A sliding mode control method and system based on improved reaching law

Abstract

The present invention discloses a sliding mode control method based on an improved reaching law, which relates to the field of numerical control technology. The method comprises obtaining a position spline reference trajectory, establishing a feed system degree of freedom model, obtaining a sliding mode surface, combining the sliding mode surface with a power reaching law with a hyperbolic tangent function term to control a sliding mode controller, inputting the position spline reference trajectory into the sliding mode controller to obtain a control law, inputting the control law into the ball screw feed system to obtain the true value of the disturbance and the output of the system, negatively feeding back the output of the system to the input of the sliding mode controller, and obtaining the control law, the true value of the disturbance, and the output of the system. The disturbance estimate is calculated through a nonlinear disturbance observer, and the sliding mode controller is adjusted to converge the tracking error to zero. The sliding mode control method based on an improved reaching law provided by the present invention achieves higher tracking performance, accurately compensates for external disturbances, improves system robustness, and solves the problem of observation errors caused by system chattering in the prior art.

Description

Technical Field

[0001] The present invention relates to the field of numerical control technology, and more particularly to a sliding mode control method and system based on an improved reaching law. Background Art

[0002] Currently, CNC technology uses digital signals to control the mechanical motion of an object. CNC equipment, a mechatronic product, is the result of the penetration of new technologies, represented by CNC technology, into both traditional and emerging manufacturing industries. The most fundamental difference between modern CNC machine tools and earlier generations lies in the dramatic changes in their processing speed and precision, each increasing nearly a thousandfold. Because high-speed, high-precision machining can significantly increase processing speed, enhance product quality and quality, and improve market competitiveness, high-speed cutting technology, characterized by high-speed cutting, high-feed speed, and high-precision machining, has become a key development trend in modern CNC machining.

[0003] However, for a long time, the feed system of CNC machine tools has mainly been "rotating motor + ball screw". This servo form has a complex structure and has a series of problems such as transmission clearance and elastic deformation that cause motion lag and other nonlinear errors. It is difficult to obtain high acceleration and positioning accuracy. The traditional PID control method has a large tracking error for the feed system and cannot guarantee high-precision tracking of the worktable position. Compared with traditional PID control, sliding mode control can greatly improve the trajectory tracking capability. However, the limitation of sliding mode control is that when the system's state trajectory or state error trajectory reaches the sliding surface, it is difficult to converge to the equilibrium point along the sliding surface. Its discontinuous switching characteristics cause the system to vibrate.

[0004] Therefore, how to reduce chattering in traditional sliding mode control is an urgent problem that those skilled in the art need to solve. Summary of the Invention

[0005] In view of this, the present invention provides a sliding mode control method and system based on an improved reaching law. By combining the sliding mode control based on the sliding surface with the improved reaching law and a control strategy combined with a nonlinear disturbance observer, higher tracking performance is achieved and the system chattering is better suppressed. The designed nonlinear disturbance observer can also observe the disturbance more ideally, accurately compensate for the external disturbance, and improve the robustness of the system.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A sliding mode control method based on an improved reaching law, comprising:

[0008] Step 1: Obtain the position spline reference trajectory of the CNC lathe;

[0009] Step 2: Establish a degree of freedom model of the feed system, obtain a sliding surface, combine the sliding surface with a power reaching law with a hyperbolic tangent function term to control a sliding mode controller, and input the position spline reference trajectory into the sliding mode controller to obtain a control law;

[0010] Step 3: Input the control law into the ball screw feed system to obtain the true value of the disturbance and the output of the ball screw feed system;

[0011] Step 4: Negatively feed back the output of the ball screw feed system to the input of the sliding mode controller, obtain the control law, the true value of the disturbance, and the output of the ball screw feed system, calculate the disturbance estimation value through the nonlinear disturbance observer, and adjust the sliding mode controller so that the tracking error converges to zero.

[0012] Preferably, the feeding system degree of freedom model includes:

[0013] The single-degree-of-freedom rigid body model is represented by

[0014]

[0015] Among them, x1 is the output displacement of the workbench, is the derivative of x1, x2 is the output speed of the workbench, is the derivative of x2, d is the external unknown interference, B s is the total damping of the feed system, J s is the total moment of inertia of the feed system, K m is the voltage-torque gain of the motor driver, u is the system control input, and l is the screw lead.

[0016] Preferably, the sliding surface is defined as:

[0017]

[0018] Where e = x r -x1 is the working error of the workbench, is the working error derivative of the workbench, x r is the displacement signal of the reference trajectory, is x r The derivative of , k is the error amplification coefficient of the sliding surface function.

