ADRC method for attenuating periodic disturbance of time-delay system

Abstract

This invention discloses an ADRC method for attenuating periodic disturbances in time-delay systems, belonging to the field of automatic control technology. The method includes: 1. Obtaining the controlled process model and disturbance frequency, and establishing an extended state model containing the periodic disturbance; 2. Constructing a reduced-order extended state observer and tuning the gain using the bandwidth parameter method; 3. Introducing a pure time-delay module into the observer's output channel to construct an ADRC structure; 4. Revealing the mechanism by which periodic disturbances in time-delay systems cannot be attenuated through two-degree-of-freedom equivalent transformation analysis; 5. Based on the frequency domain characteristic equation, analytically determining the delay of the time-delay module to achieve effective attenuation of periodic disturbances at specific frequencies. This invention solves the problem of attenuating periodic disturbances in time-delay systems without significantly increasing structural complexity. Compared to traditional schemes that require high-gain or complex prediction structures, this invention only requires adjusting the pure time delay of the observer's output channel, and the parameter tuning process is clear and intuitive, significantly lowering the threshold for engineering applications.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology, specifically relating to an ADRC method for damping periodic disturbances in a time-delay system. Background Technology

[0002] Active Disturbance Rejection Control (ADRC), with its mechanism of real-time estimation and compensation for disturbances and uncertainties, has the potential to significantly enhance the anti-interference performance of control systems. For time-delay systems, an additional time-delay compensation stage is usually required to synchronize the input signal of the core part of ADRC, namely the input signal of the Extended State Observer (ESO), thereby ensuring control performance. The current time-delay compensation mode is usually input time-delay compensation (abbreviated as time-delay ADRC, see Zhao S, Gao Z. Modified active disturbance rejection control for time-delay systems[J].ISA transactions,2014,53(4):882-888), that is, adding an artificial time-delay module to the control signal input channel of the observer, ensuring a simple structure while achieving accurate estimation of the disturbance signal.

[0003] Currently, the effectiveness of conventional ADRC is mostly limited to time-invariant (or low-frequency slowly varying) disturbances, such as constant disturbances. For time-varying periodic disturbances, conventional ADRC requires further improvement. Current research mainly focuses on two aspects. The first approach is to increase the bandwidth or order of the observer (see Godbole AA, Kolhe JP, Talole S E. Performance analysis of generalized extended state observer in tackling sinusoidal disturbances[J]. IEEE Transactions on Control Systems Technology,2012, 21(6): 2212-2223), which can effectively reduce the amplitude of output fluctuations caused by periodic disturbances compared to conventional observers. The essence of this approach is that it increases the decay rate of the disturbance by introducing a larger estimation action or more disturbance derivative information. Its advantage is that it can directly extend the order of the original observer structure. However, this treatment only reduces the peak value of the output fluctuation caused by periodic disturbances and does not completely attenuate the disturbance. In addition, due to the introduction of excessively fast observer actions, the control system becomes abnormally sensitive to measurement noise. The second approach is based on the internal model control principle, which embeds the model of the periodic disturbance into the observer stage. When the disturbance model is accurate, it can completely attenuate such disturbances. Due to the rapid development of adaptive technology, the method for frequency identification of periodic disturbance signals has gradually matured. Therefore, ADRC research currently adopts this approach to handle periodic disturbances (see Stanković MR, Rapaić MR, Manojlović SM, et al. Optimised active disturbance rejectionmotion control with resonant extended state observer[J]. International Journal of Control, 2019, 92(8): 1815-1826.).

[0004] The aforementioned design method of embedding the disturbance model into the observer can effectively attenuate periodic disturbances experienced by a time-delay-free controlled process. However, for time-delayed controlled processes, the time delay causes a mismatch between the timing of the compensation signal of the control action and the disturbance signal. This characteristic has no effect on time-invariant constant disturbances because, in this case, the ADRC control action only needs to generate the required compensation signal magnitude. Even if the time delay causes a mismatch between the compensation signal and the disturbance timing, it only results in a delay in the time required to attenuate the disturbance. However, for time-varying periodic disturbances, not only is it required that the control action generate the required compensation signal magnitude, but the timing of the compensation signal must also correspond to the disturbance timing to effectively address the time-varying nature of the disturbance. Therefore, the aforementioned design research of embedding the disturbance model into the observer is still difficult to directly apply to time-delayed controlled objects. Summary of the Invention

[0005] To address the aforementioned problems in the existing technology, the present invention aims to provide an ADRC method for attenuating periodic disturbances in a time-delay system, comprising the following steps: Step 1: Obtain the first-order inertial time-delay transfer function model of the controlled process and the frequency of the required decay periodic disturbance, and use this to establish the extended state model of the controlled object, including the periodic disturbance.

