A design method of fuzzy extended state observer in high-order control loop

Abstract

The application discloses a fuzzy extended state observer design method in a high type control loop. In view of the problem that the extended state observer in the high type control loop under the linear quadratic optimal control setting cannot change the observer bandwidth online and cannot realize accurate estimation of different disturbances, the application applies a fuzzy algorithm to parameter self-setting of the extended state observer in the high type control loop, takes system error as input of the fuzzy system and takes observer bandwidth as output of the fuzzy system. The fuzzy extended state observer designed by the application can adjust the observer bandwidth in real time according to the error size to realize accurate estimation of disturbances under different conditions, and improves the online adaptive ability of the extended state observer in the high type control loop. Compared with simulation of the fuzzy extended state observer design method in the type II control loop and the type III control loop and simulation of the traditional extended state observer, dynamic response performance indexes such as system rise time and regulation time and the anti-interference ability of the system are obviously improved under the design method of the application.

Description

Technical Field

[0001] This invention relates to the field of disturbance estimation and suppression, specifically to a design method for a fuzzy extended state observer in a high-type control loop, which mainly realizes the tuning of high-type controller parameters and the parameter self-tuning of an improved extended state observer in a high-type control loop. Background Technology

[0002] In engineering control applications, systems are often affected by external disturbances and internal uncertainties. These disturbances seriously affect the stability and control effect of the system, and may even cause instability of the closed-loop system. Taking an inertial stabilization system as an example, in reference [3] (Tian J, Yang W, Peng Z, et al. Application of MEMS accelerometers and gyroscopes in fast steering mirror control systems[J]. Sensors, 2016, 16(4): 440.), a multi-loop feedback control system is composed of fiber optic gyroscopes, accelerometers, and high-resolution position detectors. The total disturbance suppression capability of the system is the superposition of the effects of each loop. However, this method requires the installation of additional inertial sensors on the inertial stabilization platform, which is not conducive to achieving the requirements of small inertia and fast speed of the inertial stabilization platform. At the same time, it also increases the experimental space and economic cost. Reference [4] (Glück M, Pott JU, Sawodny O. Piezo-actuated vibration disturbance mirror for investigating accelerometer-based tip-tiltre construction in large telescopes[J].IFAC-PapersOnLine,2016,49(21):361-366.) uses a measurement-based direct feedforward method to suppress external vibrations measured by the base sensor, but it is necessary to accurately identify the disturbance transmission characteristics from the base to the tilt mirror. Meanwhile, Reference [5] (Tang T, Niu S, Chen X, et al. Disturbance observer-based control of tip-tilt mirror for mitigating telescope vibrations[J].IEEE Transactions on Instrumentation and Measurement,2018,68(8):2785-2791.) introduces DOB into the inertial stabilization system to enhance the system's anti-interference capability. However, the characteristics of the controlled object model are often not accurately identified, which limits the design of the compensator.

[0003] To further improve the system's disturbance suppression capability, an extended state observer, which can be used to observe external disturbances and internal system uncertainties, is widely used. Traditional linear extended state observer design methods classify external disturbances and internal system uncertainties as a total disturbance and expand it into a single state variable for observation and compensation. While this design method simplifies the system to a double-integral cascade canonical form, reducing control complexity, the traditional extended state observer design is independent of known model information. This may lead to decreased observation accuracy and consequently, a decline in the system's anti-interference capability. This invention introduces an extended state observer based on model information to further improve the system's disturbance suppression capability.