[0019] Preferably, the feeding system degree of freedom model includes:

[0020] The two-degree-of-freedom flexible body model is expressed as

[0021]

[0022] in,

[0023] d m is the equivalent interference to the moving parts in the feed system, d j is the equivalent interference of the rotating parts in the feed system, m1 is the equivalent mass of the rotating parts in the feed system, m2 is the mass of the linear motion parts in the feed system, c1 is the equivalent damping of the rotating parts of the model, c2 is the equivalent damping of the ball screw nut, c3 is the equivalent viscous damping of the linear moving parts, w1 is the constant term expression at the worktable end, w2 is the constant term expression at the motor end, g2 is the coefficient before the control input term, and g1 is the coefficient before the interference term.

[0024] Preferably, the sliding surface is defined as:

[0025] s=ρe=[ρ1 ρ2 ρ3 ρ4][e1 e2 e3 e4] T ;

[0026] Among them, ρ1[, ρ2[, ρ3, ρ4 are all sliding surface parameters; e1[ is the displacement tracking error of the workbench, e2 is the speed tracking error of the workbench, e3 is the displacement tracking error of the motor, and e4 is the speed tracking error of the motor; ρ i >0, i=1,…,4, when s=0, take ρ4=1, then e4=-ρ1e1-ρ2e2-ρ3e3, the equivalent displacement instruction of screw rotation is x 1r , is x 1r Derivative, the worktable displacement instruction is x 2r , is x 2r Derivative, define tracking error as e1 = x 2r -x2, e3=x 1r -x1,

[0027] Preferably, the power reaching law of adding the hyperbolic tangent function term specifically includes:

[0028]

[0029] Among them, k1 is the positive adjustment coefficient before the hyperbolic tangent function term, a is the positive adjustment coefficient of the hyperbolic tangent function, k2 is the positive adjustment coefficient before the power term, s is the sliding surface, It is the reaching law.

[0030] Preferably, the nonlinear disturbance observer includes:

[0031]

[0032] Among them, δ is the auxiliary parameter vector, is the interference estimate, L=Y -1 M -1 , x is the system displacement signal; is the system speed signal, Y is the reversible matrix, is the auxiliary parameter vector derivative, M is the mass matrix, u is the control input, C is the damping matrix, and K is the stiffness matrix.

[0033] A sliding mode control system based on an improved reaching law, comprising:

[0034] A reference trajectory acquisition module is used to obtain a position spline reference trajectory of a CNC lathe;

[0035] a model building module, configured to build a degree of freedom model of the feed system, obtain a sliding surface, control a sliding mode controller by combining the sliding surface with a power reaching law with a hyperbolic tangent function term, and input the position spline reference trajectory into the sliding mode controller to obtain a control law;

[0036] a value acquisition module, configured to input the control rate into a ball screw feed system and acquire a true value of interference and an output of the ball screw feed system;

[0037] The adjustment control module is used to negatively feed back the output of the ball screw feed system to the input of the sliding mode controller, obtain the control law, the true value of the disturbance, and the output of the ball screw feed system to calculate the disturbance estimation value through a nonlinear disturbance observer, and adjust the sliding mode controller so that the tracking error converges to zero.

[0038] It can be seen from the above technical solutions that, compared with the prior art, the present invention discloses a sliding mode control method and system based on an improved reaching law. Compared with the sliding mode controller designed by the traditional reaching rate, the controller designed by combining the sliding mode surface with the improved reaching law can obtain higher tracking performance and better suppress the system's chattering. The designed nonlinear disturbance observer can also observe the disturbance more ideally, accurately compensate for the external disturbance, and improve the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are merely embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying any creative work.

[0040] Figure 1 The accompanying drawing is a control block diagram of the ball screw feed system provided by the present invention;

[0041] Figure 2The accompanying drawings are function diagrams provided by the present invention;

[0042] Figure 3 The accompanying drawing is a diagram of the convergence process of the convergence law provided by the present invention;

[0043] Figure 4 The accompanying drawing is a sliding mode function curve diagram provided by the present invention;

[0044] Figure 5 The accompanying drawing is a simulation diagram of the single-degree-of-freedom double-power reaching law provided by the present invention without interference;

[0045] Figure 6 The accompanying drawing is a simulation diagram of the exponential reaching law of a single degree of freedom provided by the present invention without interference;

[0046] Figure 7 The accompanying drawing is a simulation diagram of the improved reaching law of single degree of freedom provided by the present invention without interference;

[0047] Figure 8 The accompanying drawing is a diagram of interference observation values ​​provided by the present invention;

[0048] Figure 9 The accompanying drawing is a diagram of interference observation error values ​​provided by the present invention;

[0049] Figure 10 The accompanying drawing is a simulation diagram of the single-degree-of-freedom double-power reaching law interference provided by the present invention;

[0050] Figure 11 The accompanying drawing is a simulation diagram of the interference of the single degree of freedom exponential reaching law provided by the present invention;

[0051] Figure 12 The accompanying drawing is a simulation diagram of interference with the improved reaching law of a single degree of freedom provided by the present invention;

