An improved extended state observer method based on frequency domain perturbation compression

Abstract

The application discloses an improved extended state observer method based on frequency domain disturbance compression, which comprises the following steps: an extended state observer designed for an optoelectronic tracking system is reconstructed from a frequency domain perspective, a frequency domain disturbance compression algorithm is implemented, and a disturbance compression extended state observer is obtained; the disturbance compression extended state observer is converted into a frequency domain form in combination with a linear feedback controller; the stability of the disturbance compression extended state observer and the stability of a closed loop system are analyzed to obtain stability constraint conditions; a parameter setting method of the disturbance compression extended state observer is set; and on the basis of meeting the stability constraint conditions, the influence of the disturbance compression extended state observer on disturbance suppression and noise sensitivity is analyzed. The application effectively reduces the upper limit of the disturbance and its derivative by changing the disturbance characteristics perceived by the observer, significantly improves the estimation accuracy of the state and the disturbance, and improves the anti-interference ability and noise insensitivity of the system.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of signal tracking and processing, and specifically provides an improved extended state observer method based on frequency domain disturbance compression. BACKGROUND

[0002] Disturbances and uncertainties are ubiquitous in control applications and have negative effects on control systems. Disturbance rejection has been a key area of interest for both industry and research. The extended state observer (ESO) conceptualizes disturbances and uncertainties as lumped disturbances, minimizing the reliance on a priori system knowledge, thus serving as a partial model-based observer.

[0003] Under constant disturbance conditions, the estimation error of ESO is guaranteed to converge uniformly asymptotically. However, the limitations of ESO become more and more apparent when facing time-varying disturbances. It has been shown that if the rate of change of the time-varying disturbance is bounded, the estimation error of ESO will follow the uniformly ultimately bounded (UUB) criterion within a finite period. In theory, increasing the observation gain of ESO, or equivalently, increasing the bandwidth of the observer, can potentially minimize the estimation error. Nevertheless, the increase in gain will lead to an enhancement of the differential response from the observer. This can result in a large amount of high-frequency noise being transferred from the observation output to the control signal, thus significantly impairing the control quality. Therefore, ESO has difficulty in resolving the contradiction between the need for high-gain observation accuracy and its sensitivity to noise.

[0004] There are studies Writer A quadratic extended state observer (TESO) method based on disturbance compression is proposed, which actively compresses the unknown disturbance, thus reducing the upper bound of the lumped disturbance and its derivative encountered by the observer. Compared with the traditional ESO, the TESO can significantly reduce the observation error under the same observation gain. Nevertheless, the TESO has so far only analyzed the disturbance compression technique from the perspective of state space, which is lacking in the frequency domain-based perspective and potential enhancement; in addition, the TESO still lacks analysis of the sensitivity to noise. The frequency domain is a fundamental analysis method in control applications and is also a traditional and preferred method for engineering professionals. However, the existing ESO has a relatively complex expression in the frequency domain, and the analysis perspective is limited to the inherent properties of ESO. These limitations hinder the wider adoption of ESO by engineers in industrial applications and also pose challenges to improving ESO from the frequency domain perspective. SUMMARY

[0005] To solve the above technical problems, the application provides an improved extended state observer method (DC-ESO) based on frequency domain disturbance compression, a new perspective is provided to understand the limitations and contradictions of the extended state observer by reconstructing the extended state observer in the frequency domain, the upper bound of the disturbance and its derivative is effectively reduced by changing the disturbance characteristics perceived by the observer, the estimation accuracy of the state and the disturbance is significantly improved, the contradiction between high gain observation and noise sensitivity is effectively alleviated, and the anti-interference ability and noise insensitivity of the system are improved.

[0006] To achieve the above object, the application adopts the following technical scheme:

[0007] An improved extended state observer method based on frequency domain disturbance compression comprises the following steps:

[0008] Step (1): the extended state observer designed for the photoelectric tracking system is reconstructed from the frequency domain, and the extended state observer disturbance estimation transfer function is expressed as:

[0009]

[0010] Among them, represents the lumped disturbance caused by external disturbance and modeling uncertainty, is the observation bandwidth or observation gain, is the coefficient, is the Laplace operator, is the modeling uncertainty, is the external disturbance, is the control input in the Laplace domain.