[0004] With the continuous development of classical control theory, linear quadratic optimal control has been widely applied in modern control theory. The optimal control law obtained by the linear quadratic optimal control method possesses many excellent properties, including closed-loop stability and robustness. Furthermore, by selecting the weighting matrices Q and R, the linear quadratic optimal control method can balance the control state regulation requirements with control energy consumption. This excellent property has prompted control designers to use it for PID controller parameter tuning. In high-order control loops, when traditional extended state observers adopt a large setpoint bandwidth, the system noise becomes more sensitive to the observer bandwidth value, exacerbating the noise's impact on the system. Conversely, when traditional extended state observers adopt a small setpoint bandwidth, they cannot accurately estimate the compensation disturbance value. Therefore, the bandwidth of the extended state observer is particularly important to the system. To address the limitation of traditional extended state observers, which cannot change their bandwidth online and thus cannot accurately estimate different disturbances, this invention introduces a fuzzy algorithm into an improved extended state observer in high-order control loops to achieve self-tuning of the observer bandwidth, thereby improving the observer's adaptive capability. Summary of the Invention

[0005] This invention introduces the Mandani fuzzy system to achieve online self-tuning of the parameters of the improved extended state observer in a type II control loop, thereby enhancing the noise filtering and disturbance observation compensation capabilities of the extended state observer under different conditions. The simulations in this invention primarily illustrate the effectiveness of the method using the effects of fuzzy extended state observers in type II and type III control loops. The specific implementation steps are as follows:

[0006] For an nth-order model system G(s):

[0007]

[0008] Where n>2, a1=2ζ ol ω ol , ;ζ ol ω ol...

[0009] Step 1: Tuning is performed using a combination of linear quadratic optimal control (LQR) and dominant pole techniques. Figure 1 The PID n-type controller parameters in the control framework enable control of the system after disturbance compensation. Figure 1 middle It is a PID n-type controller, as shown below:

[0010]

[0011] in, They are x1, x2…x n+1 The corresponding coefficient.

[0012] Figure 1 In this equation, e(t) represents the tracking error signal, w(t) represents the external disturbance input, and r(t) and y(t) represent the reference signal and the controlled object's position output signal, respectively. Assuming the reference signal r(t) = 0, then e(t) = -y(t). Under this condition, the controlled system equation (0.25) can be expressed as:

[0013]

[0014] Based on the form of the state space, the derivatives of the state variables in formula (0.27) of the controlled system can be written as:

[0015]

[0016] in,

[0017]

[0018] To make the system in Equation (0.29) have LQR performance, the following quadratic cost function needs to be minimized.

[0019]

[0020] Where Q is the positive semi-definite state weight matrix, R is the positive definite control weight matrix, and the standard LQR method gives the (n-2)th order differential of the optimal control vector. for:

[0021]

[0022] Where P is a symmetric positive definite Riccati coefficient matrix, which can be obtained by solving the following continuous algebraic Riccati equation:

[0023] A T P+PA+Q-PBR -1B T P = 0 (0.32)

[0024] In linear quadratic optimal control, the standard practice is to further design the high-performance controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged.

[0025] Assumption:

[0026]

[0027] Substituting formulas (0.29) and (0.33) into formula (0.31), we get:

[0028]

[0029] By comparing the coefficients of the same state variables on the right-hand side of formula (0.26) and formula (0.34), we can obtain:

[0030] K1 = r -1 bP 1(n+1) K2=r -1 bP 2(n+1) ,…,K (n+1) =r -1 bP (n+1)(n+1) (0.35)

[0031] The corresponding closed-loop system characteristic equations in the high-type control loop are as follows:

[0032]

[0033] Among them, A c =A-BR -1 B T P. Due to the system matrix A c Without any time delay, the desired closed-loop performance is obtained by directly applying the pole placement method. This is achieved by establishing the characteristic equation of the closed-loop system: Δ(s)=|sI-A c | equals the required closed-loop equation.

