ADRC control system based on intelligent variable bandwidth extended state observer and advanced internal model control system

Abstract

The present invention discloses an ADRC control system based on an intelligent variable bandwidth extended state observer and an advanced internal model control system. Compared with other internal model control methods that require the establishment of an accurate internal model, the internal model of the control system is based on the standard dynamics given by ADRC. G cl (s) acquisition, providing a new and practical solution for the design of internal model control for complex nonlinear systems. It simplifies the control system modeling process and can be extended to other control methods that require the establishment of precise models. The present invention overcomes the observation phase lag and noise sensitivity problems of existing observers, and the design process is simple, facilitating widespread application in engineering. This invention replaces the internal model in traditional internal model control with the standard dynamics given by ADRC, providing a new and practical solution for the design of internal model controllers for complex nonlinear systems.

Description

Technical Field

[0001] The present invention belongs to the field of advanced control, and in particular relates to a design method for an ADRC control system based on an intelligent variable bandwidth extended state observer and a design method for an advanced internal model control system, as well as the designed ADRC control system and advanced internal model control system based on the intelligent variable bandwidth extended state observer. Background Art

[0002] With the advancement of science and technology, the application of automatic control technology in production and daily life is becoming increasingly widespread. The use of automatic control technology can significantly reduce the demand for manpower. Currently, automatic control technology in my country is widely used in many fields, but compared with some developed countries, there is still considerable room for development. Therefore, considering the current status and future development trends of automatic control technology, it is urgent to continue researching new control methods to further improve control performance and reduce costs.

[0003] Existing control methods can be divided into three categories based on their degree of model reliance: 1. Methods that rely on minimal model information, such as PID control; 2. Methods that rely on partial model information, such as active disturbance rejection control (ADRC), fuzzy control, and sliding mode control; and 3. Methods that rely on more model information, such as feedback linearization control, internal model control (IMC), and model reference adaptive control. While PID control is simple and easy to use, it passively eliminates errors, limiting its performance in the presence of uncertainties and / or disturbances. While fuzzy control and sliding mode control are applicable to complex nonlinear systems and can achieve desirable performance, fuzzy control's rule formulation is subjective, and sliding mode control suffers from control signal jitter. Control methods that rely on more model information, while achieving good control results, are highly dependent on model information, limiting their application.

[0004] In the presence of uncertainty and interference, the active disturbance rejection controller ADRC based on the extended state observer (ESO) is not only easy to implement, but also can achieve satisfactory performance. In addition, if an accurate internal model can be obtained, IMC can also achieve the desired performance. ADRC can be explained under the framework of two-degree-of-freedom IMC. The combination of ADRC and IMC has continued to attract attention in recent years and has been applied in many fields. Invention Patent No. 2019112412522 provides a variable bandwidth active disturbance rejection control method, which uses the estimated error of the system output to dynamically adjust the observer bandwidth (abbreviated as: SESO-based PD controller). There is still a phase lag problem, which needs to be further optimized. And the existing technology also has the following problems:

[0005] 1. For IMC, even when combined with ADRC, there is still the problem of high cost of obtaining accurate internal models;

[0006] 2. Although the ESO, the core of ADRC, can effectively estimate constant disturbances, it cannot accurately estimate time-varying disturbances and suffers from observation phase lag. Performance deteriorates further in the presence of high-frequency total interference.

[0007] 3. Under a fixed observer bandwidth, the ESO cannot accurately reconstruct the total disturbance whose frequency is not within its bandwidth. Summary of the Invention

[0008] In view of the shortcomings of the existing technology, the present invention provides a design method for an ADRC control system based on an intelligent variable bandwidth extended state observer and a design method for an advanced internal model control system, as well as the designed ADRC control system and advanced internal model control system based on an intelligent variable bandwidth extended state observer.

[0009] The specific technical solutions of the present invention are as follows:

[0010] A design method for an ADRC control system based on an intelligent variable bandwidth extended state observer includes the following steps:

[0011] S1: Based on the mathematical model of the n-th order controlled object, the internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance f and used as the state variable of the system. The mathematical model of the controlled object is expressed as an extended state equation, a fixed-bandwidth linear extended state observer is constructed, and the observer bandwidth ω is set according to the bandwidth parameterization method. o and the gain of the extended state observer L = [β1,β2,…β n ,β n+1 ] T ;

[0012] S2: Output variables z1, z2, ... z based on the extended state observer n , design state feedback control law u0=k1(r-z1)-k2z2-…k n z n , and set the controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the controller gain k i (i=1,2,…n); r is the set value;

[0013] S3: Estimation of the total disturbance z by the integrated extended state observer n+1 And the state feedback control law u0, the design control law u=(u0-z n+1) / b0; b0 is the adjustable controller gain;

[0014] S4: Let ω' o =ω o +η|z n+1 -f|, construct an intelligent variable bandwidth extended state observer and estimate the deviation |z with the total disturbance n+1 -f| is the adjustment basis to dynamically adjust the bandwidth of the observer. The bandwidth of the intelligent variable bandwidth expanded state observer is expressed as follows:

[0015]

[0016] Where f = y (n) -b0u,y (n) is the nth-order differential of the controlled object output, which can be extracted by the tracking differentiator; b0 is the adjustable controller gain, u is the controlled object input; η is the scaling factor, which is used to adjust the observer bandwidth range, η>0;

[0017] The adaptive variable gain of the bandwidth parameterization configuration is as follows:

[0018]

[0019] S5: The intelligent variable bandwidth linear expansion observer is designed as follows:

[0020]

[0021] S6: Based on the output of the intelligent variable bandwidth extended state observer, the control law is adjusted accordingly:

[0022]

[0023] Among them, z 1SBW ,z 2SBW ,…z iSBW ,…z (n+1)SBW is the output of the intelligent variable bandwidth extended state observer;

[0024] S7: Use the ADRC control system based on the intelligent variable bandwidth extended state observer to complete the control of the n-order controlled object and adjust the controller parameter ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0025] A design method for an advanced internal model control system based on an intelligent variable bandwidth extended state observer includes the following steps:

[0026] S8: Based on the above method, the ADRC control system based on the intelligent variable bandwidth extended state observer is designed. When the extended state observer is reasonably designed, that is, zn+1 ≈f, the controlled object can be simplified into an n-order integral system y (n) =u0-z n+1 +f≈u0; the closed-loop transfer function of the nth-order ADRC without zero point can be obtained, that is, the standard closed-loop ADRC system dynamics G cl (s):

[0027]

[0028] Among them, k i (i=1,2,…n) is the controller gain;

[0029] S9: Based on the standard closed-loop ADRC system dynamics G cl (s), and design the internal model controller G based on the following formula c (s), internal model G m (s) and filter f c (s):

[0030]

[0031] f c (s)=1 / (1+αs) n

[0032] Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics. Compared with the standard closed-loop ADRC system dynamics G cl (s), where

[0033] S10: The internal model controller based on the intelligent variable bandwidth extended state observer completes the control of the controlled object and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0034] The design method of ADRC control system based on intelligent variable bandwidth extended state observer includes the following steps for the second-order ADRC controller:

[0035] S1: Establish a mathematical model of the second-order controlled object and obtain the model parameters of the controlled object through model identification. For a second-order controlled object, the mathematical model is:

[0036]

[0037] Among them, u is the control input of the system; y is the output of the system; is the first-order differential of the system output y; is the uncertainty within the system; ω is the external disturbance; b is the gain of the controlled object;

[0038] The internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance and as the state variables of the system. The mathematical model of the controlled object is expressed as an expanded state equation.