[0011] Step (2): based on the reconstructed extended state observer frequency domain expression, the frequency domain disturbance compression algorithm is implemented to obtain the disturbance compression extended state observer, the disturbance characteristics perceived by the observer are changed, the upper bound of the disturbance and its derivative is effectively reduced, and the estimation accuracy of the state and the disturbance is significantly improved, wherein the mathematical expression of the disturbance compression extended state observer algorithm is:

[0012]

[0013] Among them, the estimation of the extended state is is a parameter, and u is the driving voltage.

[0014] Step (3): a linear feedback controller is introduced on the basis of the disturbance compression extended state observer , and after the disturbance compression extended state observer is converted into a frequency domain form, the transfer function of the closed-loop system is:

[0015] ​​

[0016] where, , , is the characteristic polynomial, is the input reference trajectory, is the output, is the noise, denotes the plant.

[0017] Step (4): analyze the stability of the perturbed compressed extended state observer and the stability of the closed-loop system, and obtain the stability constraint condition. The stability constraint is:

[0018] If it satisfies:

[0019] and ,

[0020] then it indicates that the perturbed compressed extended state observer and the closed-loop system are stable.

[0021] Step (5): give the parameter tuning method of the perturbed compressed extended state observer , where is a constant coefficient, and the parameters are required to satisfy the stability constraint described in step (4). In addition, it is necessary to fine-tune and verify the control settings applied in the experiment.

[0022] Step (6): on the basis of satisfying the stability constraint condition, analyze the influence of the perturbed compressed extended state observer on disturbance suppression and noise sensitivity. If the stability constraint is satisfied, the perturbed compressed extended state observer satisfies

[0023]

[0024] where, and denote the residual error and the disturbance estimation transfer function of the initial lumped disturbance, respectively.

[0025] Re-defining the extended state observer from the frequency domain provides a new perspective, revealing its inherent limitations and internal contradictions. The initial lumped disturbance is , and through the lumped disturbance compression technology, the residual lumped disturbance is , and the relationship between the two is Therefore, the proposed disturbance compression extended state observer changes the perception of the observer to the disturbance characteristics compared with the traditional extended state observer, effectively reduces the upper bound of the disturbance and its derivative, and the disturbance compression extended state observer can obtain higher observation accuracy under the same observation gain. Through the lumped disturbance compression technology, the trade-off problem between the high gain observation and noise sensitivity of the disturbance compression extended state observer is solved. The step disturbance suppression error of the disturbance compression extended state observer method is , and the suppression error of the ramp disturbance is , wherein is the amplitude constant of the step disturbance and the ramp disturbance. Therefore, the disturbance compression extended state observer method can effectively suppress the step disturbance and the ramp disturbance, thereby significantly improving the accuracy of state and disturbance estimation.

[0026] Step (7): analyze whether steps (4)-(6) all meet the preset condition, if not, re-execute steps (1)-(6), if yes, end the design.

[0027] The beneficial effects of the present application are:

[0028] (1) The present application reconstructs the extended state observer from the frequency domain, provides a new perspective for understanding its limitations, and explains the contradiction between high gain observation and noise sensitivity of the extended state observer in the frequency domain.

[0029] (2) The disturbance compression extended state observer is proposed from the frequency domain, which changes the perceived disturbance characteristics of the observer, effectively reduces the upper bound of the disturbance and its derivative, and significantly improves the estimation accuracy of the state and disturbance.

[0030] (3) The disturbance compression technology is used to explain from the frequency domain how the disturbance compression extended state observer alleviates the contradiction between high gain observation and noise sensitivity, which is helpful for engineers to understand and use.

[0031] (4) The parameter tuning method of the disturbance compression extended state observer is simple and easy to use, which makes it more suitable for practical engineering applications.