[0034] When A c When the matrix is ​​2×2, the characteristic equation Δ(s) is as follows:

[0035]

[0036] in, ζ cl ,ω cl Given the desired damping ratio and natural frequency of the system;

[0037] When A cWhen the matrix is ​​(n+1)×(n+1), the characteristic equation Δ(s) using the dominant pole placement technique is as follows:

[0038]

[0039] Among them, the dominant poles p1, p2 are the same as p1, p2 in formula (0.37), and the non-dominant poles p3…p (n+1) real parts m, m1, m at distances from the dominant pole (n-2) According to the conclusion of reference [1] (Srivastava S, Misra A, Thakur SK, et al. An optimal PID controller via LQR for standard second order plus time delay systems[J].ISA transactions,2016,60:244-253.), the non-dominant pole should be far away from the real part of the other two complex conjugate closed-loop dominant poles, so as to satisfy the pole configuration of the PID high-type controller. Its value should be selected in the range of 3 or more.

[0040] By comparing the coefficients of the same variable on the right side of formula (0.38) and formula (0.36), we can obtain (P 1(n+1) P 2(n+1) … P (n-1)(n+1) P n(n+1) P (n+1)(n+1) Substituting the value of ) into formula (0.35) yields the controller parameters in the high-type control loop;

[0041] Step 2: Establish the system differential equations for the nth-order model system:

[0042]

[0043] in, For the position output, velocity output, ..., position output of the system, the (n-1)th order derivative is given. Let w be the (n-2) derivative of the system input signal, and w be the external disturbance. This is partial model information, where a0, a1, and b are known system parameters; since a0, a1, and b are often not accurately identified in practice, f is used. x This indicates the inaccurate parts of the model and the parts with internal dynamic changes; f w Indicates external disturbance; It represents the combined effect of known model dynamics and unknown perturbations.

[0044] Transforming the differential equation (0.39) into an extended state-space equation, the extended state-space equation of the system is as follows:

[0045]

[0046] in,

[0047] x1,x2...x n x4 represents the (n-1)th derivative of the position, velocity, ..., position of the control system, and the total disturbance of the system, respectively. Let denoted by , and u be the derivative of the total disturbance.

[0048] Step 3: For the nth-order model system, based on the design of the state observer in linear system theory, the continuously extended state observer ESO is as follows:

[0049]

[0050] Where z = [z1 z2 ... z n+1 ] T Let L be the observer state vector, where L = [β1 β2 ... β n+1 ] T For the observer gain matrix that needs to be determined, For the observer input combination, y c For the output of the extended state observer;

[0051] Assuming the error state variable is e(t) = x(t) - z(t), subtracting formula (0.40) from formula (0.41), the observer error matrix equation is:

[0052]

[0053] As can be seen from the above equation, (A-LC) in the observer error matrix equation determines the eigenvalues ​​of the closed-loop system. By ensuring that the eigenvalues ​​of (A-LC) are less than zero, the observer equation converges. The characteristic equation Δ(s) corresponding to the observer error matrix equation in the PID n-type control loop is shown below:

[0054] △(s)=|sI-(A-LC)| (0.43)

[0055] In reference [2] (Herbst GA simulative study on active disturbance rejection control (ADRC) as a control tool for practitioners[J]. Electronics,2013,2(3):246-279), the extended state observer is parameterized so that the poles of the corresponding characteristic equation can be placed in the same position (-w0, where w0 is the observer bandwidth), as shown below:

[0056] △(s)=|sI-(A-LC)|=(s+ω o ) n+1 (0.44)

[0057] Expanding equations (0.43) and (0.44) and comparing the coefficients of the same variables on the right, we can obtain the gain matrix L of the extended state observer in the PID n-type control loop. It can be seen that the gain matrix L of the extended state observer is related to the observer bandwidth w0.

[0058] The extended state observer can accurately estimate uncertain disturbances f′ within a certain frequency range and observe the extended state z. n+1 Provide compensation. Figure 1 (n-2) derivative of the control signal for:

[0059]

[0060] Substituting formula (0.45) into formula (0.39), we get:

[0061]

[0062] Will Substituting the available dynamic model into equation (0.46), after eliminating unnecessary disturbances, the system becomes:

[0063]

[0064] Equation (0.47) can be rewritten as the system transfer function as follows:

[0065]

[0066] After eliminating unnecessary disturbances, the system transfer function of the system using the above extended state observer is consistent with the model in formula (0.25).

