Design method of fuzzy extended state observer based on linear quadratic optimal control

Abstract

The application discloses a design method of a fuzzy extended state observer based on a linear quadratic optimal control, and applies a fuzzy algorithm to parameter self-tuning of the extended state observer, so that the adaptive capacity of the observer is improved. In order to further improve the dynamic response performance of the system such as the system rising time and the regulation time, the linear quadratic optimal control method is combined with a dominant pole placement technology to design a proportional-integral-derivative (PID) controller. Compared with the traditional extended state observer, the dynamic response performance indexes such as the system rising time and the regulation time and the anti-interference capacity of the system are significantly improved. In addition, the system error is taken as the input of the fuzzy system, and the observer bandwidth is taken as the output of the fuzzy system, so that the fuzzy extended state observer can adjust the observer bandwidth in real time according to the error size to realize accurate estimation of the disturbance under different conditions, and the online adaptive capacity of the extended state observer is improved.

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 based on linear quadratic optimal control, which further realizes the tuning of controller parameters and the self-tuning of parameters of the extended state observer. 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, the literature (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.) uses a fiber optic gyroscope, accelerometer, and high-resolution position detector to form a multi-loop feedback control system. The total disturbance suppression capability of the system is the sum 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. The literature (Xue W, Bai W, Yang S, et al. ADRC with adaptive extended stateobserver and its application to air–fuel ratio control in gasoline engines[J]. IEEE Transactions on Industrial Electronics, 2015, 62(9): 5847-5857.) uses a measurement-based direct feedforward method to suppress external vibrations measured by the base sensor, but it requires accurate identification of the disturbance transmission characteristics from the base to the tilt mirror. Meanwhile, the literature (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, limiting the design of the compensator. Therefore, when the hardware conditions of the inertial stabilization platform itself cannot be changed, it is particularly important to use high-performance control algorithms to enhance the system's anti-interference capability.

[0003] 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 characteristics, including closed-loop stability and, when the system process is single-input single-output, a phase margin of at least 60° and a gain margin of infiniteness. 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. These excellent properties have prompted control designers to use it for tuning PID controller parameters.

[0004] To further improve the disturbance suppression capability of a system, an extended state observer, which can be used to observe external disturbances and internal uncertainties of the system, is widely used. This invention introduces an extended state observer based on the linear quadratic optimal control method, classifying external disturbances and internal uncertainties into a total disturbance and expanding it into a single state variable for observation and compensation. When a traditional extended state observer uses a large setpoint bandwidth, the influence of noise on the system is exacerbated because system noise is sensitive to the observer's bandwidth value. Conversely, when a traditional extended state observer uses a small setpoint bandwidth, it cannot accurately estimate the compensation disturbance value. Therefore, the bandwidth of the extended state observer has a particularly important impact on the system.

[0005] To address the shortcomings of traditional extended state observers, which cannot change the observer bandwidth online and thus cannot accurately estimate different disturbances, this invention introduces a fuzzy algorithm into the extended state observer to achieve self-tuning of the observer bandwidth, thereby improving the observer's adaptive capability. Summary of the Invention

[0006] This invention provides a design method for a fuzzy extended state observer based on linear quadratic optimal control. By introducing the Mandani fuzzy system, the method enables online self-tuning of the extended state observer parameters, thereby improving the noise filtering and disturbance observation compensation capabilities of the extended state observer under different conditions.

[0007] The technical solution adopted in this invention is: a design method for a fuzzy extended state observer based on linear quadratic optimal control, and the specific implementation steps are as follows:

[0008] Step 1: Establish the differential equations for the inertial stable system:

[0009]

[0010] Where a0, a1, and b are unknown system parameters, y and u are system output and input, w is external disturbance, and b0 is a known system parameter. This refers to the total disturbance, which includes both external and internal disturbances.

