RBF inverse sliding mode control method for motor servo drive system considering backlash

Abstract

The application provides a motor servo driving system RBF inversion sliding mode control method considering gear gap, and the method comprises the following steps: a dynamic model of an electromechanical servo system containing gear gap is established, and a continuous approximate dead zone function is introduced; subsequently, a recursive structure of backstepping method and strong robustness of sliding mode control are fused, an inversion sliding mode controller based on RBF neural network is designed, two RBF neural networks are used to perform online approximation on unknown nonlinear dynamics in the system, and an adaptive update law and a control law are designed based on Lyapunov stability theory, so that all signals in the closed-loop system are uniformly ultimately bounded. The unknown dynamics are approximated through the online learning ability of the RBF neural network, the robustness of the system is enhanced in combination with the sliding mode surface design, torque oscillation in the gear gap crossing process is effectively inhibited, the tracking accuracy and dynamic response performance of the system are significantly improved, and the application is suitable for high-precision motor servo driving occasions.

Description

Technical Field

[0001] This invention relates to the field of electromechanical servo control technology, specifically to an RBF inversion sliding mode control method for a motor servo drive system that takes backlash into account. Background Technology

[0002] In motor servo drive control systems, high-speed motors are often used as actuators. A reducer matches the load torque and speed requirements, thereby driving the worktable to rotate. These systems offer significant advantages such as high precision, fast response, and zero pollution. However, to prevent gear jamming during meshing, a certain clearance must be maintained between the gear teeth. This clearance is difficult to control precisely in practical applications, and with long-term operation and mechanical wear, the backlash gradually increases, further exacerbating the system's nonlinear characteristics.

[0003] Backlash, a prevalent strong nonlinear factor in servo drive systems, can induce limit cycle oscillations, response hysteresis, and impact vibrations, severely limiting the system's tracking accuracy and dynamic performance. Furthermore, rigid collisions of gears generate significant oscillations and noise. Therefore, backlash compensation in high-precision servo systems has become a pressing issue. Despite significant progress in existing research, designing a controller that combines high-precision tracking capabilities with strong robustness remains a problem worthy of further exploration, especially when system parameters are unknown and time-varying disturbances coexist. Summary of the Invention

[0004] To address the problems of existing technologies, this invention provides an RBF inversion sliding mode control method for motor servo drive systems considering backlash. It combines an RBF neural network to approximate unknown nonlinear functions online, integrating the recursive structure of the backstepping method with the strong robustness of sliding mode control. This eliminates the dependence on system linear parameterization conditions found in traditional adaptive control methods, improving the tracking accuracy, response smoothness, and torque oscillation suppression performance of the motor servo system. The tracking error is stabilized within ±5×10⁻⁻⁻⁶. 5 Within a certain range, the torque transmission is more stable during the tooth gap crossing process.

[0005] This invention provides an RBF inversion sliding mode control method for a motor servo drive system considering backlash, comprising the following steps:

[0006] S1. Establish a dynamic model of an electromechanical servo system with backlash, and introduce a continuous approximate dead zone function to replace the traditional non-differentiable dead zone function. The transmission torque between the master and slave gears is described by the continuous approximate dead zone function.

[0007] S2. Define system state variables and transform the dynamic model into state equation form, which includes unknown system parameters that change due to temperature and wear.

[0008] S3. Based on the backstepping method framework, the virtual control quantity is designed step by step recursively, and a sliding mode surface is introduced to enhance the system robustness. An inverse sliding mode controller based on an RBF neural network is constructed. The specific process is to define the position tracking error. And several progressively advancing error variables, for tracking error Virtual control quantities are designed by selecting Lyapunov functions. Based on virtual control quantity Recursive error variable Select The Lyapunov function is derived by taking its derivative to obtain a nonlinear function. After introducing a sliding surface, an RBF neural network is used to approximate the nonlinear function generated in this step to design a virtual control quantity. This is used to deduce the error variable. and virtual control quantity ,Will Let T be the final system control quantity;

[0009] S4. RBF neural network is used to approximate the nonlinear function containing unknown parameters in the system online. Based on Lyapunov stability theory, an adaptive update law of RBF neural network is designed, positive design parameters are added to prevent parameter drift, and all signals of the closed-loop system are consistent and eventually bounded.

