A robust control method for piezoelectric drive based on sliding mode observer and radial basis function neural network

Abstract

The present invention proposes a piezoelectric drive robust control method based on a sliding mode observer and a radial basis function neural network. The method designs a sliding mode state observer with disturbance compensation to observe system disturbances and system states. Since the radial basis function neural network can also be used for interference suppression, the RBFNN has the characteristics and capabilities of parallel processing, self-learning and nonlinear approximation, and can generate reasonable outputs through a sufficient fitting process. Through the design of this control method, the influence of the system not being modeled and various disturbances can be compensated and the chattering effect caused by the sliding mode controller can be reduced, further improving the tracking performance of the piezoelectric drive system output.

Description

Technical Field

[0001] The present invention relates to the technical field of piezoelectric drive, and in particular to a piezoelectric drive robust control method based on a sliding mode observer and a radial basis function neural network. Background Art

[0002] With the rapid development of ultra-precision manufacturing, aerospace, biomedicine, optical precision engineering, semiconductor lithography, and detection systems, motion platforms that meet the demands of high-speed and high-precision positioning are urgently needed. Actuators using piezoelectric drive technology can achieve high-precision positioning at the micron to nanometer scale due to their advantages such as small size, simple structure, high power density, no electromagnetic interference, self-locking when powered off, and zero positioning noise.

[0003] The shortcomings and deficiencies of existing technologies: Due to the combined influence of the inherent nonlinearity of piezoelectric materials, system temperature rise, friction loss, and preload between the stator and the rotor, it is difficult to accurately model the piezoelectric drive system, which affects the effectiveness of the designed control algorithm and causes severe nonlinearity in the system output. To improve the output performance of the piezoelectric drive system, closed-loop control can be achieved by acquiring the system's state information (position and velocity signals) in real time. To this end, a robust neural network control method using a sliding mode observer with disturbance observation compensation is proposed for the piezoelectric drive system, with the hope of using this control method to track the predetermined motion trajectory. Summary of the Invention

[0004] The present invention aims to address the problem of insufficient tracking accuracy of a predetermined trajectory due to severe nonlinearity in the output characteristics of a piezoelectric drive system, which is affected by factors such as inaccurate modeling. The present invention provides a robust control method for a piezoelectric drive based on a sliding mode observer and a radial basis function neural network. The method integrates the combined feedback of a sliding mode observer with disturbance observation and a radial basis function neural network (RBFNN) as a function approximator to approximate the unknown system model and design a predetermined control voltage to meet the requirements for tracking the predetermined trajectory.

[0005] The present invention is achieved through the following technical solutions. The present invention proposes a piezoelectric drive robust control method based on a sliding mode observer and a radial basis function neural network. The method comprises the following steps:

[0006] Step 1: Combine a sliding mode observer including disturbance observation with a radial basis function neural network. The disturbance observer is used to estimate the system's centralized disturbance, thereby pre-compensating the disturbance. The sliding mode observer is used to observe the system's state variables for centralized feedback, and the radial basis function neural network is used to compensate for model uncertainty, residual disturbance, and nonlinearity.

[0007] Step 2: Design a state and disturbance observer to centrally observe the system's internal state variables and centralized disturbances;

[0008] Step 3: Design the corresponding control input u so that the piezoelectric drive system can track the predetermined trajectory, and design the control law based on the state space equation of the system.

[0009] Furthermore, in step 2, the piezoelectric drive system is a second-order system, and the system dynamic model is:

[0010]

[0011] Where u and y represent the input and output of the system respectively. and Represent the first-order and second-order derivatives of the system output respectively; the function Φ and the disturbance term D are unknown, and the input gain h can be obtained through system identification; Φ and D are concentrated into the disturbance d, and Equation (1) can be written in state space form:

[0012]

[0013] Where x1 and x2 represent the displacement and velocity of the system respectively;

[0014] The system disturbance is estimated using a first-order filter disturbance observer DOB:

[0015]

[0016] In the formula is the estimated value of the disturbance term d, β is the bandwidth of the disturbance observer;

[0017] Design a sliding mode observer:

[0018]

[0019] Where the state estimate vector Parameter Matrix It can be obtained through system identification, C = [1 0], where Θ represents the Sigmoid function;

[0020] Observation gain matrix Define the sliding surface To ensure accurate arrival at the sliding surface, the gain parameter must satisfy the following conditions: l1 > max(e2), l2 must be selected so that (A2 - l2 / l1) satisfies Hurwitz stability, and matrix A2 is the remaining rows of matrix A after removing the first row.

