Nonlinear motor system control method based on instruction filtering backstepping

Abstract

The invention relates to the technical field of motor servo system control, and discloses a nonlinear motor system control method based on instruction filtering backstepping. The method comprises the following steps: establishing a mathematical model of the motor servo system considering nonlinearity, and obtaining a motion equation of the motor servo system considering nonlinearity according to the Newton's second law; establishing a second-order system expansion state disturbance observer equation according to the motion equation; establishing a second-order instruction filter equation; establishing a nonlinear motor system control model based on instruction filtering backstepping; and controlling the motor system according to the established control model. Therefore, instruction filtering backstepping control of the motor servo system can be realized, and the anti-interference capability, the tracking performance and the robustness of the system can be effectively improved.

Description

Technical Field

[0001] The invention relates to the technical field of motor servo system control, and in particular to a nonlinear motor system control method based on command filtering backstepping. Background Art

[0002] The motor servo system is a mechatronic system used to achieve precise position and speed control. It consists of multiple key components, including position sensors, motors, motor drivers, servo controllers, and reducers. There are many factors that affect the motor control performance in the system, including friction, transmission clearance, time-varying disturbances, etc. The motor servo system is a complex nonlinear system with multiple variables, strong interference, and high order. In order to overcome these nonlinear interferences and improve the accuracy of motor control, it is necessary to design a control method for the motor servo system that takes into account complex nonlinearities.

[0003] In the prior art, an electric steering gear control system and method are proposed. First, a mathematical model of the electric steering gear system is established; second, the mathematical model is converted into a state space equation form, and a first-order filter and an extended observer are designed; finally, the control quantity of the system is designed; the influence of system interference can be reduced and the tracking ability of the command signal can be effectively improved. However, the prior art has the following disadvantages:

[0004] 1. PID control is often considered in practical engineering applications as a control method for motor servo systems. Although the PID controller is simple to use and easy to adjust, it has obvious shortcomings in dealing with stability, robustness, dynamic performance optimization, multivariable system and high-order system control in nonlinear systems.

[0005] 2. In order to reduce the command jitter and fluctuation caused by external interference or sensor noise, filters are often needed. However, the first-order filters commonly used in the prior art have some obvious shortcomings in complex and demanding signal processing applications, including poor frequency selectivity, low attenuation rate, large transition band width, phase distortion, low design flexibility and limited noise suppression ability. These shortcomings limit its applicability in high-performance filtering applications;

[0006] 3. In order to further reduce the impact of friction and load changes on the system, ensure the smooth operation of the motor servo system, improve positioning accuracy and response speed, and ensure that the servo motor can accurately and quickly track the given position or speed instructions, disturbance observation methods such as the Romberg observer, Kalman filter, and observers based on some intelligent control algorithms are usually used to observe disturbance nonlinearity. However, the observation methods of these observers all need to rely on higher-precision system models. Summary of the invention

[0007] The present invention provides a nonlinear motor system control method based on command filtering backstepping, which can solve the technical problems in the prior art.

[0008] The present invention provides a nonlinear motor system control method based on command filtering backstepping, wherein the method comprises:

[0009] Establish a mathematical model of the motor servo system considering nonlinearity, and obtain the motion equation of the motor servo system considering nonlinearity according to Newton's second law;

[0010] According to the equation of motion, the extended state disturbance observer equation of the second-order system is established;

[0011] Establish the second-order command filter equation;

[0012] Establish a nonlinear motor system control model based on command filtering and backstepping;

[0013] The motor system is controlled according to the established control model.

[0014] Preferably, the method further comprises analyzing the stability of the established control model according to Lyapunov stability theory.

[0015] Preferably, the motion equation of the motor servo system considering nonlinearity is:

[0016]

[0017] T=K(θ m / i-θ l ),

[0018] x=[x1,x2,x3,x4] T =[θ l ,ω l ,θ m ,ω m ] T ,

[0019] Transforming from the equations of motion to the state space equations:

[0020]

[0021] θ1=K / J l ,θ2=K i / J l ,θ3=b l / J l ,θ4=T l / J l ,θ5=K t / J m ,θ6=K / (i*J m ),θ7=K / Jm ,

[0022] θ8=b m / J m , D1=-Δ1 / J l , D2=-Δ2 / J m ,

[0023] Among them, J l is the load moment of inertia, ω l is the load angular velocity, J m is the motor moment of inertia, ω m is the motor mechanical angular velocity; T l is the load torque, i is the transmission ratio, T is the transmitted torque, K t is the voltage torque coefficient, Δ1 and Δ2 are system uncertainty disturbances, u is the control output, b l is the friction coefficient at the load end, b m is the friction coefficient at the motor end, K is the elastic coefficient of the motor servo system, θ m is the motor mechanical speed, θ l is the load speed, and x is the state variable of the motor servo system.