[0039]

[0040] Where x1=y(t), The total disturbance is expressed as b0 is the adjustable controller gain;

[0041] S2: Based on the extended state equation, a fixed-bandwidth extended state observer is constructed:

[0042]

[0043] And set the observer bandwidth ω according to the bandwidth parameterization method o ,ω o >0, and then design the observer gain L=[β1,β2,β3];

[0044] Where z1 is the estimate of the system output y; z2 is the estimate of the differential of the system output y; z3 is the estimate of the total disturbance;

[0045] S3: Based on the output variables z1, z2 of the extended state observer, design the PD controller u0 = k1(r-z1)-k2z2, and set the PD controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the PD controller gains k1, k2;

[0046] S4: Combine the total disturbance estimate z3 of the extended state observer and the output u0 of the PD controller to generate the control output u according to the control law;

[0047]

[0048] in, r is the set value;

[0049] S5: Let ω' o =ω o +η|z3-f|, construct an intelligent variable bandwidth extended state observer, and dynamically adjust the bandwidth of the observer based on the total disturbance estimation deviation |z3-f|. The bandwidth of the intelligent variable bandwidth extended state observer is expressed as follows:

[0050]

[0051] in, is the second-order differential of the controlled object output, which can be extracted by the tracking differentiator; b0 is the adjustable controller gain, u is the controlled object input; η is the scaling factor, which is used to adjust the observer bandwidth range, η>0;

[0052] The adaptive variable gain of the bandwidth parameterized configuration is as follows:

[0053]

[0054] S6: Design an intelligent variable bandwidth extended state observer and adjust the control law accordingly;

[0055]

[0056] Among them, z 1SBW ,z 2SBW ,z 3SBW is the output of the intelligent variable bandwidth expanded state observer; η>0 is the scaling factor, which is used to adjust the variation range of the observer bandwidth.

[0057] Accordingly, the control law is adjusted as follows:

[0058]

[0059] S7, the ADRC control system based on the intelligent variable bandwidth extended state observer completes the control of the second-order controlled object and adjusts the controller parameter ω o ,ω c , b0 and η, so that the system works normally until the desired control effect is achieved.

[0060] The design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer includes the following steps:

[0061] S8: Using the above-mentioned ADRC control system based on the intelligent variable bandwidth extended state observer, for the second-order ADRC controller, ignoring the estimation error of the extended state observer for the total disturbance, the second-order controlled object can be equivalent to the integral series type, that is, the standard closed-loop ADRC system dynamics G cl (s):

[0062]

[0063] S9: Based on the standard closed-loop ADRC system dynamics G cl (s) Design the internal model controller G based on the following formula c (s), internal model

[0064] G m (s) and filter f c (s):

[0065]

[0066] f c (s)=1 / (1+αs) 2

[0067] Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics. Compared with the standard closed-loop ADRC system dynamics G cl The deviation between (s),

[0068] S10: The internal model controller based on the intelligent variable bandwidth extended state observer completes the control of the controlled object and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0069] Furthermore, in step S10, when adjusting the controller parameters, at a given adjustment time t s , when the error tolerance is ±5%, the initial value of the filter parameter α can be determined by the following formula:

[0070] t s =3.5α

[0071] Furthermore, in step S10, when adjusting the controller parameters, under the given overshoot σ%, the PD controller bandwidth ω c The initial value of can be determined by the following formula:

[0072]

[0073] The n-th order ADRC control system is obtained by adopting the above-mentioned design method of the ADRC control system based on the intelligent variable bandwidth extended state observer.

[0074] By adopting the above-mentioned design method of the advanced internal model control system based on the intelligent variable bandwidth extended state observer, an n-order advanced internal model control system is obtained.

[0075] The second-order ADRC control system is obtained by adopting the above-mentioned design method of the ADRC control system based on the intelligent variable bandwidth extended state observer.

[0076] The second-order advanced internal model control system is obtained by adopting the above-mentioned design method of the advanced internal model control system based on the intelligent variable bandwidth extended state observer.

[0077] Compared with the prior art, the present invention has the following beneficial effects:

[0078] 1. This paper proposes an ADRC control system and design method based on an intelligent variable-bandwidth extended state observer (Linear Extended State Observer with Scalable Bandwidth, LESO-SBW). This control system dynamically adjusts the observer bandwidth according to changes in the total disturbance estimation error. Compared with an observer based on a variable-bandwidth system output estimation error, the phase of the system output estimation error lags behind the phase of the total disturbance estimation error because the system output estimation error is caused by the total disturbance estimation error. This method can obtain a more timely and phase-leading total disturbance estimate, achieve rapid response to changes, and timely adjust the observer bandwidth while satisfying the observer gain, effectively suppressing noise in the steady state.

[0079] 2. The present invention overcomes the problems of observation phase lag and system sensitivity to noise in existing observers, and the design process is simple, which is easy to promote and apply in engineering.

[0080] 3. Compared with other methods based on interference estimation and compensation, the present invention has good estimation accuracy for time-varying disturbances such as ramps and sinusoids in addition to constant disturbances; it has strong resistance to sudden load disturbances; this method is suitable for designing active disturbance rejection controllers of various orders and has good versatility; and the adjustable control parameters have clear physical meanings and are easy to adjust.

[0081] 4. The advanced internal model control system based on intelligent variable bandwidth expanded state observer (LESO-SBW-AIMC) and its design method proposed in this invention are different from other internal model control systems that require the establishment of accurate internal models. The internal model of this invention is composed of the standard closed-loop ADRC system dynamics G cl (s) instead, there is no need to establish an accurate mathematical model of the controlled object, and the model accuracy is guaranteed, which improves the overall performance of the internal model control. At the same time, the intelligent variable bandwidth expanded state observer is used to make the actual closed-loop ADRC system dynamics Compared with the standard closed-loop ADRC system dynamics G cl The deviations between the two parameters (s) are increasingly smaller, further increasing the accuracy of the internal model required for IMC. This invention provides a new and practical solution for the design of internal model control for complex nonlinear systems. It simplifies the control system modeling process and can be extended to other control applications requiring precise models.