[0032] (5) Compared with the extended state observer which can only suppress the step disturbance, the disturbance compression extended state observer method can effectively suppress the step disturbance and the ramp disturbance. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 It is the principle block diagram of the extended state observer in the frequency domain;

[0034] Figure 2 It is the principle block diagram of the disturbance compression extended state observer proposed by the present application;

[0035] Figure 3The principle block diagram of the disturbance compression extended state observer in the frequency domain is shown in the figure;

[0036] Figure 4 The simulation suppression results of different extended state observers under slope disturbance of slope change are shown in the figure, wherein (a) is a variable slope disturbance curve, (b) is a feedback error curve, (c) is a disturbance estimation error curve, and (d) is a driving voltage curve;

[0037] Figure 5 The suppression results of different ESOs under 0.5Hz sinusoidal disturbance through an experimental platform are shown in the figure, wherein (a) is a reference trajectory and system output curve, (b) is a feedback error curve, (c) is a violin plot curve of the feedback error, and (d) is a driving voltage curve. DETAILED DESCRIPTION

[0038] The application will be further described below in combination with the drawings and examples.

[0039] The application discloses an improved extended state observer method based on frequency domain disturbance compression, aiming to solve the problem that a traditional extended state observer inevitably amplifies observation noise when increasing observation gain to reduce estimation error. By reconstructing the extended state observer in the frequency domain, a new perspective is provided to understand its limitations and contradictions. The disturbance compression extended state observer (DC-ESO) effectively reduces the upper bound of disturbance and its derivative by changing the disturbance characteristics perceived by the observer, significantly improves the estimation accuracy of the state and disturbance, effectively alleviates the contradiction between high-gain observation and noise sensitivity, and improves the anti-interference ability and noise insensitivity of the system.

[0040] A second-order photoelectric tracking system is set as follows:

[0041] ,

[0042] wherein, is defined as a lumped disturbance, and are coefficients, is an angular position of the photoelectric tracking system, represents an angular velocity and an angular acceleration of the photoelectric tracking system, is a driving voltage.

[0043] By defining a state , the extended state object of the second-order photoelectric tracking system is represented as follows:

[0044] ,

[0045] wherein:

[0046] is defined .

[0047] Based on the above formula, the standard extended state observer can be expressed as:

[0048] ,

[0049] in, is the state vector Estimates, It is a parameter, and the adjustment rule follows ,in is the identity matrix, is the Laplace operator, It is called observation bandwidth or observation gain.

[0050] By definition , the disturbance estimation result of the extended state observer is obtained as:

[0051]

[0052] in, is the observation bandwidth or observation gain.

[0053] According to the above formula, the frequency domain extended state observer is as follows: Figure 1 As shown, is the output of the feedback controller, To measure noise, is an equivalent filter. Define , where ∆(s) represents the modeling uncertainty. Consider , the disturbance estimation transfer function of the extended state observer is expressed as:

[0054]

[0055] in, represents the lumped disturbance caused by external disturbances and modeling uncertainties, is the observation bandwidth or observation gain, is the coefficient, is the Laplace operator, To model uncertainty, For external interference, is the control input form in the Laplace domain.

[0056] Inspired by the frequency domain expression of ESO, an algorithm based on the concept of frequency domain perturbation compression ESO is proposed. First, the state is defined as Secondly, a disturbance compression channel is proposed in the disturbance compression expansion state observer, such as Figure 2 As shown. On this basis, the disturbance is divided into initial lumped disturbance and residual lumped disturbance, respectively defined as and In this control scheme, yes The characteristics of the residual lumped disturbance are determined by the disturbance compression channel. Based on the above definition, the following is proposed: Figure 2 The disturbance compression expansion state observer is shown in Figure 1. Its mathematical expression is as follows:

[0057] ,

[0058] The expansion state is estimated to be is a parameter and u is the driving voltage.

[0059] Based on the disturbance compression and expansion state observer, a classic linear feedback controller is introduced. Then, the disturbance compression expansion state observer is transformed into the frequency domain form, as Figure 3 As shown, is the reference trajectory.

[0060] according to Figure 3 , the transfer function of the closed-loop system is obtained as:

[0061] ,

[0062] ,

[0063] ,

[0064] Among them, is the characteristic polynomial, is the input reference trajectory, For output, is noise, Indicates the object being controlled.

