, (5) 51-54.

[24]Tan DL,Qiu ZC,Han JD.STUDY ON SENSOR-BASED CONTROL METHOD FORHARMONIC DRIVE SYSTEM[J].Chinese Journal of Mechanical Engineering,2000(3):234-103.

Disclosure of Invention

The purpose of the invention is as follows: harmonic gear drive systems are a complex class of nonlinear systems that are difficult to control to improve. In order to solve the problem, the invention designs a quadratic integral sliding mode controller aiming at a nonlinear harmonic gear transmission system, takes the nonlinear and non-matching uncertain items existing in the system into consideration, performs robust control on the system and realizes the compensation of the transmission error of the system by a control means. The invention can effectively reduce the transmission error of the system, improve the dynamic performance of the system and make the system have robustness to non-matching uncertainty.

The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that:

a design method of a quadratic integral sliding mode controller of a harmonic gear transmission system comprises the following steps:

step 1) obtaining mathematical representation of the harmonic gear system through simultaneous establishment according to a kinetic equation satisfied by the harmonic gear system and a kirchhoff law satisfied by the motor system;

step 2) modeling nonlinear friction existing in the system, nonlinear torque caused by backlash and parameter perturbation of the system, and completing mathematical modeling of a nonlinear term;

step 3) combining the mathematical models obtained in the step 1) and the step 2) to obtain a mathematical model of the harmonic gear transmission system;

step 4) designing a quadratic integral sliding mode surface by adopting a design method of the quadratic integral sliding mode surface aiming at the mathematical model obtained in the step 3);

and 5) designing a non-matching uncertain harmonic gear quadratic integral sliding mode controller based on the quadratic integral sliding mode surface in the step 4), and inhibiting the jitter by adopting a hyperbolic tangent function.

Further, in step 1), the mathematical representation of the strain wave gear system is:

wherein, J_mAnd J_lThe moment of inertia, q, of the input and output of the harmonic gear system, respectively_mAnd q is_lAngular displacement, T, of the motor and load, respectively_mIs the input torque of the motor, T_sFor harmonic gear input torque, F_mFor input of equivalent friction torque, F_lIs the equivalent friction torque of the output end, r is the reduction ratio, K_sIs the spring coefficient of the harmonic drive; k_mIs the motor torque constant, i_aIs current, u (t) is input voltage u (t), R is motor equivalent resistance, L is motor equivalent inductance, K_bIs a voltage constant, ω_mIs the motor speed.

Further, in step 2), the nonlinear friction is:

friction F on the motor side_m(x)：

Wherein,is composed ofInput torque of time;

friction F on the load side_l(x)：

Wherein,is composed ofThe input torque of the moment.

Further, in step 2), the nonlinear torque caused by the backlash is as follows:

wherein Δ e (t) is an elastic torsion angle deformation amount due to backlash, q_m(t) is the input angular displacement, q_l(t) output angular displacement, 2j system flank clearance,the torsion angle transmission error which does not participate in elastic deformation;

the nonlinear term is: t is_ul(x)＝K_sΔe(t)；

Δ a (x) is a state parameter perturbation of the system, and Δ b (x) is a control quantity gain perturbation of the system.

Further, in step 3), the mathematical model of the harmonic gear transmission system is as follows:

wherein u ∈ R is a control signal, f (x) ═ f₁(x) f₂(x) … f₄(x)]^TIn order to be a dynamic vector, the dynamic vector,

g(x)＝[g₁(x) g₂(x) … g₄(x)]^TΔ g (x) is the control quantity gain vector, Δ g is the control quantity parameter uncertainty of the system; Δ f (x) ═ Δ a (x) + [ 0-T_ul(x)+T_m(x) 0 T_ul(x)+T_l(x)]^TWhere Δ a (x) is the system parameter uncertainty.