[0011] Transforming the differential equations into extended state-space equations, the extended state-space equations of the system are as follows:

[0012]

[0013] in, C =

[100] ,

[0014] Step 2: Design a linearly extended state observer based on the system's extended state-space equations:

[0015]

[0016] Where z = [z1 z2 z3] T Let L be the observer state vector, where L = [β1β2β3]. T For the observer gain matrix that needs to be determined, u c =[uy] T For the observer input combination, y c For the output of the extended state observer;

[0017] Step 3: The extended state observer can accurately estimate the uncertain disturbance f within a certain frequency range and compensate for the extended state z3, such as... Figure 1 As shown. Figure 1 The control signal u is:

[0018] u=(u0-z3) / b0 (1.4)

[0019] Substituting formula (1.3) into formula (1.1), we can obtain

[0020]

[0021] As can be seen from the above formula, the system is simplified to a standard type with two integrators in series, thereby reducing the difficulty of control.

[0022] Step 4: Use the Linear Quadratic Optimal Control (LQR) method for tuning. Figure 1 The PID controller parameters in the control framework enable control of the system after disturbance compensation. Figure 1 In this context, u0 is a PID controller, as shown below:

[0023] u0(t)=K p x2(t)+K i x1(t)+K d x3(t) (1.6)

[0024] Where, x1(t)=ve(t)dt, x2(t)=e(t), K p ,Ki ,K d These are the coefficients before the proportional element x2, the integral element x1, and the differential element x3, respectively.

[0025] Figure 1 In this system, 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... It can be represented as:

[0026]

[0027] in,

[0028]

[0029] To obtain the LQR formula with system (1.8), the following quadratic cost function needs to be minimized.

[0030]

[0031] From the literature (Srivastava S, Misra A, Thakur SK, et al. An optimal PID controller via LQR for standard secondorder plus time delay systems[J]. ISAtransactions, 2016, 60: 244-253.), the minimized state feedback control signal in equation (1.9) is:

[0032] u(t) = -R -1 B T Px(t) (1.10)

[0033] Where P is a symmetric positive definite solution to the following Riccati equation:

[0034] A T P+PA+Q-PBR -1 B T P = 0 (1.11)

[0035] In linear quadratic optimal control, the standard practice is to further design the relevant controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged.

[0036] Assuming,

[0037]

[0038] Substituting formulas (1.8) and (1.12) into formula (1.10), we get:

[0039] u(t) = r -1 (p 13 x1(t) p 23 x2(t) p 33 x3(t)) (1.13)

[0040] By comparing the coefficients of the same state variables on the right-hand side of formula (1.13) and formula (1.6), we can obtain:

[0041] K p =r -1 p 23 ,K i =r -1 p 13 ,K d =r -1 p 33 (1.14)

[0042] The corresponding closed-loop system characteristic equations are as follows:

[0043] |sI-A c |=s 3 +r -1 p 33 s 2 +r -1 p 23 s+r -1 p 13 (1.15)

[0044] in,

[0045] Due to 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.

[0046] When A c When the matrix is ​​2x2, the characteristic equation Δ(s) is as follows:

[0047]

[0048] in, ζ cl ,ω cl Let be the damping ratio and natural frequency of the desired system.

[0049] When A cWhen the matrix is ​​3x3, using the dominant pole placement method, the characteristic equation Δ(s) is as follows:

[0050]

[0051] Among them, the dominant poles p1 and p2 are the same as above, and the non-dominant pole p3 is m times the distance from the real parts of the dominant poles p1 and p2. Meanwhile, m >> ζ. cl ω cl This ensures that the closed-loop pole is located at the dominant pole position. According to relevant literature (He JB, Wang QG, Lee T H. PI / PID controller tuning via LQR approach[J]. Chemical Engineering Science, 2000, 55(13):2429-2439. and Saha S, Das S, Das S, et al. A conformal mapping-based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(9):3628-3642.), this non-dominant pole should be far away from the real part of the other two complex (conjugate) closed-loop dominant poles. This satisfies the PID controller pole configuration, and its value should be selected to be 3 or more.