[0052] Figure 13 The accompanying drawing is a simulation diagram of the double-degree-of-freedom double-power reaching law provided by the present invention without interference;

[0053] Figure 14 The accompanying drawing is a simulation diagram of the exponential reaching law of two degrees of freedom provided by the present invention without interference;

[0054] Figure 15 The accompanying drawing is a simulation diagram of the improved reaching law of two degrees of freedom provided by the present invention without interference;

[0055] Figure 16 The accompanying drawing is a given disturbance observation diagram of a motor provided by the present invention;

[0056] Figure 17 The accompanying drawing is a given disturbance observation diagram of the workbench provided by the present invention;

[0057] Figure 18 The accompanying drawing is a simulation diagram of the double-degree-of-freedom double-power reaching law interference provided by the present invention;

[0058] Figure 19 The accompanying drawing is a simulation diagram of interference of the exponential reaching law of two degrees of freedom provided by the present invention;

[0059] Figure 20 The accompanying drawing is a simulation diagram of the interference of the improved reaching law of two degrees of freedom provided by the present invention. DETAILED DESCRIPTION

[0060] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0061] In the present invention, the power reaching rate of the hyperbolic tangent function term is added to form the improved reaching law.

[0062] like Figure 1 As shown, an embodiment of the present invention discloses a sliding mode control method based on an improved reaching law, comprising:

[0063] Step 1: Obtain the position spline reference trajectory of the CNC lathe;

[0064] Step 2: Establish a degree of freedom model for the feed system, obtain the sliding surface, combine the sliding surface with the power reaching law with the hyperbolic tangent function term to control the sliding mode controller, and input the position spline reference trajectory into the sliding mode controller to obtain the control law;

[0065] Step 3: Input the control rate into the ball screw feed system to obtain the true value of the interference and the output of the ball screw feed system;

[0066] Step 4: Negatively feed back the output of the ball screw feed system to the input of the sliding mode controller, and obtain the control law, the true value of the disturbance, and the output of the ball screw feed system. Calculate the disturbance estimate through the nonlinear disturbance observer, and adjust the sliding mode controller so that the tracking error converges to zero.

[0067] In a specific embodiment, the feeding system freedom model includes:

[0068] The single-degree-of-freedom rigid body model is represented by

[0069]

[0070] Among them, x1 is the output displacement of the workbench, is the derivative of x1, x2 is the output speed of the workbench, is the derivative of x2, d is the external unknown interference, B s is the total damping of the feed system, Js is the total moment of inertia of the feed system, K m is the voltage-torque gain of the motor driver, u is the system control input, and l is the screw lead.

[0071] In a specific embodiment, the sliding surface is defined as:

[0072]

[0073] Where e = x r -x1 is the working error of the workbench, is the working error derivative of the workbench, x r is the displacement signal of the reference trajectory, is x r The derivative of , k is the error amplification coefficient of the sliding surface function.

[0074] In a specific embodiment, the derivative of the above formula is:

[0075]

[0076] in, is the ideal speed signal of the workbench, is the actual output speed of the workbench, and x2 is the actual output speed of the workbench.

[0077] Combined with the power approach law of adding the hyperbolic tangent function term, we get:

[0078]

[0079] The control law can be obtained as:

[0080]

[0081] Where b0 = K m l / 2πJ s , a0=B s / J s , f=l / 2πJ s , f is the system interference coefficient, and d is the interference.

[0082] In the above formula, the disturbance d is unknown and the control law cannot be implemented. In order to solve this problem, a nonlinear disturbance observer is used to deal with it.

[0083] In a specific embodiment, the feeding system freedom model includes:

[0084] The two-degree-of-freedom flexible body model is expressed as

[0085]

[0086] in,

[0087] d m is the equivalent interference to the moving parts in the feed system, d j is the equivalent interference of the rotating parts in the feed system, m1 is the equivalent mass of the rotating parts in the feed system, m2 is the mass of the linear motion parts in the feed system, c1 is the equivalent damping of the rotating parts of the model, c2 is the equivalent damping of the ball screw nut, c3 is the equivalent viscous damping of the linear moving parts, w1 is the constant term expression at the worktable end, w2 is the constant term expression at the motor end, g2 is the coefficient before the control input term, and g1 is the coefficient before the interference term.

[0088] In a specific embodiment, the sliding surface is defined as:

[0089] s=ρe=[ρ1 ρ2 ρ3 ρ4][e1 e2 e3 e4] T

[0090] =ρ1e1+ρ2e2+ρ3e3+ρ4e4;

[0091] Among them, ρ1[, ρ2[, ρ3, ρ4 are all sliding surface parameters; e1[ is the displacement tracking error of the workbench, e2 is the speed tracking error of the workbench, e3 is the displacement tracking error of the motor, and e4 is the speed tracking error of the motor; ρ i >0, i=1,…,4, when s=0, take ρ4=1, then e4=-ρ1e1-ρ2e2-ρ3e3, the equivalent displacement instruction of the screw rotation is x 1r , is x 1r Derivative, the worktable displacement instruction is x 2r , is x 2r Derivative, define tracking error as e1 = x 2r -x2, e3=x 1r -x1 Let E = [e1 e2 e3] T , Then there is By designing ρ1, ρ2, and ρ3, G satisfies the Hurwitz condition; thus, when t→∞, E=[e1 e2 e3] T →0.