[0006] Step 2: Based on the extended state model obtained in Step 1, establish a reduced-order ESO and use the bandwidth parameter method to uniformly adjust the observer gain.

[0007] Step 3: Based on the ESO established in Step 2, determine the overall control action law, and introduce a pure time delay module into the observer output channel to construct the ADRC structure for the decaying time delay system periodic disturbance.

[0008] Step 4: Perform a two-degree-of-freedom equivalent transformation on the proposed ADRC system and analyze why ADRC cannot attenuate the periodic disturbance of the time-delay system without introducing a pure time delay module in the observer output channel.

[0009] Step 5: Based on the analysis in Step 4, the specific delay of the observer output channel at a specific disturbance frequency is determined analytically, thereby achieving attenuation of the periodic disturbance at that frequency in the time-delay system.

[0010] In step 1 above, the following first-order inertial time-delay transfer function model is established for the controlled time-delay process: ; In the formula These are the control variable (the input to the controlled process) and the controlled variable (the output of the controlled process), respectively. For steady-state gain; It is a time constant and ; It is a time-delay element. It is a time delay. This is the Laplace operator. For periodic perturbations that need to be processed, the system output can be directly identified. To obtain the perturbation frequency, and represent it as... Given the introduction of model information, the extended state model of the controlled process considering the disturbance state is first established as follows: ; In the formula These are the parameters after the transfer function model transformation. Indicates system output, This represents lumped disturbance. If only constant disturbances are considered, then generally we have: .

[0011] The periodic disturbance under consideration is described as follows: ,in For amplitude, For frequency, For the purposes of this invention, the periodic disturbance information used for control design is only the frequency. This time-varying perturbation is obtained by identifying the output signal using adaptive techniques. It can be viewed as the characteristic response of a resonator with a resonant frequency of , and the corresponding characteristic equation for the resonator is: If we represent it as a model In the expanded state, we have The corresponding state-space equation is then expressed as: .

[0012] In step 2 above, based on the considered extended state model Establish the following linear ESO: ; in The gain of the observer to be designed. Due to the system output... It can be directly measured without estimation. The above ESO can be further simplified into a reduced-order form to reduce the computational burden on the controller during the operation. First, the following transformation is performed: .

[0013] According to the object model , has lumped disturbance Therefore, the above equation becomes: .

[0014] To avoid micro-component signals output by the system The measurement transforms the above state-space equations as follows: .

[0015] Further introduce auxiliary variables Then, the following reduced-order ESO can be obtained: ; in: .

[0016] Based on this observer, the system state can be determined through a simple linear change. The characteristic equation of this reduced-order observer is: .

[0017] Based on the bandwidth parameter method, the roots (i.e., poles) of this characteristic equation are placed in the same location, i.e. Therefore, the optimization rules for the observer bandwidth can be derived as follows: ; Therefore, the observer gain can be determined simply by adjusting the observer bandwidth. Generally, the inner-loop observer bandwidth is 10 times that of the outer-loop. The structure of the above observer is as follows: Figure 1 As shown.

[0018] In step 3 above, the external loop feedback function of ADRC is defined as follows: ;

[0019] In the formula As a set value variable, This refers to the proportional feedback gain parameter. To address the decay period perturbation, this invention proposes introducing a pure time delay into the observer's output channel. The estimation of the lumped disturbance becomes accordingly (within the current domain) Combining the effects of the external and internal loops, the overall control action of this ADRC is: .

[0020] The overall structure of the ADRC is as follows: Figure 2 As shown.

[0021] In step 4 above, the proposed ESO is first... Perform the following Laplace transform: .