[0082] Other features and advantages of the embodiments of the present invention will be described in detail in the subsequent detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. By referring to the drawings, the features and advantages of the present invention will be more clearly understood. The drawings constitute a part of the specification and are used together with the following specific embodiments to explain the embodiments of the present invention, but do not constitute a limitation on the embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative work. Among them:

[0084] Figure 1 It is a second-order ADRC control system based on LESO-SBW;

[0085] Figure 2 It is an existing internal model control structure;

[0086] Figure 3 It is an internal model control system based on LESO-SBW-AIMC;

[0087] Figure 4 For different observer bandwidth ω o Next 1SBW With x1 and z 1SBW Frequency domain response curve of and u;

[0088] Figure 4 (a) is z 1SBW and the frequency domain response of x1; (b) is z 1SBW Frequency domain response curve of and u;

[0089] Figure 5 for Figure 3 The equivalent structure of LESO-SBW-AIMC;

[0090] Figure 6 is the time domain response curve of the system;

[0091] Figure 7 is the system frequency domain response curve;

[0092] Figure 8 It is an n-order ADRC control system based on LESO-SBW;

[0093] Figure 9 Based on Figure 8 LESO-SBW's advanced internal model control system;

[0094] Figure 10 Comparison of trajectory tracking between LADRC, PD control based on scaled bandwidth LESO with system output estimation error, and PD control based on LESO-SBW.

[0095] Figure 11Comparison diagram of control signals among LADRC, SESO-based PD control, and LESO-SBW-based PD control;

[0096] Figure 12 Comparison of the total perturbation estimates of ESO, SESO, and LESO-SBW;

[0097] Figure 13 This is the simulation result diagram output by LESO-SBW;

[0098] Figure 14 ω o =36 and ω o =12 when the trajectory tracking comparison diagram of PD control based on LESO-SBW;

[0099] Figure 15 Comparison chart of trajectory tracking of IMC, LADRC, IMC based on LADRC, and LESO-SBW-AIMC;

[0100] Figure 16 Comparison of control errors among IMC, LADRC, LADRC-based IMC, and LESO-SBW-AIMC.

[0101] Figure 17 Comparison chart of control signals of IMC, LADRC, LADRC-based IMC and LESO-SBW-AIMC. DETAILED DESCRIPTION

[0102] Example 1:

[0103] This example provides an ADRC control system based on an intelligent variable bandwidth extended state observer. The design method includes the following steps:

[0104] S1: Based on the mathematical model of the n-th order controlled object, the internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance f and used as the state variable of the system. The mathematical model of the controlled object is expressed as an extended state equation, a fixed-bandwidth linear extended state observer is constructed, and the observer bandwidth ω is set according to the bandwidth parameterization method. o and the gain of the extended state observer L = [β1,β2,…β n ,β n+1 ] T ;

[0105] S2: Output variables z1, z2, ... z based on the extended state observer n , design state feedback control law u0=k1(r-z1)-k2z2-…k n z n , and set the controller bandwidth ω according to the bandwidth parameterization method c,ω c >0, and then design the controller gain k i (i=1,2,…n); r is the set value;

[0106] S3: Estimation of the total disturbance z by the integrated extended state observer n+1 And the state feedback control law u0, the design control law u=(u0-z n+1 ) / b0; b0 is the adjustable controller gain;

[0107] S4: Let ω' o =ω o +η|z n+1 -f|, construct an intelligent variable bandwidth extended state observer and estimate the deviation |z with the total disturbance n+1 -f| is the adjustment basis to dynamically adjust the bandwidth of the observer. The bandwidth of the intelligent variable bandwidth expanded state observer is expressed as follows:

[0108] ω' o =ω o +η|z n+1 -f|=ω o +η|z n+1 -(y (n) -b0u)|

[0109] Where f = y (n) -b0u,y (n) is the nth-order differential of the controlled object output, which can be extracted by the tracking differentiator; b0 is the adjustable controller gain, u is the controlled object input; η is the scaling factor, which is used to adjust the observer bandwidth range, η>0;

[0110] The adaptive variable gain of the bandwidth parameterization configuration is as follows:

[0111]

[0112] S5: The intelligent variable bandwidth linear expansion observer is designed as follows:

[0113]

[0114] S6: Based on the output of the intelligent variable bandwidth extended state observer, the control law is adjusted accordingly:

[0115]

[0116] Among them, z 1SBW ,z 2SBW ,…z iSBW ,…z (n+1)SBW is the output of the intelligent variable bandwidth extended state observer;

[0117] S7: Use the ADRC control system based on the intelligent variable bandwidth extended state observer to complete the control of the n-order controlled object and adjust the controller parameter ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0118] Based on the above design steps S1-S7, an ADRC control system based on an intelligent variable bandwidth extended state observer is obtained.

[0119] Example 2:

[0120] Based on the first embodiment, this embodiment provides an advanced internal model control system based on an intelligent variable bandwidth extended state observer. The design method includes the following steps:

[0121] S8: Based on the ADRC control system based on the intelligent variable bandwidth extended state observer designed in Example 1, when the extended state observer is reasonably designed, that is, z n+1 ≈f, the controlled object can be simplified into an n-order integral system y (n) =u0-z n+1 +f≈u0; the closed-loop transfer function of the nth-order ADRC without zero point can be obtained, that is, the standard closed-loop ADRC system dynamics G cl (s):

[0122]

[0123] Among them, k i (i=1,2,…n) is the controller gain; all controller poles are configured to -ω c At (n) +k n s (n-1) +…+k2s+k1=(s+ω c ) n The characteristic polynomial of ;ω c is the controller bandwidth;

[0124] S9: Based on the standard closed-loop ADRC system dynamics G cl (s), and design the internal model controller G based on the following formula c (s), internal model G m (s) and filter f c (s):

[0125]

[0126] f c (s)=1 / (1+αs) n

[0127] Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics. Compared with the standard closed-loop ADRC system dynamics G cl (s), where

[0128] S10: The internal model controller based on the intelligent variable bandwidth extended state observer completes the control of the controlled object and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0129] Based on the above design steps S1-S10, an advanced internal model control system based on an intelligent variable bandwidth extended state observer is obtained.