[0010] Further improvements are made, and in step S1, the established dynamic equations for the backlash-containing electromechanical servo system are as follows:

[0011] (1)

[0012] In the formula, , These are the angular displacements of the driving and driven axes, respectively. , These are the angular velocities of the driving and driven axes, respectively. , These are the moments of inertia of the driving and driven shafts, respectively. , Here, are the coefficients of viscous friction of the driving and driven shafts, respectively, and T is the input torque of the system. The torque transmitted between the master and driven gears, where i is the transmission ratio between the master and driven gears, is expressed as:

[0013] (2)

[0014] In the formula, k represents the stiffness coefficient at the meshing point of the master and slave gears. The dead zone function for backlash:

[0015] (3)

[0016] In the formula, The tooth gap width, Relative angular displacement between master and slave gears:

[0017] (4)

[0018] In a further improvement, the continuous approximate dead-zone function introduced in step S1 is:

[0019] (5)

[0020] In the formula, , All are undetermined parameters. In order to analyze their approximation to the dead zone function, the difference between equation (3) and equation (5) is defined as:

[0021] (6)

[0022] set up The torque transmitted between gears can be expressed as:

[0023] (7)

[0024] As a further improvement, step S2 includes the following steps:

[0025] Define the system's state variables as follows:

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] The state equations of the system are as follows:

[0031] (8)

[0032] In the formula, , ;

[0033] In practical applications, , , , k will change due to temperature variations, material wear, and other changes in operating conditions, and is considered an unknown parameter.

[0034] As a further improvement, step S3 includes the following steps:

[0035] S3.1: Define position tracking error :

[0036] (9)

[0037] In the formula To determine the desired output position, select the following Lyapunov function:

[0038] (10)

[0039] Differentiating the above equation, we get:

[0040] (11)

[0041] Design virtual control quantity as follows:

[0042] (12)

[0043] In the formula Positive design parameters;

[0044] S3.2: Define the error variable :

[0045] (13)

[0046] Choose the following Lyapunov function:

[0047] (14)

[0048] Differentiating the above equation, we get:

[0049] (15)

[0050] In the formula, the nonlinear function for:

[0051] (16)

[0052] Define the sliding surface :

[0053] (17)

[0054] In the formula Positive design parameters;

[0055] The nonlinear function generated in this step is approximated using an RBF neural network. Design virtual control quantity :

[0056] (18)

[0057] In the formula, , , For positive design parameters, This is the input vector of the RBF neural network. The saturation function is used to reduce chattering, and its specific form is as follows:

[0058] (19)

[0059] S3.3: Define the error variable :

[0060] (20)

[0061] Choose the following Lyapunov function:

[0062] ;(twenty one)

[0063] Differentiating the above equation, we get:

[0064] ;(twenty two)

[0065] Define the sliding surface :

[0066] ;(twenty three)

[0067] In the formula Positive design parameters;

[0068] Design virtual control quantity :

[0069] ;(twenty four)

[0070] S3.4: Define the error variable :

[0071] (25)

[0072] Choose the following Lyapunov function,

[0073] (26)

[0074] Differentiating the above equation, we get:

[0075] (27)

[0076] The nonlinear function F2 in the formula is:

[0077] (28)

[0078] Define the sliding surface :

[0079] (29)

[0080] In the formula Positive design parameters;

[0081] The nonlinear function generated in this step is approximated using an RBF neural network. Finally, the final system control variable T was designed:

[0082] (30)

[0083] In the formula, , , For positive design parameters, This is the input vector for the RBF neural network.

[0084] Further improvements include, in step S4, designing a nonlinear function. and They are represented as follows:

[0085] (31)

[0086] The adaptive update laws are designed as follows:

[0087] (32)

[0088] In the formula, , The adaptive gain matrix is ​​for normal values. , To ensure positive design parameters and prevent parameter drift.