[0021] Furthermore, in step 3, the control law is designed based on the state space equation (2) of the system:

[0022]

[0023] in is the compensation control law provided by the disturbance observer, u t The control law to be designed should take into account the error between the disturbance observation and the actual disturbance and the error caused by the unmodeled part of the system and the inaccurate model; the tracking error of the system:

[0024]

[0025] Where x r Represents the intended tracking trajectory of the system.

[0026] Furthermore, u t A radial basis function neural network is used to learn the upper bound of the uncertain system model. The neural network structure is a local approximation network with a three-layer feedforward structure and a single hidden layer. It can approximate any continuous function with arbitrary precision. The input of the neural network is The output of the neural network is an estimate of the upper bound of the uncertain system model parameters in is the weight of the radial basis function neural network, and φ(θ) is the Gaussian function:

[0027]

[0028] Where m i and σ i is the center position and width of the i-th neuron; the adaptive algorithm is used to adjust the weights, taking η>0,

[0029] Determine the estimated value of the upper bound of the model through the neural network The control input to track the predetermined trajectory can then be output:

[0030]

[0031] The present invention has the following beneficial effects:

[0032] In this paper, the piezoelectric drive system itself is subject to multiple coupling factors, making it difficult to accurately model the system. Consequently, the accuracy of motion control is affected by this unknown model. The piezoelectric drive system's performance is also affected by hysteresis, nonlinearity, and external disturbances, making precise control of the piezoelectric drive system difficult. To address this issue, a sliding mode state observer with disturbance compensation is designed to observe both system disturbances and the system state. Furthermore, since radial basis function neural networks (RBFNNs) can also be used for disturbance suppression, RBFNNs possess the characteristics and capabilities of parallel processing, self-learning, and nonlinear approximation, and can be used to produce reasonable outputs through a comprehensive fitting process. Neural networks have shown promising application prospects in the field of piezoelectric drive systems. This paper proposes a motion controller capable of compensating for parameter uncertainty and disturbances, namely, a robust neural network control with a disturbance observation strategy. These two techniques aim to overcome parameter uncertainty, nonlinearity, and disturbances. The proposed motion control method aims to improve the tracking performance of the piezoelectric drive system. A disturbance observer is used to estimate nonlinear dynamics and external disturbances, and actively compensate for concentrated disturbances during control. The RBFNN with a disturbance observer and a sliding mode controller is used to eliminate model uncertainty, residual disturbances, and nonlinearities. The thresholds and weights of the RBFNN are estimated and updated during real-time control using an online tuning method. This control method compensates for the effects of unmodeled system conditions and various disturbances, reduces chattering effects caused by the sliding mode controller, and further improves the tracking performance of the piezoelectric drive system output. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 A block diagram of the overall controller design of the present invention;

[0034] Figure 2 This is a neural network structure diagram of the present invention. DETAILED DESCRIPTION

[0035] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0036] The present invention proposes a piezoelectric drive robust control method based on a sliding mode observer and a radial basis function neural network, the method comprising the following steps:

[0037] Step 1: Due to the inherent nonlinearity and the coupling of thermal, electrical, and force multi-physics fields, it is challenging to establish an accurate mathematical model of the piezoelectric drive system. Therefore, traditional control methods are not suitable for piezoelectric drive control. In order to achieve precise control of the motion of the piezoelectric drive system, a sliding mode observer including disturbance observation is combined with a radial basis function neural network to address the problem of insufficient tracking trajectory accuracy. The block diagram of the overall controller design is shown in the figure below. Figure 1 As shown in the figure, the disturbance observer is used to estimate the centralized disturbance of the system and thus pre-compensate the disturbance. The sliding mode observer is used to observe the system state variables and provide centralized feedback. The radial basis function neural network is used to compensate for model uncertainty as well as residual disturbances and nonlinearity to further improve the control performance of the system.

[0038] Step 2: Design a state and disturbance observer to centrally observe the system's internal state variables and centralized disturbances;

[0039] In step 2, the piezoelectric drive system is a second-order system, and the system dynamic model is:

[0040]

[0041] Where u and y represent the input and output of the system respectively. and Represent the first-order and second-order derivatives of the system output respectively; the function Φ and the disturbance term D are unknown, and the input gain h can be obtained through system identification; Φ and D are concentrated into the disturbance d, and Equation (1) can be written in state space form:

[0042]

[0043] Where x1 and x2 represent the displacement and velocity of the system respectively;

[0044] The system disturbance is estimated using a first-order filter disturbance observer DOB:

[0045]

[0046] In the formula is the estimated value of the disturbance term d, β is the bandwidth of the disturbance observer;

[0047] Design a sliding mode observer:

[0048]

[0049] Where the state estimate vector Parameter Matrix It can be obtained through system identification, C = [1 0], where Θ represents the Sigmoid function;

[0050] Observation gain matrix Define the sliding surface To ensure accurate arrival at the sliding surface, the gain parameter selection must satisfy: l1>max(e2), l2 is selected so that (A2-l2 / l1) satisfies Hurwitz stability, and the matrix A2 is the remaining rows after removing the first row of the matrix A.