Further, in step 4), the quadratic integral sliding mode surface is:

wherein, some corresponding functions or symbols in the formula, the original intersection material has no corresponding explanation, please add: x (t) is the state vector of the system, u ═ u₀+u₁，u₁To control the non-linear part of the quantity, u₀Equivalent control is adopted;

b(x)∈R，b(x)＝σ+ζ₀||x||+ζ₁||x||²+||x^T(t) g (x) l, and σ > 0, ζ₀＞0，ζ₁> 0, | | | | · | |, denotes the euclidean norm. b (x) is an auxiliary function related to x (t) designed to give the sliding-mode surface s the desired properties; zeta₀And ζ₁The two parameters are positive and real numbers, and are adjusted according to the performance requirement of the controller, so that the controller has satisfactory performance.

Further, the design method of the quadratic integral sliding mode surface is as follows:

according to the harmonic gear transmission system:

the following assumptions were introduced:

suppose 1| | Δ f (x) | ≦ ξ₁||x||+ξ₀Wherein ξ₀＞0，ξ₁Greater than 0, | | | | · | |, represents the euclidean norm;

suppose 2| |. Δ g (x) | | ≦ ζ₁||x||+ζ₀Therein ζ of₀＞0，ζ₁Greater than 0, | | | | · | |, represents the euclidean norm;

defining a quadratic integral sliding mode function of the harmonic gear transmission system as follows:

wherein x₀For initial values of state variables, b (x) ∈ R is defined as:

b(x)＝σ+ζ₀||x||+ζ₁||x²+||x^T(t)g(x)|| (9)

and σ > 0, so | | | b (x) | non-woven phosphor^-1≤，＝(σ+ζ₀||x||+ζ₁||x||²)^-1(ii) a Because the quadratic integral sliding mode function (8) of the harmonic gear transmission system has a quadratic structure, the corresponding sliding mode surface is a quadratic integral sliding mode surface;

without assuming that the quadratic integral sliding mode surface s is 0, there is

As can be inferred from the harmonic gear transmission system (7), the quadratic integral sliding mode surface is as follows:

derivation is carried out on the quadratic integral sliding mode surface (11), a harmonic gear transmission system (7) and a quadratic integral sliding mode control law are considered, and the following conditions are obtained by assuming 1 and 2:

are synthesized to obtainThe condition for entering the sliding mode is satisfied, so that the quadratic integral sliding-mode surface s (x) ═ 0 can be reached in a limited time and remains on the sliding-mode surface.

Further, in step 5), the control law of the harmonic gear quadratic form integral sliding mode controller is designed as follows:

u＝u₀-b^-1(x)[(λ₀+λ₁||x||)s+(η₀+η₁||x||)sgn(s)](16)

wherein

λ₁≥0 (18)

η₁≥(σ)^-1[ξ₀+ζ₀β₀+(ξ₁+ζ₀β₁+ζ₁β₀)||x||+ζ₁β₁||x||²](20)

Wherein₁And₂for positive approach coefficients, Cx is an auxiliary variable,and k is more than 0;

the quadratic integral sliding mode control law of the harmonic gear transmission system is that u is equal to u₁+u₀Wherein u is₁To control the non-linear part of the law, u₀For equivalent control of the control law u, i.e. ensuring control of a nominal system free from uncertainties and disturbances, and u₀Satisfy u₀≤β₀+β₁||x||，β₀＞0，β₁Is greater than 0; when the sliding mode surface is reached, the quadratic integral sliding mode control law is equal to that of the equal sliding modeEffective control law, i.e. u-u₀。

Further, in step 5), the hyperbolic tangent function is:

wherein S is a sliding mode surface S and is a real number parameter.

