A piezoelectric drive optimal trajectory tracking control method based on adaptive step length

Abstract

The application discloses a piezoelectric driving optimal trajectory tracking control method based on an adaptive step length, and comprises the following steps: a piezoelectric ceramic driver is mathematically modeled by using an electromechanical coupling nonlinear dynamic system model to obtain a piezoelectric ceramic driver model; the piezoelectric ceramic driver model is converted into a linear controllable canonical subsystem; an expected displacement x * (0) at time t=0 is taken as an initial actual tracking displacement x(0); a feedback control u is obtained by using the linear controllable canonical subsystem; the initial actual tracking displacement x(0) is substituted into the linear controllable canonical subsystem, and an actual tracking displacement is obtained by using a Runge-Kutta-Felberg method with an adaptive step length; a performance index function of the linear controllable canonical subsystem is determined, the feedback control u and the actual tracking displacement are substituted into the performance index function, and an optimal performance index of the piezoelectric ceramic driver model is obtained. In the calculation process, the piezoelectric ceramic driver can be improved in calculation efficiency while maintaining high precision.

Description

Technical Field

[0001] The present invention relates to the technical field of piezoelectric ceramic driver control, and in particular to a piezoelectric driver optimal trajectory tracking control method based on adaptive step length. Background Art

[0002] Piezoelectric ceramic actuators are widely used in ultra-precision machining, biomedical engineering, nanotechnology, and other fields due to their high stiffness, high-frequency response, small size, and light weight. However, the nonlinear characteristics of piezoelectric ceramic actuators, particularly hysteresis and creep, limit their application in high-precision displacement control. These nonlinear characteristics not only affect the control accuracy of the system but can also lead to system instability, posing a key challenge for further improving the performance of piezoelectric actuators. Among previous control methods, PID control, while widely used, struggles to meet high-precision requirements given the complex nonlinear hysteresis characteristics. To accurately describe the nonlinear hysteresis characteristics of piezoelectric ceramic actuators, researchers have proposed various models, such as the Preisach model, the Wiener model, and the hysteresis-creep model based on the Prandtl-Ishlinskii operator. While these models can effectively describe the hysteretic behavior of piezoelectric ceramic actuators, most of them can only handle static hysteresis characteristics and require extensive experimental data to support them.

[0003] The CNLP (electromechanically coupled nonlinear lumped parameter) model, proposed by Bin Yang et al. in the Journal of Automatica Sinica, integrates the nonlinear hysteresis characteristics of a piezoelectric stack and the dynamics of the actuator. This model, based on the input-output linearization principle of differential geometry, transforms the stack into a linearly controllable subsystem. Combined with the linear quadratic form (LQ) method, it effectively improves the actuator's displacement tracking accuracy, providing a new approach for modeling the nonlinear dynamics of piezoelectric ceramic actuators. However, these control algorithms still suffer from computational complexity and limited control accuracy in practical applications. To address these deficiencies, Mai Heng et al., in their patent application CN108777553B, entitled "A Control Method for a Piezoelectric Ceramic Actuator Based on the Runge-Kutta Method," employed the traditional fourth-order Runge-Kutta method (hereinafter referred to as "RK4") to calculate the tracking displacement in this system. However, due to the fixed step size used in the traditional RK4 method, the accuracy of the computationally complex piezoelectric ceramic actuator remains to be improved within a short timeframe.

[0004] Therefore, in the field of micro-nano precision control, there is still a need for an algorithm that can improve the displacement tracking accuracy of piezoelectric actuators to adapt to complex ECNLP models. Summary of the Invention

[0005] The purpose of the present invention is to overcome the shortcomings of the existing technology and provide a piezoelectric drive optimal trajectory tracking control method based on adaptive step size. This method improves the computational efficiency and control accuracy of the piezoelectric ceramic drive by adaptively adjusting the step size while maintaining high precision during the calculation process. Compared with the RK4 algorithm proposed by Mai Heng et al. with a fixed step size, the Runge Kutta-Feldberg method can dynamically adjust the step size according to the system state, thereby better coping with the complexity of nonlinear dynamic systems, especially in the field of micro-nano precision control. It shows superior performance. This method significantly improves the displacement tracking capability of the piezoelectric ceramic drive and provides a new solution for the application of high-precision positioning technology.