[0051] Step 3: Design the corresponding control input u so that the piezoelectric drive system can track the predetermined trajectory, and design the control law based on the state space equation of the system.

[0052] In step 3, the control law is designed based on the state space equation (2) of the system:

[0053]

[0054] in is the compensation control law provided by the disturbance observer, u t The control law to be designed should take into account the error between the disturbance observation and the actual disturbance and the error caused by the unmodeled part of the system and the inaccurate model; the tracking error of the system:

[0055]

[0056] Where x r Represents the intended tracking trajectory of the system.

[0057] u t The radial basis function neural network is used to learn the upper bound of the uncertain system model. The neural network structure is as follows: Figure 2 As shown, the neural network structure is a local approximation network with a three-layer feedforward structure and a single hidden layer, which can approximate any continuous function with arbitrary precision. The input of the neural network is The output of the neural network is an estimate of the upper bound of the uncertain system model parameters in is the weight of the radial basis function neural network, and φ(θ) is the Gaussian function:

[0058]

[0059] Where m i and σ i is the center position and width of the i-th neuron; the adaptive algorithm is used to adjust the weights, taking η>0,

[0060] Determine the estimated value of the upper bound of the model through the neural network The control input to track the predetermined trajectory can then be output:

[0061]

[0062] The above description only expresses the preferred embodiments of the present invention and does not limit the present invention in any other form. Any technician familiar with the present invention may use the above disclosure to change or modify it into an equivalent embodiment with equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiment based on the technical essence of the present invention without departing from the content of the technical solution of the present invention shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A piezoelectric drive robust control method based on sliding mode observer and radial basis function neural network, characterized in that: The method comprises the following steps: Step 1: Combine a sliding mode observer including disturbance observation with a radial basis function neural network. The disturbance observer is used to estimate the system concentrated disturbance, so as to pre-compensate the disturbance. The sliding mode observer is used to observe the system state variables for centralized feedback. The radial basis function neural network is used to compensate for model uncertainty as well as residual disturbance and nonlinearity. Step 2: Design state and disturbance observers to centrally observe the internal state variables and central disturbances of the system; Step 3: Design the corresponding control input u so that the piezoelectric drive system can track the predetermined trajectory, and design the control law based on the state space equation of the system; In step 2, the piezoelectric drive system is a second-order system, and the system dynamic model is: Where u and y represent the input and output of the system respectively. and Represent the first-order and second-order derivatives of the system output respectively; the function Φ and the disturbance term D are unknown, and the input gain h can be obtained through system identification; Φ and D are concentrated into the disturbance d, and equation (1) can be written in state space form: Where x1 and x2 represent the displacement and velocity of the system respectively; The system disturbance is estimated using a first-order filter disturbance observer DOB: In the formula is the estimated value of the disturbance term d, β is the bandwidth of the disturbance observer; Design a sliding mode observer: The state estimation vector Parameter Matrix It can be obtained through system identification, C = [10], where Θ represents the Sigmoid function; Observation gain matrix Define the sliding surface To ensure accurate arrival at the sliding surface, the selection of gain parameters must satisfy: l1>max(e2), l2 must be selected so that (A2-l2 / l1) satisfies Hurwitz stability, and matrix A2 is the remaining rows of matrix A after removing the first row; In step 3, the control law is designed based on the state space equation (2) of the system: in is the compensation control law provided by the disturbance observer, u t The control law to be designed should take into account the error between the disturbance observation and the actual disturbance and the error caused by the unmodeled part of the system and the inaccurate model; the tracking error of the system: Where x r represents the predetermined tracking trajectory of the system; u t A radial basis function neural network is used to learn the upper bound of the uncertain system model. The neural network structure is a local approximation network with a three-layer feedforward structure with a single hidden layer, which can approximate any continuous function with arbitrary accuracy. The input of the neural network is The output of the neural network is an estimate of the upper bound of the uncertain system model parameters. in is the weight of the radial basis function neural network, and φ(θ) is the Gaussian function: Where m i and σ i is the center position and width of the i-th neuron; the weights are adjusted using an adaptive algorithm. η>0, Determine the estimated value of the upper bound of the model through a neural network The control input that tracks the predetermined trajectory can then be output:

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