[0130] Example 3:

[0131] like Figure 1 As shown in FIG, the design method of the second-order ADRC control system based on the intelligent variable bandwidth extended state observer includes the following steps:

[0132] S1: Establish a mathematical model of the second-order controlled object and obtain the model parameters of the controlled object through model identification. For a second-order controlled object, the mathematical model is:

[0133]

[0134] Among them, u is the control input of the system; y is the output of the system; is the first-order differential of the system output y; is the uncertainty within the system; ω is the external disturbance; b is the gain of the controlled object;

[0135] The internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance and as the state variables of the system. The mathematical model of the controlled object is expressed as an expanded state equation.

[0136]

[0137] Where x1=y(t), The total disturbance is expressed as b0 is the adjustable controller gain;

[0138] S2: Based on the extended state equation, a fixed-bandwidth extended state observer is constructed:

[0139]

[0140] And set the observer bandwidth ω according to the bandwidth parameterization method o ,ω o>0, and then design the observer gain L=[β1,β2,β3];

[0141] Where z1 is the estimate of the system output y; z2 is the estimate of the differential of the system output y; z3 is the estimate of the total disturbance;

[0142] S3: Based on the output variables z1, z2 of the extended state observer, design the PD controller u0 = k1(r-z1)-k2z2, and set the PD controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the PD controller gains k1, k2;

[0143] S4: Combine the total disturbance estimate z3 of the extended state observer and the output u0 of the PD controller to generate the control output u according to the control law;

[0144]

[0145] in, r is the set value;

[0146] S5: Let ω' o =ω o +η|z3-f|, construct an intelligent variable bandwidth extended state observer, and dynamically adjust the bandwidth of the observer based on the total disturbance estimation deviation |z3-f|. The bandwidth of the intelligent variable bandwidth extended state observer is expressed as follows:

[0147]

[0148] in, is the second-order differential of the controlled object output, which can be extracted by the tracking differentiator; b0 is the adjustable controller gain, u is the controlled object input; η is the scaling factor, which is used to adjust the observer bandwidth range, η>0;

[0149] The adaptive variable gain of the bandwidth parameterized configuration is as follows:

[0150]

[0151] S6: Design an intelligent variable bandwidth extended state observer and adjust the control law accordingly;

[0152]

[0153] Among them, z 1SBW ,z 2SBW ,z 3SBW is the output of the intelligent variable bandwidth expanded state observer; η>0 is the scaling factor, which is used to adjust the variation range of the observer bandwidth.

[0154] Accordingly, the control law is adjusted as follows:

[0155]

[0156] S7, the ADRC control system based on the intelligent variable bandwidth extended state observer completes the control of the second-order controlled object and adjusts the controller parameter ω o ,ω c , b0 and η, so that the system works normally until the desired control effect is achieved.

[0157] Based on the above design steps S1-S7, an ADRC control system based on an intelligent variable bandwidth extended state observer is obtained.

[0158] Example 4:

[0159] Based on the third embodiment, this example further designs an advanced internal model control system based on an intelligent variable bandwidth extended state observer (LESO-SBW-AIMC). The existing internal model control structure is as follows: Figure 2 As shown, the advanced internal model control system of this application is as follows Figure 3 As shown, the design steps for the second-order system are as follows:

[0160] S8: Ignoring the estimation error of the extended state observer for the total disturbance, the second-order controlled object can be equivalent to the integral series type, that is, the standard closed-loop ADRC system dynamics G cl (s):

[0161]

[0162] S9: Based on the standard closed-loop ADRC system dynamics G cl (s), and design the internal model controller G based on the following formula c (s), internal model G m (s) and filter f c (s):

[0163]

[0164] f c (s)=1 / (1+αs) 2

[0165] Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics G p (s) and the standard closed-loop ADRC system dynamics G cl The deviation between (s),

[0166] S10: The internal model controller based on the intelligent variable bandwidth extended state observer completes the control of the controlled object and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0167] Based on the above design steps S1-S10, an advanced internal model control system based on an intelligent variable bandwidth extended state observer is obtained.

[0168] Analysis of the design principle of the advanced internal model control system in this example:

[0169] like Figure 2 、 Figure 3 It can be seen that the present invention uses ADRC to control the controlled object and can obtain the closed-loop dynamics based on ADRC. That is, the red dotted box indicates that To represent the entire closed-loop system, the actual model in the internal model control is That is

[0170] When the present invention designs the inner mold structure, G cl (s) as the internal model G m (s), combined with the filter f c (s)=1 / (1+αs) 2 The internal model controller G is designed c (s), i.e.

[0171] for and G cl (s) In simple terms, The actual closed-loop ADRC dynamics are obtained by controlling the controlled object through ADRC. However, due to the uncertainty and disturbance in the system, the ESO has estimation deviations and cannot fully conform to the standard closed-loop ADRC system dynamics. Actual closed-loop ADRC system dynamics Compared with the standard closed-loop ADRC system dynamics G cl There will be a deviation Δ between (s), that is,

[0172] Internal model control relies on an accurate model of the controlled object. The G c (s) and G m (s) are based on G cl (s), and the actual model of the controlled object is The closer to G cl (s), the more accurate the actual model of the controlled object is.

[0173] The intelligent variable bandwidth ESO designed by the present invention can reduce the deviation Δ. As Δ decreases, Closer to G cl (s), i.e. The higher the model accuracy of the internal model control.

[0174] This application uses G when designing the inner mold structure. cl (s) as the internal model G m (s), ensuring model accuracy without the need to establish a precise mathematical model of the controlled object, thereby improving the overall performance of internal model control. This shows that the present application can make the internal model required for IMC more accurate without having to establish a precise mathematical model of the controlled object, providing a new and practical solution for the design of internal model control for complex nonlinear systems.

[0175] Embodiment 5:

[0176] This example is based on the advanced internal model control system based on the intelligent variable bandwidth extended state observer designed in the fourth embodiment. For the second-order system, the controller parameters include α, ω o ,ω c , b0 and eta.

[0177] Parameters that need to be adjusted in LESO-SBW-AIMC, LESO-SBW observer bandwidth ω o , scaling factor η, IMC filter parameter α, controller bandwidth ω c and the controller gain b0.

[0178] For the second-order ADRC control system based on intelligent variable bandwidth extended state observer, when adjusting the controller parameters, the observer bandwidth ω o and the scaling factor η are tuned online according to the following formulas:

[0179] (1) LESO-SBW observer bandwidth and scaling factor η: From the expression of LESO-SBW, we can deduce z 1SBW The transfer function is:

[0180]

[0181] From the above formula, we can get z 1SBW With x1 and z 1SBW The transfer function between and u:

[0182]

[0183] in: Essentially, the actual observer bandwidth ω' o is the initial value of the observer bandwidth ω o and Composition. Different observer bandwidth ω o Next 1SBW With x1 and z 1SBW The frequency domain response of u is as follows Figure 4 shown.