[0052] By comparing the coefficients of the same state variables on the right-hand side of formulas (1.15) and (1.17), we can obtain...

[0053]

[0054] The remaining elements of matrices P and Q can be obtained by solving the Riccati equation (1.11), as shown below:

[0055]

[0056] Step 5: Design the extended state observer gain matrix. Assuming the error state variable is e(t) = x(t) - z(t), subtracting equation (1.2) from equation (1.3), the observer error matrix equation is:

[0057]

[0058] 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 corresponding to the observer error matrix equation is shown below:

[0059] |sI-(A-LC)|=s 3 +β1s 2 +β2s+β3 (1.21)

[0060] According to the conclusion of the literature (Chen Z, Jia H. Design of flight control system for a noveltilt-rotor UAV[J]. Complexity, 2020, 2020.), after parameterization, the poles of the corresponding characteristic equation of the extended state observer can be placed in the same position (-w0, where w0 is the observer bandwidth), as shown below:

[0061] |sI-(A-LC)|=(s+ω o ) 3 =s 3 +3ω o s 2 +3ω o 2 s+ω o 3 (1.22)

[0062] By comparing the coefficients of the same variables on the right-hand side of formula (1.21) and formula (1.22), we can obtain:

[0063] β1=3ω o β2=3ω o 2 β3=ω o 3 (1.23)

[0064] Observer gain matrix and observer bandwidth ω o The only relevant one, i.e., ω o This is the only parameter that the observer needs to determine.

[0065] Step 6: Design a single-input fuzzy system and perform online self-tuning of the extended state observer parameters. 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.

[0066] Furthermore, such as Figure 2 As shown, the fuzzy system 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 both 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 established fuzzy rules, a minimum inference approach is used for inference calculation. Finally, the extended state observer parameter tuning rules are obtained through fuzzy tuning inference, and the fuzzy system output observer bandwidth ω is obtained using the centroid defuzzification method. o This enables online self-tuning of the observer parameters.

[0067] Based on the above technical solution, the following beneficial effects can be achieved:

[0068] 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;

[0069] 2. Compared with the traditional extended state observer, this method introduces a fuzzy algorithm, which improves the disturbance estimation and compensation capabilities of the extended state observer and realizes the self-tuning of the extended state observer parameters under different system states;

[0070] 3. Compared with traditional disturbance observation compensation methods such as disturbance observers, the extended state observer involved in this invention can simultaneously achieve observation compensation for external disturbances and internal uncertainties of the system, and does not depend on the controlled object model. Attached Figure Description

[0071] Figure 1 This is a control block diagram of the fuzzy extended state observer design method based on linear quadratic optimal control in this invention;

[0072] Figure 2 This is a graph showing the input and output membership functions of the fuzzy system in this invention.

[0073] Figure 3 Comparison of step responses of inertial stable systems under sinusoidal disturbances with a frequency of 1 Hz;

[0074] Figure 4 Comparison of step responses of inertial stable systems under sinusoidal disturbances with a frequency of 3Hz;

[0075] Figure 5Comparison of step responses of inertial stable systems under sinusoidal disturbances with a frequency of 5 Hz;

[0076] Figure 6 A comparison of the step response of an inertial stable system under a step disturbance;

[0077] Figure 7 This is a graph showing the parameter changes of the fuzzy extended state observer for the inertial stable system under a step disturbance in this invention.

[0078] Figure 8 A comparison of the step response of an inertial stable system (step disturbance) under different methods;

[0079] Figure 9 This is a comparison of the step response of an inertial stable system (sinusoidal disturbance) under different methods. Detailed Implementation

[0080] 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:

[0081] Step 1: The inertial stabilization 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 is then obtained through fitting.

