A neural network-based PID vibration active control and reliability analysis method

Abstract

The application discloses a PID vibration active control and reliability analysis method based on a neural network.In the method, for a low-frequency vibration control problem of a two-degree-of-freedom spring mass damping system, the uncertainty model of the two-degree-of-freedom spring mass damping system is constructed by considering the uncertainty of the mass of the mass block, the spring stiffness and the damping.The incremental PID algorithm is combined with the BP neural network, the self-learning and the adjustment weight function of the neural network are realized, and the optimization of three adjustable parameters of the PID and the vibration control are realized.A reliability analysis method for the vibration control system is provided, and the analysis method can be used in reliability evaluation and parameter optimization of various vibration control systems.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of vibration control system design of two-degree-of-freedom spring mass system, and particularly relates to a PID vibration active control and reliability analysis method based on neural network, which considers the uncertainty of system parameters, and a control system reliability analysis method based on dynamic reliability and Monte Carlo method, thereby providing a reference and evaluation approach for the reliability analysis and design of active control system. BACKGROUND

[0002] Vibration control methods can be generally divided into passive control and active control. The former changes the dynamic characteristics of the controlled object through additional structure, thereby realizing effective control of vibration. The passive control method can effectively suppress high-frequency vibration, but has low dissipation efficiency for low-frequency vibration energy. The vibration active control system calculates driving force by analyzing the displacement signals collected by the sensor and combining the preset control law, thereby realizing vibration suppression. The vibration active control can effectively suppress low-frequency vibration of the system, but the design of the controller generally depends on the accurate dynamic model of the controlled object. When the system parameters have uncertainty, the effect of vibration active control is greatly reduced. With the rapid development of the field of vibration active control, the design of vibration active controller that can adapt to system parameter uncertainty is increasingly important.

[0003] In recent decades, researchers have designed various active control methods for uncertain systems, such as adaptive control, robust control, etc. The adaptive control method is divided into adaptive feedforward control, self-correcting control and model reference adaptive control, which essentially combines online identification of system parameters and controller parameter setting. This method can solve the vibration problem of low-uncertainty systems, but does not have robustness itself. The design of robust control selects linear feedback law, so that the stability or performance of the closed-loop system has certain resistance to disturbance. Since the dispersion of the system is considered in the state space expression of the controlled object, the robust controller can completely suppress vibration within the preset parameter fluctuation range, but the robust control method aims to solve the conservative controller that meets the conditions, and a large amount of other performance is lost to ensure robustness.

[0004] The artificial intelligence control method based on machine learning and reinforcement learning has been gradually applied to temperature control, attitude control and other fields in the past two decades, and has shown good control effect. Artificial neural network as a powerful nonlinear fitting tool in machine learning, it is initially used for identification of controlled object, because only input and output data are needed for training, without analyzing the structure and physical characteristics of the controlled object, thus effectively improving the efficiency of system modeling. In addition, neural network has strong nonlinear fitting ability, so it has even higher accuracy than traditional numerical methods in the representation of nonlinear systems. The design of neural network controller generally refers to the existing control idea, such as using two neural networks as system identifier and controller, realizing more efficient adaptive control through data transmission of weight coefficient, or combining neural network with fuzzy control, realizing conventional fuzzy control through forward propagation and updating fuzzy rules through back propagation.

[0005] So far, BP neural network has certain development in temperature control and other control fields, but in the field of vibration control, there is no perfect active vibration control method considering the uncertainty of system parameters and matching reliable analysis method. In order to fill this gap, the present application takes a two-degree-of-freedom spring mass damping system as the research object, constructs a dynamic model and quantifies the uncertainty of system parameters, combines BP neural network with incremental PID control algorithm to realize adaptive low-frequency active vibration controller design. The present application proposes a matching control system reliability analysis method based on dynamic reliability and Monte Carlo method, which provides a reference and evaluation way for the reliability analysis and design of active control system. SUMMARY

[0006] In order to overcome the shortcomings of the prior art, the present application provides a PID active vibration control and reliability analysis method based on neural network, which fully considers the parameter uncertainty widely existing in engineering practical problems, and provides a feasible design and analysis method for vibration controller design and evaluation field. The results obtained are in line with the actual situation and have strong engineering applicability.