Further, various parameters of the harmonic gear transmission system are as follows:

TABLE 1 harmonic Gear drive System parameters

Has the advantages that: the invention provides a novel design method of a sliding mode controller of a harmonic gear transmission system by adopting a novel quadratic integral sliding mode surface aiming at a nonlinear harmonic gear transmission system. In consideration of uncertainty and nonlinearity existing in a harmonic gear system, the controller provided by the invention can effectively reduce steady-state error of the system, improve dynamic performance of the system, has strong robustness to the uncertainty, realizes robust control over a harmonic gear transmission system, and achieves the purpose of compensating the transmission error through a control means and improving the performance of the system.

Drawings

FIG. 1 is a view of a harmonic gear structure;

FIG. 2 is a schematic illustration of backlash;

FIG. 3 is a block diagram of a harmonic gear drive system control system;

FIG. 4 shows the state quantity x without Δ f (x)₁A graph;

FIG. 5 shows the state quantity x without Δ f (x)₂A graph;

FIG. 6 shows the state quantity x without Δ f (x)₃A graph;

FIG. 7 shows the state quantity x without Δ f (x)₄A graph;

FIG. 8 is a graph of the state quantity u without Δ f (x);

FIG. 9 shows the state quantity x when Δ f (x) is contained₁A graph;

FIG. 10 shows the state quantity x when Δ f (x) is contained₂A graph;

FIG. 11 shows the state quantity x when Δ f (x) is contained₃A graph;

FIG. 12 shows the state quantity x when Δ f (x) is contained₄A graph;

FIG. 13 is a graph showing the control amount u when Δ f (x) is contained.

Detailed Description

The present invention will be further described with reference to the accompanying drawings.

Examples

The technical scheme of the invention is explained in detail below by taking a certain type of harmonic gear transmission system as an example:

step 1) obtaining mathematical representation of the harmonic gear system in a simultaneous manner according to a kinetic equation satisfied by the harmonic gear system and a kirchhoff law satisfied by the motor system

As shown in fig. 1, the harmonic gear is composed of three basic components, which are a rigid gear, a flexible gear and a wave generator.

The whole harmonic gear model satisfies the following equation:

in the formula (1), q_mAnd q is_lAngular displacement of the motor and load, respectively; t is_mIs the input torque of the motor, T_sFor harmonic gear input torque, F_mFor input of equivalent friction torque, F_lThe equivalent friction torque at the output end. And T_mAnd T_sThe following relationship is satisfied:

T_m＝K_mi_a(3)

wherein r is the reduction ratio, K_sIs the spring coefficient of the harmonic drive; k_mIs the motor torque constant, i_aIs current, u (t) is input voltage u (t), R is motor equivalent resistance, L is motor equivalent inductance, K_bIs a voltage constant. Omega_mIs the motor speed, L is negligible since it is generally small.

Derived from the formula (4)Will i_aIn formula (3), there are

Will T_mAnd T_sWhen the formula (1) is substituted, the equation becomes

And 2) modeling nonlinear friction existing in the system, nonlinear torque caused by backlash and parameter perturbation of the system, and completing mathematical modeling of a nonlinear term.

Consider the literature^[4]The friction model in (1) is that since the angle range of control is small in the precise control system, the friction generated by the harmonic gear in work is in average coulomb friction f_averInstead, this model is also sufficiently accurate for most engineering applications. Coulomb friction may be expressed as

The static friction and the coulomb friction of the friction material are consistent and are increased by about 3.88 percent compared with the coulomb friction, so the friction material has the advantages of high friction coefficient, low friction coefficient, high friction coefficient and low friction coefficient

f_sm＝1.0388×f_aver(8)

So friction F on the motor side_m(x) Can be expressed as

In the formula (9)Is composed ofThe input torque of the moment. When the input torque is smaller than the maximum static friction force, the friction on the motor side is the input torque. When in useIn this case, the motor side friction is coulomb friction. The same can be said of the friction F on the load side_l(x) Can be expressed as

In the formula (10)Is composed ofThe input torque of the moment.

There is a drive gap in the servo system as shown in fig. 2. These clearances are increasing due to repeated start and stop, frictional wear caused by gear change, deterioration of working environment, etc., resulting in problems of motion lag, inaccurate position tracking, etc.