[0089] The beneficial effects of this invention are as follows:

[0090] 1. The strategy proposed in this invention combines the recursive structure of backstepping with the strong robustness of sliding mode control. It adopts a four-layer backstepping method plus a two-layer RBF, which eliminates the dependence on the linear parameterization conditions of the system in traditional adaptive control methods.

[0091] 2. The backstepping framework uses a saturation function to suppress chattering, which is smoother than the sign function.

[0092] 3. This invention designs a new adaptive update law for RBF neural networks to ensure that the network weights can be fully trained and effectively approximate the unknown dynamics in the system. Furthermore, a correction term is added to the adaptive update law to prevent weight drift.

[0093] 4. Compared with traditional PID controllers and existing fuzzy adaptive controllers, the controller designed in this invention shows significant advantages in tracking accuracy, response smoothness, and torque oscillation suppression, with the tracking error stabilized at ±5×10⁻⁻⁻⁴. 5 Within a certain range, the torque transmission is more stable during the tooth gap crossing process.

[0094] 5. This invention simultaneously considers the problems of unknown system parameters and nonlinear dynamic approximation. By introducing a continuous approximate dead-zone function, it overcomes the limitation of the non-differentiability of the traditional backlash model, and combines it with an RBF neural network to approximate the unknown nonlinear function online, laying the foundation for achieving high-precision tracking control. Attached Figure Description

[0095] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0096] Figure 1 This is a flowchart of the RBF-based inversion sliding mode control method of the present invention;

[0097] Figure 2 This is a comparison chart of the tracking error of the present invention with PID control and existing adaptive fuzzy control (AFC);

[0098] Figure 3 This is a comparison diagram of the relative angular displacement of the master and slave gears of the present invention, PID control, and AFC;

[0099] Figure 4 This is a comparison diagram of the torque transmission of the present invention with PID control and AFC. Detailed Implementation

[0100] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0101] The present invention will now be described in further detail with reference to the accompanying drawings.

[0102] like Figure 1 As shown, this invention proposes an RBF inversion sliding mode control method for a backlash-considered motor servo drive system, comprising the following four steps:

[0103] Step 1: Establish a dynamic model of the electromechanical servo system with backlash, and introduce a continuous approximate dead zone function to replace the traditional non-differentiable dead zone function in order to describe the transmitted torque between the master and slave gears;

[0104] Step 2: Define system state variables and transform the dynamic model into state equation form, which includes unknown system parameters that change due to temperature and wear.

[0105] Step 3: Based on the inverse stepping framework, design virtual control quantities step by step, and introduce sliding mode surfaces to enhance system robustness, and construct an inverse sliding mode controller based on RBF neural network;

[0106] Step 4: Use an RBF neural network to process the nonlinear function containing unknown parameters in the system. and Online approximation is performed, and based on Lyapunov stability theory, an adaptive update law for the RBF neural network is designed to ensure that all signals in the closed-loop system are consistent and eventually bounded.

[0107] Furthermore, the dynamic equations of the backlash-containing electromechanical servo system established in step one are as follows:

[0108] (1)

[0109] In the formula, , These are the angular displacements of the driving and driven axes, respectively. , These are the angular velocities of the driving and driven axes, respectively. , These are the moments of inertia of the driving and driven shafts, respectively. , Here, are the coefficients of viscous friction of the driving and driven shafts, respectively, and T is the input torque of the system. The torque transmitted between the master and slave gears, where i is the transmission ratio between the master and slave gears, can be expressed as:

[0110] (2)

[0111] In the formula, k represents the stiffness coefficient at the meshing point of the master and slave gears. The dead zone function for backlash:

[0112] (3)

[0113] In the formula, The tooth gap width, Relative angular displacement between master and slave gears:

[0114] (4)

[0115] Since the non-differentiable nature of the traditional dead-zone function affects controller design, the following continuous approximate dead-zone function is introduced to replace the traditional non-differentiable dead-zone function:

[0116] (5)

[0117] In the formula, , All are undetermined parameters. In order to analyze their approximation to the dead zone function, the difference between equation (3) and equation (5) is defined as:

[0118] (6)

[0119] set up The torque transmitted between gears can be expressed as:

[0120] (7)

[0121] Furthermore, step two defines the system's state variables as follows:

[0122] ;

[0123] ;

[0124] ;

[0125] ;

[0126] The state equations of the system are as follows:

[0127] (8)

[0128] In the formula, , ;

[0129] In practical applications, , , , k can change due to temperature variations, material wear, and other changes in operating conditions, so it is considered an unknown parameter in the controller design process.

