Piezoelectric micro-positioning platform preset performance control method based on extended state observer

Abstract

The invention discloses a piezoelectric micro-positioning platform preset performance control method based on an extended state observer, and the method mainly comprises the steps: firstly, building a piezoelectric micro-positioning platform system model considering input hysteresis and unknown disturbance; secondly, designing an extended state observer based on a radial basis function neural network to estimate an unmeasurable state of the system and lumped disturbance containing input hysteresis; then, a preset performance function is used, so that the overshoot performance of the system is effectively improved; thirdly, in combination with a preset performance function, a first-order sliding mode differentiator and an expansion state observer, providing a virtual control law and an adaptive control law; and finally, a preset performance controller is utilized, a Lyapunov stability theory is combined, proper parameters are selected to ensure that the closed-loop system is kept stable within preset time, and tracking errors are controlled within a set error range.

Description

Technical Field

[0001] This invention belongs to the field of micro-nano-level tracking and control technology of intelligent material-driven devices, and more specifically, relates to a preset performance control method for a piezoelectric micro-positioning platform based on an extended state observer. Background Technology

[0002] As industrial technology continues to advance towards greater precision and intelligence, the demand for ultra-precision machining technology in advanced manufacturing is increasing. Traditional actuators struggle to meet these requirements, leading to the growing prevalence of micro / nano actuators based on smart materials in cutting-edge industrial fields. Among these micro / nano actuators, piezoelectric micro-positioning platforms using piezoelectric ceramic materials play a crucial role in ultra-precision machining due to their rapid response, strong driving force, and high positioning accuracy. However, the inherent hysteresis nonlinearity of piezoelectric ceramic materials severely limits the platform's application range. Therefore, effectively compensating for hysteresis and further improving the platform's tracking and control performance has become a challenging task.

[0003] Due to the unique nature of ultra-precision machining technology, the performance of piezoelectric micro-positioning platforms requires precise control in practical applications. In controller design, a preset performance function is typically introduced to clearly define system performance. However, the creep characteristics of piezoelectric micro-positioning platforms and the difficulty in measuring their initial state limit the application of traditional preset performance functions. To address this issue, researchers have introduced a transformation function to extend the constraints of the preset performance function to infinity in the initial stage, ensuring that any initial state of the system satisfies the constraints. However, it should be noted that such infinite initial constraints may lead to significant overshoot during system operation, which is unacceptable in some machining scenarios. Therefore, finding a method to improve the overshoot performance of the system while relaxing the limitations imposed by initial state uncertainties on the application of preset performance functions is of significant practical importance. Summary of the Invention

[0004] Considering the problems of uncertain initial state, unmeasurable internal state, and unknown input hysteresis, this invention proposes a pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer to overcome the shortcomings of existing technologies and provide a control strategy to improve the overshoot performance of the system. Note: (·) represents the first derivative of the physical quantity, (^) represents the estimated value of the physical quantity, and (~) represents the estimation error;

[0005] The specific implementation steps of this invention are as follows:

[0006] A pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer, comprising the following steps:

[0007] Step 1: Describe the piezoelectric micro-positioning platform system as a second-order system with input hysteresis and unknown disturbance. For the hysteresis nonlinearity of the piezoelectric actuator, decompose the hysteresis nonlinearity using the PI model, decomposing it into a linear term and a hysteresis nonlinear term with respect to the control signal.

[0008] Step 2: Define the generalized perturbation of the system, which includes the unknown parts of the hysteresis nonlinear term, external perturbation, and input gain. Treat it as the extended state of the original system. Construct an extended state observer to estimate this state and the state of the original system. For the unknown nonlinear function in the system, it is approximated by a radial basis function neural network.

[0009] Step 3: Construct a preset performance function to improve the system's overshoot performance while ensuring the system's steady-state performance;

[0010] Step 4: Based on the extended state observer in Step 2 and the preset performance function in Step 3, calculate the virtual error, and design the virtual control law and adaptive law by combining the adaptive method;

[0011] Step 5: Combining Lyapunov stability theory, a preset performance controller is obtained to control the piezoelectric micro-positioning platform, so that the tracking error of the closed-loop system is kept within the preset error range.