[0024] Preferably, the second-order system extended state disturbance observer equation is:

[0025]

[0026] L=[l1,l2,l3] T ,

[0027] L=[3ω0,3ω0 2 ,ω0 3 ] T

[0028] Where ω0 is the bandwidth of the extended state observer.

[0029] Preferably, the second-order command filter equation is:

[0030]

[0031] Among them, χ i and is the intermediate quantity, α i is the virtual control rate input of the filter, w n is the filter bandwidth, is the damping ratio of the filter, α i c =x i c and is the output of the filter.

[0032] Preferably, establishing a nonlinear motor system control model based on command filtering backstepping includes:

[0033] Determine motor servo system tracking error;

[0034] Derivative the tracking error of the motor servo system to obtain the derivative result of the tracking error;

[0035] determining a filter error of the filter and a virtual control rate input of the filter;

[0036] According to the tracking error derivative results, filtering error and virtual control rate, a nonlinear motor system control model based on command filtering backstepping is established.

[0037] Preferably, the tracking error is:

[0038]

[0039] The derivative of the tracking error is:

[0040]

[0041] The filtering error is:

[0042]

[0043] The virtual control rate input is:

[0044]

[0045] The control model of the nonlinear motor system based on command filtering backstepping is:

[0046] u=(u a +u s1 +u s2 ) / θ 5n ,

[0047] Among them, x 1d is the given command signal, k1 is the first adjustment gain coefficient, α 2a , α 3a 、u a is the feedforward compensation term, α 2s1 , α 3s1 、u s1 is a linear robust term, u s2 , α 2s2 is the nonlinear robust term, and u is the actual control output.

[0048] Preferably, the Lyapunov stability theory analyzes the stability of the established control model including:

[0049] Determine the Lyapunov function;

[0050] Derivative the Lyapunov function to obtain the derivative result of the Lyapunov function;

[0051] The stability of the established control model is analyzed based on the derivative results of the Lyapunov function.

[0052] Preferably, the Lyapunov function of the Lyapunov stability theory is:

[0053]

[0054] Among them, V is the output of the Lyapunov function, ε is the state observer error, and P1 and P2 are positive definite matrices.

[0055] Preferably, the derivative of the Lyapunov function is:

[0056]

[0057] Wherein, k1, k2, k3, k4 are respectively the first adjustment gain coefficient, the second adjustment gain coefficient, the third adjustment gain coefficient and the fourth adjustment gain coefficient, λ min (Q1),λ min (Q2) is the minimum eigenvalue of the matrix, B2~B7 are all matrices, ε1 and ε2 are residuals.

[0058] Through the above technical scheme, uncertain nonlinear factors such as friction, transmission clearance, time-varying disturbances, etc. existing in the motor servo system can be considered. First, by improving the model of the motor servo system, a mathematical model of the motor considering transmission clearance, friction and time-varying disturbances is further established. Secondly, the control model is designed based on the backstepping method, and the complex nonlinearities existing in the system are unified as disturbances, which are estimated by the extended state observer (ESO); then, the command filter is used to transform the high-order derivative process into an integral process, and signal filtering is realized to suppress system noise; finally, the command filter backstepping control of the motor servo system is realized, which can effectively improve the anti-interference ability, tracking performance and robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] The included drawings are used to provide a further understanding of the embodiments of the present invention, which constitute a part of the specification, are used to illustrate the embodiments of the present invention, and together with the text description, explain the principles of the present invention. Obviously, the drawings in the following description are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0060] Figure 1A flow chart of a nonlinear motor system control method based on command filtering backstepping according to an embodiment of the present invention is shown;

[0061] Figure 2 It is a structural schematic diagram of the motor servo system in the present invention;

[0062] Figure 3 It is the observation error diagram of state x1 under the sinusoidal command signal in the present invention;

[0063] Figure 4 It is the state x2 observation error diagram under the sinusoidal command signal in the present invention;

[0064] Figure 5 It is the observation error diagram of the expansion state φ1 under the sinusoidal command signal in the present invention;