[0130] Furthermore, step three first defines the position tracking error:

[0131] (9)

[0132] In the formula To determine the desired output position, select the following Lyapunov function:

[0133] (10)

[0134] Differentiating the above equation, we get:

[0135] (11)

[0136] Design virtual control quantity as follows:

[0137] (12)

[0138] In the formula These are positive design parameters.

[0139] Furthermore, we continue to define the error variable. :

[0140] (13)

[0141] Choose the following Lyapunov function:

[0142] (14)

[0143] Differentiating the above equation, we get:

[0144] (15)

[0145] In the formula, the nonlinear function for:

[0146] (16)

[0147] Define the sliding surface :

[0148] (17)

[0149] In the formula These are positive design parameters.

[0150] The nonlinear function generated in this step is approximated using an RBF neural network. Design virtual control quantity :

[0151] (18)

[0152] In the formula, , , For positive design parameters, This is the input vector of the RBF neural network. The saturation function is used to reduce chattering, and its specific form is as follows:

[0153] (19)

[0154] Furthermore, we continue to define the error variable. :

[0155] (20)

[0156] Choose the following Lyapunov function:

[0157] ;(twenty one)

[0158] Differentiating the above equation, we get:

[0159] ;(twenty two)

[0160] Define the sliding surface :

[0161] ;(twenty three)

[0162] In the formula These are positive design parameters.

[0163] Design virtual control quantity :

[0164] .(twenty four)

[0165] Furthermore, we continue to define the error variable. :

[0166] (25)

[0167] Choose the following Lyapunov function,

[0168] (26)

[0169] Differentiating the above equation, we get:

[0170] (27)

[0171] The nonlinear function F2 in the formula is:

[0172] (28)

[0173] Define the sliding surface :

[0174] (29)

[0175] In the formula These are positive design parameters.

[0176] The nonlinear function generated in this step is approximated using an RBF neural network. Finally, the final system control variable T was designed:

[0177] (30)

[0178] In the formula, , , For positive design parameters, This is the input vector for the RBF neural network.

[0179] Furthermore, the nonlinear function designed in step four... and They are represented as follows:

[0180] (31)

[0181] The adaptive update laws are designed as follows:

[0182] (32)

[0183] In the formula, , The adaptive gain matrix is ​​for normal values. , To ensure positive design parameters and prevent parameter drift.

[0184] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. In particular, for the device embodiments, the above descriptions are merely preferred embodiments of the present invention. Since they are fundamentally similar to the method embodiments, the descriptions are relatively simple, and relevant parts can be referred to the descriptions of the method embodiments. The above descriptions are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention, without departing from the principle of the present invention, should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A backlash-considering RBF inversion sliding mode control method for a motor servo drive system, characterized in that... Includes the following steps: S1. Establish a dynamic model of an electromechanical servo system with backlash, and introduce a continuous approximate dead zone function to replace the traditional non-differentiable dead zone function. The transmission torque between the master and slave gears is described by the continuous approximate dead zone function. S2. Define system state variables and transform the dynamic model into state equation form, which includes unknown system parameters that change due to temperature and wear. S3. Based on the backstepping method framework, the virtual control quantity is designed step by step recursively, and a sliding mode surface is introduced to enhance the system robustness. An inverse sliding mode controller based on an RBF neural network is constructed. The specific process is to define the position tracking error. And several progressively advancing error variables, for tracking error Virtual control quantities are designed by selecting Lyapunov functions. Based on virtual control quantity Recursive error variable Select The Lyapunov function is derived by taking its derivative to obtain a nonlinear function. After introducing a sliding surface, an RBF neural network is used to approximate the nonlinear function generated in this step to design a virtual control quantity. This is used to deduce the error variable. and virtual control quantity ,Will Let T be the final system control quantity; S4. RBF neural network is used to approximate the nonlinear function containing unknown parameters in the system online. Based on Lyapunov stability theory, an adaptive update law of RBF neural network is designed, positive design parameters are added to prevent parameter drift, and all signals of the closed-loop system are consistent and eventually bounded.

