阅读说明：本技术 一种直线电机系统的预设性能鲁棒运动控制方法 (Preset performance robust motion control method of linear motor system ) 是由 郭一军 于 2021-09-09 设计创作，主要内容包括：本发明公开了一种直线电机系统的预设性能鲁棒运动控制方法,包括以下具体步骤：建立直线电机系统的运动方程；系统预设性能鲁棒运动控制输入信号设计；基于设计的运动控制信号控制直线电机系统。本发明通过引入预设性能函数及障碍李亚普诺夫函数进行直线电机系统的预设性能鲁棒运动控制输入信号的设计,实现了系统控制过程中跟踪误差的暂态性能和稳态性能的预先设置,解决了系统高频扰动不易估计以及系统初始调节阶段超调过大的问题,且使得系统稳态控制精度满足预设性能要求。(The invention discloses a preset performance robust motion control method of a linear motor system, which comprises the following specific steps: establishing a motion equation of a linear motor system; designing a performance robust motion control input signal preset by a system; and controlling the linear motor system based on the designed motion control signal. According to the invention, the design of the preset performance robust motion control input signal of the linear motor system is carried out by introducing the preset performance function and the obstacle Lyapunov function, so that the preset of the transient performance and the steady-state performance of the tracking error in the system control process is realized, the problems that the high-frequency disturbance of the system is difficult to estimate and the overshoot of the initial adjustment stage of the system is overlarge are solved, and the steady-state control precision of the system meets the preset performance requirement.)

1. A preset performance robust motion control method of a linear motor system is characterized by comprising the following steps:

step 1, establishing a preset performance robust motion control input signal of a linear motor system based on a preset performance function method and introducing a barrier Lyapunov function as shown in a formula (14):

in the formula (14), u is a preset performance robust motion control input signal of the linear motor system; m is₀The nominal value of the mass m of the moving part of the linear motor system is obtained;a first derivative of a virtual control signal to be designed; controlling input signal design parameterse₁Position tracking error of the linear motor system; e.g. of the type₂Is an auxiliary error variable; mu is a time-varying preset performance function;

and 2, controlling the linear motor system to work based on the preset performance robust motion control input signal established in the step 1.

2. The preset performance robust motion control method of the linear motor system according to claim 1, wherein the step 1 process is as follows:

(1) establishing a motion equation of the linear motor system as shown in formula (1):

in the formula (1), m is the mass of the moving part of the linear motor system; y is the position of the moving part of the system,is the first derivative of y and is,is the second derivative of y; u is a control input signal; f is friction; d is the bounded uncertainty suffered by the system; k is a radical of_vIs the coefficient of viscous friction; k is a radical of_cIs the coulomb friction coefficient; sign (·) represents a sign function;

the uncertainty of the system parameters is assumed to satisfy the following relationship:

in the formula (2), m₀，k_v0，k_c0Respectively corresponding to the mass m and the viscous friction coefficient k of the moving part of the linear motor system_vCoefficient of viscous friction k_vNominal value of (d); Δ m, Δ k_v，Δk_cRespectively representing the uncertain parts of the corresponding parameters;

due to the fact thatCan be expressed as:

in the formula (3), the first and second groups,is an indeterminate portion;

equation (1) can be rewritten according to equations (2), (3):

in the formula (4), the first and second groups,represents the sum perturbation of the system, and f ═ f₀+Δf，

Since the control input signal to the system is a bounded signal in a real system, the total disturbance of the system is also bounded, i.e. the relationship:wherein the content of the first and second substances,is an unknown normal number;

(2) designing a linear motor system to preset a performance robust motion control input signal:

first, the position tracking error of the linear motor system is defined as shown in equation (5):

e₁＝y-y_d (5)，

in the formula (5), e₁As position tracking error of the system, y_dA desired position signal that is second order derivable;

the auxiliary error variable is defined as shown in equation (6):

in the formula (6), e₂To assist the error variable, u₀Inputting a signal for virtual control to be designed;

the time-varying preset performance function μ (t) is designed as shown in equation (7):

μ(t)＝(μ₀-μ_∞)e^-kt+μ_∞ (7)，

in the formula (7), μ (t) is a time-varying preset performance function; mu.s₀Is an initial value of a preset performance function mu (t); mu.s_∞A steady state value for a time varying preset performance function μ (t); k > 0 is a parameter related to the decay rate of a time-varying predetermined performance function mu (t), 0 < mu_∞＜μ₀；