[0022] Solve the above system of linear equations and consider The expression for the lumped perturbation in the complex field can be derived as follows: .

[0023] In conjunction with the overall control function of ADRC, ADRC is performed as follows: Figure 3 The equivalent two-degree-of-freedom transformation is shown. First, without considering the introduction of the observer output channel delay module (i.e., ... ), combined and The equivalent two-degree-of-freedom controller shown below is obtained: ; in .

[0024] Based on this equivalent controller, the following transfer function of the disturbance to the output channel can be derived: .

[0025] For the transfer function of this channel, its response characteristics depend on the numerator term. Expanding it in the frequency domain yields: ; Based on this expression, it can be concluded that in the low-frequency range there are And thus This indicates that constant disturbances in the low-frequency range can be attenuated. Therefore, as mentioned above, the existence of time delay of the controlled object does not affect the attenuation of constant disturbances by ADRC.

[0026] For frequency For periodic disturbances, the first term of the above frequency domain characteristic equation is 0, but the second term cannot be guaranteed to be 0, therefore the condition cannot be satisfied. The conditions prevent the periodic disturbance at that frequency from decaying. Furthermore, it can be seen that if the controlled process has no time delay, i.e. Therefore, the second term of the above frequency domain equation naturally remains 0, and the existence of only the first term is sufficient to guarantee the operation of the frequency domain equation. The attenuation of periodic disturbances. Therefore, based on the frequency domain characteristic equation analysis of this transfer function, it can be clearly explained at the theoretical level why ADRC can attenuate periodic disturbances in systems without time delay, but cannot attenuate periodic disturbances in systems with time delay.

[0027] In step 5 above, after introducing the observer channel delay module proposed in this invention, there is... Combined and The two-degree-of-freedom equivalent controller of the control system becomes: .

[0028] The transfer function from the perturbation to the system output channel then becomes: .

[0029] Furthermore, the frequency domain characteristic equation of the corresponding denominator term can be derived as follows: .

[0030] It can be seen that, compared to the formula After introducing a time delay module into the observer output channel, the formula... The second parameter now has adjustable degrees of freedom. The introduced delay module will be adjusted in the following ways: ; in For integers greater than 0, their values ​​should be guaranteed to be within the range of 0. Under the condition of minimizing it as much as possible. Under this condition, the frequency response equation The second term has a frequency of When the time is 0, it guarantees that Thus, the conditions are met at this theoretical analysis level. This achieves the attenuation of periodic disturbances.

[0031] Therefore, for the ADRC proposed in this invention, after obtaining the transfer function of the controlled object and the frequency of the periodic disturbance, the attenuation of the periodic disturbance can be achieved according to the above design process. The structure proposed in this invention is fixed; compared to conventional ADRC, it only requires adding an adjustable delay module to the observer output channel, without significantly increasing complexity. Furthermore, very few parameters need to be adjusted; besides the general proportional gain and observer bandwidth, only the following parameters need to be adjusted according to the formula... Re-adjust the time delay introduced into the observer output channel That is, while the proportional gain and observer bandwidth can be tuned according to the current mainstream and mature tuning rules.

[0032] By employing the above-described technology, the beneficial effects of the present invention compared to the prior art are as follows: (1) The method of the present invention establishes a reduced-order ESO driven by model information and disturbance characteristics, which has a low-level mechanism for decaying periodic disturbances and does not require the design of a high-gain module to enhance the control effect; (2) The method of this invention has developed a theoretical analysis framework for disturbance attenuation, which explains in detail from the principle why the current ADRC can attenuate constant disturbances but cannot attenuate periodic disturbances; (3) The method of the present invention proposes a pure time delay compensation module for the observer output channel and formulates corresponding tuning rules, which can realize the attenuation of periodic disturbances at a specific frequency of the time delay system. Attached Figure Description

[0033] Figure 1 This is a schematic diagram of the ESO structure driven by model information and perturbation features in this invention; Figure 2 This is a schematic diagram of the ADRC structure for the periodic perturbation of the decaying time-delay system in this invention; Figure 3 This is a schematic diagram of the equivalent two-degree-of-freedom structure of ADRC in this invention; Figure 4 This is a schematic diagram of the amplitude-frequency characteristic curve of the perturbation to the output channel transfer function in this invention; Figure 5 This is a simulation diagram of the control system setpoint tracking and disturbance suppression in this invention. Detailed Implementation

[0034] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0035] Conversely, this invention encompasses any substitutions, modifications, equivalent methods, and solutions made within the spirit and scope of the invention as defined in the claims. Furthermore, to provide a better understanding of the invention, certain specific details are described in detail below. However, those skilled in the art will fully understand the invention even without these detailed descriptions.

