A LQR tracking control method based on RBF

Abstract

The present invention discloses an RBF-based LQR tracking control method. The method comprises the following steps: designing an LQR-based PID controller using existing methods; designing an RBF controller and using an RBF neural network to approximate and compensate for the nonlinear portion of the controlled object in PID type I, II, and III control loops, thereby achieving LQR tracking of fast signals by the system and improving the system's tracking dynamic response performance. To address the issue of the LQR method's reliance on model linearity, the present invention uses an RBF neural network to approximate and compensate for the nonlinear portion of the controlled object to achieve improved system performance. The present invention uses different industrial systems, such as underdamped, overdamped, and critically damped, in PID type I, II, and III control loops, as examples to illustrate the effectiveness of the method. Specifically, dynamic response performance indicators such as system rise time and settling time are significantly improved under the design method.

Description

Technical Field

[0001] In response to the problems of nonlinearity and unmodeled errors in practical applications, the present invention proposes an RBF-based LQR tracking control method to further improve the system's dynamic response performance indicators such as tracking accuracy, rise time, and adjustment time. Background Art

[0002] In engineering applications, PID control methods are widely used to enable systems to track and measure moving targets. With the advancement of high-precision machining technology and the continuous expansion of its application areas, systems are affected by friction during shaft rotation, sensor measurement noise, system modeling errors, and various nonlinear factors. Traditional PID control methods ignore some of the nonlinear factors in the system during mathematical modeling, making it difficult to significantly improve the system's tracking performance using the linear controller designed in this way. Furthermore, with the increasing maneuverability of tracked targets, the tracking accuracy and tracking capabilities of the system under traditional PID control methods no longer meet the actual system requirements. These issues pose significant challenges to control system design.

[0003] Improving the system type is a very effective approach to improving system tracking capabilities and accuracy. Theoretically, a PID I control loop can achieve zero steady-state position error, a PID II control loop can achieve zero steady-state position and velocity errors, and a PID III control loop can achieve zero steady-state position, velocity, and acceleration errors. Therefore, higher-order control loops offer the advantages of tracking faster reference signals and eliminating steady-state higher-order errors. However, tuning controller parameters in high-order control loops remains a challenging problem in both academia and industry. With the continuous development of classical control theory, linear quadratic optimal control methods are considered an ideal approach for providing linear controller parameter gains. By adjusting the weighting matrix, the trade-off between state regulation requirements and control energy consumption can be controlled. This advantageous property has prompted control designers to utilize it in PID controller parameter tuning.

[0004] In response to problems such as friction during shaft rotation, sensor measurement noise, system modeling errors, and various nonlinear factors in the system, neural network control has greatly expanded its application in nonlinear system identification and control due to its highly parallel structure, powerful learning ability, ability to approximate continuous nonlinear functions, and fault tolerance. Summary of the Invention

[0005] In order to solve the problems of friction during shaft rotation, sensor measurement noise, modeling errors and various nonlinear factors in the system, the present invention adopts RBF neural network (i.e. radial basis function neural network) to approximate and compensate the nonlinear part of the controlled object in PID type I, II and III control loops, and ultimately realizes LQR tracking (i.e. linear quadratic regulation tracking) of fast signals and improves the system's tracking dynamic response performance.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] An LQR tracking control method based on RBF includes the following steps:

[0008] Step 1: Design an LQR-based PID controller using existing methods;

[0009] Step 2: Design an RBF controller; use an RBF neural network to approximate and compensate for the nonlinear part of the controlled object in the PID type I, II, and III control loops, to achieve LQR tracking of fast signals and improve the system's tracking dynamic response performance.

[0010] Furthermore, the system of LQR tracking control based on RBF in PID type I, II and III control loops is stable.

[0011] Furthermore, the step 2 includes:

[0012] Step 2-1: Based on the second-order time-delay transfer function in the PID I control loop, the second-order time-delay transfer function plus an integral link in the PID II control loop, and the second-order time-delay transfer function plus two integral links in the PID III control loop;

[0013] Step 2-2: Model the controlled object in the control loop as a state-space equation, define the position tracking error, velocity tracking error, acceleration tracking error, jerk tracking error, and error function, and obtain the model uncertainty based on this. Expressions of

[0014] Step 2-3: Use RBF network to approximate model uncertainty , so that the error function r satisfies ,in is the coefficient before the error function r; satisfy ;at the same time, in , is the ideal weight of the network, is the ideal weight of the network estimated value of; is the output of the radial basis function of the RBF network; To overcome the neural network approximation error Robust terms; It is an unknown disturbance.