[0184] Figure 4 (a) is z 1SBW and x1 frequency domain response curve; (b) is z 1SBW Frequency domain response curve of and u. Figure 4 (a) shows that as ω o As , the frequency domain characteristic curve shifts to the right, indicating that the phase lag of the LESO-SBW output observation curve is reduced, the estimation error converges faster, and it is beneficial to improve the dynamic performance of the system. Figure 4 (b) shows that the disturbance gain increases with ω o The increase of ω decreases with the increase of ω, indicating that the anti-interference performance of the system has improved. LESO-SBW has a high ability to suppress the interference of the control input u, and the noise introduced by the input signal x1 is the main factor affecting the performance of LESO-SBW. In addition, choosing too large ω o It will cause high-frequency noise pollution, which will cause system oscillation or even instability. Therefore, the actual selection should be adjusted from small to large until the noise impact can no longer meet the system requirements.

[0185] For the scaling factor η: It can be seen that η is used to adjust the range of variation of the observer bandwidth and can be selected by considering the rate of change of the observed system behavior and the accuracy requirements.

[0186] (2) Controller bandwidth ω c And the filter parameter α:

[0187] Figure 5 is the equivalent structure of LESO-SBW-AIMC, in which the feedback controller is:

[0188] C(s)=G c (s)[1-G c (s)G m (s)] -1

[0189] The open-loop transfer function is:

[0190]

[0191] Therefore, the given adjustment time is: t s =3.5α.

[0192] The overshoot is: In this case, it can be seen from the above open-loop transfer function that the damping ratio and natural angular frequency are: 1 / ω c and ω c / α.

[0193] Figure 6 is the time domain response curve, Figure 7 is the frequency domain response curve; Figure 6 (a) and Figure 7 As can be seen from (a), when ω c When fixed, adjust the time t s As α decreases, it decreases. Similarly, from Figure 6 (b) and Figure 7 As can be seen from (b), when α is fixed, the overshoot σ% increases with ω. c In other words, reducing α and ω c It can improve the closed-loop performance of the control system. Figure 6 (c) and Figure 7 As can be seen in (c), reducing α and ω at the same time c More noise will be introduced in the high frequency band. Therefore, the PD controller bandwidth ω c The initial value of and the initial value of the filter parameter α can be adjusted according to the required time t s and overshoot σ%, and then further select appropriate α and ω according to actual needs. c .

[0194] (3) Adjustable controller gain b0

[0195] b0 is an estimate of the system gain b, i.e., b0 ≈ b. This value affects the system's interference rejection and response speed. Calculating b0 is necessary for controller design. If b0 is unknown, an estimate of b can be used, and the term (b - b0)u is then included in the total disturbance f. Typically, the system's identified value b is selected to ensure basic dynamic performance and stability, and further adjustments can be made based on actual needs.

[0196] Among these parameters to be tuned, b0 can use the system identification value b as the initial value; the controller bandwidth ω c The filter parameter α can be calculated based on the preset adjustment time and overshoot indicators. Therefore, during the controller parameter adjustment process, only the observer bandwidth ω needs to be focused on. o and online tuning of the scaling factor η.

[0197] Example 6:

[0198] like Figure 8 As shown in the figure, this example provides a design method for an ADRC control system based on an intelligent variable bandwidth extended state observer. The design process is as follows:

[0199] S1: Consider a nonlinear time-varying system with a single input and a single output:

[0200] y (n) (t) = F(y (n-1) (t),…,y(t))+bu+d (1)

[0201] Among them, u is the control input of the system; y is the output of the system; F(y (n-1) (t),…,y(t)) is the internal dynamics of the system; let b0 be the estimate of the control gain b, and record f(y (n-1) (t),…,y(t),d,u)=F(y (n-1) (t),…,y(t))+d+(b-b0)u is the total disturbance of the system, and we define , write equation (1) into the state equation of the controlled object (2):

[0202]

[0203] Where X=[x1…x n+1 ] T is the system state variable,

[0204] S2: Based on the extended state equation of the controlled object, the fixed-bandwidth linear extended state observer is designed as:

[0205]

[0206] Where L = [β1,β2,…β n ,β n+1 ] T is the observer gain, and the observer gain is selected so that s (n+1) +β1s (n-1) +…β i s (i-1) +…+β n Satisfy the Hurwitz stability condition; place all observer poles at -ω o At , we get the characteristic polynomial of formula (3); at the observer bandwidth ω o When the values ​​are reasonable, z1→x1,z2→x2,…,z i →x i ,…,z n-1 →x n-1 ,z n →x n ,z n+1 →f(y (n-1) (t),…,y(t),d,u);

[0207] s (n+1)+β1s (n) +…+β n s+β n+1 =(s+ω o ) n+1 (3)

[0208] S3: Estimate z by compensating for the total disturbance n+1 , the control law is designed as follows:

[0209]

[0210] S4: Substitute equation (4) into equation (2), when the total disturbance estimate z n+1 Better estimate of the total disturbance Sometimes, there are

[0211] y (n) =u0-z n+1 +f≈u0 (5)

[0212] S5: The extended state observer actively compensates and the control law compensates the total disturbance in real time. System (2) can be dynamically linearized into an integrator series form. The state feedback control law u0 is designed as follows:

[0213] u0=k1(r-z1)-k2z2-…k n z n (6)

[0214] Among them, k i (i=1,2,…n) is the controller gain, and the controller poles are all configured to -ω c At (n) +k n s (n-1) +…+k2s+k1=(s+ω c ) n Characteristic polynomial of ; r is the set value;

[0215] S6: If we ignore the total disturbance estimate z n+1 For total disturbance The estimated deviation can be simplified into an n-order integral system y (n) =u0-z n+1 +f≈u0; the closed-loop transfer function of the nth-order ADRC without zero point can be obtained, that is, the standard closed-loop ADRC system dynamics G cl (s):

[0216]

[0217] S7: To further reduce the total disturbance estimation deviation, the total disturbance estimation deviation |z n+1-f| is used as the basis for adjusting the bandwidth of the extended state observer to dynamically adjust the bandwidth of the extended state observer, that is, ω' o =ω o +η|z n+1 -f|=ω o +η|z n+1 -(y (n) -b0u)|, the bandwidth parameterization is as follows:

[0218]

[0219] The intelligent variable bandwidth linear expansion observer is designed as follows:

[0220]

[0221] Among them, z 1SBW ,z 2SBW ,…z iSBW ,…z (n+1)SBW is the output of the intelligent variable bandwidth extended state observer; η>0 is the scaling factor, which is used to adjust the range of the observer bandwidth;

[0222] S7: Accordingly, the control law is adjusted to:

[0223]

[0224] S8: Based on the internal model controller of the intelligent variable bandwidth extended state observer, the controlled object is controlled and the controller parameters α, ω are adjusted. o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0225] Based on the above design steps S1-S8, an ADRC control system based on an intelligent variable bandwidth extended state observer is obtained.

