RBF-based LQR tracking control method

Abstract

The invention discloses an RBF (Radial Basis Function)-based LQR (Linear Quality Response) tracking control method. The method comprises the following steps: designing an LQR-based PID controller by adopting an existing method; an RBF controller is designed, an RBF neural network is adopted to carry out approximation and compensation on nonlinear parts of controlled objects in PID I-type, II-type and III-type control loops, and LQR tracking of the system on fast signals and improvement of tracking dynamic response performance of the system are achieved. In order to solve the problem that an LQR method depends on model linearity, an RBF neural network is adopted to approach and compensate a nonlinear part in a controlled object so as to improve system performance. Different industrial systems such as under-damping, over-damping and critical damping in PID I-type, II-type and III-type control loops are taken as examples to illustrate the effectiveness of the method, that is, dynamic response performance indexes such as system rise time and adjustment time under the design method are remarkably improved.

Description

Technical Field

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

[0002] In engineering applications, PID control methods are widely used to achieve the tracking and measurement of moving targets by the system. With the development of high-precision machining technology and the continuous expansion of application fields, the system will be affected by friction during shaft rotation, sensor measurement noise, system modeling errors and various nonlinear factors. The traditional PID control method ignores some nonlinear factors in the system during the mathematical modeling process, and the linear controller designed in this way is difficult to significantly improve the tracking performance of the system. At the same time, with the continuous improvement of the maneuverability of the tracking target, the tracking accuracy and tracking ability of the system under the traditional PID control method can no longer meet the actual needs of the system. These problems pose great challenges to the design of control systems.

[0003] Regarding how to improve the tracking capability and tracking accuracy of the system, the method of improving the system type is very effective. Theoretically, the PID I control loop can achieve zero steady-state position error, the PID II control loop can achieve zero steady-state position and velocity errors, and the PID III control loop can achieve zero steady-state position, velocity and acceleration errors. Therefore, high-order control loops have the advantages of tracking faster reference signals and eliminating steady-state high-order errors. However, in academia and industry, the problem of tuning controller parameters in high-order control loops is very difficult. With the continuous development of classical control theory, the linear quadratic optimal control method is considered to be an ideal method for providing linear controller parameter gains. The trade-off between state adjustment requirements and control energy consumption can be controlled by adjusting the weighting matrix. This excellent property encourages control designers to use it in the work of 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, sensor measurement noise, modeling error and various nonlinear factors existing in the system during shaft rotation, 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, type II and type III control loops, and finally realizes LQR tracking (i.e. linear quadratic regulation tracking) of fast signals by the system and improves the dynamic response performance of the system.

[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 the 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 comprises:

[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 in the PID II control loop plus an integral link, and the second-order time-delay transfer function in the PID III control loop plus two integral links;

[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, jerk tracking error and error function, and obtain the model uncertainty based on this Expression 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 An estimated value of is the output of the radial basis function of the RBF network; To overcome the neural network approximation error The robust term of It is an unknown disturbance.

[0015] Furthermore, the nonlinear part is generated by the shaft system 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 consistent with 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, It represents the second-order derivative of the desired acceleration. The meanings of other network inputs are consistent with those in the above PID type I and type II control loops. The RBF control rates in the three types of control loops are ,in is the uncertainty in the model estimates; is the coefficient before the error function r; To overcome the neural network approximation error The outputs of the RBF neural networks in the three types of control loops are ,in, is the ideal weight of the network An estimated value of is the output of the radial basis function of the RBF network.

[0017] Furthermore, 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 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 method of the present invention uses 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 It is the block diagram of LQR control based on RBF;

[0022] Figure 2 It 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;

[0023] Figure 3 It 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;

[0024] Figure 4 It 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;

[0025] Figure 5 It is a comparison diagram of the step response of the system under 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 in conjunction with 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 a PID type I, type II, and type III control loop as an example to illustrate the effectiveness of the method of the present invention.

[0029] Step 1: Design a LQR-based PID controller.

[0030] The LQR-based PID controller part in the PID I type control loop can be designed according 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 part in the PID II and III control loops can be designed according 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] For the 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 delay factor of the system; 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 Respectively represent the desired position and velocity of the system; Represent the actual position and velocity of the system respectively.

[0044] Define the error function:

[0045] (5)

[0046] in .

[0047] According to formula (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] Model uncertainty in practical applications Usually unknown, so the uncertainty needs to be Make an approximation.