[0015] Furthermore, the nonlinear part is generated by shaft rotation friction, sensor measurement noise, modeling error and various nonlinear factors in the system.

[0016] Furthermore, the RBF neural network input in the PID type I control loop is , respectively represent the system error, the derivative of the system error, the second-order derivative of the system error, the desired position, the desired velocity, and the desired acceleration; the RBF neural network input in the PID II control loop is ,in represents the third-order derivative of the system error, represents the derivative of the desired acceleration. The meanings of other network inputs are the same as those in the PID I control loop. The RBF neural network input in the PID III control loop is , in represents the fourth-order derivative of the system error, Represents the second-order derivative of the desired acceleration. The meanings of other network inputs are consistent with those in the above-mentioned PID type I and type II control loops. The RBF control rates in the three types of control loops are ,in is the uncertainty of the model estimates; is the coefficient before the error function r; To overcome the neural network approximation error The output of the RBF neural network in the three types of control loops is ,in, is the ideal weight of the network estimated value of; is the output of the radial basis function of the RBF network.

[0017] Furthermore, the control signal of the control method consists of two parts, one of which is the control signal generated by RBF control and the other is the control signal generated by LQR-PID control; when the error function When the error function When the system control signal is controlled by the LQR-PID control part, They are the unknown disturbances in RBF and neural network approximation error The maximum boundary of .

[0018] Compared with the prior art, the present invention has the following beneficial effects:

[0019] 1. The system of the present invention adopts RBF neural network to approximate and compensate the nonlinear part of the controlled object to achieve the tracking of fast signals and improve the dynamic response performance;

[0020] 2. The method of the present invention significantly improves the dynamic response performance indicators such as the rise time and adjustment time of the system in different industrial systems such as underdamping, overdamping, and critical damping. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 This is the block diagram of RBF-based LQR control;

[0022] Figure 2 The comparison diagram of the step response of the system under RBF-based LQR tracking control of the controlled object in PID type I, II, and III control loops;

[0023] Figure 3 The comparison diagram of the step response of the system under RBF-based LQR tracking control of the controlled object in PID type I, II, and III control loops;

[0024] Figure 4 The comparison diagram of the step response of the system under RBF-based LQR tracking control of the controlled object in PID type I, II, and III control loops;

[0025] Figure 5 The figure is a comparison diagram of the step response of the system under the RBF-based LQR tracking control of the controlled object in PID type I, type II, and type III control loops. DETAILED DESCRIPTION

[0026] The present invention will be further described in detail below with reference to the accompanying drawings.

[0027] The present invention provides an RBF-based LQR tracking control method, which is specifically implemented according to the following steps:

[0028] The present invention takes the design of a composite controller in PID type I, type II, and type III control loops as an example to illustrate the effectiveness of the method of the present invention.

[0029] Step 1: Design a PID controller based on LQR;

[0030] The LQR-based PID controller in the PID type I control loop can be designed by referring to the parameter tuning method in Reference 1 (Srivastava, Saurabh, et al. "An optimal PID controller via LQR for standard second orderplus time delay systems." ISA transactions 60 (2016): 244-253.); the LQR-based PID controller in the PID type II and type III control loops can be designed by referring to the parameter tuning method in Reference 2 (Liu, Chao, et al. "Extending the LQR to the design of PID type-ii and type-iii controlloops." IET Control Theory \& Applications 17.6 (2023): 713-743.).

[0031] Step 2: Design RBF controller;

[0032] Design an RBF controller in a PID type I control loop.

[0033] Second-order time-delay transfer function in PID type I control loop , as follows

[0034] (1)

[0035] in ; are the damping ratio and natural frequency of the open-loop system respectively; is the open-loop gain of the system; is the system delay factor; the controlled object in the above PID I type control loop is modeled as follows

[0036] State space equations:

[0037] (2)

[0038] in denote the deflection angle and velocity of the system respectively; is the control signal of the system.

[0039] Assumptions: , then:

[0040] (3)

[0041] The position tracking error and velocity tracking error of the system are defined as

[0042] (4)

[0043] in They represent the desired position and velocity of the system respectively; Represent the actual position and velocity of the system respectively.

[0044] Define the error function:

[0045] (5)

[0046] in .