[0226] Based on the ADRC control system of intelligent variable bandwidth extended state observer, the advanced internal model control system of LESO-SBW is further obtained by the design method of the present invention. Figure 9 shown.

[0227] Embodiment seven:

[0228] As a specific embodiment, this example takes the control system of the van der Pol oscillator as an example to design an advanced internal model control system based on an intelligent variable bandwidth extended state observer. The specific steps are as follows:

[0229] S1: Establish a mathematical model of the second-order controlled object and obtain the model parameters of the controlled object through model identification.

[0230] The control problem of the van der Pol oscillator can be described by the following differential equation:

[0231]

[0232] in, are position, velocity and acceleration respectively; u is the control input signal.

[0233] The controlled object model has given parameters, [η1, η2, η3] = [1.5, 1.5, 1.0] T , initial state

[0234] S2: Consider the internal uncertainty and external disturbance in the van der Pol oscillator control system as a total disturbance and rewrite its differential equation into a form that includes the total disturbance. The total disturbance of the system is defined as: Where b0 is the adjustable controller gain. Let x1=y(t), The expanded state equation of the van der Pol oscillator is expressed as follows:

[0235]

[0236] S3: Based on the extended state equation, the extended state observer ESO is constructed and the observer bandwidth ω is set according to the bandwidth parameterization method o , we can get:

[0237]

[0238] Where z1 is the estimate of the system output y; z2 is the estimate of the differential of the system output y; z3 is the estimate of the total disturbance;

[0239] S4: Based on the ESO output variables z1, z2 and the set value r, the PD controller is designed as: u0 = k1(r-z1)-k2z2. According to the bandwidth parameterization idea, we can get:

[0240] S5: Combining the ESO’s estimate of the total disturbance z3 and the PD controller’s output u0, the control output u is generated according to the control law as follows:

[0241]

[0242] S6: Ignoring the estimation error of the total disturbance f by the extended state observer, the van der Pol oscillator control system can be equivalent to an integral series type, and the closed-loop transfer function of the n-th order ADRC without zero points can be obtained:

[0243]

[0244] S7: In order to further reduce the total disturbance estimation deviation, the total disturbance estimation deviation |z3-f| is used as the basis for adjusting the observer bandwidth to dynamically adjust the observer bandwidth, that is, The bandwidth parameterization configuration of the intelligent variable bandwidth extended state observer is as follows:

[0245]

[0246] S8: The constructed intelligent variable bandwidth expansion observer is as follows:

[0247]

[0248] Among them, z 1SBW ,z 2SBW ,z 3SBW is the output of the intelligent variable bandwidth expanded state observer; η>0 is the scaling factor, which is used to adjust the variation range of the observer bandwidth.

[0249] Correspondingly, the control law is:

[0250]

[0251] The above design of S1-S8 obtains the active disturbance rejection controller ADRC (PD+LESO-SBW) based on intelligent variable bandwidth extended state observer.

[0252] S9: Based on the standard closed-loop ADRC system dynamics G cl (s) to design the internal model G of IMC m (s), based on the internal model and selecting the appropriate filter f c (s) Design the internal model controller G c (s), we can get:

[0253]

[0254] f c (s)=1 / (1+αs) 2

[0255] Among them, G m (s) is the internal model, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics Compared with the standard closed-loop ADRC system dynamics G cl The deviation between (s),

[0256] S10: The advanced internal model controller based on intelligent LESO-SBW completes the control of the van der Pol oscillator and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

[0257] This results in an advanced internal model control system based on an intelligent variable bandwidth expanded state observer (LESO-SBW-AIMC).

[0258] Test example:

[0259] The following two cases are used to evaluate the performance of PD+LESO-SBW and LESO-SBW-AIMC designed in Example 7.

[0260] Case 1 is used to study trajectory tracking control in the presence of time-varying disturbances to evaluate the performance of the PD+LESO-SBW designed in this application. The comparison controllers include: the existing LADRC controller (LESO-based PD control, referred to as LADRC(PD+LESO)); and the SESO-based PD control (Patent No. 2019112412522 provides a variable-bandwidth auto-disturbance rejection control method that dynamically adjusts the observer bandwidth using the estimated error of the system output, referred to as PD+SESO). For a fair comparison, the three control methods mentioned above are identical except for the different observers.

[0261] Case 2 is used to evaluate the performance of the advanced internal model controller based on LESO-SBW-AIMC. The comparison controllers include: the existing IMC, the existing LADRC controller (same as Case 1), and the LADRC-based IMC (IMC+LADRC).

[0262] For the design of the existing IMC, in this example, the internal model G is first designed based on the differential equation of the controlled object Van der Pol oscillator of the seventh embodiment. m (s), then based on the internal model G m (s), combined with the filter f c (s) Design the internal model controller G c (s), for the internal model G in IMC m (s) and the internal model controller G c (s), which is designed based on an accurate model of the controlled object.

[0263] For LADRC-based IMC, the existing LADRC controller (PD+LESO) in Case 1 is adopted, and the IMC design uses the standard closed-loop ADRC system dynamics G cl (s) to replace the internal model G m (s), combined with G cl (s) and filter f c (s), design the internal model controller G c (s) design. The IMC in this part is different from the IMC introduced above. The internal model here is based on the standard closed-loop ADRC system dynamics Gcl (s) provided.

[0264] The controller parameters selected for the test are shown in Table 1:

[0265] Table 1 Controller parameters

[0266]

[0267] Case 1: Performance test of PD+LESO-SBW

[0268] Let the setting value be 1, that is, r(t) = 1(t), and add a time-varying disturbance from 3s to 5s: d(t) = 10sin(6t) + 10cos(6t). The trajectory tracking, control signal, total disturbance estimation and LESO-SBW output are respectively as follows: Figure 10-13 As shown. Among them, from Figure 10 It can be seen that under time-varying disturbances, all three control methods achieve the specified tracking performance. However, compared with the existing linear active disturbance rejection controller LADRC and the existing SESO-based PD controller, the PD+LESO-SBW controller of the present invention has a shorter recovery time and significantly less fluctuation. The control signal is as follows Figure 11 As shown in Figure 2, the PD+LESO-SBW controller of the present invention responds faster to time-varying disturbances than other methods. The total disturbance is estimated as Figure 12 As shown in Figure 2, the total disturbance estimation of the LESO-SBW of the present invention always outperforms the existing LADRC controller and the existing SESO-based PD controller. Figure 13 (a) and Figure 13 In (b), we can see that the PD+LESO-SBW controller of the present invention has the following characteristics: 1SBW and z 2SBW The system output y and .also, Figure 13 (c) confirms that the PD+LESO-SBW controller has the following advantages: 3SBW The time-varying total disturbance f can be estimated accurately and timely. Therefore, the PD+LESO-SBW controller of the present invention achieves better tracking response.