[0036] This invention provides an ADRC method for attenuating periodic disturbances in a time-delay system, the specific implementation of which is as follows: (1) Consider the following transfer function model of the controlled dynamic process of a wet electrostatic precipitator (hereinafter referred to as wet electrostatic precipitator) in a coal-fired power plant, where the input is the high-frequency secondary current (unit: A) and the output is the dust concentration at the outlet of the precipitator (unit: mg / Nm³). 3 ): .

[0037] Therefore, the model parameters can be obtained as follows: The corresponding model parameters after conversion to state space are: For this embodiment, consider the following periodic perturbation: ; The perturbation frequency is Based on the model and disturbance information of the controlled process, the following extended state model is established according to step 1 of the technical solution: .

[0038] (2) Based on the specific extended state model in this embodiment, the following reduced-order ESO is established according to step 2 of the technical solution:

[0039] in: ; .

[0040] The system expansion state can be achieved through The characteristic equation of the reduced-order ESO is obtained as follows: ; According to the description of step 2 in the technical solution, the roots of the above equations are placed in the same position, that is... The tuning rule for the observer bandwidth can be obtained as follows: .

[0041] Therefore, it is only necessary to determine the observer bandwidth. The observer gain can then be determined. In this embodiment, the ideal closed-loop time constant of the system is set to the time delay of the controlled object, then the closed-loop bandwidth (expressed as...) is... Then it is its reciprocal, that is The bandwidth of the inner loop observer is typically 5 to 10 times that of the outer loop closed-loop bandwidth. This embodiment adopts... Therefore, the observer bandwidth in this embodiment is... It should be noted that the selection of the observer bandwidth is not mandatory and can be tuned using current conventional methods. Generally speaking, the larger the observer bandwidth, the stronger the ability to suppress disturbances, but the more sensitive it is to noise.

[0042] (3) Based on the controlled object model in this embodiment, the external loop control action of the control system is as follows: ; For proportional gain This embodiment adopts a tuning method based on low-frequency approximation (see "An ADRC Design Method for Thermal Time-Delay Processes with Time-Delay Robustness" by Zhejiang Zheneng Technology Research Institute Co., Ltd.: 202511064489.3 [P]. 2025-11-07) as an illustration. The calculation results are shown below: .

[0043] In this embodiment, when applying the ADRC decay period perturbation, the parameters are... There are no mandatory requirements for setting the parameters, and other design methods can be used to adjust the control parameters, as long as they are within a stable and reasonable range.

[0044] Based on the explanation of step 3 in the technical solution, the overall control function of ADRC in this embodiment is as follows: .

[0045] (4) For this embodiment, the ADRC lumped disturbance in the complex domain can be expressed as: .

[0046] According to step 4 of the technical solution, the module that does not consider the delay of the observer output channel in this embodiment (i.e.) The equivalent two-degree-of-freedom controller: ; in .

[0047] Based on this equivalent controller, the following transfer function of the disturbance to the output channel can be derived: .

[0048] For the transfer function of this channel, its response characteristics depend on the terms in the denominator. Expanding it in the frequency domain yields: .

[0049] It can be seen that there is in the low frequency range And thus This indicates that constant disturbances in the low-frequency range can be attenuated. Therefore, as mentioned above, the existence of time delay of the controlled object does not affect the attenuation of constant disturbances by ADRC.