[0012] The specific steps for step 1 are as follows:

[0013] First, the piezoelectric micro-positioning platform system is described as a second-order system with input hysteresis and unknown disturbances, and the state-space equation model is established as follows:

[0014] (1)

[0015] In equation (1), and It is an unmeasurable system state. For state vectors, It is a control input. It is a non-zero control gain coefficient. It is a measurable system output. and Represents an unknown smooth continuous function. This is a bounded interference.

[0016] For the hysteresis nonlinearity of piezoelectric actuators, the PI model is used to decompose the hysteresis nonlinearity:

[0017] (2)

[0018] In equation (2), Indicates the model input, Let be the density function. It is an unknown positive constant determined by the density function. The threshold representing hysteresis. It is a hysteresis operator.

[0019] The hysteresis model, i.e., formula (2), can be rewritten as:

[0020] (3)

[0021] In equation (3),

[0022] Substituting formula (3) into formula (1), we get:

[0023] (4)

[0024] In equation (4), It is worth noting that the control method used does not require constructing an accurate hysteresis model; it only requires decomposing the hysteresis into linear terms and hysteresis nonlinear terms with respect to the control signal. The decomposition process is as follows:

[0025] (5)

[0026] In equation (5), The prior part obtained from the actual system, and The following unknown positive constants exist. and , making and .

[0027] Therefore, the system, i.e., formula (4), can be rewritten as:

[0028] (6).

[0029] The specific steps for step 2 are as follows:

[0030] First, define The system's generalized perturbation includes the unknown components of the hysteresis nonlinearity, external perturbation, and input gain. Then... Considered a new state of the system, soon The extended state is then used to estimate the state of the original system. Based on this, an extended state observer is designed to estimate the state and the state of the original system. The system, i.e., equation (6), can be rewritten as:

[0031] (7)

[0032] In equation (7), let An unknown nonlinear function is approximated by a radial basis function neural network as an unknown continuous function. Right now:

[0033] (8)

[0034] In equation (8), It is the input vector of the neural network. For the weights of the neural network, These are the basis function vectors of the neural network. It is the approximation error of the neural network.

[0035] Based on formula (8), the following extended state observer is constructed:

[0036] (9)

[0037] In equation (9), For gain, for The estimated value.

[0038] Observation error is defined as The observation error dynamics are as follows:

[0039] (10)

[0040] In equation (10), , .

[0041] Rewrite equation (10) as:

[0042] (11)

[0043] In equation (11), , , , , , ; can be achieved by designing a reasonable gain Make Established, among which and All are positive definite matrices.

[0044] Consider the following Lyapunov function for estimating the error of the extended state observer. :

[0045] The conclusion is The derivative with respect to time:

[0046] (12)

[0047] There are unknown constants make According to Young's inequality, we have:

[0048] (13)

[0049] In equation (13), , , .

[0050] There are known constants ( This makes the following inequality always true: ,then:

[0051] (14)

[0052] In equation (14), It is an identity matrix.

[0053] The specific steps for step 3 are as follows:

[0054] To achieve the preset control of the radial basis function neural network, the tracking error is first defined as:

[0055] (15)

[0056] In equation (15), Indicates tracking error. This indicates the desired signal.

[0057] The preset performance for tracking error is described as follows:

[0058] (16)

[0059] In equation (16), and For design parameters, the performance function is chosen to be exponential:

[0060] (17)

[0061] In equation (17), It is a natural constant. , and It is a positive number, and ; ,satisfy .

[0062] To satisfy the condition, select... ,in If is a natural constant, then:

[0063] (18)

[0064] In equation (18), To account for the error after transformation, we further have:

[0065] (19)

[0066] Therefore:

[0067] (20)

[0068] In equation (20), the definition is... ;

[0069] Introduce the following coordinate transformations:

[0070] (twenty one)

[0071] In equation (21), This is a virtual control law; to avoid differential explosion and singularity problems, a first-order sliding mode differentiator is used to obtain the derivative of the virtual control law, i.e.:

[0072] (twenty two)

[0073] In equation (22), For the input of the differentiator, , and The state of the differentiator; , The parameters of the designed differentiator; if the initial error , If it is bounded, then the differentiator can output with any precision. estimation error It is bounded and has a positive constant. Make Established.