[0065] Mathematical expressions and feedback controllers for the compression-expansion state observer considering disturbances , the following conditions exist:

[0066] (1)

[0067] (2)

[0068] When (1) is satisfied, the disturbance compression expansion state observer is stable; when (1) and (2) are satisfied at the same time, it indicates that the disturbance compression expansion state observer closed-loop system is stable.

[0069] After stability is met, anti-interference and noise insensitivity are considered.

[0070] Considering the mathematical expression of the disturbance compression and expansion state observer, if condition (1) is satisfied, the disturbance compression and expansion state observer satisfies:

[0071] ,

[0072] ,

[0073] wherein, and represent the residual and the disturbance estimation transfer function of the initial lumped disturbance, respectively. Compared with the expansion state observer, the disturbance compression and expansion state observer has an additional disturbance observation and compensation channel, i.e. . And the disturbance and noise transfer function of the disturbance compression and expansion state observer satisfies:

[0074] ,

[0075] .

[0076] The parameter tuning method of the disturbance compression and expansion state observer is given as , wherein is a constant coefficient, and the parameters are required to satisfy the stability constraint. In addition, it is necessary to fine-tune and verify the control settings applied in the experiment.

[0077] After the disturbance compression, the disturbance suppression effect of the disturbance compression and expansion state observer is , which is a linear superposition of , that is, the disturbance compression channel first suppresses the initial lumped disturbance, which can be represented as , and then the disturbance compression and expansion state observer suppresses the residual lumped disturbance again, which can be represented as , and the relationship between the two is . Therefore, the proposed disturbance compression and expansion state observer changes the perception of the observer to the disturbance characteristics compared with the traditional expansion state observer, effectively reduces the upper bound of the disturbance and its derivative, and can obtain higher observation accuracy under the same observation gain. Although the disturbance compression and expansion state observer enhances its disturbance suppression capability, it does not lead to a significant increase in noise sensitivity, that is, the disturbance compression and expansion state observer shows high noise insensitivity, providing a new means to alleviate the inherent contradiction between high gain observation and noise sensitivity.

[0078] When there are step disturbances and ramp disturbances, the step disturbance suppression error of the disturbance compression and expansion state observer method is , and the suppression error of the ramp disturbance is , wherein are constant. Therefore, the perturbation compression and expansion state observer method can effectively suppress the step disturbance and the ramp disturbance, thereby significantly improving the accuracy of state and disturbance estimation.

[0079] Next, the relevant simulation verification is given, and the suppression test under the slope disturbance of slope change is carried out, and the simulation results are as shown in Figure 4 (a), wherein the number after the method name represents the bandwidth of the observer , and the larger the number is, the higher the bandwidth of the observer is. As can be seen from the results in Figure 4 (b), for ESO 10Hz, ESO 20Hz and DC-ESO 10Hz, the estimation error of the step disturbance is approximately zero. However, for the ramp disturbance with a slope of 0.5° / s, the estimation errors of ESO 20Hz and ESO 10Hz are 0.003° and 0.009°, respectively. In addition, for the ramp disturbance with a slope of 1.5° / s, their estimation errors are 0.01° and 0.026°, respectively. As shown in Figure 4 (c), the estimation output of the ESO 20Hz method shows obvious noise amplification, in which the noise in the driving voltage is very obvious in Figure 4 (d). In contrast, the estimation errors of DC-ESO for the ramp disturbances with slopes of 0.5° / s and 1.5° / s are close to zero. Therefore, the proposed DC-ESO method can effectively estimate the ramp disturbance without obvious noise amplification.

[0080] Next, the relevant experimental verification is given, and the suppression test results of ESO under 0.5Hz sinusoidal disturbance are as shown in Figure 5 (a) and (b). The results show that compared with ESO 10Hz and ESO 20Hz, the feedback error of DC-ESO 10Hz is reduced by 89.77% and 76.19%, respectively. As can be seen from Figure 5 (c), most of the errors of DC-ESO are concentrated between-0.02° and 0.02°, and the error distribution is more concentrated compared with ESO. Figure 5 (d) shows that compared with ESO 10Hz and DC-ESO 10Hz, ESO 20Hz significantly amplifies the driving noise, indicating a serious noise sensitivity problem.