[0047] According to formulas (4) and (5), we can get

[0048] (6)

[0049] Through formulas (3), (4), (5), and (6), we can get

[0050] (7)

[0051] in is an unknown disturbance; The model is uncertain and Satisfies the following relationship:

[0052] (8)

[0053] In practical applications, model uncertainty Usually unknown, so the uncertainty needs to be Make an approximation.

[0054] Since neural networks can approximate any continuous nonlinear function with arbitrary precision, they have adaptive and self-learning capabilities for complex uncertain problems, have strong information synthesis capabilities, and can well coordinate the relationship between multiple input information. The ideal RBF network algorithm in Reference 3 (Zabihifar, Seyed Hassan, Arkady Semenovich Yushchenko, and Hamed Navvabi. "Robust control based on adaptive neural network for Rotary inverted pendulumwith oscillation compensation. " Neural Computing and Applications 32 (2020):14667-14679.) and Reference 4 (Liu, Qiong, et al. "Adaptive bias RBF neural network control for a robotic manipulator." Neurocomputing 447 (2021): 213-223.) is

[0055] (9)

[0056] (10)

[0057] in is the input signal of the network; For the Nodes in the hidden layer of the network; is the radial basis function output of the network, and ; For the first The center point vector value of each hidden layer neuron; , For the The width of the Gaussian basis function of each hidden layer neuron; is the model uncertainty; is the ideal weight of the network; is the neural network approximation error.

[0058] Therefore, the RBF network is used to approximate the uncertainty of the model When the network input in the PID I type control loop is

[0059]

[0060] The designed RBF control rate is

[0061] (11)

[0062] in is the uncertainty of the model estimates; is the error function The coefficient before To overcome the neural network approximation error Robust item.

[0063] Substituting formula (11) into formula (7), we can get

[0064] (12)

[0065] in .

[0066] Use RBF network to approximate model uncertainty , the output of the RBF neural network is

[0067] (13)

[0068] in is the ideal weight of the network estimated value.

[0069] Pick , and substitute formula (13) into formula (12), then the error function satisfies

[0070] (14)

[0071] in .

[0072] Next, the RBF controller in the PID II type control loop is designed.

[0073] Adding an integral link to the second-order time-delay transfer function in the PID II control loop It can be expressed as

[0074] (15)

[0075] in The meaning of is the same as that in formula (1). The controlled object in the above PID II control loop is modeled as the state space equation shown below:

[0076] (16)

[0077] in denote the deflection angle, angular velocity and angular acceleration of the system respectively; is the control signal of the system. Assume that: , then:

[0078] (17)

[0079] The position tracking error, velocity tracking error and acceleration tracking error of the system are defined as

[0080] (18)

[0081] in Represent the desired position, velocity, and acceleration of the system respectively; They represent the actual position, velocity, and acceleration of the system respectively.

[0082] Define the error function:

[0083] (19)

[0084] in .

[0085] According to formulas (18) and (19), we can get

[0086] (20)

[0087] Through formulas (17), (18), (19), and (20), we can get

[0088] (twenty one)

[0089] in is an unknown disturbance; The model is uncertain and

[0090] (twenty two)

[0091] The RBF network input in the PID II control loop is , using RBF neural network to approximate the model uncertainty .

[0092] Finally, the RBF controller in the PID III type control loop is designed.

[0093] Second-order time-delay transfer function plus two integral links in a PID III control loop It can be expressed as

[0094] (twenty three)

[0095] in The meaning of is the same as that in formula (1).

[0096] The controlled plant in the above PID III control loop is modeled as the state space equation shown below:

[0097] (twenty four)

[0098] in They represent the system's deflection angle, angular velocity, angular acceleration, and the derivative of angular acceleration, respectively.

[0099] Assumptions: ,but

[0100] (25)

[0101] Define position tracking error, velocity tracking error, acceleration tracking error, and jerk tracking error as

[0102] (26)

[0103] in Represent the desired position, velocity, acceleration, and jerk of the system respectively; They represent the actual position, velocity, acceleration and jerk of the system respectively.

[0104] The error function is defined as follows:

[0105] (27)

[0106] in .

[0107] According to formulas (58) and (59), we can get

[0108] (28)

[0109] According to formulas (57), (58), (59), and (60), the error function satisfies

[0110] (29)

[0111] in is an unknown disturbance, The model is uncertain and satisfies the following relationship:

[0112] (30)

[0113] The RBF network input in the PID III control loop is , using RBF neural network to approximate the model uncertainty .