[0007] The technical scheme adopted by the present application to solve the above technical problems is:

[0008] A PID active vibration control and reliability analysis method based on neural network, comprising the following steps:

[0009] First step: establishing the dynamic differential equation of a two-degree-of-freedom spring mass damping system;

[0010] Second step: constructing a PID feedback control framework based on classical active control theory;

[0011] Third step: designing a neural network PID controller based on BP neural network algorithm;

[0012] The fourth step is to analyze the time-varying reliability of the neural network PID controller.

[0013] Further,

[0014] The first step includes:

[0015] The force analysis of the mass m2 is performed to obtain its dynamic differential equation as follows:

[0016]

[0017] Similarly, the dynamic differential equation of m1 is obtained as:

[0018]

[0019] The assembled dynamic differential equation of the two-degree-of-freedom spring-mass-damper system is:

[0020]

[0021] where m i represents the mass, k i represents the spring stiffness, c i represents the damper damping, i = 1, 2; f1 represents the control force applied to the mass m1, and f2 represents the disturbance force for the mass m2; M, C, and K are the total mass matrix, the total damping matrix, and the total stiffness matrix, respectively, x(t) represents the acceleration, velocity, and displacement vectors of the controlled structure, respectively, x(t0) and x0 and are the initial displacement and velocity vectors corresponding to the initial time t0, and x0 and a are the given input conditions of the initial displacement and initial velocity, respectively, f(t) is the external disturbance force vector, and f r (t) represents the control force vector.

[0022] Further, the second step includes:

[0023] According to the motion differential equation, the Laplace transform is performed to obtain:

[0024]

[0025] That is,

[0026]

[0027]

[0028] where x2 represents the displacement vector of the mass m2; x rThe target signal representing vibration suppression, i.e. zero vector; F1 represents the control force vector; F2 represents the external disturbance force vector; G1 represents the PID controller, expressed as:

[0029]

[0030] Wherein, K P , K I , K D represent the gain, the differential, the integral element coefficient respectively;

[0031] Or the control law is expressed as:

[0032]

[0033] Wherein, u(t) represents the control force output by the control system, k p is the proportional coefficient, T i is the integral time constant, T d is the differential time constant, and e(t) represents the control deviation formed by the given value and the actual output value.

[0034] Further, the third step comprises:

[0035] The integral is replaced by summation, and the differential is replaced by difference quotient, and the formula (8) is approximated and transformed as follows:

[0036]

[0037] In the formula, k is the sampling serial number, k = 1, 2, …, T is the sampling period, t is the sampling time, j represents the discrete time point, i.e. j = 1, 2, …, k; e(j) represents the error signal corresponding to the time point j.

[0038] Thus, the position formula PID algorithm is as follows:

[0039]

[0040] Or

[0041]

[0042] Wherein, k d = k P T d , e(k-1) and e(k) are the deviation signals obtained at the (k-1) and k time, respectively, and u(k) is the computer output value at the kth sampling time;

[0043] In order to prevent the error from accumulating in the calculation process, according to the recursive principle, we obtain:

[0044]

[0045] Subtracting equation (9) from equation (10), the incremental PID algorithm is obtained:

[0046]

[0047] The above equation is expressed as:

[0048] u(k) = f[u(k-1), K P ,K I ,K D , e(k), e(k-1), e(k-2)] (14)

[0049] In the equation, f[·] is a nonlinear function, so the optimal control law can be obtained by training the neural network with the help of the nonlinear fitting ability of the BP neural network;

[0050] A three-layer BP neural network is designed, including m input nodes, q hidden layer nodes and 3 output nodes.

[0051] The input of the input layer node O (1) of the BP neural network is:

[0052]

[0053] The output of the input layer node of the network is:

[0054]

[0055] The input and output of the hidden layer node O (2) of the network are:

[0056]

[0057] In the equation, w ij (2) is the hidden layer weighting coefficient, the superscripts (1), (2), and (3) represent the input layer, the hidden layer, and the output layer respectively, and f(·) represents the hidden layer activation function.