[0065] Figure 6 It is the state x3 observation error diagram under the sinusoidal command signal in the present invention;

[0066] Figure 7 It is the state x4 observation error diagram under the sinusoidal command signal in the present invention;

[0067] Figure 8 It is the observation error diagram of expansion state φ2 under the sinusoidal command signal in the present invention;

[0068] Fig. 9 is the load tracking error of the PID controller under the sinusoidal command signal in the present invention;

[0069] Fig.10 is the load tracking error of the ADRC controller under the sinusoidal command signal in the present invention;

[0070] Fig.11 is the load tracking error of the ARC+ESO controller under the sinusoidal command signal in the present invention;

[0071] Fig.12 It is the observation error diagram of state x1 under the step command signal in the present invention;

[0072] Fig.13 It is the state x2 observation error diagram under the step command signal in the present invention;

[0073] Fig.14 It is the observation error diagram of the expansion state φ1 under the step command signal in the present invention;

[0074] Fig.15 It is the state x3 observation error diagram under the step command signal in the present invention;

[0075] Fig.16 It is the state x4 observation error diagram under the step command signal in the present invention;

[0076] Fig.17 It is the observation error diagram of the expansion state φ2 under the step command signal in the present invention;

[0077] Fig.18 is the load tracking error of the PID controller under the step command signal in the present invention;

[0078] Fig.19 is the load tracking error of the ADRC controller under the step command signal in the present invention;

[0079] Fig. 20 It is the load tracking error of the ARC+ESO controller under the step command signal in the present invention. DETAILED DESCRIPTION

[0080] It should be noted that, in the absence of conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is by no means intended to limit the present invention and its application or use. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0081] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0082] Unless otherwise specifically stated, the relative arrangement of the parts and steps described in these embodiments, numerical expressions and numerical values ​​do not limit the scope of the present invention. At the same time, it should be understood that, for ease of description, the sizes of the various parts shown in the accompanying drawings are not drawn according to the actual proportional relationship. The technology, method and equipment known to ordinary technicians in the relevant field may not be discussed in detail, but in appropriate cases, the technology, method and equipment should be regarded as a part of the authorization specification. In all examples shown and discussed here, any specific value should be interpreted as being merely exemplary, rather than as a limitation. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters represent similar items in the following drawings, so once a certain item is defined in an accompanying drawing, it does not need to be further discussed in subsequent drawings.

[0083] Figure 1 A flow chart of a nonlinear motor system control method based on command filtering backstepping according to an embodiment of the present invention is shown.

[0084] In the present invention, the motor servo system may include a plurality of key components, including a position sensor, a permanent magnet DC motor, a motor driver, a servo controller, a reducer and an inertial load, such as Figure 2 shown.

[0085] like Figure 1 As shown, an embodiment of the present invention provides a nonlinear motor system control method based on command filtering backstepping, wherein the method includes:

[0086] Establish a mathematical model of the motor servo system considering nonlinearity, and obtain the motion equation of the motor servo system considering nonlinearity according to Newton's second law;

[0087] According to the equation of motion, the extended state disturbance observer equation of the second-order system is established;

[0088] Among them, the state observer can track the internal state estimation based on the input and output of the observed system. The extended state observer adds a dimension to the observer, which can estimate the internal and external disturbances that affect the system output. After obtaining the estimated value of the disturbance, the disturbance is eliminated through the compensation link, thereby achieving the effect of anti-disturbance. The extended state observer suppresses high-frequency noise, which makes it play a certain filtering role.

[0089] Establish the second-order command filter equation;

[0090] The use of a second-order tracking differentiator helps to better suppress high-frequency noise and external disturbances and improve the system's resistance to interference. Therefore, a second-order command filter can be established for the fluctuation of the command signal in the system.

[0091] Establish a nonlinear motor system control model (controller) based on command filtering backstepping;

[0092] The motor system is controlled according to the established control model.

[0093] Through the above technical scheme, uncertain nonlinear factors such as friction, transmission clearance, time-varying disturbances, etc. existing in the motor servo system can be considered. First, by improving the model of the motor servo system, a mathematical model of the motor considering transmission clearance, friction and time-varying disturbances is further established. Secondly, the control model is designed based on the backstepping method, and the complex nonlinearities existing in the system are unified as disturbances, which are estimated by the extended state observer (ESO); then, the command filter is used to transform the high-order derivative process into an integral process, and signal filtering is realized to suppress system noise; finally, the command filter backstepping control of the motor servo system is realized, which can effectively improve the anti-interference ability, tracking performance and robustness of the system.