[0067] Step 4: Design a single-input fuzzy system in the above PID n-type control loop, perform online self-tuning of the extended state observer parameters, and design a single-input single-output fuzzy system, where the input is the error e and the output is the observer bandwidth ω. o When the error e is large, the observer bandwidth is increased to improve the observer's overall disturbance compensation capability. When the error e is small, the observer bandwidth is decreased to improve the observer's filtering effect and reduce the impact of noise on the observer. Figure 2 The relationship between the input and output membership functions of a fuzzy system is given.

[0068] Furthermore, the fuzzy system in the PID n-type control loop mainly consists of four parts: a fuzzification interface, a rule base, an inference engine, and a defuzzification interface. First, the current error e is fuzzified, and seven linguistic variables are used to describe the error e: [negative large (NB), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive large (PB)]. Three linguistic variables are used to describe the observer bandwidth ω. o The system is described as follows: [Small (PS), Medium (PM), Large (PB)], and the membership functions adopted for input and output are trigonometric functions. The membership function value for each input is calculated based on the current system error e. Secondly, based on the set fuzzy rules, a minimum inference approach is used for inference calculation. Finally, through fuzzy tuning inference, the parameter tuning rules for the extended state observer in the PID n-type control loop are obtained, and the centroid defuzzification method is used to obtain the fuzzy system output observer bandwidth ω. o This enables online self-tuning of the observer parameters.

[0069] Based on the above technical solution, the following beneficial effects can be achieved:

[0070] 1. Compared with traditional extended state observers, the dynamic response performance indicators such as system rise time and settling time, as well as the system's anti-interference capability, are significantly improved under this invention;

[0071] 2. Compared with the fixed-bandwidth extended state observer in the high-type control loop and the traditional fixed-bandwidth extended state observer, this method introduces a fuzzy algorithm, which improves the disturbance estimation and compensation capability of the extended state observer and realizes the self-tuning of the extended state observer parameters under different system states. Attached Figure Description

[0072] Figure 1 This is the control block diagram for a PID high-level control loop;

[0073] Figure 2 This is a graph showing the input and output membership functions of the fuzzy system in this invention.

[0074] Figure 3A comparison of the step response under a sinusoidal disturbance with an amplitude of 20 and a frequency of 1Hz in a PID II control loop;

[0075] Figure 4 A comparison of the step response under a sinusoidal disturbance with an amplitude of 20 and a frequency of 3Hz in a PID III control loop;

[0076] Figure 5 The top figure shows a comparison of the step responses of the PID II controller in the fixed-value extended state observer of the inertial stable system under a step disturbance of magnitude 20 in the PID II type control loop, and the PID II controller in the fuzzy extended state observer of this invention; the bottom figure shows a comparison of the step responses of the PID III controller in the fixed-value extended state observer of the inertial stable system under a step disturbance of magnitude 20 in the PID III type control loop, and the PID III controller in the fuzzy extended state observer of this invention.

[0077] Figure 6 The top figure shows the parameter changes of the fuzzy extended state observer in the PID Type II control loop under a step disturbance; the bottom figure shows the parameter changes of the fuzzy extended state observer in the PID Type III control loop under a step disturbance. Detailed Implementation

[0078] The specific implementation steps of the present invention will be described in detail below with reference to the accompanying drawings and an inertial stabilization system as an example:

[0079] This invention provides a design method for a fuzzy extended state observer in a high-performance control loop, comprising:

[0080] Step 1: The system operates at a sampling frequency of 5000Hz. The frequency response curve of the controlled object is obtained through frequency response testing. The position transfer function of the controlled object (second order) is then obtained through fitting.