[0092] In order for G to satisfy Hurwitz, the real part of the root in the following formula must be negative

[0093]

[0094] The corresponding eigenvalue λ is selected according to the system, so that the corresponding ρ value can be obtained. The constant matrix ρ can be appropriately adjusted to enable the workbench to obtain good trajectory tracking performance.

[0095] Taking the derivative of the tracking error, we can get:

[0096]

[0097]

[0098]

[0099]

[0100] Derivative of the sliding surface of the two-degree-of-freedom flexible body model, the sliding surface derivative is obtained

[0101]

[0102] Among them, m1 is the equivalent mass of the rotating parts in the feed system, and m2 is the mass of the linear motion parts in the feed system.

[0103] Combined with the power approach law of adding the hyperbolic tangent function term, we get:

[0104]

[0105] The control law can be obtained as:

[0106]

[0107] where F = m1ρ2d m / m2d m +d j , represents the total disturbance to the feed system.

[0108] In a specific embodiment, the power approximation rate of adding the hyperbolic tangent function term specifically includes:

[0109]

[0110] Among them, k1 is the positive adjustment coefficient before the hyperbolic tangent function term, a is the positive adjustment coefficient of the hyperbolic tangent function, k2 is the positive adjustment coefficient before the power term, s is the sliding surface, It is the reaching law.

[0111] Mainly considering the hyperbolic tangent function The function itself and its derivative values ​​are both in the range of [-1,1]. In addition, the value of tanh(s) changes very slowly. According to its characteristics, when s is 0, its value will not have a breakpoint compared to the sign(s) function. In the process of s approaching 0, the function value can be infinitely close to 0. This solves the problem of derivation of the sign function. Figure 2 As shown in the figure, the image contains the sign function sign(s), the hyperbolic tangent function tanh(s) under different a values, and its derivative function Dtanh(s). By adjusting the a value, the sign function can be well replaced, which can effectively reduce the jitter problem in sliding mode control.

[0112] 1) Stable accessibility

[0113] Define the Lipschitz function:

[0114]

[0115] Combined with the power approximation rate of the hyperbolic tangent function term, we can get

[0116]

[0117] Theorem 1: For any given s, there exists a>0, and there is an inequality

[0118] stanh(as)=|stanh(as)|=|s||tanh(as)|≥0

[0119] prove:

[0120]

[0121] because

[0122]

[0123] We can get: s(e 2as -1)≥0, Theorem 1 holds.

[0124] Combined When s=0,

[0125] When s>0, tanh(as)>0, |s| r+1 sign(s)>0, we can get

[0126] When s<0, tanh(as)<0, |s| r+1 sign(s)<0, we can get

[0127] In summary, the designed power reaching law with the addition of hyperbolic tangent function term satisfies the sliding mode reachability condition If and only if s=0,

[0128] 2) The time to reach the sliding surface is limited

[0129] Theorem 2: Given any initial value s0, s will eventually converge to zero in a finite time.

[0130] Proof: Assuming that the initial state s0>0, the system can be divided into two stages from the initial state to the sliding surface: from the initial state s0=1 and from s0=1 to the sliding surface. The time for s to finally converge to zero is:

[0131]

[0132]

[0133]

[0134] like Figure 3 As shown in the above formula It can be seen that s can reach 1 from the initial value s0 in a finite time T2. As s gradually tends from 1 to 0, It also gradually tends to 0, and eventually tends to be parallel to the time axis, so T1 is an infinite time. When we define the slope k of the convergence curve s When s is small enough (such as 0.01, 0.001), s actually converges, and the convergence time is a finite value. Although there may be a small steady-state error, it will not affect its convergence accuracy.

[0135] Assume there is a sufficiently small slope k s , we can get

[0136]

[0137] From the above formula, we can get

[0138]

[0139] At this time, T1 can be expressed as

[0140]

[0141] Combining the above formula, we can get:

[0142]

[0143] So it is proved that s can converge to zero in finite time.

[0144] When s0<0, the system can be divided into two stages: from the initial state to s0=-1 and from s0=-1 to the sliding surface. The analysis method is the same as when s0>0.

[0145] Take k1=k2=20, a=5, r=2, r1=0.5, such as Figure 4 As shown in Figure 1, the time-varying curves of the sliding mode function s under the action of the above five reaching laws are given, with the initial value s0 = 5. It can be seen that the improved reaching law has a faster convergence speed, higher convergence accuracy and no obvious chattering, which not only ensures that the system can reach the sliding mode surface quickly and stably, but also achieves a smooth transition with the sliding mode surface.