[0082]

[0083] The controlled object is transformed into an extended state-space equation form. The extended state-space equations of the system are as follows:

[0084]

[0085] in, C =

[100] ,

[0086] Step 2: Design a traditional linear extended state observer based on the system's extended state-space equations:

[0087]

[0088] According to formula (1.46), the gain matrix L of the traditional extended state observer is:

[0089]

[0090] Step 3: Design a single-input fuzzy system and perform online self-tuning of the observation bandwidth of the extended state observer. For example... Figure 2As shown, the fuzzy system 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 both 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 established fuzzy rules, a minimum inference approach is used for inference calculation. Finally, the extended state observer parameter tuning rules are obtained through fuzzy tuning inference, and the fuzzy system output observer bandwidth ω is obtained using the centroid defuzzification method. o This enables online self-tuning of the observer parameters.

[0091] The parameters ω of the extended state observer o The detailed fuzzy rules are shown in Table 1.

[0092] Table 1. Parameters ω of the Extended State Observer o Fuzzy rule table

[0093]

[0094] Step 4: Based on the extended state observer, design a PID controller to control the system after disturbance compensation. For example... Figure 1 As shown, the LQR method is used to tune the PID controller parameters. Typically, the weighting matrix R = 1. The desired system damping ratio ζ is taken as... cl Natural frequency ω cl And the relative advantage degree m is ζ cl =1.8,ω cl =2, m=100, satisfying m>>ζ cl ω cl This ensures that the closed-loop poles are located at the dominant pole positions.

[0095] Using formula (1.37), the PID controller parameters tuned by the LQR method are as follows:

[0096] [K p K i K d ]=[25961440367.2](1.24)

[0097] To verify the effectiveness of the PID controller tuned by the LQR method in this invention, a PD controller from a traditional extended state observer was designed as a comparison. Based on the conclusions of the literature (Xue W, Bai W, Yang S, et al. ADRC with adaptive extended state observer and its application to air-fuel ratio controling asoline engines[J]. IEEE Transactions on Industrial Electronics, 2015, 62(9): 5847-5857.), the PD controller was 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 The bandwidth ω of the constant observer o =30Hz PD controller, observer bandwidth ω o The control effects of a PID controller with a frequency of 30Hz (controlled by LQR method with parameters tuned) and a PID controller under a fuzzy extended state observer are as follows: Figure 8 , Figure 9 As shown. From Figure 8 , Figure 9 As can be seen, the dynamic response performance indicators such as rise time and settling time, as well as the anti-interference capability of the system, are significantly improved under the PID controller tuned by the LQR method in this invention. Furthermore, the effectiveness of the fuzzy extended state observer design method based on linear quadratic optimal control can be further demonstrated.

[0098] Similarly, to verify the effectiveness of the fuzzy extended state observer in this invention, under the same controller bandwidth ω c Based on the two constant bandwidths ω o =30Hz,ω o The control effects of the 50Hz extended state observer and the fuzzy extended state observer are compared.

[0099] The comparison diagrams of the system step responses of the three extended state observers under sinusoidal perturbations of 1Hz, 3Hz, and 5Hz are shown below. Figure 3 , Figure 4 , Figure 5 As shown in the figure, the fuzzy extended state observer exhibits better disturbance suppression performance for sinusoidal disturbances at these three frequencies. Figure 6The diagram shows a comparison of the step response of an inertial stable system under a step disturbance. It can be seen that the fuzzy extended state observer also exhibits better disturbance suppression performance. Figure 7 The variation of the observer bandwidth of the fuzzy extended state observer is presented. It can be seen that when the system is in a stable tracking state and no step disturbance is introduced, the fuzzy extended state observer uses a smaller bandwidth, achieving stable tracking while exhibiting better filtering performance and reducing the impact of noise on the system. When a step disturbance is introduced into the system, the bandwidth of the fuzzy extended state observer increases to improve the observer's disturbance observation and compensation capabilities, reducing the impact of the disturbance on the system's control performance.