[0081]

[0082] The simulations in this invention primarily illustrate the effectiveness of the method using the effects of fuzzy extended state observers in Type II and Type III control loops. In the Type II control loop, the transfer function of the controlled object is:

[0083]

[0084] The above third-order controlled object is transformed into an extended state-space equation form, and the gain matrix L of the linear extended state observer based on model information is designed according to formulas (0.43) and (0.44).

[0085]

[0086] Step 2: In the above PID type II control loop, a PID type II controller is designed using a combination of the LQR method and the dominant pole placement technique to achieve control of the disturbance-compensated system. The weighting matrix R = 1 is taken, and the desired system damping ratio ζ is... cl Natural frequency ω cl And the relative advantage degree m is ζ cl =1.8,ω cl =2,m=100,m1=3.

[0087] Using formula (0.35), the parameters of the PID type II controller tuned by the LQR method are as follows:

[0088] [K1K2K3K4]=[75.13142.426.141.7](0.49)

[0089] Step 3: The transfer function of the controlled object in the PID III type control loop is:

[0090]

[0091] The above fourth-order controlled object is transformed into an extended state-space equation form, and the gain matrix L of the linear extended state observer based on model information is designed according to formulas (0.43) and (0.44):

[0092]

[0093] Step 4: In the above PID type III control loop, a PID type III controller is designed using a combination of the LQR method and the dominant pole placement technique to achieve control of the disturbance-compensated system. The weighting matrix R = 1, and the desired system damping ratio ζ... cl Natural frequency ω cl And the relative advantage degree m is ζ cl =1.8,ω cl =2, m=100, m1=3, m2=10.

[0094] Using formula (0.35), the parameters of the PID type III controller tuned by the LQR method are as follows:

[0095] [K1 K2 K3 K4 K5]=[2704.7 5201.5 1283.6 91.88 1.875] (0.50)

[0096] Step 5: Design a corresponding single-input fuzzy system in the above PID type II and type III control loops, and perform online self-tuning of the observation bandwidth of the extended state observer.

[0097] The fuzzy system in the aforementioned PID Type II and Type III control loops mainly consists of four parts: a fuzzification interface, a rule base, an inference engine, and a defuzzification interface. First, the current error e is fuzzified, and seven linguistic variables are used to describe it: [negative large (NB), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive large (PB)]. Three linguistic variables are used to describe the observer bandwidth ω. o The system is described as follows: [Small (PS), Medium (PM), Large (PB)], and the membership functions adopted for input and output are trigonometric functions. The membership function value for each input is calculated based on the current system error e. Secondly, based on the set fuzzy rules, a minimum inference approach is used for inference calculation. Finally, through fuzzy tuning inference, the tuning rules for the extended state observer parameters in the corresponding control loop are obtained, and the centroid defuzzification method is used to obtain the fuzzy system output observer bandwidth ω. o This enables online self-tuning of the observer parameters. Figure 2 The relationship between the input and output membership functions of the fuzzy system is given, and Table 1 shows the parameters ω of the extended state observer. o Detailed fuzzy rules.

[0098] Table 1. Parameters ω of the Extended State Observer o Fuzzy rule table

[0099]

[0100] Step 6: To verify the effectiveness of the PID controller parameter tuning and fuzzy extended state observer in the PID Type II and Type III control loops of this invention, a PD controller from a traditional extended state observer is designed for comparison. The PD controller is designed as follows: K d =2ω c , where ω c For controller bandwidth. In engineering applications, the bandwidth ω of the extended state observer... o With controller bandwidth ω c The relationship is generally ω o =3~5ω c .