[0094] According to an embodiment of the present invention, the method further comprises analyzing the stability of the established control model according to Lyapunov stability theory.

[0095] According to one embodiment of the present invention, the motion equation of the motor servo system considering nonlinearity is:

[0096]

[0097] T=K(θ m / i-θ l ), (2)

[0098] x=[x1,x2,x3,x4] T =[θ l ,ω l ,θ m ,ω m ] T ,

[0099] Transforming from the equations of motion to the state space equations:

[0100]

[0101] θ1=K / J l ,θ2=K i / J l ,θ3=b l / J l ,θ4=T l / J l ,θ5=K t / J m ,θ6=K / (i*J m ),θ7=K / J m ,θ8=b m / J m , D1=-Δ1 / J l , D2=-Δ2 / J m ,

[0102] Among them, J l is the load moment of inertia, ω l is the load angular velocity, J m is the motor moment of inertia (including reducer), ω m is the motor mechanical angular velocity; T l is the load torque (including friction torque and torque generated by model parameter uncertainty, etc.), i is the transmission ratio, T is the transmitted torque, K t is the voltage torque coefficient, Δ1 and Δ2 are system uncertainty disturbances, u is the control output, b l is the friction coefficient at the load end, b m is the friction coefficient at the motor end, K is the elastic coefficient of the motor servo system, θ m is the motor mechanical speed, θ l is the load speed, and x is the state variable of the motor servo system.

[0103] Assume that the derivatives of D1 and D2 have a maximum value, that is θ i(i＝1,,,8) .

[0104] According to one embodiment of the present invention, the extended state disturbance observer equation of the second-order system is:

[0105]

[0106] L=[l1,l2,l3] T ,

[0107] L=[3ω0,3ω0 2 ,ω0 3 ] T

[0108] Where ω0 is the bandwidth of the extended state observer and L is the gain bandwidth parameter.

[0109] For the gain bandwidth parameter L = [l1,l2,l3] T , whose value is L=[3ω0,3ω0 2 ,ω0 3 ] T .

[0110] According to one embodiment of the present invention, the second-order instruction filter equation is:

[0111]

[0112] Among them, χ i and is the intermediate quantity, α i is the virtual control rate input of the filter, w n is the filter bandwidth (w n >0), is the damping ratio of the filter α i c =x i c and is the output of the filter.

[0113] According to an embodiment of the present invention, establishing a nonlinear motor system control model based on command filtering backstepping includes:

[0114] Determine motor servo system tracking error;

[0115] Derivative the tracking error of the motor servo system to obtain the derivative result of the tracking error;

[0116] determining a filter error of the filter and a virtual control rate input of the filter;

[0117] According to the tracking error derivative results, filtering error and virtual control rate, a nonlinear motor system control model based on command filtering backstepping is established.

[0118] For example, the virtual control rate of each step can be determined by the backstepping method to establish a command filtering backstepping controller.

[0119] According to one embodiment of the present invention, the tracking error is:

[0120]

[0121] The derivative of the tracking error is:

[0122]

[0123] The filtering error is:

[0124]

[0125] The virtual control rate input is:

[0126]

[0127] The control model of the nonlinear motor system based on command filtering backstepping is:

[0128] u=(u a +u s1 +u s2 ) / θ 5n , (10)

[0129] Among them, x 1d is the given command signal, k1 is the first adjustment gain coefficient, α 2a , α 3a 、u a is the feedforward compensation term, α2s1 , α 3s1 、u s1 is a linear robust term, u s2 , α 2s2 is the nonlinear robust term, and u is the actual control output.

[0130] According to an embodiment of the present invention, the Lyapunov stability theory analyzes the stability of the established control model including:

[0131] Determine the Lyapunov function;

[0132] Derivative the Lyapunov function to obtain the derivative result of the Lyapunov function;

[0133] The stability of the established control model is analyzed based on the derivative results of the Lyapunov function.

[0134] According to one embodiment of the present invention, the Lyapunov function of the Lyapunov stability theory is:

[0135]

[0136] Among them, V is the output of the Lyapunov function, ε is the state observer error, and P1 and P2 are positive definite matrices.