[0074] The specific steps of step 4 are as follows:

[0075] First, based on backstepping and adaptive techniques, the virtual control law is designed as follows:

[0076] (twenty three)

[0077] In equation (23), ;

[0078] The actual control law is designed as follows:

[0079] (twenty four)

[0080] The adaptive law is designed as follows:

[0081] (25)

[0082] In equation (25), and For the design parameters, and These are positive design parameters.

[0083] The specific steps for step 5 are as follows:

[0084] Choose the following Lyapunov functions for the entire system:

[0085] (26)

[0086] In equation (26), the Lyapunov function for the estimation error of the extended state observer is... Lyapunov function for controller error Lyapunov function for neural network weight error ;

[0087] Stability analysis is performed based on Lyapunov's stability theory, as follows:

[0088] (1) Select the Lyapunov function as:

[0089] (27)

[0090] right Taking the derivative, we further obtain:

[0091] (28)

[0092] And:

[0093] (29)

[0094] (30)

[0095] Combining the virtual control law and the adaptive law, we get:

[0096] (31)

[0097] (2) The Lyapunov function is selected as:

[0098] (32)

[0099] right Taking the derivative and combining it with the above equation, we get:

[0100] (33)

[0101] And:

[0102] (34)

[0103] Substituting the actual control law and the adaptive law, we get:

[0104] (35)

[0105] In equation (35), , ;

[0106] By selecting appropriate parameters, the desired result can be achieved. , , , .

[0107] Preferably, the selection principles for relevant controller parameters are as follows:

[0108] (1) Extended state observer parameters: , , , ;

[0109] (2) Parameters of the preset performance function , , , , The parameter selection meets the requirements of actual working scenarios for system overshoot performance and steady-state accuracy;

[0110] (3) Differentiator parameters , The selected parameters can ensure that the output signal of the differentiator is effective in the control process of the closed-loop system;

[0111] (4) Virtual control law, adaptive law and controller parameters , , , , , The closed-loop system needs to converge quickly and remain stable after the convergence process, and its performance needs to meet the requirements of ultra-precision machining scenarios; the parameters are selected as follows: , , , .

[0112] The beneficial effects of this invention are as follows:

[0113] This invention addresses the tracking control problem of piezoelectric micro-positioning platforms with uncertain initial states and input hysteresis. It proposes a pre-set performance control method for piezoelectric micro-positioning platforms based on an extended state observer to improve platform performance. The method offers two main advantages: first, it designs an extended state observer capable of estimating unmeasurable states and input hysteresis, accurately estimating the system state and effectively eliminating the impact of hysteresis on tracking performance; second, it incorporates a pre-set performance function into the controller design, effectively improving the system's overshoot performance.

[0114] The method of this invention takes into account the tracking performance and input hysteresis problems of piezoelectric micro-positioning platform systems in practical applications, and provides a practical and feasible technical solution for the practical application of piezoelectric micro-positioning platforms in the field of ultra-precision machining technology. Attached Figure Description

[0115] Figure 1 This is a control block diagram of the pre-set performance control method for the piezoelectric micro-positioning platform based on an extended state observer according to the present invention.

[0116] Figure 2 This is a schematic diagram of the experimental setup for the piezoelectric micro-positioning platform system of the present invention;

[0117] Figure 3 This is a block diagram illustrating the experimental principle of the piezoelectric micro-positioning platform system of the present invention;

[0118] Figure 4 The tracking frequency conversion signal of the present invention A comparison diagram of the reference trajectory and actual displacement of the piezoelectric micro-positioning platform at that time;

[0119] Figure 5 The tracking frequency conversion signal of the present invention The curve of the tracking error of the piezoelectric micro-positioning platform versus the preset performance constraint boundary;

[0120] Figure 6 The tracking frequency of this invention is 100Hz signal. A comparison diagram of the reference trajectory and the actual displacement at that time;

[0121] Figure 7 The tracking frequency of this invention is 100Hz signal. The curve of the tracking error of the piezoelectric micro-positioning platform versus the preset performance constraint boundary. Detailed Implementation

[0122] The invention will now be described in detail with reference to the accompanying drawings and examples, and the effectiveness of the technical solution will be further verified.