[0006] To achieve the above objectives, the technical solutions provided by the present invention are:

[0007] A piezoelectric drive optimal trajectory tracking control method based on adaptive step size, comprising:

[0008] The piezoelectric ceramic actuator is mathematically modeled using an electromechanical coupling nonlinear dynamic system model to obtain a piezoelectric ceramic actuator model.

[0009] Convert the piezoelectric ceramic actuator model into a linear controllable canonical subsystem;

[0010] The expected displacement x at time t = 0 * (0) as the initial actual tracking displacement x(0);

[0011] Determine the feedback control u through the linear controllable canonical subsystem;

[0012] Substitute the initial state value x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Feuerberg method with adaptive step size to calculate the actual tracking displacement;

[0013] The performance index function of the linear controllable canonical subsystem is determined, and the feedback control u and the actual tracking displacement are substituted into the performance index function to obtain the optimal performance index of the piezoelectric ceramic actuator model.

[0014] Furthermore, after mathematically modeling the piezoelectric ceramic actuator using the electromechanical coupling nonlinear dynamic system model, the obtained piezoelectric ceramic actuator model is as follows:

[0015]

[0016] in, is the derivative of X, is the state variable, U in is the control variable, F is the state variable X and the control variable U of the piezoelectric ceramic actuator model in Related function matrix, G is the weight matrix, x is the actual tracking displacement, is the derivative of x.

[0017] Furthermore, the transformed linear controllable canonical subsystem is as follows:

[0018]

[0019] in, u is the feedback control.

[0020] Furthermore, the formula for calculating the feedback control u is as follows:

[0021] u=-R -1 B T P LQ Z+R -1 B T G g (3)

[0022] R is the weight matrix, constant matrix P LQ It is the only symmetric positive definite solution to the Riccati algebraic equation;

[0023] A T P LQ +P LQ AP LQ BR -1 B T P LQ +C T QC=0 (4)

[0024] C is the piezoelectric ceramic driver parameter, and its value is 1.2×10 -6 F, matrix G g for:

[0025] G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5)

[0026] x * is the expected displacement.

[0027] Furthermore, the adaptive step-size Runge-Kutta-Feuerberg method is used to obtain the actual tracking displacement, including:

[0028] A1. Set the initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the system initial state value x0 = x(0), the independent variable t = t0, i = 1;

[0029] A2. Calculate the current state value and the resulting error;

[0030] A3. Determine whether the error generated in the previous step A2 is equal to 0. If so, proceed directly to step A5; otherwise, proceed to step A4.

[0031] A4. Determine whether i is equal to temp. If so, proceed to step A5. Otherwise,

[0032] Update the scaling factor s, step size h, time t, and i:

[0033] in,

[0034] If min_scale<s<max_scale, then err i is the current error, h = h·s, t = t+h, i = i+1;

[0035] After updating, return to step A2;

[0036] A5. Use the latest state value as the actual tracking displacement x(t).

[0037] Furthermore, the calculation process of the state value and the error corresponding to the i-th iteration includes:

[0038] k1=hf(t i ,x i )

[0039]

[0040] Where h is the step size.

[0041] Furthermore, the performance index function of the determined linear controllable canonical subsystem is as follows:

[0042]

[0043] Among them, x * is the desired displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

[0044] Furthermore, the optimal performance indicators of the piezoelectric ceramic driver model are obtained, including:

[0045] Transform the performance indicator function into:

[0046]

[0047] Let M tend to positive infinity, according to

[0048]

[0049] Right now

[0050] When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

[0051] Compared with the existing technology, the principles and advantages of this technical solution are as follows:

[0052] 1. By adaptively adjusting the step size based on the Runge-Kutta-Feldberg method (RKF45), the computational efficiency and control accuracy of the piezoelectric ceramic actuator are improved while maintaining high accuracy during the calculation process.