[0058] The input and output of the output layer node O (3) of the network are:

[0059]

[0060] In the equation, w ij (2) is the hidden layer weighting coefficient, and g(·) represents the output layer activation function. The corresponding output of the output layer node is the three adjustable parameters K p ,K i ,K d; l = 1, 2, 3; by connecting the above-mentioned 3-layer BP neural network with the incremental PID algorithm, the function of the PID controller is realized; the control effect of the PID controller is determined by parameters K p ,K i ,K d in the three-layer BP neural network, K p ,K i ,K d the value of depends on the weight coefficient w ij (2) and w ij (2) ;

[0061] The performance index is taken as:

[0062]

[0063] Wherein, J represents the performance index, r(k+1) represents the given target value at k+1 time, and y represents the actual output of the control system at k+1 time;

[0064] According to the steepest descent method, the weight function of the network is corrected, and the adjustment formula of the output layer weight coefficient is obtained as:

[0065]

[0066] In the formula, η is a learning rate, representing the speed of network correction, η>0, and α is an inertia coefficient, 0<α<1;

[0067] The adjustment formula of the hidden layer weight coefficient is:

[0068]

[0069] Further, the fourth step comprises:

[0070] The uncertainty of m i is represented as Wherein, m represents the nominal value of the mass block, represents the dispersion degree of m i , and satisfies The following is established: Wherein, c represents the nominal value of the damper, represents the dispersion degree of c i , and satisfies k represents the nominal value of the spring stiffness, represents the dispersion degree of k i , and satisfies

[0071] For the uncertainty of the mass block m1, the uncertainty of the closed-loop system response is represented by Taylor expansion as:

[0072]

[0073] In formula (22), a small perturbation representing parameter m1;

[0074] To measure the reliability of the control system, it is assumed that the system response only once may exceed the threshold value in time [t1, t2], then the limit state equation is defined as follows:

[0075]

[0076] Where, R cr The maximum response allowed by the closed-loop control system, the maximum value of the actual response of the system in time [t1, t2], in order to express simply, define When g>0, the system is safe; when g<0, the system fails; when g=0, the system is in a critical state; therefore, the reliability of the system is defined as:

[0077] P s =Prob(g>0, t∈[t1, t2]) (24)

[0078] Where, Prob(·) represents the probability of system failure, when the system response has multiple times to exceed the threshold value in time [t1, t5], the reliability of time [t1, t5] is defined as:

[0079] P s =P s1 ·P s2 ·····P sp (25)

[0080] Where, p is the number of times that the system response may exceed the given value in [t1, t5], P si (i=1, 2, …, p) represents the probability of system response exceeding the given value in time [t i-1 ,t i ], divided into [t1, t2], [t2, t3], [t3, t4], [t4, t5] four parts, at this time, the reliability of the control system can be obtained by interval method, that is:

[0081]

[0082] In the formula,

[0083] Beneficial effects:

[0084] The application provides a new idea for the vibration controller design of a two-degree-of-freedom spring mass damper system considering system parameter uncertainty, and makes up for and perfects the limitation that the traditional vibration controller design cannot adapt to system parameter changes. The reliability analysis strategy of the constructed control system considers and quantifies system parameter uncertainty on one hand, and defines a limit state function according to actual failure conditions on the other hand, thereby providing a new idea for the vibration active controller design of a two-degree-of-freedom spring mass damper system with parameter uncertainty, and providing a theoretical basis for further reliability evaluation, model verification and active control. BRIEF DESCRIPTION OF DRAWINGS

[0085] Figure 1 is a two-degree-of-freedom spring mass damper system schematic diagram;

[0086] Figure 2 is a two-degree-of-freedom spring mass damper system feedback control framework diagram;

[0087] Figure 3 is a BP neural network schematic diagram;

[0088] Figure 4 is an uncertainty schematic diagram of a closed-loop system vibration response;

[0089] Figure 5 is a schematic diagram of the possible number of times that the system dynamic response crosses a threshold value;

[0090] Figure 6 is a constant parameter PID feedback control framework diagram;

[0091] Figure 7 is a neural network PID feedback control framework diagram;

[0092] Figure 8 is a control effect comparison diagram;

[0093] Figure 9a , Figure 9b , Figure 9c is a PID parameter tuning diagram; wherein, Figure 9a is a k p tuning diagram, Figure 9b is a k i tuning diagram, Figure 9c is a k d tuning diagram;

[0094] Figure 10 is a neural network PID controller dynamic response distribution;

[0095] Figure 11 is a constant parameter PID controller dynamic response distribution;

[0096] Figure 12It is a neural network PID vibration active control design and reliability analysis method flow chart of a two-degree-of-freedom spring mass damper system of the application. DETAILED DESCRIPTION

[0097] The technical solutions in the embodiments of the application will be apparently and completely described in combination with the drawings in the embodiments of the application. Obviously, the described embodiments are only part of the embodiments of the application, rather than all the embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by a person of ordinary skill in the art without any creative effort fall within the protection scope of the application.