[0081] Based on the above simulation and experimental results, compared with ESO, DC-ESO significantly reduces the disturbance estimation error and feedback error without amplifying the driving noise. This confirms that DC-ESO can alleviate the contradiction between high gain observation and noise sensitivity. In other words, DC-ESO has better noise tolerance. In applications where there is obvious noise, DC-ESO can more effectively play a role in disturbance suppression.

[0082] The above-described specific embodiments further illustrate the objects, technical solutions and advantages of the present application. It should be understood that the above-described specific embodiments are merely for the purpose of illustrating the present application and are not intended to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. An improved extended state observer method based on frequency domain disturbance compression, characterized in that: The steps include: Step (1): Reconstruct the extended state observer designed for the optoelectronic tracking system from the frequency domain perspective; Step (2): Based on the reconstructed frequency domain expression of the extended state observer, implement the frequency domain disturbance compression algorithm to obtain the disturbance compression extended state observer; Step (3): Combined with linear feedback controller Convert the disturbance compression and expansion state observer into frequency domain form; Step (4): Analyze the stability of the disturbance compression expansion state observer and the stability of the closed-loop system to obtain the stability constraints; Step (5): Setting the parameter tuning method of the disturbance compression expansion state observer; Step (6): On the basis of satisfying the stability constraints, analyze the impact of the disturbance compression expansion state observer on disturbance rejection and noise sensitivity; Step (7): Analyze whether steps (4) to (6) all meet the preset conditions. If not, re-execute steps (1) to (6). If they do, end the design. The extended state observer disturbance estimation transfer function obtained by reconstructing from the frequency domain perspective in step (1) is expressed as: ， in, represents the lumped disturbance caused by external disturbances and modeling uncertainties, is the observation gain, is the coefficient, is the Laplace operator, To model uncertainty, For external interference, is the control input form in the Laplace domain; The extended state observer designed for the optoelectronic tracking system is a standard extended state observer, which can be expressed as: ， in, is the state vector Estimates, It is an intermediate parameter, and the adjustment rule follows ,in is the identity matrix, and the intermediate matrices ; The mathematical expression of the frequency domain perturbation compression algorithm in step (2) is: ， in: The expansion state is estimated to be is the intermediate parameter, u is the driving voltage; In step (3), a linear feedback controller is introduced , and after converting the disturbance compression expansion state observer into frequency domain form, the transfer function of the closed-loop system is obtained as follows: in: , , is the characteristic polynomial, is the input reference trajectory, For output, is noise, Indicates the object being controlled.

2. The improved extended state observer method based on frequency domain disturbance compression according to claim 1, characterized in that: The stability constraint in step (4) is that if: and This indicates that the disturbance compression expansion state observer and the closed-loop system are stable.

3. The improved extended state observer method based on frequency domain disturbance compression according to claim 2, characterized in that: The parameter setting method in step (5) is: , ,in is a constant coefficient, and the parameters are required to satisfy the stability constraints.

4. The improved extended state observer method based on frequency domain disturbance compression according to claim 3 is characterized in that: In step (6), if the stability constraint is satisfied, the disturbance compression expansion state observer satisfies: in, and denote the disturbance estimation transfer functions of the residual and initial lumped disturbance, respectively.

5. The improved extended state observer method based on frequency domain disturbance compression according to claim 4 is characterized in that: In step (6), the disturbance suppression effect of the disturbance compression expansion state observer is and The linear superposition of , that is, the disturbance compression channel first suppresses the initial lumped disturbance, which can be expressed as , and then the disturbance compression expansion state observer suppresses the residual lumped disturbance again, which is expressed as , the relationship between the initial lumped disturbance and the residual lumped disturbance is .

6. The improved extended state observer method based on frequency domain disturbance compression according to claim 4, characterized in that: In step (6), when there are step disturbances and ramp disturbances, the step disturbance suppression error of the disturbance compression expansion state observer method is , and the suppression error of the ramp disturbance is ,in is the amplitude constant of step disturbance and ramp disturbance.

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