[0053] 2. The error tolerance ε is adjusted according to the accuracy requirements, so that the calculation accuracy is better than the fourth-order Runge-Kutta method and the calculation speed is better than the fifth-order Runge-Kutta method, thereby further improving the calculation accuracy and speed of the piezoelectric actuator displacement tracking, so that the system obtains better control effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the services required for use in the embodiments or the prior art descriptions will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0055] Figure 1 This is a principle flow chart of a piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to the present invention;

[0056] Figure 2 This is a schematic diagram of solving the actual tracking displacement in a piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to the present invention;

[0057] Figure 3 Flowchart for solving the actual tracking displacement using the Runge-Kutta-Feilberg method with adaptive step size;

[0058] Figure 4 Schematic diagram comparing the errors between RKF45 and ode45 and the errors between RK4 and ode45;

[0059] Figure 5 Schematic diagram for the comparison of ECNLP models RK4 and RKF45 in displacement tracking control systems. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with specific embodiments:

[0061] As Figures 1 to 2 shown, the piezoelectric drive optimal trajectory tracking control method based on adaptive step length in the embodiment includes the following steps:

[0062] S1, the piezoelectric ceramic driver is mathematically modeled by using the electromechanical coupling nonlinear dynamic system model, and the following piezoelectric ceramic driver model is obtained:

[0063]

[0064] Among them, is the derivative of X, is the state variable, U in is the control variable, F is the function matrix related to the state variable X and the control variable U of the piezoelectric ceramic driver model, G is the weight matrix, x is the actual tracking displacement, in is the derivative of x.

[0065] S2, the piezoelectric ceramic driver model is converted into a linear controllable canonical subsystem:

[0066]

[0067] Among them, u is the feedback control.

[0068] S3, the expected displacement x * (0) at time t=0 is taken as the initial actual tracking displacement x(0);

[0069] S4, the feedback control u is obtained by the linear controllable canonical subsystem, and the formula is as follows:

[0070] u=-R -1 B T P LQ Z+R -1 B T G g (3)

[0071] R is the weight matrix, the constant matrix P LQ is the unique symmetric positive definite solution of Riccati algebraic equation;

[0072] A T P LQ +P LQ A-P LQ BR -1 B T P LQ +C T QC=0 (4)

[0073] C is the parameter of the piezoelectric ceramic driver, and the parameter value is 1.2×10​-6 F, matrix G g for:

[0074] G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5)

[0075] x * is the expected displacement.

[0076] S5. Substitute the initial state value x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Felberg method with adaptive step size to calculate the actual tracking displacement, as follows: Figure 3 As shown, the specific process includes:

[0077] A1. Set the initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the step size h, the system initial state value x0 = x(0), the independent variable t = t0, i = 1;

[0078] A2. Calculate the current state value and the resulting error;

[0079] The calculation process of the state value and the error corresponding to the i-th iteration includes:

[0080] k1=hf(t i ,x i )

[0081]

[0082] A3. Determine whether the error generated in the previous step A2 is equal to 0. If so, proceed directly to step A5; otherwise, proceed to step A4.

[0083] A4. Determine whether i is equal to temp. If so, proceed to step A5. Otherwise,

[0084] Update the scaling factor s, step size h, time t, and i:

[0085] in,

[0086] If min_scale<s<max_scale, then (h here is before update), err i is the current error, h = h·s, t = t+h (here h is the updated one), i = i+1;

[0087] After updating, return to step A2;

[0088] A5. Use the latest state value as the actual tracking displacement x(t).

[0089] S6. Determine the performance index function of the linear controllable specification subsystem, and substitute the feedback control u and the actual tracking displacement into the performance index function, so as to obtain the optimal performance index of the piezoelectric ceramic driver model.

[0090] In this step, the performance index function of the linear controllable specification subsystem is determined as follows:

[0091]

[0092] Among them, x * is the desired displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

[0093] Determine the optimal performance indicators of the piezoelectric ceramic actuator model, including:

[0094] Transform the performance indicator function into:

[0095]

[0096] Let M tend to positive infinity, according to

[0097]

[0098] Right now

[0099] When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

[0100] The comparison between the method described in this embodiment and the existing method is as follows:

[0101] In the past, the fixed-step RK4 algorithm required calculating k four times. i Function, the local error is O(h 5 ); while the fixed-step fifth-order Runge-Kutta (RK5) algorithm needs to calculate k 10 times i Function, the local error is O(h 6 ), although RK5 improves the calculation accuracy compared to RK4, it brings about the consumption of calculation cost. In order to integrate the above advantages and solve the related shortcomings, the method described in this embodiment combines the advantages of the fourth-order Runge Kutta method with the high calculation speed and the fifth-order Runge Kutta method with high calculation accuracy, and proposes the RKF45 algorithm with adaptive step size, which only needs to calculate k six times. ifunction, and adjust the error tolerance ε according to the need of accuracy, so that it is better than the RK4 algorithm with a fixed step size in terms of calculation accuracy and better than the RK5 algorithm in terms of calculation speed, thereby further improving the accuracy of piezoelectric actuator displacement tracking.