The design obstacle Lyapunov function is shown in equation (8):

wherein, V₁Is the barrier Lyapunov function; mu is a time-varying preset performance function;

derivation of equation (8) yields equation (9):

in the formula (9), the reaction mixture,a first derivative of a time-varying preset performance function;the first derivative of the position tracking error of the linear motor system is obtained;is the first derivative of the desired position signal;

virtually controlling the input signal u by equation (9)₀Can be designed as follows:

in the formula (10), k₁＞0，

Substituting equation (10) into equation (9) yields:

to determine the actual control input signal, equation (6) is derived and equation (12) is obtained by combining equation (4):

designing Lyapunov functionsDerivation of the Lyapunov function yields:

the preset performance robust motion control input signal u is designed by equation (13) as:

wherein u is a preset performance robust motion control input signal of the linear motor system; m is₀The nominal value of the mass m of the moving part of the linear motor system is obtained;a first derivative of a virtual control signal to be designed; controlling input signal design parameterse₁Position tracking error of the linear motor system; e.g. of the type₂Is an auxiliary error variable; μ is a time-varying preset performance function.

3. The preset performance robust motion control method of a linear motor system according to claim 2, wherein: the bounded uncertainty d experienced by the linear motor system includes at least system unmodeled dynamics, external disturbances, and measurement noise of the sensor.

Technical Field

The invention relates to the field of motor control methods, in particular to a preset performance robust motion control method of a linear motor system.

Background

The linear motor has the characteristics of simple structure, almost no mechanical friction loss, capability of realizing direct transmission, high transmission efficiency and the like, and in recent years, the linear motor is widely applied to equipment such as a numerical control machine tool, 3D printing equipment, a robot and the like.

Because the frictional force in the linear motor system model is coupled with the sign function, high-frequency disturbance is easily caused when the speed of a moving part of the system tends to be close to a balance point, and the high-frequency disturbance signal is difficult to effectively estimate by designing a related disturbance observer, so that the control precision of the system is seriously influenced, and the problem is more obvious particularly in the high-precision application occasion of the linear motor system. In addition, when the initial error of the system is large, the problem that the overshoot amount is too large is often existed in the initial adjustment stage of the system.

Disclosure of Invention

The invention aims to provide a preset performance robust motion control method of a linear motor system, and aims to solve the problems that the control precision of the linear motor system in the prior art is easily disturbed and the overshoot is overlarge.

In order to achieve the purpose, the technical scheme adopted by the invention is as follows:

a preset performance robust motion control method of a linear motor system comprises the following steps:

step 1, establishing a preset performance robust motion control input signal of a linear motor system based on a preset performance function method and introducing a barrier Lyapunov function as shown in a formula (14):

in the formula (14), u is a preset performance robust motion control input signal of the linear motor system; m is₀The nominal value of the mass m of the moving part of the linear motor system is obtained;a first derivative of a virtual control signal to be designed; controlling input signal design parameterse₁Position tracking error of the linear motor system; e.g. of the type₂Is an auxiliary error variable; μ is a time-varying preset performance function.

And 2, controlling the linear motor system to work based on the preset performance robust motion control input signal established in the step 1.

Further, the step 1 process is as follows:

(1) establishing a motion equation of the linear motor system as shown in formula (1):

in the formula (1), m is the mass of the moving part of the linear motor system; y is the position of the moving part of the system,is the first derivative of y and is,is the second derivative of y; u is a control input signal; f is friction; d is the bounded uncertainty suffered by the system; k is a radical of_vIs the coefficient of viscous friction; k is a radical of_cIs the coulomb friction coefficient; sign (·) represents a sign function;

the uncertainty of the system parameters is assumed to satisfy the following relationship:

in the formula (2), m₀，k_v0，k_c0Respectively corresponding to the mass m and the viscous friction coefficient k of the moving part of the linear motor system_vCoefficient of viscous friction k_vNominal value of (d); Δ m, Δ k_v，Δk_cRespectively representing the uncertain parts of the corresponding parameters;

due to the fact thatCan be expressed as:

in the formula (3), the first and second groups,is an indeterminate portion;

equation (1) can be rewritten according to equations (2), (3):

in the formula (4), the first and second groups,represents the sum perturbation of the system, and f ═ f₀+Δf，

Since the control input signal to the system is a bounded signal in a real system, the total disturbance of the system is also bounded, i.e. the relationship:wherein the content of the first and second substances,is an unknown normal number;

(2) designing a linear motor system to preset a performance robust motion control input signal:

first, the position tracking error of the linear motor system is defined as shown in equation (5):

e₁＝y-y_d (5)，

in the formula (5), e₁As position tracking error of the system, y_dA desired position signal that is second order derivable;

the auxiliary error variable is defined as shown in equation (6):

in the formula (6), e₂To assist the error variable, u₀Inputting a signal for virtual control to be designed;

the time-varying preset performance function μ (t) is designed as shown in equation (7):

μ(t)＝(μ₀-μ_∞)e^-kt+μ_∞ (7)，

in the formula (7), μ (t) is a time-varying preset performance function; mu.s₀Is an initial value of a preset performance function mu (t); mu.s_∞A steady state value for a time varying preset performance function μ (t); k > 0 is a parameter related to the decay rate of a time-varying predetermined performance function mu (t), 0 < mu_∞＜μ₀；

The design obstacle Lyapunov function is shown in equation (8):

in the formula (8), V₁Is the barrier Lyapunov function; μ is a time-varying preset performance function.