2. The RBF inversion sliding mode control method for a motor servo drive system considering backlash as described in claim 1, characterized in that: In step S1, the established dynamic equations for the backlash-containing electromechanical servo system are as follows: ；（1） In the formula, , These are the angular displacements of the driving and driven axes, respectively. , These are the angular velocities of the driving and driven axes, respectively. , These are the moments of inertia of the driving and driven shafts, respectively. , Here, are the coefficients of viscous friction of the driving and driven shafts, respectively, and T is the input torque of the system. The torque transmitted between the master and driven gears, where i is the transmission ratio between the master and driven gears, is expressed as: ；（2） In the formula, k represents the stiffness coefficient at the meshing point of the master and slave gears. The dead zone function for backlash: ；（3） In the formula, The tooth gap width, Relative angular displacement between master and slave gears: （4）。 3. The RBF inversion sliding mode control method for a motor servo drive system considering backlash, as described in claim 1 or 2, is characterized in that: In step S1, the introduced continuous approximate dead-zone function is: ；（5） In the formula, , All are undetermined parameters. In order to analyze their approximation to the dead zone function, the difference between equation (3) and equation (5) is defined as: ；（6） set up The torque transmitted between gears can be expressed as: （7）。 4. The RBF inversion sliding mode control method for a motor servo drive system considering backlash as described in claim 3, characterized in that: Step S2 includes the following steps: Define the system's state variables as follows: ； ； ； ； The state equations of the system are as follows: ；（8） In the formula, , ; In practical applications, , , , k will change due to temperature variations, material wear, and other changes in operating conditions, and is considered an unknown parameter.

5. The RBF inversion sliding mode control method for a motor servo drive system considering backlash as described in claim 4, characterized in that: Step S3 includes the following steps: S3.1: Define position tracking error : ； （9） In the formula To determine the desired output position, select the following Lyapunov function: ；（10） Differentiating the above equation, we get: ；（11） Design virtual control quantity as follows: ；（12） In the formula Positive design parameters; S3.2: Define the error variable : ；（13） Choose the following Lyapunov function: ；（14） Differentiating the above equation, we get: ； （15） In the formula, the nonlinear function for: ；（16） Define the sliding surface : ；（17） In the formula Positive design parameters; The nonlinear function generated in this step is approximated using an RBF neural network. Design virtual control quantity : ；（18） In the formula, , , For positive design parameters, This is the input vector of the RBF neural network. The saturation function is used to reduce chattering, and its specific form is as follows: ；（19） S3.3: Define the error variable : ；（20） Choose the following Lyapunov function: ；（21） Differentiating the above equation, we get: ；（22） Define the sliding surface : ； （23） In the formula Positive design parameters; Design virtual control quantity : ； （24） S3.4: Define the error variable : ；（25） Choose the following Lyapunov function, ；（26） Differentiating the above equation, we get: ；（27） The nonlinear function F2 in the formula is: ； （28） Define the sliding surface : ；（29） In the formula Positive design parameters; The nonlinear function generated in this step is approximated using an RBF neural network. Finally, the final system control variable T was designed: ；（30） In the formula, , , For positive design parameters, This is the input vector for the RBF neural network.

6. The RBF inversion sliding mode control method for a motor servo drive system considering backlash as described in claim 5, characterized in that: In step S4, the designed nonlinear function and They are represented as follows: ；（31） The adaptive update laws are designed as follows: ； （32） In the formula, , The adaptive gain matrix is ​​for normal values. , To ensure positive design parameters and prevent parameter drift.

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