[0050] In this embodiment, the frequency is... For periodic disturbances, the first term of the above frequency domain characteristic equation is 0, but the second term cannot be guaranteed to be 0, therefore the condition cannot be satisfied. The condition is that the periodic disturbance at that frequency cannot be attenuated. Furthermore, it can be seen that if the controlled process has no time delay, the second term of the frequency domain equation containing the trigonometric function polynomial will naturally remain 0, while the first term can guarantee that for frequencies of... The attenuation of periodic disturbances. Therefore, the frequency domain characteristic equation of this embodiment can theoretically explain why ADRC can effectively attenuate periodic disturbances in systems without time delay, but cannot attenuate periodic disturbances in systems with time delay.

[0051] (5) Introduce an observer channel delay module, which has The two-degree-of-freedom equivalent controller in this embodiment is: ; The transfer function from the disturbance to the system output channel is: .

[0052] Furthermore, the frequency domain characteristic equation of the corresponding denominator term can be derived as follows: .

[0053] According to the description of step 5 in the technical solution, the time delay module introduced into the observer output channel will be adjusted in the following manner: ; Therefore, the second term in the frequency response equation contains the frequency within the brackets. At that time This ensures Thus, the conditions are met at this theoretical level. This achieves the attenuation of periodic disturbances.

[0054] (6) Based on the specific parameters of this embodiment, plot the delay module of the system with and without the observer output channel. At that time, the amplitude-frequency curve of the system perturbation to the output channel transfer function is as follows: Figure 4 As shown, for low-frequency disturbances, both amplitudes approach 0, indicating that disturbances within this range can be attenuated. However, for the frequency in this embodiment... The periodic perturbation can only be addressed by introducing a delay module into the observer output channel. The magnitude of the transfer function only approaches zero under certain conditions, indicating that the delay module of the observer output channel proposed in this invention is only effective when this is introduced. Only in this way can the attenuation of the periodic disturbance be achieved.

[0055] (7) Based on the specific parameters of this embodiment, the system is simulated with / without the observer output channel delay module, such as... Figure 5As shown. Since both systems use the same external loop, their tracking responses are identical, and both can suppress constant disturbances when they are introduced into the system. However, for periodic disturbances, without a time delay module in the observer output channel, the system's control action does not correspond to the timing of the disturbance signal, i.e., there is a phase deviation. Applying the method proposed in this invention, after adding a time delay module, the system's control action corresponds to the timing of the disturbance signal; that is, the peak of the control action corresponds to the trough of the disturbance, achieving attenuation of periodic disturbances in the time-delay system.

[0056] As can be seen from the above embodiments and simulation results, the method proposed in this invention reveals the mechanism of ADRC decay time delay system from a mechanistic perspective. It does not require setting up a complex prediction structure, nor does it use high gain parameters to enhance the control action. It only uses the current mainstream tuning method, so it has great potential to solve the challenges of periodic disturbances in industrial time delay systems.

[0057] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An ADRC method for decaying periodic disturbances in a time-delay system, characterized in that, Includes the following steps: Step 1: Obtain the first-order inertial time-delay transfer function model of the controlled process and the frequency of the required decaying periodic disturbance, and establish an extended state model of the controlled object including the periodic disturbance based on the obtained function model and frequency; Step 2: Based on the extended state model obtained in Step 1, establish a reduced-order ESO and use the bandwidth parameter method to uniformly adjust the observer gain; Step 3: Based on the ESO established in Step 2, determine the overall control action law, and introduce a pure time delay module into the observer output channel to construct the ADRC structure for the decaying time delay system periodic disturbance; Step 4: Perform a two-degree-of-freedom equivalent transformation on the constructed ADRC system and analyze why ADRC cannot attenuate the periodic disturbance of the time-delay system without introducing a pure time delay module in the observer output channel; Step 5: Based on the analysis in Step 4, the specific delay of the observer output channel at a specific disturbance frequency is determined analytically, thereby achieving attenuation of the periodic disturbance at that frequency in the time-delay system.

2. The ADRC method for attenuating periodic disturbances in a time-delay system according to claim 1, characterized in that, The specific process of step 1 is as follows: 1.1) Based on the first-order inertial time-delay process model, an extended state model including lumped disturbances is established; 1.2) The periodic disturbance is described as the resonator output, the resonant frequency of which is obtained based on adaptive identification of the system output; 1.3) Introduce the periodic disturbance as an extended state into the extended state model to construct a state-space model for control design. The model includes the state variables of the controlled output of the system, the periodic disturbance, and the derivative of the periodic disturbance.