[0146] Since the design method of the nonlinear disturbance observer can use the error between the estimated value and the true value inside the system to readjust the estimated value, reduce the error as much as possible, and reach the correct range of the estimated value, it can handle the above-mentioned disturbance F well. Taking into account the mathematical model of the feed system, a nonlinear disturbance observer is adopted, which is shown in the following formula.

[0147] In a specific embodiment, the nonlinear disturbance observer includes:

[0148]

[0149] Among them, δ is the auxiliary parameter vector, is the interference estimate, L=Y -1 M -1 , x is the system displacement signal; is the system speed signal, Y is the reversible matrix, is the auxiliary parameter vector derivative, M is the mass matrix, u is the control input, C is the damping matrix, and K is the stiffness matrix.

[0150] make Generally, there is no prior knowledge of the differential of the true value F of the disturbance. Assuming that the change of the disturbance is slow relative to the dynamic characteristics of the observer, it is advisable to

[0151] Design the Lyapunov function as

[0152]

[0153] in, is the interference estimation error, M=M T >0, T is the matrix transpose.

[0154] then

[0155]

[0156] According to the two-degree-of-freedom flexible body model, we can get

[0157]

[0158] Therefore, the observation error equation is

[0159]

[0160] Thus we get

[0161]

[0162] then

[0163]

[0164] Constructing inequalities

[0165]

[0166] Among them, Γ>0 is a symmetric positive definite matrix, then there exists Γ>0, satisfying

[0167] It can be seen that the nonlinear disturbance observer exhibits exponential convergence, and the value of its parameter Γ determines the convergence accuracy. The larger the value, the faster the convergence speed and the higher the accuracy.

[0168] Since there are nonlinear terms in the inequality, we first convert it into a linear matrix inequality and then solve it, setting ψ=Y- 1 , change ψ T =(Y -1 ) T and ψ=Y- 1 Multiply the left and right sides of the inequality respectively and simplify to get

[0169] ψ T +ψ-ψ T Γψ≥M due to because The sufficient condition for the above formula to be true is

[0170]

[0171] According to Schur's complement theorem: Let C be a positive definite matrix, then A-BC -1 B T ≥0 is equivalent to

[0172]

[0173] Then the formula Can be equivalent to

[0174]

[0175] By solving, we can get Y, which The smaller the Γ value, the easier it is to obtain an effective solution.

[0176] A sliding mode control system based on improved reaching rate, comprising:

[0177] A reference trajectory acquisition module is used to obtain a position spline reference trajectory of a CNC lathe;

[0178] A model building module is used to build a degree of freedom model of the feed system, obtain a sliding surface, combine the sliding surface with a power approach rate control sliding mode controller with a hyperbolic tangent function term, and input the position spline reference trajectory into the sliding mode controller to obtain the control rate;

[0179] A value acquisition module is used to input the control rate into the ball screw feed system and obtain the real value of the interference and the output of the ball screw feed system;

[0180] The regulation control module is used to negatively feed back the output of the ball screw feed system to the input of the sliding mode controller, obtain the control law, the true value of the disturbance, and the output of the ball screw feed system to calculate the disturbance estimation value through the nonlinear disturbance observer, and adjust the sliding mode controller to make the tracking error converge to zero.

[0181] For the entire single-degree-of-freedom rigid body model, take the ψ Lyapunov function

[0182]

[0183] right Derivative

[0184]

[0185] Combining the two-degree-of-freedom flexible body model with We can get:

[0186]

[0187] Since the disturbance observer converges exponentially, its convergence accuracy and speed are determined by the positive definite symmetric matrix Γ. When t→∞, s≡0, so when t→∞, s→0, thus ensuring that the tracking error e converges to zero and making the closed-loop system asymptotically stable.

[0188] For the entire two-degree-of-freedom flexible body model, take the Lyapunov function

[0189]

[0190] Pair Taking the derivative we get

[0191]

[0192] in, is the motor reference acceleration, is the reference acceleration of the workbench.

[0193] Substitution Japanese style have to:

[0194]

[0195] make When |k1stanh(as)-k2|s| r+1 sign(s)|=0, thus s=0, According to the Lasalla invariance principle, when t→∞, s→0 so that the tracking error e converges to zero.