[0137] According to an embodiment of the present invention, the derivative result of the Lyapunov function is:

[0138]

[0139] Wherein, k1, k2, k3, k4 are respectively the first adjustment gain coefficient, the second adjustment gain coefficient, the third adjustment gain coefficient and the fourth adjustment gain coefficient, λ min (Q1),λ min (Q2) is the minimum eigenvalue of the matrix, B2~B7 are all matrices, ε1 and ε2 are residuals.

[0140] By simplifying formula (12), we can get:

[0141]

[0142]

[0143] Among them, γ is the iteration step size, η is e1 2 +e2 2 +e3 2 +e4 2 +||ε|| 2 +||ξ|| 2 .

[0144] According to the final result of formula (13), the established controller is bounded and stable.

[0145] The present invention is further described in detail below with reference to the accompanying drawings and specific examples. The present invention provides a complex nonlinear motor system control method based on command filtering backstepping, comprising the following steps:

[0146] Step 1. Establish a mathematical model of the motor servo system considering complex nonlinearity. According to Newton's second law, the motion equation of the motor servo system considering complex nonlinearity is:

[0147]

[0148] Where: J l is the load moment of inertia, ω l is the load angular velocity; J m is the motor moment of inertia (including reducer), ω m is the motor mechanical angular velocity; T l is the load torque (including friction torque and torque generated by model parameter uncertainty, etc.); i is the transmission ratio; T is the transmitted torque; K t is the voltage torque coefficient; Δ1 and Δ2 are system uncertainty disturbances;

[0149] The transmission torque in the motor servo system is:

[0150] T=K(θ m / i-θ l ) (15)

[0151] Where: K is the elastic coefficient of the motor servo system, θ m is the motor mechanical speed, θ l is the load speed;

[0152] Define system state variables: x = [x1, x2, x3, x4] T =[θ l ,ω l ,θ m ,ω m ] T , then the motion equation of formula (14) is converted into the state space equation form as follows:

[0153]

[0154] Where: θ1 = K / J l ,θ2=K i / J l ,θ3=b l / J l ,θ4=T l / J l,θ5=K t / J m ,θ6=K / (i*J m ),θ7=K / J m ,θ8=b m / J m , D1=-Δ1 / J l , D2=-Δ2 / J m Assume that the derivatives of D1 and D2 have a maximum value, that is θ i(i＝1,,,8) ;

[0155] Step 2. Establish the extended state disturbance observer equation. The specific establishment process is as follows. To facilitate the subsequent controller design, the state space equation (16) can be transformed into:

[0156]

[0157] Where: θ in is the nominal value of the uncertain parameter, φ1 and φ2 are parameter errors The new state that expands with the system uncertainty disturbances D1 and D2;

[0158] Take the expanded state variables φ1, φ2, Assuming that |d1(t)| and |d2(t)| have upper bounds, the state equation of the system can be converted to:

[0159]

[0160] Where: w0, w1 is the bandwidth of the dual ESO observer, and the state equation of the observer error is obtained:

[0161]

[0162] Where: is the state error of the observer, and the error equation of the extended state observer is simplified as follows:

[0163]

[0164] Where: ε T =[ε1,ε2,ε3] T is the state observer error state, which can be simplified as Δ0=θ 1n ξ1-θ 2n ε1;

[0165] Simplifying formula (20) we can get:

[0166]

[0167] Where:

[0168] Where w0>0, reasonably design w0 so that the matrix A1 is a Hurwitz matrix, then there is a theorem: for any given symmetric positive definite matrix Q1, there exists a positive definite matrix P1 that satisfies the following equation:

[0169] A1 T P1+P1A1=-Q1 (22)

[0170] Similarly, if w1 is properly designed so that the matrix A2 is a Hurwitz matrix, then there is a theorem: for any given symmetric positive definite matrix Q2, there exists a positive definite matrix P2 that satisfies the following equation:

[0171] A2 T P2+P2A2=-Q2 (23)

[0172] Step 3. Establish a complex nonlinear motor system controller based on command filtering backstepping:

[0173] Derivative of the tracking error:

[0174]

[0175] Design: α1=-k1e1+x 1d (25)

[0176] Where: the input of x2 is the virtual control rate α1, -k1>0;

[0177] To avoid subsequent high-order derivatives, α1 is subjected to second-order filtering:

[0178] e f1 =x1 c -α1, e2 = x2-x1 c (26)

[0179] Where: e2 is the filtering error e f1 and the tracking error of x2, x1 c and derivatives Perform a second-order filter on α1 to obtain the filtered estimate;