Derivation of equation (8) yields equation (9):

in the formula (9), the reaction mixture,a first derivative of a time-varying preset performance function;the first derivative of the position tracking error of the linear motor system is obtained;the first derivative of the desired position signal.

Virtually controlling the input signal u by equation (9)₀Can be designed as follows:

in the formula (10), k₁＞0，

Substituting equation (10) into equation (9) yields:

to determine the actual control input signal, equation (6) is derived and equation (12) is obtained by combining equation (4):

designing Lyapunov functionsDerivation of the Lyapunov function yields:

the preset performance robust motion control input signal u is designed by equation (13) as:

wherein u is a preset performance robust motion control input signal of the linear motor system; m is₀The nominal value of the mass m of the moving part of the linear motor system is obtained;a first derivative of a virtual control signal to be designed; controlling input signal design parameterse₁Position tracking error of the linear motor system; e.g. of the type₂Is an auxiliary error variable; μ is a time-varying preset performance function.

Further, the linear motor system suffers from a bounded uncertainty d that includes at least the unmodeled dynamics of the system, external disturbances, and measurement noise of the sensors.

The robust motion control input signal of the system is designed by introducing the barrier Lyapunov function and the preset performance function, so that the influence of high-frequency disturbance on the control accuracy of the linear motor motion system can be eliminated, the robustness of the system is improved, and the transient response performance and the steady-state control accuracy of the system control process can be preset through the designed performance function. Therefore, the problem that high-frequency disturbance signals in a linear motor system are difficult to effectively estimate is solved, and the high-precision requirement on system output is ensured while the design process of control input signals is simplified.

Compared with the prior art, the invention has the advantages that:

(1) the motion control signal design of the linear motor system is carried out by introducing a preset performance function and an obstacle Lyapunov function, so that the preset of the transient performance and the steady-state performance of tracking errors in the system control process is realized.

(2) The output performance of the system can still be ensured to meet the high-precision control requirement on the premise of not estimating the uncertainty of the system.

(3) The control algorithm is simple to realize, the calculation efficiency of the control input signal is improved, and the real-time performance of control is ensured.

Drawings

Fig. 1 is a schematic diagram of a situation that a desired position signal is tracked by a position of a moving part of a system in case 1 of the embodiment of the present invention.

Fig. 2 is a schematic diagram of a position tracking error of a linear motor system in case 1 of the embodiment of the present invention.

Fig. 3 is a schematic diagram of the situation of tracking the expected position signal by the position of the moving part of the system in case 2 according to the embodiment of the present invention.

Fig. 4 is a schematic diagram of a position tracking error of a linear motor system in case 2 of the embodiment of the present invention.

Fig. 5 is a schematic diagram of control input signals of the control method under the situation 1 according to the embodiment of the present invention.

Fig. 6 is a schematic diagram of control input signals of the control method under the situation 2 according to the embodiment of the present invention.

FIG. 7 is a control schematic block diagram of the control method of the present invention.

Detailed Description

The invention is further illustrated with reference to the following figures and examples.

As shown in fig. 7, a method for controlling a preset performance robust motion of a linear motor system includes the following steps:

s1, establishing a motion equation of the linear motor system, wherein the process is as follows:

s1.1 the equation of motion for a linear motor system can be described as:

wherein m is the mass of the moving part of the linear motor system; y is the position of the moving part of the system,is the first derivative of y and is,is the second derivative of y; u is a control input signal; f is friction; d is the bounded uncertainty suffered by the system; k is a radical of_vIs the coefficient of viscous friction; k is a radical of_cIs the coulomb friction coefficient; sign (·) represents a sign function.

S1.2 assumes that the uncertainty of the system parameters satisfies the following relationship:

wherein m is₀，k_v0，k_c0Is a nominal value of the corresponding parameter; Δ m, Δ k_v，Δk_cMeans for indicating the corresponding parameterA portion is determined.

Due to the fact thatCan be expressed as:

wherein the content of the first and second substances,is an indeterminate portion.