[0196] Analysis of the simulation results of the single-degree-of-freedom rigid body model. In the control simulation of the single-degree-of-freedom rigid body model of the feed system: F(t) = 0.08sin(2πt)cos(6πt) is used to replace the random interference signal in the system. The sliding mode controller designed by the traditional reaching law and the controller designed by the improved reaching law in this paper are used to track the given quintic position spline reference trajectory and record the displacement tracking error and control input signal. In the improved reaching law, the parameters are k1 = 50, k2 = 100, a = 5, r = 2. In the exponential reaching law, the parameters are k1 = 0.01, k2 = 50. In the double power reaching law, the parameters are k1 = 100, k2 = 50, r = 2, r1 = 0.6. In the absence of interference, the simulation results of the single-degree-of-freedom double power reaching law are as follows: Figure 5 As shown; the simulation results of the single degree of freedom exponential reaching law are as follows Figure 6 As shown; the simulation results of the improved reaching law of single degree of freedom are shown as Figure 7 shown.

[0197] from Figure 5 、 Figure 6 、 Figure 7 It can be seen that when there is no interference in the system, the root mean square error of the workbench tracking position under the traditional double power and exponential reaching laws are 3.247μm and 2.523μm respectively, and there is obvious chattering in the control signal; in the sliding mode controller designed based on the improved reaching law, the root mean square error of the workbench position tracking is 2.331μm, which is the smallest value, and there is no severe chattering in the control signal, which effectively suppresses the chattering vibration in the sliding mode control. The results show that the sliding mode controller based on the improved reaching law designed in this paper has a significant effect in eliminating chattering and improving tracking accuracy. In the disturbance observer, take Y = Γ = 0.1, and when adding random disturbance d(t) = F(t) = 0.08sin(2πt)cos(6πt), the simulation curve is as follows Figure 8 Interference Observation Values ​​and Figure 9 Interference observation error value diagram; Figure 10 Single degree of freedom double power reaching law interference simulation diagram, Figure 11 Single degree of freedom exponential reaching law interference simulation diagram and Figure 12 The improved reaching law interference simulation diagram of single degree of freedom is shown in the figure.

[0198] pass Figure 8 Interference observation value map and Figure 9 The interference observation error value diagram shows that the interference observer designed in this paper can better observe the external interference with high observation accuracy. Figure 5 and Figure 10 , Figure 6 and Figure 11 , Figure 7 and Figure 12 A comparison reveals that when a disturbance is introduced into the system, the steady-state error increases compared to the uninterrupted state, but the peak error remains unaffected. Under sliding mode control using traditional bi-power reaching and exponential reaching laws, the platform's tracking root mean square error increases to 3.325μm and 2.676μm, respectively. Under the improved reaching law, the root mean square error is 2.493μm, a minimal change compared to the uninterrupted state. This ensures the platform's tracking accuracy and, to a certain extent, suppresses chattering, resulting in no noticeable vibration.

[0199] The simulation results of the two-degree-of-freedom model are analyzed. The sliding mode controller designed by the traditional reaching law and the controller designed by the improved reaching law in this paper are used to track the given quintic position spline reference trajectory and record the displacement tracking error and control input signal. In the improved reaching law, the parameters are k1=125, k2=150, a=5, r=2, in the exponential reaching law, the parameters are k1=0.05, k2=150, in the bi-power reaching law, the parameters are k1=125, k2=150, r=2, r1=0.6, and the interference d j and d m They are: F(t)=ksin(2πt)cos(6πt), k j =0.05, k m =0.08, two-degree-of-freedom control model parameters m1 = 2.3016, m2 = 0.1484, k = 4.1814 × 10 4 , c1=8.0954×10 -4 , c2=5.3550,c3=1.6103,the invertible matrix in the observer In the absence of interference, the simulation results of the double-degree-of-freedom double-power reaching law are as follows Figure 13 As shown; the simulation results of the double-degree-of-freedom exponential reaching law are as follows Figure 14 As shown; the simulation results of the improved reaching law of two degrees of freedom are as follows Figure 15 shown.

[0200] from Figure 13 、 Figure 14 、 Figure 15 It can be seen from the figure that when there is no interference in the system, the root mean square error of the workbench tracking under the traditional double power and exponential reaching laws are 1.757μm and 1.660μm respectively; the root mean square error of the tracking under the improved reaching law is 1.385μm. The workbench tracking error is significantly smaller than the controller designed based on the rigid body model, and it has a significant effect on the chattering suppression of the control signal. There is no severe chattering phenomenon in the control signal. When the random interference F(t) is added, it is obtained Figure 16 Motor given disturbance observation diagram, Figure 17 The workbench gives the disturbance observation diagram; Figure 18 Simulation diagram of double-degree-of-freedom double-power reaching law interference, Figure 19 Simulation diagram of dual-degree-of-freedom exponential reaching law interference, Figure 20 Simulation diagram of the improved reaching law interference of two degrees of freedom.