[0180] Design the virtual control rate α2 to make the tracking error e2 approach 0, and take the derivative of the tracking error e2 to obtain:

[0181]

[0182] Design: α2=(α 2a +α 2s1 +α 2s2 ) / θ 1n (28)

[0183] Where: The input of x3 is the virtual control rate α2, α 2s1 = -k2e2, k2>0;

[0184] The nonlinear robust term satisfies the following conditions:

[0185]

[0186] Where: ε1 is an arbitrarily small and always positive number;

[0187]

[0188] Where: ε2 is an arbitrarily small and always positive number;

[0189] To avoid subsequent high-order derivatives, α2 is subjected to second-order filtering:

[0190] e f2 =x2 c -α2, e3 = x3-x2 c (31)

[0191] Where: e3 is the filtering error e f2 and tracking error of x3, x2 c and derivatives Perform a second-order filter on α2 to obtain the filtered estimate;

[0192] Design the virtual control rate α3 to make the tracking error e3 approach 0, and take the derivative of the tracking error e3 to obtain:

[0193]

[0194] Design: α3=α 3a +α 3s1 (33)

[0195] Where: The input of x4 is the virtual control rate α3, α 3s1 =-k3e3, k3>0;

[0196] To avoid subsequent high-order derivatives, α3 is subjected to second-order filtering:

[0197] e f3 =x3 c -α3, e4 = x4-x3 c (34)

[0198] Where: e4 is the filtering error e f3 and tracking error of x4, x3 c and derivatives The estimated value of α3 is obtained by filtering α3 through instructions;

[0199] Design the actual control variable u so that the tracking error e4 approaches 0, and take the derivative of the tracking error e4 to obtain:

[0200]

[0201] The actual control quantity of the system:

[0202] u=(u a +u s1 +u s2 ) / θ 5n (36)

[0203] Where: u s1 =-k4e4, k4>0;

[0204] The nonlinear robust term satisfies the following conditions:

[0205]

[0206] Where: ε2 is an arbitrarily small and positive number, Step 4. Proof of the stability of the established command filter backstepping controller:

[0207] Define the Lyapunov function as:

[0208]

[0209] Using Young's inequality:

[0210] e1e f1 ≤(e1 2 +e f1 2 ) / 2 (39)

[0211] θ 1n e2e f2 ≤(e2 2 +(θ 1n e f2 ) 2 ) (40)

[0212] e3e f3 ≤(e3 2 +e f3 2 ) / 2 (41)

[0213]

[0214] According to the properties of equations (22) and (23), the state observer can be obtained:

[0215]

[0216] ε T P1B2θ 3n ε2≤|θ 3n |||P1B2||||ε|| 2 (45)

[0217]

[0218] ξ T P2B5θ 8n ξ2≤|θ 8n |||P2B5||||ξ|| 2 (47)

[0219] According to formulas (39) to (47), we can obtain:

[0220]

[0221] Simplifying formula (48), we can get:

[0222]

[0223] Where: η T =[|e1|,|e2|,|e3|,|e4|,|ε|,|ξ|] T ,

[0224]

[0225] Continuing to simplify formula (48), we can obtain:

[0226]

[0227] Where: κ = 2 min [λ min (Λ),γη 2 / 2,λ min (Λ) / λ min (P1),λ min (Λ) / λ min (P2)] is the exponential convergence rate, λ min (·) and λ max (·) are the minimum and maximum eigenvalues ​​of the matrix respectively. Solving equation (50) yields:

[0228] V(t)≤V(0)exp(-κt)+Δ[1-exp(-κt)] / κ (51)

[0229] According to formula (51), the controller is bounded and stable, and the observer is bounded and stable.

[0230] Step 5: Experimentally verify and analyze the designed controller. The specific parameters of the motor servo system in the example are as follows: System stiffness coefficient K = 3.5*10 2 N / m, voltage torque coefficient K t =0.8806, load moment of inertia J l =0.02kg·m 2 , motor moment of inertia J m =0.031kg·m 2 , motor end viscous friction coefficient b m =0.45, load end viscous friction coefficient b l =0.91, transmission ratio i=2.4, load moment T l =10: The controller parameters under the two working conditions are selected as follows:

[0231] PID: Parameter K p Increase the K value to speed up the response and reduce the steady-state error of the system. i Can reduce system static error; increase K d It can speed up the system response and reduce overshoot, but too much will increase the adjustment time. According to the actual debugging, the parameter setting K p =55,K i =48,K d =35;