Considering the backlash phenomenon occurring in gear transmissions, define:

in the formula (11), Δ e (t) is an elastic torsion angle deformation amount due to backlash, q_m(t) is the input angular displacement, q_l(t) output angular displacement, 2j system flank clearance,the torsion angle transmission error which does not participate in elastic deformation can be approximated by a random digital model.

Due to the existence of delta e (T), a nonlinear term T appears in a model of the harmonic drive system_ul(x)＝K_sΔe(t)。

Because the harmonic gear is influenced by factors such as abrasion, environment and the like in the operation process, system parameters and characteristics can slowly drift, wherein Δ A (x) is taken as a state parameter perturbation of the system, and Δ B (x) is taken as a control quantity gain perturbation of the system. Δ a (x) and Δ B (x) are different depending on the system and in different states, and Δ a (x) and Δ B (x) selected in the simulation are 0.1A and 0.1B, respectively.

And 3) combining the mathematical models obtained in the step 1 and the step 2 to obtain the mathematical model of the harmonic gear transmission system.

Selecting a system state variable ofThe harmonic gear drive system may be represented as

Wherein

The system can be written as

Wherein u ∈ R is a control signal, f (x) ═ f₁(x) f₂(x) ... f₄(x)]^TIn order to be a dynamic vector, the dynamic vector,

g(x)＝[g₁(x) g₂(x) ... g₄(x)]^Tfor the control gain vector, Δ g (x) is the uncertainty of the control parameter of the system, Δ f (x) ═ Δ a (x) + [ 0-T ]_ul(x)+T_m(x) 0 T_ul(x)+T_l(x)]^TAnd Δ A (x) is the system parameter uncertainty.

And 4) designing a quadratic integral sliding mode surface by adopting a novel design method of the quadratic integral sliding mode surface aiming at the mathematical model obtained in the step 3.

Taking into account the harmonic gear drive system (14), the assumptions required herein are introduced

Suppose 1| | Δ f (x) | ≦ ξ₁||x||+ξ₀Wherein ξ₀＞0，ξ₁> 0, | | | | · | | represents the euclidean norm

Suppose 2| |. Δ g (x) | | ≦ ζ₁||x||+ζ₀Therein ζ of₀＞0，ζ₁> 0, | | | | · | | represents the euclidean norm

Defining a quadratic integral sliding mode function of a harmonic gear drive system (14) as

Wherein x₀Is the initial value of the state variable, b (x) ∈ R is defined as

b(x)＝σ+ζ₀||x||+ζ₁||x||²+||x^T(t)g(x)|| (16)

And σ > 0, we can deduce | | b (x) | non-woven cells^-1≤，＝(σ+ζ₀||x||+ζ₁||x||²)^-1. Because (15) has a quadratic structure, the sliding mode surface corresponding to (15) is a quadratic integral sliding mode surface.

Without assuming that the quadratic integral sliding mode surface s is 0, there is

Derived from harmonic gear drive systems (14)

So that the quadratic integral sliding mode surface is

Derivation of quadratic integral sliding mode surfaces (18) and consideration of harmonic gear drive (14), quadratic integral sliding mode control law (19), and assumptions 1 and 2, yields

Thus, it is possible to provide

From (20) to (23) below, it can be obtainedThe condition for entering the sliding mode is satisfied, so that the quadratic integral sliding-mode surface s (x) ═ 0 can be reached in a limited time and remains on the sliding-mode surface.

And 5) designing a non-matching uncertain harmonic gear quadratic integral sliding mode controller based on the quadratic integral sliding mode surface in the step 4. Jitter is suppressed by using a hyperbolic tangent function.

A sliding mode controller is designed for a harmonic gear transmission system, and the structure diagram is shown in figure 3.