[0269] In addition, noise sensitivity is another key issue. In the low frequency range, ESO can perform well, but in the high frequency range, ESO is very sensitive to noise. The controller parameters are set to: b0 = 1, ω o =12,ω c =5,η=10. Figure 14 It can be seen that even if ω oEven when reduced to one-third of its original value, it still achieves better performance. In other words, a relatively low initial observer bandwidth can be set, making the system less susceptible to noise. The LESO-SBW-based PD controller of the present invention can better balance noise sensitivity and total disturbance estimation.

[0270] Case 2: Performance test of the advanced internal model controller based on LESO-SBW-AIMC of the present invention.

[0271] In this set of simulation tests, the same reference values ​​as in case (1) are selected. The same time-varying disturbance is added from 5s to 7s. The trajectory tracking, control error, and control signal are respectively as follows: Figure 15-17 As shown. Among them, from Figure 15 As can be seen, adverse factors severely degrade the tracking performance of the existing IMC controller. However, compared to the existing IMC controller, the existing LADRC controller and the existing LADRC-based IMC controller exhibit shorter rise and recovery times and significantly less ripple. With the help of the LESO-SBW-based AIMC controller of the present invention, the closed-loop system performance is further improved. Figure 16 It also proves that the trajectory tracking error fluctuation of the LESO-SBW-AIMC controller of the present invention is minimal. Figure 17 As shown in the figure, the LESO-SBW-AIMC controller of the present invention responds faster to time-varying disturbances than other methods.

[0272] The present invention provides an advanced internal model control system based on an intelligent variable-bandwidth extended state observer, achieving satisfactory control results in complex nonlinear systems with time-varying disturbances. Through ingenious design, the present invention successfully replaces the internal model in the IMC with the standard dynamics given by the ADRC, providing a new and practical solution for the IMC design of complex nonlinear systems. The total disturbance estimation deviation, an important performance indicator reflecting the real-time estimation effect of the observer, is used as the basis for observer bandwidth adjustment, which is more timely and phase-advanced than adjusting the bandwidth using the output estimation deviation. Dynamically adjusting the observer bandwidth according to the different states of the system (transition process or steady-state process) achieves the goals of both rapid response to changes and noise suppression in steady state, achieving a good compromise between tracking speed and noise sensitivity. In terms of tracking performance and disturbance suppression performance, the closed-loop system constructed by the present invention achieves better tracking performance, has stronger anti-interference ability, and can more effectively suppress the impact of external disturbances on system performance, providing a new and practical solution for the IMC design of complex nonlinear systems.

[0273] The above describes in detail the optional implementation methods of the embodiments of the present invention in conjunction with the accompanying drawings, but it should be noted that the present invention is not limited to the above specific details. Within the scope of the technical concept of the present invention, a variety of simple modifications can be made to the technical solution, and these modifications all fall within the scope of protection of the present invention. Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from its spirit and scope. Therefore, if these changes and modifications fall within the scope of the claims of the present invention and their equivalent technologies, they are also intended to be included in the protection of the present invention.

Claims

1. A design method for an ADRC control system based on an intelligent variable bandwidth extended state observer, characterized in that: The following steps are involved: S1: Based on the mathematical model of the n-th order controlled object, the internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance f and used as the state variable of the system. The mathematical model of the controlled object is expressed as an extended state equation, a fixed-bandwidth linear extended state observer is constructed, and the observer bandwidth ω is set according to the bandwidth parameterization method. o and the gain of the extended state observer L = [β1,β2,…β n ,β n+1 ] T ; S2: Output variables z1, z2, ... z based on the extended state observer n , design state feedback control law u0=k1(r-z1)-k2z2-…k n z n , and set the controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the controller gain k i (i=1,2,…n); r is the set value; S3: Estimation of the total disturbance z by the integrated extended state observer n+1 And the state feedback control law u0, the design control law u=(u0-z n+1 ) / b0; b0 is the adjustable controller gain; S4: Let ω' o =ω o +η|z n+1 -f|, construct an intelligent variable bandwidth extended state observer and estimate the deviation |z with the total disturbance n+1 -f| is the adjustment basis to dynamically adjust the bandwidth of the observer. The bandwidth of the intelligent variable bandwidth expanded state observer is expressed as follows: oh o =ω o +η|z n+1 -f|=ω o +η|z n+1 -(y (n) -b0u)| Where f = y (n) -b0u,y (n) is the nth-order differential of the controlled object output, b0 is the adjustable controller gain, u is the controlled object input; η is the scaling factor, which is used to adjust the observer bandwidth range, η>0; The adaptive variable gain of the bandwidth parameterization configuration is as follows: S5: The intelligent variable bandwidth linear expansion observer is designed as follows: S6: Based on the output of the intelligent variable bandwidth extended state observer, the control law is adjusted accordingly: Among them, z 1SBW ,z 2SBW ,…z iSBW ,…z (n+1)SBW is the output of the intelligent variable bandwidth extended state observer; S7: Use the ADRC control system based on the intelligent variable bandwidth extended state observer to complete the control of the n-order controlled object and adjust the controller parameter ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