[0232] ADRC: A nonlinear disturbance observer is introduced to estimate the system state and disturbance, which can realize active interference suppression. According to actual debugging, the parameter setting K p =55,K i =48,K d =35, observer bandwidth w0=10, w1=50;

[0233] ARC+ESO: Adaptive robust controller based on extended state observer. According to actual debugging, the parameters are set as K1=22.5, K2=4.2, K3=20.7, K4=0.43. Observer bandwidth w0=10, w1=50;

[0234] Depend on Figure 3 to Figure 7 It can be seen that under working condition 1, the designed ESO makes a relatively accurate estimate of the system state quantity and the extended state term, and the estimation error is small. It can be seen that the application of ESO effectively overcomes the problem of introducing uncertainty in the system.

[0235] Depend on Figures 9 to 11It can be seen that the PID controller does not consider the disturbance in the system, and there is state uncertainty, and the tracking accuracy is 0.035, which is the worst; the ADRC controller observes the disturbance and state uncertainty in the system, reducing the tracking error to 0.01; the ARC+ESO controller continues the advantages of the ADRC controller in observing the system disturbance and state uncertainty, and introduces a feedforward term for disturbance suppression and error compensation, and the tracking error is further reduced to 0.002, with the highest tracking accuracy;

[0236] Depend on Figure 12 to Figure 17 It can be seen that under working condition 2, the designed ESO effectively overcomes the problem of uncertainty introduction in the system.

[0237] Depend on Figure 18 to Figure 20 It can be seen that the designed controller response time is 140ms, which is the fastest in response time and convergence speed compared with the other two controllers. The response time of PID controller is 3.4s; the response time of ADRC is 1.84s; and the response time of ARC+ESO is 0.57s, which has the fastest response speed.

[0238] It can be seen from the above embodiments that the nonlinear motor system control method based on command filtering backstepping described in the above embodiments of the present invention has at least the following advantages:

[0239] 1. Considering the shortcomings of PID control in complex nonlinear system control, the backstepping method based on Lyapunov stability theory can gradually optimize the dynamic performance of the system by designing multiple virtual controllers, and achieve high robustness to system parameter changes and external disturbances by adjusting parameter design. For high-order systems, the backstepping method can gradually decompose the high-order system, making each step of control relatively simple, thereby effectively controlling the complex system;

[0240] 2. Considering the unadaptability of the first-order filter in high-performance filtering applications, the second-order command filter based on the first-order filter can achieve different characteristics by adjusting its parameters, such as different quality factors and damping ratios. Its transition band is usually smoother than that of the first-order filter, making the filter smoother and continuous in frequency response. The second-order filter can also achieve a steeper frequency response curve, thereby more effectively distinguishing different frequency components of the signal, and is more effective than the first-order filter in attenuating unwanted frequencies (such as noise);

[0241] 3. Taking into account the disadvantage of the commonly used nonlinear observer that is highly dependent on model accuracy, and the extended state observer that is less dependent on the accurate mathematical model of the control object, the system's internal disturbance, external disturbance and overall unmodeled error are often unified into the total disturbance of the system as a whole. This can effectively estimate the system disturbance and eliminate the influence of the disturbance on the control performance of the overall system in the closed-loop system feedback, thereby realizing the disturbance observation of the system and improving the control performance of the electromechanical servo system.

[0242] In the description of the present invention, it is necessary to understand that the directions or positional relationships indicated by directional words such as "front, back, up, down, left, right", "lateral, vertical, perpendicular, horizontal" and "top, bottom" are usually based on the directions or positional relationships shown in the drawings. They are only for the convenience of describing the present invention and simplifying the description. Unless otherwise specified, these directional words do not indicate or imply that the devices or elements referred to must have a specific direction or be constructed and operated in a specific direction. Therefore, they cannot be understood as limiting the scope of protection of the present invention. The directional words "inside and outside" refer to the inside and outside relative to the contours of each component itself.

[0243] For ease of description, spatially relative terms such as "above", "above", "on the upper surface of", "above", etc. may be used here to describe the spatial positional relationship between a device or feature and other devices or features as shown in the figure. It should be understood that spatially relative terms are intended to include different orientations of the device in use or operation in addition to the orientation described in the figure. For example, if the device in the accompanying drawings is inverted, the device described as "above other devices or structures" or "above other devices or structures" will be positioned as "below other devices or structures" or "below other devices or structures". Thus, the exemplary term "above" can include both "above" and "below". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatially relative descriptions used here are interpreted accordingly.