[0102] In order to prove the advantages of the method described in this embodiment, the system was simulated using the mathematical software MATLAB and experimental data were collected. The simulation time was set to 0.05 seconds. Figure 4 The error comparison between the present invention and the RK4 method when applied to the implementation example is shown below: Figure 5 As can be seen from the figure, the present invention significantly improves the displacement tracking accuracy of the system when solving the displacement tracking of the ECNLP model, and effectively optimizes the dynamic performance of the system.

[0103] The embodiments described above are only preferred embodiments of the present invention and are not intended to limit the scope of implementation of the present invention. Therefore, any changes made based on the shape and principle of the present invention should be included in the scope of protection of the present invention.

Claims

1. A piezoelectric drive optimal trajectory tracking control method based on adaptive step size, characterized in that: include: The piezoelectric ceramic actuator is mathematically modeled using an electromechanical coupling nonlinear dynamic system model to obtain a piezoelectric ceramic actuator model. Convert the piezoelectric ceramic actuator model into a linear controllable canonical subsystem; The expected displacement x at time t = 0 * (0) as the initial actual tracking displacement x(0); Determine the feedback control u through the linear controllable canonical subsystem; Substitute the initial actual tracking displacement x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Feldberg method with adaptive step size to calculate the actual tracking displacement; Determine the performance index function of the linear controllable canonical subsystem, and substitute the feedback control u and the actual tracking displacement into the performance index function to obtain the optimal performance index of the piezoelectric ceramic actuator model; The actual tracking displacement is obtained using the Runge-Kutta-Feldberg method with adaptive step size, including: A1. Set the initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the step size h, the system initial state value x0 = x(0), the independent variable t = t0, i = 1; A2. Calculate the current state value and the resulting error; A3. Determine whether the error generated in the previous step A2 is equal to 0. If so, proceed directly to step A5; otherwise, proceed to step A4. A4. Determine whether i is equal to temp. If so, proceed to step A5. Otherwise, Update the scaling factor s, step size h, time t, and i: in, If min_scale<s<max_scale, then err i is the current error, h = h·s, t = t+h, i = i+1; After updating, return to step A2; A5. Use the latest state value as the actual tracking displacement x(t).

2. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 1, characterized in that: After mathematically modeling the piezoelectric ceramic actuator using the electromechanical coupling nonlinear dynamic system model, the resulting piezoelectric ceramic actuator model is as follows: in, is the derivative of X, is the state variable, U in is the control variable, F is the state variable X and the control variable U of the piezoelectric ceramic actuator model in Related function matrix, G is the weight matrix, x is the actual tracking displacement, is the derivative of x.

3. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 2, characterized in that: The transformed linear controllable canonical subsystem is as follows: in, u is the feedback control.

4. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 3, characterized in that: The formula for calculating feedback control u is as follows: u=-R -1 B T P LQ Z+R -1 B T G g (3) R is the weight matrix, constant matrix P LQ It is the only symmetric positive definite solution to the Riccati algebraic equation; A T P LQ +P LQ A-P LQ BR -1 B T P LQ +C T QC=0 (4) C is the piezoelectric ceramic driver parameter, and its value is 1.2×10 -6 F, matrix G g for: G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5) x * is the expected displacement.

5. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 1, characterized in that: The calculation process of the state value and error corresponding to the i-th iteration include: k1=hf(t i ,x i ) Where h is the step size.

6. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 1, characterized in that: The performance index function of the determined linear controllable canonical subsystem is as follows: Among them, x * is the desired displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

7. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 6, characterized in that: Determine the optimal performance indicators of the piezoelectric ceramic actuator model, including: Transform the performance indicator function into: Let M tend to positive infinity, according to Right now When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

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