[0201] Table 1 Comparison of simulation results

[0202]

[0203]

[0204] pass Figure 16 、 17 It can be found that the disturbance observer can better observe the given disturbance at both ends of the motor and the workbench; Figure 18 、 Figure 19 、 Figure 20 It can be seen from the control signal that there is no obvious and violent chattering in the sliding mode controller based on the improved reaching law, which greatly suppresses the chattering phenomenon in the sliding mode control. In terms of tracking error, when the disturbance is added to the system, compared with the simulation results without disturbance, it is consistent with the rigid body model simulation results. It has an impact on the steady-state error, but its error peak is not affected. The root mean square error of the workbench under the improved approach is reduced to 1.502μm. The root mean square error results in Table 1 also show that the controller designed with the improved reaching law of the present invention has higher workbench tracking accuracy than the sliding mode controller designed with the traditional reaching law. Compared with the absence of disturbance, the root mean square error of the workbench tracking in the rigid body model and the two-degree-of-freedom model does not increase much, increasing from 2.331μm to 2.493μm and 1.385μm to 1.502μm, respectively. This shows the effectiveness of the improved reaching law and the feasibility and correctness of the control strategy based on the combination of observer and sliding mode control. The addition of the disturbance observer is beneficial to improving the robustness of the feed system.

[0205] Based on the traditional reaching law, this paper introduces a saturation function and designs an improved reaching law with low chattering and fast convergence. Theoretically, the finite convergence time of the improved reaching law is proved. Subsequently, a sliding mode controller based on the improved reaching law is designed for the single-degree-of-freedom rigid body control model and the two-degree-of-freedom control model. For the uncertain interference in the system, it is input into the system as a random interference signal and observed by a nonlinear disturbance observer. The exponential convergence of the observer is analyzed. Finally, the stability of the proposed control strategy based on the improved reaching law sliding mode control combined with the nonlinear disturbance observer is proved using the Lyapunov function and verified by simulation. The simulation results show that compared with the sliding mode controller designed by the traditional reaching law, the controller designed based on the improved reaching law proposed in this chapter can achieve higher tracking performance and better suppress the system chattering. The designed nonlinear disturbance observer can also observe the interference more ideally, accurately compensate for the external interference, and improve the robustness of the system.

[0206] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. Reference can be made to the common and similar parts between the various embodiments. For the devices disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the method description.

[0207] The above description of the disclosed embodiments is intended to enable one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention is not limited to the embodiments shown herein but is intended to conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A sliding mode control method based on an improved reaching law, characterized in that: include: Step 1: Obtain the position spline reference trajectory of the CNC lathe; Step 2: Establish a degree of freedom model of the feed system, obtain a sliding surface, combine the sliding surface with a power reaching law with a hyperbolic tangent function term to control a sliding mode controller, and input the position spline reference trajectory into the sliding mode controller to obtain a control law; Step 3: Input the control law into the ball screw feed system to obtain the true value of the disturbance and the output of the ball screw feed system; Step 4: Negatively feeding back the output of the ball screw feed system to the input of the sliding mode controller, obtaining the control law, the true value of the disturbance, and the output of the ball screw feed system, calculating the disturbance estimate through a nonlinear disturbance observer, and adjusting the sliding mode controller so that the tracking error converges to zero; The feeding system freedom model includes: The two-degree-of-freedom flexible body model is expressed as in, d m is the equivalent interference to the moving parts in the feed system, d j is the equivalent interference of the rotating parts in the feed system, m1 is the equivalent mass of the rotating parts in the feed system, m2 is the mass of the linear motion parts in the feed system, c1 is the equivalent damping of the rotating parts of the model, c2 is the equivalent damping of the ball screw nut, c3 is the equivalent viscous damping of the linear motion parts, w1 is the constant term expression of the worktable end, w2 is the constant term expression of the motor end, g2 is the coefficient before the control input term, and g1 is the coefficient before the interference term; The sliding surface is defined as: in, ρ3 and ρ4 are sliding surface parameters; is the displacement tracking error of the workbench, e2 is the speed tracking error of the workbench, e3 is the displacement tracking error of the motor, and e4 is the speed tracking error of the motor; ρ i >0, i=1,…,4, when s=0, take ρ4=1, then e4=-ρ1e1-ρ2e2-ρ3e3, the equivalent displacement instruction of screw rotation is x 1r , is x 1r Derivative, the worktable displacement instruction is x 2r , is x 2r Derivative, define tracking error as e1 = x 2r -x2, e3=x 1r -x1, Let E = [e1 e2 e3] T , Then there is By designing ρ1, ρ2, and ρ3, G satisfies the Hurwitz condition; thus, when t→∞, E=[e1 e2 e3] T →0; In order for G to satisfy Hurwitz, the real part of the root in the following formula must be negative According to the system, the corresponding eigenvalue λ value is selected to obtain the corresponding ρ value; Taking the derivative of the tracking error, we can get: Derivative of the sliding surface of the two-degree-of-freedom flexible body model, the sliding surface derivative is obtained: Among them, m1 is the equivalent mass of the rotating parts in the feed system, and m2 is the mass of the linear motion parts in the feed system; The power approaching law of adding the hyperbolic tangent function term specifically includes: Among them, k1 is the positive adjustment coefficient before the hyperbolic tangent function term, a is the positive adjustment coefficient of the hyperbolic tangent function, k2 is the positive adjustment coefficient before the power term, s is the sliding surface, is the law of approach; The control law corresponding to the two-degree-of-freedom flexible body model is: where F = m1ρ2d m / m2d m +d j , represents the total disturbance to the feed system.