[0101] Figure 3 In a PID type II control loop, the bandwidth ω of the setpoint observer under a sinusoidal disturbance with an amplitude of 20 and a frequency of 1Hz is... o =30Hz PD controller, observer bandwidth ω o A comparison of the step responses of a PID Type II controller with a 30Hz frequency and a PID Type II controller under a fuzzy extended state observer. Figure 4In a PID III control loop, the bandwidth ω of the setpoint observer under a sinusoidal disturbance with an amplitude of 20 and a frequency of 3Hz is given. o =30Hz PD controller, observer bandwidth ω o A comparison of the step responses of a PID III controller with a 30Hz frequency and a PID III controller under a fuzzy extended state observer. From... Figure 3 , Figure 4 As can be seen, the fuzzy extended state observer in this invention exhibits better disturbance suppression performance for sinusoidal disturbances at these two frequencies. Furthermore, compared with traditional extended state observer design methods, the method described in this paper significantly improves the system's dynamic response performance indicators such as rise time and settling time, as well as its anti-interference capability.

[0102] Figure 5 This is a comparison of the step responses of an inertial stable system under step disturbances using a fixed-value extended state observer and a fuzzy extended state observer in PID type II and type III control loops. It can also be seen that the fuzzy extended state observer exhibits better disturbance suppression and superior anti-interference capabilities under high-type control.

[0103] Figure 6 The observer bandwidth variations of the fuzzy extended state observer in PID type II and type III control loops are presented. It can be seen that when the system is in a stable tracking state, the fuzzy extended state observer uses a smaller bandwidth, achieving stable tracking while exhibiting better filtering effects and reducing the impact of noise on the system. When the system tracks a step signal or introduces a step disturbance, the fuzzy extended state observer bandwidth increases to improve the observer's disturbance observation and compensation capabilities, reducing the impact of disturbances on the system's control performance.

[0104] The specific embodiments, processes, and effects of the present invention have been described in detail above with reference to the accompanying drawings and examples. However, the content described is only one embodiment of the method and should not be used to limit the scope of implementation of the method.