[0100] 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 based on linear quadratic optimal control, applied to inertial stable systems, characterized by: The specific steps are as follows: Step 1: Establish the differential equations for the inertial stable system: (1.1) in, For unknown system parameters, For system outputs and inputs, External disturbances For system known parameters, The disturbance includes both external and internal disturbances; the inertial stabilization system operates at a sampling frequency of 5000Hz, and the frequency response curve of the controlled object is obtained through frequency response testing. The position transfer function of the controlled object is obtained by fitting the curve. Transforming the differential equations into extended state-space equations, the extended state-space equations of the system are as follows: (1.2) in, , , , , ; Step 2: Design a linearly extended state observer based on the system's extended state-space equations: (1.3) in, Let be the observer state vector. For the observer gain matrix that needs to be determined, Input combinations for the observer, For the output of the extended state observer; Step 3: The extended state observer can accurately estimate uncertain disturbances within a certain frequency range. and for extended states Compensation and control signals for: (1.4) Substituting formula (1.4) into formula (1.1), we get: (1.5) As can be seen from equation (1.5), the system is simplified to a standard type with dual integrators in series, thereby reducing the control difficulty; Step 4: Use the Linear Quadratic Optimal Control (LQR) method to tune the PID controller parameters in the control framework to achieve control of the system after disturbance compensation. The controller is a PID controller, as shown below: (1.6) in, , These are proportional elements. Integral elements Differential element The coefficient before; To track error signals, For external disturbance input, and These are the reference signal and the controlled object position output signal, respectively. Let's assume the reference signal... ,but Under these conditions, the controlled system It can be represented as: (1.7) in, (1.8) To obtain a systematic LQR formula, the following quadratic cost function needs to be minimized: (1.9) The minimized state feedback control signal in equation (1.9) is: (1.10) in, Here are the symmetric positive definite solutions to the following Riccati equation: (1.11) In linear quadratic optimal control, the standard practice is to further design the relevant controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged; Assuming, (1.12) Substituting formulas (1.8) and (1.12) into formula (1.10), we get: (1.13) By comparing the coefficients of the same state variables on the right-hand side of formula (1.13) and formula (1.6), we can obtain: (1.14) The corresponding closed-loop system characteristic equations are as follows: (1.15) 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 When it is a 2x2 matrix, the characteristic equation is... As shown below: (1.16) in, ; Given the desired damping ratio and natural frequency of the system; when When the matrix is ​​3x3, the characteristic equation is obtained using the dominant pole placement method. As shown below: (1.17) Among them, the dominant pole Similar to the above, non-dominant poles Distance from dominant pole real part Times, at the same time This ensures that the closed-loop pole is located at the dominant pole position. This non-dominant pole should be far away from the real part of the other two complex (conjugate) closed-loop dominant poles. This satisfies the pole configuration of the PID controller. Its value should be selected to be 3 or more. By comparing the coefficients of the same state variables on the right-hand side of formulas (1.15) and (1.17), we can obtain: (1.18) The remaining elements of matrices P and Q can be obtained by solving the Riccati equation (1.11), as shown below: (1.19) Step 5: Design the extended state observer gain matrix, assuming the error state variable is... Subtracting equation (1.2) from equation (1.3), the observer error matrix equation is: (1.20) 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 characteristic equation corresponding to the observer error matrix equation is shown below: (1.21) The extended state observer, after parameterization, can place the poles of the corresponding characteristic equation in the same location. , The observer bandwidth is as follows: (1.22) Comparing the coefficients of the same variables on the right-hand side of formula (1.21) and formula (1.22), we obtain: (1.23) Observer gain matrix and observer bandwidth Uniquely relevant, that is This is the only parameter that the observer needs to determine; Step 6: Design a single-input fuzzy system and perform online self-tuning of the extended state observer parameters. 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.

2. The design method for a fuzzy extended state observer based on linear quadratic optimal control according to claim 1, characterized in that: A fuzzy system mainly 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, based on the current system error. First, calculate the membership function value for each input. Second, perform inference calculations using a minimum inference approach based on the established fuzzy rules. Finally, obtain the extended state observer parameter tuning rules 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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