2. A sliding mode control method based on an improved reaching law according to claim 1, characterized in that: The feeding system freedom model includes: The single-degree-of-freedom rigid body model is represented by Among them, x1 is the output displacement of the workbench, is the derivative of x1, x2 is the output speed of the workbench, is the derivative of x2, d is the external unknown interference, B s is the total damping of the feed system, J s is the total moment of inertia of the feed system, K m is the voltage-torque gain of the motor driver, u is the system control input, and l is the screw lead.

3. A sliding mode control method based on an improved reaching law according to claim 2, characterized in that: The sliding surface is defined as: Where e = x r -x1 is the working error of the workbench, is the working error derivative of the workbench, x r is the displacement signal of the reference trajectory, is x r The derivative of , k is the error amplification coefficient of the sliding surface function.

4. A sliding mode control method based on an improved reaching law according to claim 1, characterized in that: The nonlinear disturbance observer comprises: Among them, δ is the auxiliary parameter vector, is the interference estimate, L=Y -1 M -1 , x is the system displacement signal; is the system speed signal, Y is the reversible matrix, is the auxiliary parameter vector derivative, M is the mass matrix, u is the control input, C is the damping matrix, and K is the stiffness matrix.

5. A sliding mode control system based on an improved reaching law, characterized in that: include: A reference trajectory acquisition module is used to obtain a position spline reference trajectory of a CNC lathe; a model building module, configured to build a degree of freedom model of the feed system, obtain a sliding surface, control a sliding mode controller by combining the sliding surface with a power reaching law with a hyperbolic tangent function term, and input the position spline reference trajectory into the sliding mode controller to obtain a control law; a value acquisition module, configured to input the control law into the ball screw feed system and acquire the true value of the interference and the output of the ball screw feed system; an adjustment control module, configured to negatively feed back the output of the ball screw feed system to an input of a sliding mode controller, obtain the control law, the true value of the disturbance, and the output of the ball screw feed system, calculate a disturbance estimate through a nonlinear disturbance observer, and adjust the sliding mode controller so that the tracking error converges to zero; The feeding system freedom model includes: The two-degree-of-freedom flexible body model is expressed as in, d m is the equivalent interference to the moving parts in the feed system, d j is the equivalent interference of the rotating parts in the feed system, m1 is the equivalent mass of the rotating parts in the feed system, m2 is the mass of the linear motion parts in the feed system, c1 is the equivalent damping of the rotating parts of the model, c2 is the equivalent damping of the ball screw nut, c3 is the equivalent viscous damping of the linear motion parts, w1 is the constant term expression of the worktable end, w2 is the constant term expression of the motor end, g2 is the coefficient before the control input term, and g1 is the coefficient before the interference term; The sliding surface is defined as: in, ρ3 and ρ4 are sliding surface parameters; is the displacement tracking error of the workbench, e2 is the speed tracking error of the workbench, e3 is the displacement tracking error of the motor, and e4 is the speed tracking error of the motor; ρ i >0, i=1,…,4, when s=0, take ρ4=1, then e4=-ρ1e1-ρ2e2-ρ3e3, the equivalent displacement instruction of screw rotation is x 1r , is x 1r Derivative, the worktable displacement instruction is x 2r , is x 2r Derivative, define tracking error as e1 = x 2r -x2, e3=x 1r -x1, Let E = [e1 e2 e3] T , Then there is By designing ρ1, ρ2, and ρ3, G satisfies the Hurwitz condition; thus, when t→∞, E=[e1 e2 e3] T →0; In order for G to satisfy Hurwitz, the real part of the root in the following formula must be negative According to the system, the corresponding eigenvalue λ value is selected to obtain the corresponding ρ value; Taking the derivative of the tracking error, we can get: Derivative of the sliding surface of the two-degree-of-freedom flexible body model, the sliding surface derivative is obtained: Among them, m1 is the equivalent mass of the rotating parts in the feed system, and m2 is the mass of the linear motion parts in the feed system; The power approaching law of adding the hyperbolic tangent function term specifically includes: Among them, k1 is the positive adjustment coefficient before the hyperbolic tangent function term, a is the positive adjustment coefficient of the hyperbolic tangent function, k2 is the positive adjustment coefficient before the power term, s is the sliding surface, is the law of approach; The control law corresponding to the two-degree-of-freedom flexible body model is: where F = m1ρ2d m / m2d m +d j , represents the total disturbance to the feed system.

Images