S1.3 is represented by formula (2) and formula (3), wherein formula (1) can be rewritten as:

wherein the content of the first and second substances,represents the sum perturbation of the system, and f ═ f₀+Δf，Since the control input signal to the system is a bounded signal in a real system, the total disturbance of the system is also bounded, i.e. the relationship:wherein the content of the first and second substances,as unknown normal numbers.

S2, designing a system preset performance robust motion control input signal, wherein the specific design process is as follows:

s2.1 defines the position tracking error of the system:

e₁＝y-y_d (5)，

wherein e is₁Tracking an error for a position of the system; y is_dIs of second orderA derivable desired position signal.

S2.2 defines the auxiliary error variable:

wherein e is₂To assist the error variable, u₀The input signal is a virtual control to be designed.

S2.3 the time-varying preset performance function mu (t) is designed as follows:

μ(t)＝(μ₀-μ_∞)e^-kt+μ_∞ (7)，

wherein μ (t) is a time-varying preset performance function; mu.s₀Is an initial value of a preset performance function mu (t); mu.s_∞A steady state value for a time varying preset performance function μ (t); k > 0 is a parameter related to the decay rate of a time-varying predetermined performance function mu (t), 0 < mu_∞＜μ₀。

S2.4 design obstacle Lyapunov function:

wherein, V₁Is the barrier Lyapunov function; μ is a time-varying preset performance function.

S2.5 can be derived from equation (8):

in the formula (9), the reaction mixture,a first derivative of a time-varying preset performance function;the first derivative of the position tracking error of the linear motor system is obtained;the first derivative of the desired position signal.

Virtually controlling the input signal u by equation (9)₀Can be designed as follows:

wherein k is₁＞0，

S2.6 substitution of formula (10) for formula (9) gives:

s2.7 to determine the actual control input signal, the derivation of equation (6) is required, and equation (4) is combined to obtain:

s2.8 design of Lyapunov functionThe derivation of this function yields:

the preset performance robust motion control input signal u is designed by equation (13) as:

wherein u is a preset performance robust motion control input signal of the linear motor system; m is₀For linear motor systemsNominal value of the mass m of the moving part;a first derivative of a virtual control signal to be designed; controlling input signal design parameterse₁Position tracking error of the linear motor system; e.g. of the type₂Is an auxiliary error variable; μ is a time-varying preset performance function.

S2.9 substitution of formula (14) for formula (13) gives:

further, in the present invention,can be expressed as:

wherein the content of the first and second substances,

by solving the inequality (16):

s2.10 therefore, V when t → ∞₂Is bounded by V₂Is defined by V₁Is bounded. In view of equation (8) this is available:

therefore, the following formulae (18) and V₁Is the location of the systemTrace error e₁Will vary according to a preset property, i.e. e₁A predetermined control performance can be obtained.

In order to verify the effectiveness of the control method, simulation research is performed on the control performance under two situations of system parameter uncertainty. Case 1: Δ m is 0.09m₀，Δk_v＝0.09k_v0，Δk_c＝0.09k_c0(ii) a Case 2: Δ m ═ 0.18m₀，Δk_v＝0.18k_v0，Δk_c＝0.18k_c0。

Relevant physical parameters of the linear motor system: m is₀＝6kg，k_v0＝10Ns/m，k_c013N; desired position signal: y is_dSin (t) (mm); the initial position of the moving part of the system is y (0) ═ 10(mm), and d ═ 0.9+0.3cos (5 t).

The time-varying preset performance function parameters are designed as follows: mu.s₀＝0.015，μ_∞0.001 and 4. The control signal parameters are designed as: k is a radical of₁＝25，k₂＝17，k₃＝82。

The results of the simulation study are shown in fig. 1-6. FIG. 1 is a schematic diagram of the tracking of a desired position signal by the position of a moving part of a system in case 1; FIG. 2 is a schematic diagram of the position tracking error of the linear motor system in case 1; FIG. 3 is a schematic diagram of the tracking of the desired position signal by the position of the moving part of the system in case 2; fig. 4 is a schematic diagram of the position tracking error of the linear motor system in the case of case 2. As can be seen from fig. 1 to 4, the control method can achieve high-precision position tracking control under two different conditions, and the transient performance and the steady-state performance of the tracking error meet the preset control requirements. Fig. 5 and fig. 6 are schematic diagrams of control input signals of the control method under two different situations, respectively.

The embodiments of the present invention are described only for the preferred embodiments of the present invention, and not for the limitation of the concept and scope of the present invention, and various modifications and improvements made to the technical solution of the present invention by those skilled in the art without departing from the design concept of the present invention shall fall into the protection scope of the present invention, and the technical content of the present invention which is claimed is fully set forth in the claims.