[0244] In addition, it should be noted that the use of terms such as "first" and "second" to limit components is only for the convenience of distinguishing the corresponding components. If not otherwise stated, the above terms have no special meaning and therefore cannot be understood as limiting the scope of protection of the present invention.

[0245] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A nonlinear motor system control method based on command filtering backstepping, characterized in that: The method includes: Establish a mathematical model of the motor servo system considering nonlinearity, and obtain the motion equation of the motor servo system considering nonlinearity according to Newton's second law; According to the equation of motion, the extended state disturbance observer equation of the second-order system is established; Establish the second-order command filter equation; Establish a nonlinear motor system control model based on command filtering and backstepping; The motor system is controlled according to the established control model.

2. The method according to claim 1, characterized in that The method also includes analyzing the stability of the established control model based on the Lyapunov stability theory.

3. The method according to claim 2, characterized in that The motion equation of the motor servo system considering nonlinearity is: T=K(θ m / i-θ l ), x=[x1,x2,x3,x4] T =[θ l ,oh l ,i m ,oh m ] T , Transforming from the equations of motion to the state space equations: θ1=K / J l ,θ2=K i / J l ,θ3=b l / J l ,θ4=T l / J l ,θ5=K t / J m ,θ6=K / (i*J m ),θ7=K / J m ,θ8=b m / J m ,D1=-Δ1 / J l ,D2=-Δ2 / J m , Among them, J l is the load moment of inertia, ω l is the load angular velocity, J m is the motor moment of inertia, ω m is the motor mechanical angular velocity; T l is the load torque, i is the transmission ratio, T is the transmitted torque, K t is the voltage torque coefficient, Δ1 and Δ2 are system uncertainty disturbances, u is the control output, b l is the friction coefficient at the load end, b m is the friction coefficient at the motor end, K is the elastic coefficient of the motor servo system, θ m is the motor mechanical speed, θ l is the load speed, and x is the state variable of the motor servo system.

4. The method according to claim 3, characterized in that The extended state disturbance observer equation for the second-order system is: L＝[l1,l2,l3] T ， L=[3ω0,3ω0 2 ,ω0 3 ] T Where ω0 is the bandwidth of the extended state observer.

5. The method according to claim 4, characterized in that The second-order command filter equation is: Among them, χ i and is the intermediate quantity, α i is the virtual control rate input of the filter, w n is the filter bandwidth, is the damping ratio of the filter, and is the output of the filter.

6. The method according to claim 5, characterized in that The establishment of a nonlinear motor system control model based on command filtering backstepping includes: Determine motor servo system tracking error; Derivative the tracking error of the motor servo system to obtain the derivative result of the tracking error; determining a filter error of the filter and a virtual control rate input of the filter; According to the tracking error derivative results, filtering error and virtual control rate, a nonlinear motor system control model based on command filtering backstepping is established.

7. The method according to claim 6, characterized in that The tracking error is: The derivative of the tracking error is: The filtering error is: The virtual control rate input is: The control model of the nonlinear motor system based on command filtering backstepping is: u=(u a +in s1 +in s2 ) / θ 5n , Among them, x 1d is the given command signal, k1 is the first adjustment gain coefficient, α 2a , α 3a 、u a is the feedforward compensation term, α 2s1 , α 3s1 、u s1 is a linear robust term, u s2 , α 2s2 is the nonlinear robust term, and u is the actual control output.

8. The method according to claim 7, characterized in that The Lyapunov stability theory analyzes the stability of the established control model including: Determine the Lyapunov function; Derivative the Lyapunov function to obtain the derivative result of the Lyapunov function; The stability of the established control model is analyzed based on the derivative results of the Lyapunov function.

9. The method according to claim 8, characterized in that The Lyapunov function of the Lyapunov stability theory is: Among them, V is the output of the Lyapunov function, ε is the state observer error, and P1 and P2 are positive definite matrices.

10. The method according to claim 9, characterized in that The derivative of the Lyapunov function is: Wherein, k1, k2, k3, k4 are respectively the first adjustment gain coefficient, the second adjustment gain coefficient, the third adjustment gain coefficient and the fourth adjustment gain coefficient, λ min (Q1),λ min (Q2) is the minimum eigenvalue of the matrix, B2~B7 are all matrices, ε1 and ε2 are residuals.

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