3. The method according to claim 2, characterized in that, The characteristic equation corresponding to the resonator is: ,in This indicates a periodic disturbance. The frequency of the periodic perturbation.

4. The ADRC method for attenuating periodic disturbances in a time-delay system according to claim 1, characterized in that, In step 2, the process of establishing the reduced-order ESO is as follows: 2.1) Based on the system model that includes the extended state, construct an initial linear extended state observer with the same order as the system model; 2.2) The estimation error term of the lumped disturbance is substituted based on the object model relationship, and the system output differential term in the equation is eliminated by introducing a state transformation based on auxiliary variables; 2.3) Based on the state transformation, reconstruct the state space equations to obtain the reduced-order form of the linear extended state observer; 2.4) Using the bandwidth parameter method, the poles of the characteristic equation of the reduced-order observer are configured to the same expected value to uniformly determine its observer gain.

5. The ADRC method for damped time-delay system periodic disturbances according to claim 4, characterized in that, In the bandwidth parameter method, the poles of the characteristic equation of the reduced-order ESO are configured as follows: ， The observer gain satisfies: β 1=3ω o , β 2=3ω o 2 −ω p 2 , β 3=ω o 3 - β 1ω p 2 ,in ω o This represents the observer bandwidth.

6. The ADRC method for damped time-delay system periodic disturbances according to claim 1, characterized in that, In step 3, the specific process of generating the control action is as follows: 3.1) Calculate the external loop feedback term of the ADRC system. : ； in, r As a set value variable, y This is the actual output of the system. k p For proportional feedback gain, a 0 For object model parameters; 3.2) In the output channel of the extended state observer, an adjustable pure time delay module is introduced into the estimated lumped disturbance signal to obtain the time delay corrected lumped disturbance estimate. 3.3) Subtract the external loop feedback action from the time-delay corrected lumped disturbance estimate, divide the result by the model parameters, and obtain the final control action and output it to the controlled object.

7. The ADRC method for attenuating periodic disturbances in a time-delay system according to claim 6, characterized in that, The proportional feedback gain k p According to the formula Adjustment is performed, among which ω c The desired closed-loop bandwidth of the system. L is the time delay constant.

8. The ADRC method for damped time-delay system periodic disturbances according to claim 1, characterized in that, Step 4, the process of analyzing the periodic disturbance attenuation characteristics based on an equivalent two-degree-of-freedom controller, includes: 4.1) Establish the disturbance estimation expression of the extended state observer in the complex domain without introducing a time delay module, and solve for the equivalent two-degree-of-freedom controller structure and the corresponding transfer function of the disturbance to the system output channel in the case without a time delay module; 4.2) Based on the characteristic equation of the numerator of the transfer function in the frequency domain, it is confirmed that the system has the ability to attenuate low-frequency constant disturbances; 4.3) Based on the same frequency domain characteristic equation, when the controlled object has a time delay, for a periodic disturbance at a specific frequency, the second term of the characteristic equation is not zero, and the periodic disturbance cannot be attenuated; when the controlled object has no time delay, the second term of the characteristic equation is zero, and the periodic disturbance can be attenuated.

9. The ADRC method for attenuating periodic disturbances in a time-delay system according to claim 1, characterized in that, In step 5, the adjustment process of the delay module includes: 5.1) Obtain the time delay constant of the controlled object and the frequency of the periodic disturbance to be suppressed; 5.2) Based on the time delay constant and the frequency of the periodic disturbance, calculate the time delay of the time delay module according to the preset time delay adjustment formula, so that after the time delay is introduced, the amplitude gain of the disturbance transfer function at the specific frequency is zero, thereby achieving effective attenuation of the periodic disturbance.

10. The ADRC method for attenuating periodic disturbances in a time-delay system according to claim 9, characterized in that, The time delay adjustment formula is: ； in, L p For the delay amount of the delay module, L The time delay constant of the controlled object, ω p The frequency of the periodic disturbance to be suppressed. k It is an integer greater than 0.

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