[0054] Since neural networks can approximate any continuous nonlinear function with arbitrary precision, have adaptive and self-learning capabilities for complex uncertain problems, have strong information synthesis capabilities, and can coordinate multiple input information relationships well. 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 networkcontrol 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 the hidden layer neurons; 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 in 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 The estimated value of .

[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 They represent the deflection angle, angular velocity and angular acceleration of the system respectively; is the control signal of the system. Assume: , 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 They 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 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 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 object in the above PID III type 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 They 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 the 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 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 proof of the system under LQR-based PID control in PID type I, II, and III control loops can be carried out by referring to the methods in reference 1 (Srivastava, Saurabh, et al. "An optimal PID controller via LQR forstandard 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 proof of the system under the corresponding RBF controller are as follows:

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

[0118] Defining the Lyapunov function for:

[0119] (31)

[0120] Then the derivative of the Lyapunov function is

[0121] (32)

[0122] Substituting 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) can be 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 design neural network adaptation rate is 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. "Adaptive bias 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] Substituting formula (40) into formula (41), we get

[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 simulation effects of different industrial systems such as underdamping, overdamping and critical damping in PID type I, type II and type 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 , the expected closed-loop system damping ratio and the system natural frequency for .

[0152] Step 1: According to the premise of the above system and the methods in reference 1 (Srivastava, Saurabh, et al. "Anoptimal PID controller via LQR for standard second order plus time delaysystems." 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.), design the LQR-based PID controller in PID type I, II, and III control loops;

[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 network is used to approximate and compensate the nonlinear part of the controlled object in PID type I, II and III control loops. The value of the radial basis function parameter of the RBF network is very important for the control of the neural network. If the parameter value is not appropriate, the radial basis function cannot be effectively mapped and the RBF network is invalid.

[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 , the expected closed-loop system damping ratio and the system natural frequency for .

[0168] Step 1: Design LQR-based PID controllers in PID type I, II, and III control loops under the premise of the above system;

[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 similar to the above overdamped system Consistent in.

[0177] The other RBF controller parameters in the PID I 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 critical damping system , its transfer function is as follows:

[0179] (51)

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

[0181] Step 1: Under the premise of the above system, design the LQR-based PID controller in PID type I, type II, and type 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 similar to the above overdamped system Consistent in.

[0190] The other RBF controller parameters in the PID I 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 , the expected closed-loop system damping ratio and the system natural frequency for .

[0194] Step 1: Under the premise of the above system, design the LQR-based PID controller in PID type I, type II, and type 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 similar to the above overdamped system Consistent in.

[0203] The other RBF controller parameters in the PID I 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 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 RBF control stability and convergence proof section.

[0205] The control effects of the PID I, II, and III controllers of the LQR tracking control method based on RBF in 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 After only using the PID I type controller in the embodiment, the system response curve diverges, therefore, Figure 5 The response curve of the system under the PID I type controller is not given. However, when the PID I type 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 It can be seen that after adding RBF to the controller of the PID I control loop, the dynamic response indicators of the system, such as the adjustment time and overshoot, are significantly improved. In other words, the superiority of adding RBF neural network to the LQR-based PID control method in this paper is verified through the simulation of different process objects. Similarly, by comparing the system response curves with and without RBF in the controller of the PID II control loop and the PID III control loop, it can be clearly seen that after adding RBF neural network to the system, the dynamic response indicators of the system, such as the adjustment time and overshoot, are further improved.

[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, such as the system response becomes faster, the system adjustment time becomes shorter, and the tracking of fast signals is achieved.

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

Claims

1. An RBF-based LQR tracking control method, characterized in that: The steps include: Step 1: Design an LQR-based PID controller using existing methods; Step 2: Design an RBF controller; use the 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.

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 step 2 comprises: 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 in the PID II control loop plus an integral link, and the second-order time-delay transfer function in the PID III control loop plus two integral links; 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, jerk tracking error and error function, and obtain the model uncertainty based on this Expression 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 An estimated value of is the output of the radial basis function of the RBF network; To overcome the neural network approximation error The robust term of It is an unknown disturbance.

4. 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.

5. The RBF-based LQR tracking control method according to claim 3, characterized in that: 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 consistent with 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, It represents the second-order derivative of the desired acceleration. The meanings of other network inputs are consistent with those in the above PID type I and type II control loops. The RBF control rates in the three types of control loops are ,in is the uncertainty in the model estimates; is the coefficient before the error function r; To overcome the neural network approximation error The outputs of the RBF neural networks in the three types of control loops are ,in, is the ideal weight of the network An estimated value of is the output of the radial basis function of the RBF network.

6. 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 .

Images