[0123] The control block diagram of the preset performance control method for a piezoelectric micropositioning platform based on an extended state observer described in this embodiment is as follows: Figure 1 As shown. The specific implementation steps are as follows:

[0124] Step 1: Describe the piezoelectric micro-positioning platform system as a second-order system with input hysteresis and unknown disturbance. For the hysteresis nonlinearity of the piezoelectric actuator, decompose the hysteresis nonlinearity using the PI model, decomposing it into a linear term and a hysteresis nonlinear term with respect to the control signal.

[0125] First, the piezoelectric micro-positioning platform system is described as a second-order system with input hysteresis and unknown disturbances, and the state-space equation model is established as follows:

[0126] (1)

[0127] In equation (1), and It is an unmeasurable system state. For state vectors, It is a control input. It is a non-zero control gain coefficient. It is a measurable system output. and Represents an unknown smooth continuous function. This is a bounded interference.

[0128] For the hysteresis nonlinearity of piezoelectric actuators, the PI model is used to decompose the hysteresis nonlinearity:

[0129] (2)

[0130] In equation (2), Indicates the model input, Let be the density function. It is an unknown positive constant determined by the density function. The threshold representing hysteresis. It is a hysteresis operator.

[0131] The hysteresis model, i.e., formula (2), can be rewritten as:

[0132] (3)

[0133] In equation (3),

[0134] Substituting formula (3) into formula (1), we get:

[0135] (4)

[0136] In equation (4), It is worth noting that the control method used does not require constructing an accurate hysteresis model; it only requires decomposing the hysteresis into linear terms and hysteresis nonlinear terms with respect to the control signal. The decomposition process is as follows:

[0137] (5)

[0138] In equation (5), The prior part obtained from the actual system, and The following unknown positive constants exist. and , making and .

[0139] Therefore, the system, i.e., formula (4), can be rewritten as:

[0140] (6).

[0141] Step 2: Define the generalized perturbation of the system, which includes the unknown parts of the hysteresis nonlinear term, external perturbation, and input gain. Treat it as the extended state of the original system. Construct an extended state observer to estimate this state and the state of the original system. For the unknown nonlinear function in the system, it is approximated by a radial basis function neural network.

[0142] First, define The system's generalized perturbation includes the unknown components of the hysteresis nonlinearity, external perturbation, and input gain. Then... Considered a new state of the system, soon The extended state is then used to estimate the state of the original system. Based on this, an extended state observer is designed to estimate the state and the state of the original system. The system, i.e., equation (6), can be rewritten as:

[0143] (7)

[0144] In equation (7), let An unknown nonlinear function can be approximated by a radial basis function neural network (RBFNN) to approximate an unknown continuous function. Right now:

[0145] (8)

[0146] In equation (8), It is the input vector of the RBFNN neural network. The weights of the RBFNN neural network are... These are the basis function vectors of the RBFNN neural network. It is the approximation error of the RBFNN neural network.

[0147] Based on formula (8), the following extended state observer is constructed:

[0148] (9)

[0149] In equation (9), For gain, for The estimated value.

[0150] Observation error is defined as The observation error dynamics are as follows:

[0151] (10)

[0152] In equation (10), , .

[0153] Rewrite equation (10) as:

[0154] (11)

[0155] In equation (11), , , , , , It can be achieved by designing a reasonable gain. Make Established, among which and All are positive definite matrices.

[0156] Consider the following Lyapunov function for estimating the error of the extended state observer. :

[0157] The conclusion is The derivative with respect to time:

[0158] (12)

[0159] There are unknown constants make According to Young's inequality, we have:

[0160] (13)

[0161] In equation (13), , , .