Designing a quadratic integral sliding mode control law as follows:

u＝u₀-b^-1(x)[(λ₀+λ₁||x||)s+(η₀+η₁||x||)sgn(s)](19)

wherein

λ₁≥0 (21)

η₁≥(σ)^-1[ξ₀+ζ₀β₀+(ξ₁+ζ₀β₁+ζ₁β₀)||x||+ζ₁β₁||x||²](23)

In the formulae (20) and (22)₁And₂in the formula (24), Cx is an auxiliary variable for the positive approximation coefficient,and k > 0.

The quadratic integral sliding mode control law of the system (14) is that u is u₁+u₀Wherein u is₁To control the non-linear part of the law, u₀To control the equivalent of law u, i.e. to guarantee control of a nominal system free from uncertainties and disturbances. And u is₀Satisfy u₀≤β₀+β₁||x||，β₀＞0，β₁Is greater than 0. When the sliding mode surface is reached, the quadratic integral sliding mode control law is equal to an equivalent control law, namely u is equal to u₀。

Due to factors such as switching time lag, space lag, system inertia influence and the like, buffeting is inevitably generated in sliding mode control, control precision is influenced, control quality of the system is reduced, and control elements are seriously damaged even.

In many documents, a quasi-slip mode method in which a saturation function replaces an original discontinuous switching function is used to suppress chattering. The method can effectively overcome sliding mode buffeting, but belongs to a discontinuous function, and is not suitable for occasions needing derivation of the conversion function.

Because the hyperbolic tangent function is continuous and smooth, the hyperbolic tangent function is adopted to replace a discontinuous switching function, and buffeting in sliding mode control can be effectively reduced

The hyperbolic tangent function is as follows:

wherein S is the sliding mode surface S described above, and is an adjustable real number parameter, and different suppression effects on jitter are selected.

The present embodiment uses the harmonic gear drive system parameter table of table 1.

TABLE 1 harmonic Gear drive System parameters

Using the data in the table and the harmonic gear drive system model (12) established above, the following system is obtained:

consider that

f_sm＝1.0388×(1.5738×10^-6x₁ ²-3.7901×10^-4x₁+0.0720)

f_sl＝1.0388×(1.5738×10^-6x₃ ²-3.7901×10^-4x₃+0.0720)

When the initial condition of the system is x₀＝[0 0 0.05 0]When, T_mAnd T_lThe interference is added into a harmonic gear model as a disturbance variable, and a system is simulated under the action of a second-order integral sliding mode controller (QSMC). By revising the parameters for a plurality of times, zeta is found when sigma is 1₀＝0，ζ₁＝0，λ₀＝60，λ₀＝0，η₀＝0，η₀The system has better performance when the nonlinear mismatch term Δ f (x) of the system is not considered, and the following simulation results are obtained, as shown in fig. 4 to 8.

The simulation result shows that the system can reach stability within 0.1s, the steady-state error is rapidly reduced, the overshoot, the stabilization time and the error are small, and the requirements of the system on high precision, quick response and stable operation can be met.

When Δ f (x) in which the system exists is considered, the following simulation images are obtained as in fig. 9 to 13.

The results show that when the system has Δ f (x), the state of the system at 0.1s is taken for comparative analysis, and it can be found that the system can still achieve stability, and the curve is smooth, and the rise time, overshoot and steady-state error are slightly increased, which indicates that the system has robustness to Δ f (x), and still has the advantages of fast response, short rise time, small overshoot, small error and the like, and indicates that the controller has strong robustness to non-matching uncertainty.