[0114] After adopting the RBF control rate shown in formula (11) and the RBF neural network output shown in formula (13), the error function also satisfies the relationship in formula (14).

[0115] Step 3: Prove the stability of the system;

[0116] The stability of the system under LQR-based PID control in PID type I, II, and III control loops can be proved by referring to the methods in Reference 1 (Srivastava, Saurabh, et al. "An optimal PID controller via LQR for standard second order plus time delay systems." ISA transactions 60 (2016): 244-253.) and Reference 2 (Liu, Chao, et al. "Extending the LQR to the design of PIDtype-ii and type-iii control loops." IET Control Theory \& Applications 17.6(2023): 713-743.). The stability and convergence of the system under the corresponding RBF controller are proved as follows:

[0117] (1) Take ,exist and situation;

[0118] Define the Lyapunov function for:

[0119] (31)

[0120] Then the derivative of the Lyapunov function is

[0121] (32)

[0122] Substitute formula (14) into formula (32),

[0123] (33)

[0124] We take

[0125] (34)

[0126] That is, the adaptive rate of the RBF neural network is

[0127] (35)

[0128] Then formula (33) is further simplified to

[0129] (36)

[0130] in .

[0131] When the following convergence conditions are met, , that is, the system is stable.

[0132] (37)

[0133] Pick , and situation;

[0134] Define the Lyapunov function in formula (31), then the derivative of the Lyapunov function is

[0135] (38)

[0136] The neural network adaptation rate is designed as shown in formula (35), then

[0137] (39)

[0138] because ,but is bounded. According to the Barbalat lemma in the literature 4 (Liu, Qiong, et al. "Adaptivebias RBF neural network control for a robotic manipulator." Neurocomputing447 (2021): 213-223.), , the system is stable.

[0139] (2) Existence and , considering the robust term situation;

[0140] Define the Lyapunov function of formula (31) and replace the robust term Designed for

[0141] (40)

[0142] Taking the neural network adaptation rate in formula (35) and the RBF control rate in formula (11), and substituting them into the differential of the Lyapunov function, we can get

[0143] (41)

[0144] Substitute formula (40) into formula (41), then

[0145] (42)

[0146] That is, the system is stable.

[0147] At this point, the stability and convergence of the system under the RBF controller have been proven.

[0148] The effectiveness of the method of the present invention is mainly demonstrated by simulating the effects of different industrial systems such as underdamped, overdamped, and critically damped in PID type I, II, and III control loops.

[0149] 1. For over-damped systems , its transfer function is as follows:

[0150] (43)

[0151] In this overdamped system, the relative advantage of the system is taken as , expected closed-loop system damping ratio and the system's natural frequency for .

[0152] Step 1: Design LQR-based PID controllers for PID type I, II, and III control loops based on the above system assumptions and the methods in Reference 1 (Srivastava, Saurabh, et al. "Anoptimal PID controller via LQR for standard second order plus time delay systems." ISA transactions 60 (2016): 244-253.) and Reference 2 (Liu, Chao, et al. "Extending the LQR to the design of PID type-ii and type-iii control loops." IET Control Theory \& Applications 17.6 (2023): 713-743.).

[0153] In a PID type I control loop, the LQR-based PID controller is as follows:

[0154] (44)

[0155] In a PID type II control loop, the LQR-based PID controller is as follows:

[0156] (45)

[0157] In a PID type III control loop, the LQR-based PID controller is as follows:

[0158] (46)

[0159] Step 2: Design RBF controllers in PID type I, II, and III control loops.

[0160] RBF neural networks are used to approximate and compensate for the nonlinearity of the controlled object in PID type I, II, and III control loops. The values ​​of the radial basis function parameters of the RBF network are very important for the control of the neural network. If the parameter values ​​are inappropriate, the radial basis function cannot be effectively mapped, and the RBF network is ineffective.

[0161] In formula (9), the radial basis function of the RBF network The value of is the range of network input value. The initial value of RBF network input value is 0, and the RBF robust term .

[0162] PID I type control loop The other RBF controller parameters are set to .

[0163] PID II type control loop The other RBF controller parameters are set to .

[0164] PID III type control loop The other RBF controller parameters are set to .

[0165] 2. For over-damped systems , its transfer function is as follows:

[0166] (47)

[0167] In this overdamped system, the relative advantage of the system is taken as , expected closed-loop system damping ratio and the system's natural frequency for .