[0162] There are known constants ( This makes the following inequality always true: ,then:

[0163] (14)

[0164] In equation (14), It is an identity matrix.

[0165] Step 3: Construct a preset performance function to improve the system's overshoot performance while ensuring the system's steady-state performance;

[0166] To achieve the preset control of the radial basis function neural network, the tracking error is first defined as:

[0167] (15)

[0168] In equation (15), Indicates tracking error. This indicates the desired signal.

[0169] The preset performance for tracking error is described as follows:

[0170] (16)

[0171] In equation (16), and For design parameters, the performance function is chosen to be exponential:

[0172] (17)

[0173] In equation (17), It is a natural constant. , and It is a positive number, and ; ,satisfy .

[0174] To satisfy the condition, select... ,in If is a natural constant, then:

[0175] (18)

[0176] In equation (18), To account for the error after transformation, we further have:

[0177] (19)

[0178] Therefore:

[0179] (20)

[0180] In equation (20), the definition is... .

[0181] Introduce the following coordinate transformations:

[0182] (twenty one)

[0183] In equation (21), This is a virtual control law; to avoid differential explosion and singularity problems, a first-order sliding mode differentiator is used to obtain the derivative of the virtual control law, i.e.:

[0184] (twenty two)

[0185] In equation (22), For the input of the differentiator, , and The state of the differentiator; , The parameters of the designed differentiator; if the initial error , If it is bounded, then the differentiator can output with any precision. Estimation error It is bounded and has a positive constant. Make Established.

[0186] Step 4: Based on the extended state observer in Step 2 and the preset performance function in Step 3, calculate the virtual error, and design the virtual control law and adaptive law by combining the adaptive method;

[0187] First, based on backstepping and adaptive techniques, the virtual control law is designed as follows:

[0188] (twenty three)

[0189] In equation (23), .

[0190] The actual control law is designed as follows:

[0191] (twenty four)

[0192] The adaptive law is designed as follows:

[0193] (25)

[0194] In equation (25), and For the design parameters, and These are positive design parameters.

[0195] Step 5: Design a preset performance controller based on Lyapunov stability theory;

[0196] Choose the following Lyapunov functions for the entire system:

[0197] (26)

[0198] In equation (26), the Lyapunov function for the estimation error of the extended state observer is... Lyapunov function for controller error Lyapunov function for neural network weight error .

[0199] Stability analysis is performed based on Lyapunov's stability theory, as follows:

[0200] (1) Select the Lyapunov function as:

[0201] (27)

[0202] right Taking the derivative, we further obtain:

[0203] (28)

[0204] And:

[0205] (29)

[0206] (30)

[0207] Combining the virtual control law and the adaptive law, we get:

[0208] (31)

[0209] (2) The Lyapunov function is selected as:

[0210] (32)

[0211] right Taking the derivative and combining it with the above equation, we get:

[0212] (33)

[0213] And:

[0214] (34)

[0215] Substituting the actual control law and the adaptive law, we get:

[0216] (35)

[0217] In equation (35), , .

[0218] By selecting appropriate parameters, the desired result can be achieved. , , , .

[0219] because:

[0220] (36)

[0221] It can be obtained It is exponentially convergent, that is... Therefore, the closed-loop system signal , , , ,in It is bounded. Proof complete.

[0222] The beneficial effects of the present invention are verified by the following specific implementation examples.

[0223] Implementation Case:

[0224] The pre-defined performance control method based on the extended state observer designed in this invention is applied to... Figure 2 The piezoelectric micro-positioning platform system shown was used to conduct trajectory tracking experiments.