The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (10)

1. A design method of a quadratic integral sliding mode controller of a harmonic gear transmission system is characterized by comprising the following steps: the method comprises the following steps:

step 1) obtaining mathematical representation of the harmonic gear system through simultaneous establishment according to a kinetic equation satisfied by the harmonic gear system and a kirchhoff law satisfied by the motor system;

step 2) modeling nonlinear friction existing in the system, nonlinear torque caused by backlash and parameter perturbation of the system, and completing mathematical modeling of a nonlinear term;

step 3) combining the mathematical models obtained in the step 1) and the step 2) to obtain a mathematical model of the harmonic gear transmission system;

step 4) designing a quadratic integral sliding mode surface by adopting a design method of the quadratic integral sliding mode surface aiming at the mathematical model obtained in the step 3);

and 5) designing a non-matching uncertain harmonic gear quadratic integral sliding mode controller based on the quadratic integral sliding mode surface in the step 4), and inhibiting the jitter by adopting a hyperbolic tangent function.

2. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: step 1), the mathematical representation of the strain wave gear system is:

$\begin{array}{c}{J}_m{\stackrel{\·\·}{q}}_m=\frac{K_m}{R}u\left(t\right)-K_s(\frac{q_m}{r}-q_l)-\frac{K_m{K}_b}{R}{\stackrel{\·}{q}}_m+F_m\\ J_1\stackrel{\·\·}{q}=K_s(\frac{q_m}{r}-q_l)-F_l\end{array}---\left(1\right)$

wherein, J_mAnd J_lThe moment of inertia, q, of the input and output of the harmonic gear system, respectively_mAnd q is_lAngular displacement, T, of the motor and load, respectively_mIs the input torque of the motor, T_sFor harmonic gear input torque, F_mFor input of equivalent friction torque, F_lIs an output endEquivalent friction torque, r is the reduction ratio, K_sIs the spring coefficient of the harmonic drive; k_mIs the motor torque constant, i_aIs current, u (t) is input voltage u (t), R is motor equivalent resistance, L is motor equivalent inductance, K_bIs a voltage constant, ω_mIs the motor speed.

3. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in step 2), the nonlinear friction is as follows:

friction F on the motor side_m(x)：

$\left\{\begin{array}{cc}{F}_m=-{\mathrm{\ψ}}_m,& {\stackrel{\·}{q}}_m=0,\left|{\mathrm{\ψ}}_m\right|\≤f_{sm}\\ F_m=-sgn\left({\mathrm{\ψ}}_m\right)f_{sm},& {\stackrel{\·}{q}}_m=0,\left|{\mathrm{\ψ}}_m\right|>f_{sm}\\ F_m=-sgn\left(q_m\right)f_{cm},& \left|{\stackrel{\·}{q}}_m\right|>0\end{array}\right.---\left(2\right)$

Wherein,is composed ofInput torque of time;

friction F on the load side_l(x)：

$\left\{\begin{array}{cc}{F}_l=-{\mathrm{\ψ}}_l,& {\stackrel{\·}{q}}_l=0,\left|{\mathrm{\ψ}}_l\right|\≤f_{sl}\\ F_l=-\mathrm{sgn}\left({\mathrm{\ψ}}_l\right)f_{sl},& {\stackrel{\·}{q}}_l=0,\left|{\mathrm{\ψ}}_l\right|>f_{sl}\\ F_l=-\mathrm{sgn}\left(q_l\right)f_{cl},& \left|{\stackrel{\·}{q}}_l\right|>0\end{array}\right.---\left(3\right)$

Wherein,is composed ofThe input torque of the moment.

4. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in step 2), the nonlinear torque caused by the backlash is:

wherein Δ e (t) is an elastic torsion angle deformation amount due to backlash, q_m(t) is the input angular displacement, q_l(t) output angular displacement, 2j system flank clearance,the torsion angle transmission error which does not participate in elastic deformation;

the nonlinear term is: t is_ul(x)＝K_sΔe(t)；

Δ a (x) is a state parameter perturbation of the system, and Δ b (x) is a control quantity gain perturbation of the system.

5. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in step 3), the mathematical model of the harmonic gear transmission system is as follows:

$\stackrel{\·}{x}=f\left(x\right)+\mathrm{\Δ}f\left(x\right)+\[g\left(x\right)+\mathrm{\Δ}g\left(x\right)\]u---\left(5\right)$

wherein u ∈ R is a control signal, f (x) ═ f₁(x) f₂(x) ... f₄(x)]^TIs a motion vector, g (x) ═ g₁(x)g₂(x) … g₄(x)]^TΔ g (x) is the control quantity gain vector, Δ g is the control quantity parameter uncertainty of the system; Δ f (x) ═ Δ a (x) + [ 0-T_ul(x)+T_m(x) 0 T_ul(x)+T_l(x)]^TWhere Δ a (x) is the system parameter uncertainty.

6. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in the step 4), the quadratic integral sliding mode surface is as follows:

$s={\∫}_0^t\{x^T\left(t\right)\[\mathrm{\Δ}f\left(x\right)+\mathrm{\Δ}g\left(x\right)u\]+b\left(x\right)u_1\}dt=0---\left(6\right)$

where x (t) is the state vector of the system, and u ═ u₀+u₁，u₁To control the non-linear part of the quantity, u₀Equivalent control is adopted;

b(x)∈R，b(x)＝σ+ζ₀||x||+ζ₁||x||²+||x^T(t) g (x) l, and σ > 0, ζ₀＞0，ζ₁> 0, | | | | · | |, denotes the euclidean norm.

7. The harmonic gear drive system quadratic form integral sliding mode controller design method according to claim 1 or 6, characterized in that: the design method of the quadratic integral sliding mode surface comprises the following steps:

according to the harmonic gear transmission system:

$\stackrel{\·}{x}=f\left(x\right)+\mathrm{\Δ}f\left(x\right)+\[g\left(x\right)+\mathrm{\Δ}g\left(x\right)\]u---\left(7\right)$

the following assumptions were introduced:

suppose 1| | Δ f (x) | ≦ ξ₁||x||+ξ₀Wherein ξ₀＞0，ξ₁Greater than 0, | | | | · | |, represents the euclidean norm;

suppose 2| |. Δ g (x) | | ≦ ζ₁||x||+ζ₀Therein ζ of₀＞0，ζ₁Greater than 0, | | | | · | |, represents the euclidean norm;

defining a quadratic integral sliding mode function of the harmonic gear transmission system as follows:

$s=\frac{1}{2}\[x^T\left(t\right)x\left(t\right)-{x_0}^T{x}_0\]-{\∫}_0^t\{x^T\left(t\right)\[f\left(x\right)+g\left(x\right)u\]-b\left(x\right)u_1\}dt---\left(8\right)$

wherein x₀For initial values of state variables, b (x) ∈ R is defined as:

b(x)＝σ+ζ₀||x||+ζ₁||x||²+||x^T(t)g(x)|| (9)

and σ > 0, so | | | b (x) | non-woven phosphor^-1≤，＝(σ+ζ₀||x||+ζ₁||x||²)^-1(ii) a Because the quadratic integral sliding mode function (8) of the harmonic gear transmission system has a quadratic structure, the corresponding sliding mode surface is a quadratic integral sliding mode surface;

without assuming that the quadratic integral sliding mode surface s is 0, there is

$\frac{1}{2}\[x^T\left(t\right)x\left(t\right)-{x_0}^T{x}_0\]-{\∫}_0^t\{x^T\left(t\right)\[f\left(x\right)+g\left(x\right)u\]-b\left(x\right)u_1\}dt=0---\left(10\right)$

As can be inferred from the harmonic gear transmission system (7), the quadratic integral sliding mode surface is as follows:

$s={\∫}_0^t\{x^T\left(t\right)\[\mathrm{\Δ}f\left(x\right)+\mathrm{\Δ}g\left(x\right)u\]+b\left(x\right)u_1\}dt=0---\left(11\right)$

derivation is carried out on the quadratic integral sliding mode surface (11), a harmonic gear transmission system (7) and a quadratic integral sliding mode control law are considered, and the following conditions are obtained by assuming 1 and 2:

$\begin{array}{c}s\stackrel{\·}{s}\≤\{{\mathrm{\ξ}}_0+{\mathrm{\ζ}}_0{\mathrm{\β}}_0+({\mathrm{\ξ}}_1+{\mathrm{\ζ}}_0{\mathrm{\β}}_1+{\mathrm{\ζ}}_1{\mathrm{\β}}_0)|\left|x\right||+{\mathrm{\ζ}}_1{\mathrm{\β}}_1|\left|x\right|{|}^2\left\}\right|\left|x\right||\·|s|\\ -\frac{\mathrm{\σ}+\left|\right|x^T\left(t\right)g\left(x\right)\left|\right|}{b\left(x\right)}({\mathrm{\λ}}_0+{\mathrm{\λ}}_1|\left|x\right|\left|\right)s^2-\frac{\mathrm{\σ}+\left|\right|x^T\left(t\right)g\left(x\right)\left|\right|}{b\left(x\right)}({\mathrm{\η}}_0+{\mathrm{\η}}_1|\left|x\right|\left|\right)\left|s\right|\end{array}---\left(12\right)$

are synthesized to obtainThe condition for entering the sliding mode is satisfied, so that the quadratic integral sliding-mode surface s (x) ═ 0 can be reached in a limited time and remains on the sliding-mode surface.

8. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in the step 5), designing a control law of the harmonic gear quadratic form integral sliding mode controller as follows:

u＝u₀-b^-1(x)[(λ₀+λ₁||x||)s+(η₀+η₁||x||)sgn(s)](16)

wherein

${\mathrm{\λ}}_0\≥\frac{{\mathrm{\ϵ}}_1\left({\mathrm{\ζ}}_1\right|\left|x\right|{|}^2+{\mathrm{\ζ}}_0\left|\right|x\left|\right|+\mathrm{\σ})}{\mathrm{\σ}}---\left(17\right)$

λ₁≥0 (18)

${\mathrm{\η}}_0\≥\frac{{\mathrm{\ϵ}}_2\left({\mathrm{\ζ}}_1\right|\left|x\right|{|}^2+{\mathrm{\ζ}}_0\left|\right|x\left|\right|+\mathrm{\σ})}{\mathrm{\σ}}---\left(19\right)$

η₁≥(σ)^-1[ξ₀+ζ₀β₀+(ξ₁+ζ₀β₁+ζ₁β₀)||x||+ζ₁β₁||x||²](20)

$u_0={\left(CB\right)}^{-1}(-CAx+\stackrel{\·}{\mathrm{\δ}})---\left(21\right)$

Wherein₁And₂for positive approach coefficients, Cx is an auxiliary variable,and k is more than 0;

the quadratic integral sliding mode control law of the harmonic gear transmission system is that u is equal to u₁+u₀Wherein u is₁To control the non-linear part of the law, u₀For equivalent control of the control law u, i.e. ensuring control of a nominal system free from uncertainties and disturbances, and u₀Satisfy u₀≤β₀+β₁||x||，β₀＞0，β₁Is greater than 0; when the sliding mode surface is reached, the quadratic integral sliding mode control law is equal to an equivalent control law, namely u is equal to u₀。

9. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: in step 5), the hyperbolic tangent function is:

$\tanh \left(\frac{s}{\mathrm{\ϵ}}\right)=\frac{e^{\frac{s}{\mathrm{\ϵ}}}-e^{-\frac{s}{\mathrm{\ϵ}}}}{e^{\frac{s}{\mathrm{\ϵ}}}+e^{-\frac{s}{\mathrm{\ϵ}}}}---\left(22\right)$

wherein S is a sliding mode surface S and is a real number parameter.

10. The design method of a quadratic form integral sliding mode controller of a harmonic gear transmission system according to claim 1, characterized in that: the parameters of the harmonic gear transmission system are as follows:

TABLE 1 harmonic Gear drive System parameters