[0168] Step 1: Design LQR-based PID controllers for PID type I, II, and III control loops based on the above system conditions.

[0169] In a PID type I control loop, the LQR-based PID controller is as follows:

[0170] (48)

[0171] In a PID type II control loop, the LQR-based PID controller is as follows:

[0172] (49)

[0173] In a PID type III control loop, the LQR-based PID controller is as follows:

[0174] (50)

[0175] Step 2: Design RBF controllers in PID type I, II, and III control loops.

[0176] The initial value of the RBF network input value is 0, and the RBF robust term , PID type I, II, and III control loops The value of is the same as that of the above overdamped system Consistent in.

[0177] The other RBF controller parameters in the PID I type control loop are set to The other RBF controller parameters in the PID II control loop are set to The other RBF controller parameters in the PID III control loop are set to .

[0178] 3. For critically damped systems , its transfer function is as follows:

[0179] (51)

[0180] In this overdamped system, the relative advantage of the system is taken as , expected closed-loop system damping ratio and the system's natural frequency for .

[0181] Step 1: Based on the above system conditions, design LQR-based PID controllers for PID type I, II, and III control loops according to the methods in Reference 1 (Srivastava, Saurabh, et al. "An optimal PID controller via LQR for standard second order plus time delay systems." ISA transactions 60 (2016): 244-253.) and Reference 2 (Liu, Chao, et al. "Extending the LQR to the design of PID type-ii and type-iii control loops." IET Control Theory \& Applications 17.6 (2023): 713-743.);

[0182] In a PID type I control loop, the LQR-based PID controller is as follows:

[0183] (52)

[0184] In a PID type II control loop, the LQR-based PID controller is as follows:

[0185] (53)

[0186] In a PID type III control loop, the LQR-based PID controller is as follows:

[0187] (54)

[0188] Step 2: Design RBF controllers in PID type I, II, and III control loops.

[0189] The initial value of the RBF network input value is 0, and the RBF robust term , PID type I, II, and III control loops The value of is the same as that of the above overdamped system Consistent in.

[0190] The other RBF controller parameters in the PID I type control loop are set to The other RBF controller parameters in the PID II control loop are set to The other RBF controller parameters in the PID III control loop are set to .

[0191] 4. For underdamped systems , its transfer function is as follows:

[0192] (55)

[0193] In this overdamped system, the relative advantage of the system is taken as , expected closed-loop system damping ratio and the system's natural frequency for .

[0194] Step 1: Based on the above system conditions, design LQR-based PID controllers for PID type I, II, and III control loops according to the methods in Reference 1 (Srivastava, Saurabh, et al. "An optimal PID controller via LQR for standard second order plus time delay systems." ISA transactions 60 (2016): 244-253.) and Reference 2 (Liu, Chao, et al. "Extending the LQR to the design of PID type-ii and type-iii control loops." IET Control Theory \& Applications 17.6 (2023): 713-743.);

[0195] In a PID type I control loop, the LQR-based PID controller is as follows:

[0196] (56)

[0197] In a PID type II control loop, the LQR-based PID controller is as follows:

[0198] (57)

[0199] In a PID type III control loop, the LQR-based PID controller is as follows:

[0200] (58)

[0201] Step 2: Design RBF controllers in PID type I, II, and III control loops.

[0202] The initial value of the RBF network input value is 0, and the RBF robust term , PID type I, II, and III control loops The value of is the same as that of the above overdamped system Consistent in.

[0203] The other RBF controller parameters in the PID I type control loop are set to The other RBF controller parameters in the PID II control loop are set to The other RBF controller parameters in the PID III control loop are set to .

[0204] In the present invention Figure 1 This is the block diagram of LQR tracking control based on RBF. The control signal in the figure is It consists of two parts, one of which is the control signal generated by RBF control and the other is the control signal generated by LQR-PID control. When the system control signal The LQR-PID control and RBF control work together; when the error function When the system control signal The LQR-PID control part plays a role. Error function The derivation of the boundary value is specifically explained in the above proof of RBF control stability and convergence.