[0225] First, the relevant program was written and built in the computer using MATLAB / Simulink, with the sampling frequency set to 10kHz. The program was run in real time using Desktop Real-time External mode. Then, the data acquisition card converted the digital signal output by the computer into an analog signal via D / A conversion and input it into the integrated positioning controller. The drive amplification module in the integrated positioning controller amplified the signal and input it into the piezoelectric micro-positioning platform to drive the platform to generate displacement. The piezoelectric micro-positioning platform has a built-in displacement sensor that can measure the output displacement of the platform and transmit the output displacement information to the displacement sensing module in the integrated positioning controller. The displacement signal output by the integrated positioning controller is converted into an analog signal by the data acquisition card via A / D conversion, converting the analog signal into a digital signal that the computer can recognize and inputting it into the computer, thereby realizing real-time data feedback and processing.

[0226] To verify the effectiveness of this invention, the following reference trajectory was selected: frequency conversion signal. ; Single-frequency 100Hz signal .

[0227] The controller parameters for the design are selected as follows:

[0228] Extended State Observer Parameters: , , , ; Parameters of the preset performance function , , , , Differentiator parameters , The virtual control law, adaptive law, and controller parameters are selected as follows: , , ,

[0229] Experimental results are as follows Figures 4-7 As shown. From Figure 4 It can be seen that the preset performance control method proposed in this invention can effectively compensate for the hysteresis related to the compensation rate, enabling the platform's actual output to accurately follow the frequency conversion signal. From Figure 5 As can be observed from the tracking error curve, the tracking error increases with the increase of the desired trajectory frequency due to the significant impact of hysteresis nonlinearity on the platform's performance at high frequencies. Despite this, the expected control performance is still achieved. Combined with the preset performance function constraint curve in the figure, it can be seen that in the transient phase, thanks to the design of the preset performance function, the overshoot performance of the system's tracking control is further improved, and the rapid convergence of the system is ensured. Figure 6 and Figure 7 The tracking curve and error curve can be seen when tracking a 100Hz single-frequency signal. Based on... Figures 6 to 7 Based on the data, the control method described above can accurately track the desired signal. In summary, the control method of the present invention can improve the overshoot performance of the system.

Claims

1. A pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer, characterized in that, The steps of this method are as follows: Step 1: Describe the piezoelectric micro-positioning platform system as a second-order system with input hysteresis and unknown disturbance. For the hysteresis nonlinearity of the piezoelectric actuator, decompose the hysteresis nonlinearity using the PI model, decomposing it into a linear term and a hysteresis nonlinear term with respect to the control signal. Step 2: Define the generalized perturbation of the system, which includes the unknown parts of the hysteresis nonlinear term, external perturbation, and input gain. Treat it as the extended state of the original system. Construct an extended state observer to estimate this state and the state of the original system. For the unknown nonlinear function in the system, it is approximated by a radial basis function neural network. Step 3: Construct a preset performance function to improve the system's overshoot performance while ensuring the system's steady-state performance; Step 4: Based on the extended state observer in Step 2 and the preset performance function in Step 3, calculate the virtual error, and design the virtual control law and adaptive law by combining the adaptive method; Step 5: Combining Lyapunov stability theory, a preset performance controller is obtained to control the piezoelectric micro-positioning platform, so that the tracking error of the closed-loop system is kept within the preset error range.

2. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer according to claim 1, characterized in that, The specific steps of step 1 are as follows: First, the piezoelectric micro-positioning platform system is described as a second-order system with input hysteresis and unknown disturbances, and the state-space equation model is established as follows: （1） In equation (1), and It is an unmeasurable system state. For state vectors, It is a control input. It is a non-zero control gain coefficient. It is a measurable system output. and Represents an unknown smooth continuous function. This is a bounded interference; For the hysteresis nonlinearity of piezoelectric actuators, the PI model is used to decompose the hysteresis nonlinearity: （2） In equation (2), Indicates the model input, Let be the density function. It is an unknown positive constant determined by the density function. The threshold representing hysteresis. It is a hysteresis operator; The hysteresis model, i.e., formula (2), can be rewritten as: （3） In equation (3), Substituting formula (3) into formula (1), we get: （4） In equation (4), It is worth noting that the control method used does not require constructing an accurate hysteresis model; it only requires decomposing the hysteresis into linear terms and hysteresis nonlinear terms related to the control signal. The decomposition process is as follows: （5） In equation (5), The prior part obtained from the actual system, and ; The following unknown positive constants exist. and , making and ; Therefore, the system, i.e., formula (4), can be rewritten as: （6）。 3. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer according to claim 2, characterized in that, The specific steps of step 2 are as follows: First, define The system's generalized perturbation includes the unknown components of the hysteresis nonlinearity, external perturbation, and input gain; then... Considered a new state of the system, soon The extended state is then used to estimate the state of the original system. Based on this, an extended state observer is designed to estimate the state and the state of the original system. The system, i.e., formula (6), can be rewritten as: （7） In equation (7), let An unknown nonlinear function is approximated by a radial basis function neural network as an unknown continuous function. Right now: （8） In equation (8), It is the input vector of the neural network. For the weights of the neural network, These are the basis function vectors of the neural network. It is the approximation error of the neural network; Based on formula (8), the following extended state observer is constructed: （9） In equation (9), For gain, for The estimated value; Observation error is defined as The observation error dynamics are as follows: （10） In equation (10), , ; Rewrite equation (10) as: （11） In equation (11), , , , , , ; can be achieved by designing a reasonable gain Make Established, among which and All are positive definite matrices; Consider the following Lyapunov function for estimating the error of the extended state observer. : The conclusion is The derivative with respect to time: （12） There are unknown constants make According to Young's inequality, we have: （13） In equation (13), , , ; There are known constants ( This makes the following inequality always true: ,then: （14） In equation (14), It is an identity matrix.

4. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer according to claim 3, characterized in that, The specific steps of step 3 are as follows: To achieve the preset performance control of the radial basis function neural network, the tracking error is first defined as: （15） In equation (15), Indicates tracking error. Indicates the desired signal; The preset performance for tracking error is described as follows: （16） In equation (16), and For design parameters, the performance function is chosen to be exponential: （17） In equation (17), It is a natural constant. , and It is a positive number, and ; ,satisfy ; To satisfy the condition, select... ,in If is a natural constant, then: （18） In equation (18), To account for the error after transformation, we further have: （19） Therefore: （20） In equation (20), the definition is... ; Introduce the following coordinate transformations: （21） In equation (21), This is a virtual control law; to avoid differential explosion and singularity problems, a first-order sliding mode differentiator is used to obtain the derivative of the virtual control law, i.e.: （22） In equation (22), For the input of the differentiator, , and The state of the differentiator; , The parameters of the designed differentiator; if the initial error , If it is bounded, then the differentiator can output with any precision. estimation error It is bounded and has a positive constant. Make Established.

5. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer according to claim 4, characterized in that, The specific steps of step 4 are as follows: First, based on backstepping and adaptive techniques, the virtual control law is designed as follows: （23） In equation (23), ; The actual control law is designed as follows: （24） The adaptive law is designed as follows: （25） In equation (25), and For the design parameters, and These are positive design parameters.

6. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer as described in claim 5, characterized in that, The specific steps of step 5 are as follows: Choose the following Lyapunov functions for the entire system: （26） In equation (26), the Lyapunov function for the estimation error of the extended state observer is... Lyapunov function for controller error Lyapunov function for neural network weight error ; Stability analysis is performed based on Lyapunov's stability theory, as follows: (1) Select the Lyapunov function as: （27） right Taking the derivative, we further obtain: （28） And: （29） （30） Combining the virtual control law and the adaptive law, we get: （31） (2) The Lyapunov function is selected as: （32） right Taking the derivative and combining it with the above equation, we get: （33） And: （34） Substituting the actual control law and the adaptive law, we get: （35） In equation (35), , ; By selecting appropriate parameters, the desired result can be achieved. , , , .

7. The pre-set performance control method for a piezoelectric micro-positioning platform based on an extended state observer according to claim 6, characterized in that, The selection principles for relevant controller parameters are as follows: (1) Extended state observer parameters: , , , ; (2) Parameters of the preset performance function: , , , , ; (3) Differentiator parameters: , ; (4) The virtual control law, adaptive law, and controller parameters are selected as follows: , , , .

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