Claims

1. A design method for a fuzzy extended state observer in a high-precision control loop, characterized in that: The specific steps are as follows: For n-order model systems : (0.1) in, , , ; , ... Step 1: A combination of linear quadratic optimal control (LQR) and dominant pole techniques is used to tune the parameters of the PID n-type controller in the control framework, achieving control of the disturbance-compensated system. It is a PID n-type controller, as shown below: (0.2) in, They are respectively The corresponding coefficient; To track error signals, For external disturbance input, and These are the reference signal and the controlled object's position output signal, respectively; assuming the reference signal... ,but Under these conditions, the controlled system formula (0.1) can be expressed as: (0.3) Based on the form of the state space, the derivatives of the state variables in formula (0.3) of the controlled system can be written as: (0.4) in, (0.5) To achieve LQR performance in the system described in equation (0.5), the following quadratic cost function needs to be minimized: (0.6) in, Let R be the semi-definite state weight matrix and R be the positive definite control weight matrix. The standard LQR method gives the optimal control vector. First differential for: (0.7) in, The symmetric positive definite Riccati coefficient matrix can be obtained by solving the following continuous algebraic Riccati equation: (0.8) In linear quadratic optimal control, the standard practice is to further design the high-performance controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged. Assumption: (0.9) Substituting formulas (0.5) and (0.9) into formula (0.7), we get: (0.10) By comparing the coefficients of the same state variables on the right-hand side of formula (0.2) and formula (0.10), we can obtain: (0.11) The corresponding closed-loop system characteristic equations in the high-type control loop are as follows: (0.12) in, Due to the system matrix Without any time delay, the desired closed-loop performance is obtained by directly applying the pole placement method and establishing the characteristic equation of the closed-loop system. It equals the required closed-loop equation; when for When dealing with matrices, the characteristic equation As shown below: (0.13) in, ; Given the desired damping ratio and natural frequency of the system; when for When dealing with matrices, the dominant pole placement technique is used to determine the characteristic equation. As shown below: (0.14) Among them, the dominant pole In formula (0.13) Same, non-dominant pole real part of the distance from the dominant pole The non-dominant pole should be far from the real parts of the other two complex conjugate closed-loop dominant poles to satisfy the pole configuration of the PID high-type controller. Its value should be selected to be 3 or more. By comparing the coefficients of the same variable on the right side of formula (0.14) and formula (0.12), we can obtain... Substituting the value of into formula (0.11) yields the controller parameters in the high-type control loop; Step 2: Establish the system differential equations for the nth-order model system: (0.15) in, For the system's position output, velocity output... position output First derivative, Input signals to the system differential, External disturbances This is partial model information. These are known parameters of the system; because in practice... The identification was inaccurate, so it was adopted. This indicates the inaccurate parts of the model and the parts with internal dynamic changes; Indicates external disturbance; This represents the combined effect of known model dynamics and unknown perturbations; Transforming the differential equation (0.15) into an extended state-space equation, the system's extended state-space equation is as follows: (0.16) in, , ; These represent the position, velocity, ... position of the control system, respectively. The first derivative and the total perturbation of the system, Let be the derivative of the total disturbance, and u be the control input; Step 3: For the nth-order model system, based on the design of the state observer in linear system theory, the continuously extended state observer ESO is as follows: (0.17) in, Let be the observer state vector. For the observer gain matrix that needs to be determined, Input combination for the observer, For the output of the extended state observer; Assume the error state variable is Subtracting formula (0.16) from formula (0.17), the observer error matrix equation is: (0.18) As can be seen from the above equation, in the observer error matrix equation, The characteristic values ​​of the closed-loop system are determined by ensuring If the eigenvalues ​​are less than zero, then the observer equation converges, and the corresponding characteristic equation of the observer error matrix in the PID n-type control loop is obtained. As shown below: (0.19) The extended state observer, after parameterization, can place the poles of the corresponding characteristic equation in the same location. , (where the observer bandwidth is), as shown below: (0.20) Expanding equations (0.19) and (0.20) and comparing the coefficients of the same variables on the right side, we can obtain the gain matrix of the extended state observer in the PID n-type control loop. The gain matrix of the extended state observer can be seen. With observer bandwidth Related; Extended state observers can accurately estimate uncertain disturbances within a certain frequency range. and for extended states Compensation is performed on the control signal. differential for: (0.21) Substituting formula (0.21) into formula (0.15), we get: (0.22) Will Substituting the available dynamic model into equation (0.22), after eliminating unnecessary disturbances, the system becomes: (0.23) Equation (0.23) can be rewritten as the system transfer function as follows: (0.24) After eliminating unnecessary disturbances, the system transfer function of the system using the above extended state observer is consistent with the model in formula (0.1); Step 4: Design a single-input fuzzy system in the above PID n-type control loop, perform online self-tuning of the extended state observer parameters, and design a single-input single-output fuzzy system, where the input is the error. The output is the observer bandwidth. When the error When the error is large, increase the observer bandwidth to improve the observer's overall disturbance compensation capability. When the bandwidth is small, reduce the observer bandwidth, improve the observer's filtering effect, and reduce the impact of noise on the observer; The fuzzy system in the PID n-type control loop consists of four parts: a fuzzification interface, a rule base, an inference engine, and a defuzzification interface. First, by analyzing the current error... Fuzzification was performed, and seven linguistic variables were used to assess the error. The following statistic is used to describe the following values: [Negative Large (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PM), Positive Large (PB)]. Three linguistic variables are used to measure the observer bandwidth. The system is described as follows: [Small Positive (PS), Medium Positive (PM), Large Positive (PB)], and the membership functions used for input and output are trigonometric functions. This is based on the current system error. First, calculate the membership function value for each input. Second, based on the established fuzzy rules, perform inference calculations using a minimum inference approach. Finally, obtain the tuning rules for the extended state observer parameters in the PID n-type control loop through fuzzy tuning inference, and obtain the fuzzy system output observer bandwidth using a centroid defuzzification method. This enables online self-tuning of the observer parameters.

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