2. The design method of ADRC control system based on intelligent variable bandwidth extended state observer according to claim 1 is characterized in that: For a second-order ADRC control system, the design approach includes the following steps: B1: Establish a mathematical model of the second-order controlled object and obtain the model parameters of the controlled object through model identification. For a second-order controlled object, the mathematical model is: Among them, u is the control input of the system; y is the output of the system; is the first-order differential of the system output y; is the uncertainty within the system; ω is the external disturbance; b is the gain of the controlled object; the internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance and used as the state variables of the system. The mathematical model of the controlled object is expressed as an expanded state equation; Where x1=y(t), The total disturbance is expressed as b0 is the adjustable controller gain; B2: Based on the extended state equation, a fixed-bandwidth extended state observer is constructed: And set the observer bandwidth ω according to the bandwidth parameterization method o ,ω o >0, and then design the observer gain L=[β1,β2,β3]; Where z1 is the estimate of the system output y; z2 is the estimate of the differential of the system output y; z3 is the estimate of the total disturbance; B3: Based on the output variables z1, z2 of the extended state observer, design the PD controller u0 = k1(r-z1)-k2z2, and set the PD controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the PD controller gains k1, k2; B4: Combine the total disturbance estimate z3 of the extended state observer and the output u0 of the PD controller to generate the control output u according to the control law; in, r is the set value; B5: Let ω' o =ω o +η|z3-f|, construct an intelligent variable bandwidth extended state observer, and dynamically adjust the bandwidth of the observer based on the total disturbance estimation deviation |z3-f|. The bandwidth of the intelligent variable bandwidth extended state observer is expressed as follows: in, is the second-order differential of the controlled object output, b0 is the adjustable controller gain, u is the controlled object input; η is the expansion factor, which is used to adjust the observer bandwidth range, η>0; The adaptive variable gain of the bandwidth parameterized configuration is as follows: B6: Design an intelligent variable bandwidth extended state observer and adjust the control law accordingly; Among them, z 1SBW ,z 2SBW ,z 3SBW is the output of the intelligent variable bandwidth extended state observer; η>0 is the scaling factor, which is used to adjust the range of the observer bandwidth; Accordingly, the control law is adjusted as follows: B7, the ADRC control system based on the intelligent variable bandwidth extended state observer completes the control of the second-order controlled object and adjusts the controller parameter ω o ,ω c , b0 and η, so that the system works normally until the desired control effect is achieved.

3. A design method for an advanced internal model control system based on an intelligent variable bandwidth extended state observer, characterized in that: The method comprises steps S1 to S7 in claim 1, further comprising the following steps: S8: Using the ADRC control system based on the intelligent variable bandwidth extended state observer, when the extended state observer is reasonably designed, that is, z n+1 ≈f, the controlled object can be simplified into an n-order integral system y (n) =u0-z n+1 +f≈u0; the closed-loop transfer function of the nth-order ADRC without zero point can be obtained, that is, the standard closed-loop ADRC system dynamics G cl (s): Among them, k i (i=1,2,…n) is the controller gain; S9: Based on the standard closed-loop ADRC system dynamics G cl (s), and design the internal model controller G based on the following formula c (s), internal model G m (s) and filter f c (s): f c (s)=1 / (1+αs) n Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics. Compared with the standard closed-loop ADRC system dynamics G cl (s), where S10: The internal model controller based on the intelligent variable bandwidth extended state observer completes the control of the controlled object and adjusts the controller parameters α, ω o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

4. The design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer according to claim 3 is characterized in that: For a second-order ADRC control system, the design approach includes the following steps: B1: Establish a mathematical model of the second-order controlled object and obtain the model parameters of the controlled object through model identification. For a second-order controlled object, the mathematical model is: Among them, u is the control input of the system; y is the output of the system; is the first-order differential of the system output y; is the uncertainty within the system; ω is the external disturbance; b is the gain of the controlled object; the internal uncertainty and external disturbance of the controlled object are regarded as the total disturbance and used as the state variables of the system. The mathematical model of the controlled object is expressed as an expanded state equation; Where x1=y(t), The total disturbance is expressed as b0 is the adjustable controller gain; B2: Based on the extended state equation, a fixed-bandwidth extended state observer is constructed: And set the observer bandwidth ω according to the bandwidth parameterization method o ,ω o >0, and then design the observer gain L=[β1,β2,β3]; Where z1 is the estimate of the system output y; z2 is the estimate of the differential of the system output y; z3 is the estimate of the total disturbance; B3: Based on the output variables z1, z2 of the extended state observer, design the PD controller u0 = k1(r-z1)-k2z2, and set the PD controller bandwidth ω according to the bandwidth parameterization method c ,ω c >0, and then design the PD controller gains k1, k2; B4: Combine the total disturbance estimate z3 of the extended state observer and the output u0 of the PD controller to generate the control output u according to the control law; in, r is the set value; B5: Let ω' o =ω o +η|z3-f|, construct an intelligent variable bandwidth extended state observer, and dynamically adjust the bandwidth of the observer based on the total disturbance estimation deviation |z3-f|. The bandwidth of the intelligent variable bandwidth extended state observer is expressed as follows: in, is the second-order differential of the controlled object output, b0 is the adjustable controller gain, u is the controlled object input; η is the expansion factor, which is used to adjust the observer bandwidth range, η>0; The adaptive variable gain of the bandwidth parameterized configuration is as follows: B6: Design an intelligent variable bandwidth extended state observer and adjust the control law accordingly; Among them, z 1SBW ,z 2SBW ,z 3SBW is the output of the intelligent variable bandwidth extended state observer; η>0 is the scaling factor, which is used to adjust the range of the observer bandwidth; Accordingly, the control law is adjusted as follows: B7, the ADRC control system based on the intelligent variable bandwidth extended state observer completes the control of the second-order controlled object and adjusts the controller parameter ω o ,ω c , b0 and η, make the system work normally until the desired control effect is achieved; B8: Using the ADRC control system based on the intelligent variable bandwidth extended state observer designed in B7, ignoring the estimation error of the extended state observer for the total disturbance, the second-order controlled object can be equivalent to the integral series type, and the standard closed-loop ADRC system dynamics G is obtained. cl (s): B9: Based on the standard closed-loop ADRC system dynamics G cl (s), and design the internal model controller G based on the following formula c (s), internal model G m (s) and filter f c (s): f c (s)=1 / (1+αs) 2 Where n is the order of the controlled object, α is the time constant, and Δ is the actual closed-loop ADRC system dynamics. Compared with the standard closed-loop ADRC system dynamics G cl The deviation between (s), B10: Based on the internal model controller of the intelligent variable bandwidth extended state observer, the controlled object is controlled and the controller parameters α, ω are adjusted. o ,ω c , b0 and η make the system work normally until the desired control effect is achieved.

5. The design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer according to claim 4 is characterized in that: In B10, when adjusting the controller parameters, at a given adjustment time t s , when the error tolerance is ±5%, the initial value of the filter parameter α can be determined by the following formula: t s ＝3.5α。 6. The design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer according to claim 4, characterized in that: In B10, when adjusting the controller parameters, under the given overshoot σ%, the PD controller bandwidth ω c The initial value of can be determined by the following formula:

7. An n-order ADRC control system obtained based on the design method of the ADRC control system based on the intelligent variable bandwidth extended state observer according to claim 1.

8. A second-order ADRC control system obtained based on the design method of the ADRC control system based on the intelligent variable bandwidth extended state observer according to claim 2.

9. An n-order advanced internal model control system obtained based on the design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer as claimed in claim 3.

10. A second-order advanced internal model control system obtained based on the design method of an advanced internal model control system based on an intelligent variable bandwidth extended state observer according to claim 4.

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