[0205] The control effects of the PID type I, type II, and type III controllers of the RBF-based LQR tracking control method of the present invention are as follows: Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 shown. Figure 2 For overdamped controlled objects in PID type I, II, and III control loops Comparison of the system's step response under RBF-based LQR tracking control. Figure 3 For overdamped controlled objects in PID type I, II, and III control loops Comparison of the system's step response under RBF-based LQR tracking control. Figure 4 For critically damped controlled objects in PID type I, II, and III control loops Comparison of the system's step response under RBF-based LQR tracking control. Figure 5 For underdamped controlled objects in PID type I, II, and III control loops Comparison of the step response of the system under RBF-based LQR tracking control. It should be noted that: Figure 5 When only the PID I type controller in the embodiment is used, the system response curve diverges. Therefore, Figure 5 The response curve of the system under the PID I controller is not given. However, when the PID I controller is combined with RBF control, it can be clearly seen that the system response tends to be stable, which is also the advantage of RBF neural network in this process object.

[0206] from Figures 2 to 5 As can be seen from the figure, the addition of an RBF neural network to the controller of the PID I control loop significantly improves the system's dynamic response indicators, such as settling time and overshoot. This means that the superiority of incorporating an RBF neural network into this LQR-based PID control method has been verified through simulations of different process objects. Similarly, by comparing the system response curves of the PID II and PID III control loops with and without the RBF in the controller, it is clear that the addition of the RBF neural network further improves the system's dynamic response indicators, such as settling time and overshoot.

[0207] On the other hand, by comparing the simulation results of different types of control loops, it can be clearly seen that when the system type is improved, the dynamic response performance of the system is also greatly improved. For example, the system response becomes faster, the system adjustment time becomes shorter, and the tracking of fast signals is achieved.

[0208] The above describes the specific implementation methods, processes and effects of the present invention in detail with reference to the accompanying drawings and examples, but the content described is only an embodiment of the method and cannot limit the scope of implementation of the method.

Claims

1. An RBF-based LQR tracking control method, characterized by: The steps include: Step 1: Design an LQR-based PID controller using existing methods; Step 2: Design an RBF controller; use an RBF neural network to approximate and compensate for the nonlinear part of the controlled object in PID type I, II, and III control loops, achieving LQR tracking of fast signals and improving the system's tracking dynamic response performance; The step 2 includes: Step 2-1: Based on the second-order time-delay transfer function in the PID I control loop, the second-order time-delay transfer function plus an integral link in the PID II control loop, and the second-order time-delay transfer function plus two integral links in the PID III control loop; Step 2-2: Model the controlled object in the control loop as a state-space equation, define the position tracking error, velocity tracking error, acceleration tracking error, jerk tracking error, and error function, and obtain the model uncertainty based on this. Expressions of Step 2-3: Use RBF network to approximate model uncertainty , so that the error function r satisfies ,in is the coefficient before the error function r; satisfy ;at the same time, in , is the ideal weight of the network, is the ideal weight of the network estimated value of; is the output of the radial basis function of the RBF network; To overcome the neural network approximation error Robust terms; is an unknown disturbance; Among them, the RBF neural network input in the PID I type control loop is , respectively represent the system error, the derivative of the system error, the second-order derivative of the system error, the desired position, the desired velocity, and the desired acceleration; the RBF neural network input in the PID II control loop is ,in represents the third-order derivative of the system error, represents the derivative of the desired acceleration. The meanings of other network inputs are the same as those in the PID I control loop. The RBF neural network input in the PID III control loop is , in represents the fourth-order derivative of the system error, Represents the second-order derivative of the desired acceleration. The meanings of other network inputs are consistent with those in the above-mentioned PID type I and type II control loops. The RBF control rates in the three types of control loops are ,in is the uncertainty of the model estimates; is the coefficient before the error function r; To overcome the neural network approximation error The output of the RBF neural network in the three types of control loops is ,in, is the ideal weight of the network estimated value of; is the output of the radial basis function of the RBF network.

2. The RBF-based LQR tracking control method according to claim 1, characterized in that: The system of LQR tracking control based on RBF in PID type I, II and III control loops is stable.

3. The RBF-based LQR tracking control method according to claim 1, characterized in that: The nonlinear part is generated by the shaft rotation friction, sensor measurement noise, modeling error and various nonlinear factors in the system.

4. The RBF-based LQR tracking control method according to claim 1, characterized in that: The control signal of the control method is composed of two parts, one of which is the control signal generated by RBF control and the other is the control signal generated by LQR-PID control. When the error function When the system control signal is controlled by the LQR-PID control part, They are the unknown disturbances in RBF and neural network approximation error The maximum boundary of .

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