[0098] As shown in Figure 12 , the application provides a neural network PID vibration active control design and reliability analysis method of a two-degree-of-freedom spring mass damper system, comprising the following steps:

[0099] Step (1) establishes a dynamic differential equation of the two-degree-of-freedom spring mass damper system. As shown in Figure 1 , the two-degree-of-freedom spring mass damper system is considered, wherein m i represents a mass block, k i represents a spring stiffness, c i represents a damper damping, i=1,2, f1 represents a control force applied to the mass block m1, f2 represents a disturbance force for the mass block m2, x1 represents a displacement of the mass block m1, and x2 represents a displacement of the mass block m2:

[0100] Force analysis is performed on the mass block m2, and a dynamic differential equation thereof is obtained as follows:

[0101]

[0102] Similarly, the dynamic differential equation of m1 is obtained as follows:

[0103]

[0104] The dynamic differential equation of the two-degree-of-freedom spring mass damper system is assembled as follows:

[0105]

[0106] wherein, m i represents a mass block, k i represents a spring stiffness, c i represents a damper damping, i=1,2; f1 represents a control force applied to the mass block m1, f2 represents a disturbance force for the mass block m2; M, C, and K are respectively a total mass matrix, a total damping matrix, and a total stiffness matrix, x(t) represents the acceleration, velocity and displacement vectors of the controlled structure, respectively, x(t0) and x0 and are the given input conditions of initial displacement and initial velocity, respectively, f(t) is the external disturbance force vector, f a (t) represents the control force vector;

[0107] Step (2) establishes an incremental PID feedback control framework based on classical active control theory.

[0108] A two-degree-of-freedom spring-mass-damper system feedback control framework is constructed based on classical control theory, as shown in Figure 2 where x2 represents the displacement vector of the mass m2; x r represents the target signal for vibration suppression, i.e., a zero vector; F1 represents the control force vector; F2 represents the external disturbance force vector; G1 represents the PID controller, which is expressed as: 21 22 represents the transfer function of the disturbance force F2 to the controlled target x2.

[0109] According to the motion differential equation, Laplace transformation thereof can obtain:

[0110]

[0111] where s is a complex variable.

[0112] i.e.

[0113]

[0114]

[0115] where x2 represents the displacement vector of the mass m2; x r represents the target signal for vibration suppression, i.e., a zero vector; F1 represents the control force vector; F2 represents the external disturbance force vector; G1 represents the PID controller, which is expressed as:

[0116]

[0117] where K P , K I , and K D represent the gain, differential, and integral coefficients, respectively, and the control law can also be expressed as:

[0118]

[0119] where u(t) represents the control force output by the control system, k p is the proportional coefficient, and T​i T is integral time constant d e(t) represents control deviation between given value and actual output value.

[0120] Step (3) establishes neural network PID controller based on BP neural network algorithm

[0121] The controller in the present application combines incremental PID algorithm with BP neural network algorithm. Since computer control is a kind of sampling control, control quantity can only be calculated according to deviation value at sampling time. When sampling period is quite short, summation can be used to replace integral, and difference quotient can be used to replace differential, and formula (8) can be transformed as follows:

[0122]

[0123] In the formula, k is sampling serial number, k = 1, 2, …, T is sampling period, t is sampling time, j represents discrete time point, i.e. j = 1, 2, …, k; e(j) represents error signal corresponding to time point j.

[0124] Thus, position PID algorithm is obtained as follows:

[0125]

[0126] Or

[0127]

[0128] Wherein, k d = k P T d , e(k-1) and e(k) are deviation signals obtained at (k-1) and k time, and u(k) is computer output value at k sampling time.

[0129] In order to prevent error from accumulating in calculation process, according to recursion principle, the following formula is obtained:

[0130]

[0131] Subtracting formula (9) from formula (10), incremental PID algorithm is obtained:

[0132]

[0133] The above formula is expressed as:

[0134] u(k) = f[u(k-1), K P , K I , K D , e(k), e(k-1), e(k-2)] (14)

[0135] where f[·] is a nonlinear function, and thus the optimal control law can be obtained by training the neural network with the nonlinear fitting ability of the BP neural network;

[0136] The three-layer BP neural network is designed, including m input nodes, q hidden layer nodes and 3 output nodes;

[0137] The input of the input layer node O (1) of the BP neural network is:

[0138]

[0139] The output of the input layer node of the network is:

[0140]

[0141] The input and output of the hidden layer node O (2) of the network are:

[0142]

[0143] where w ij (2) is a hidden layer weighting coefficient, and the superscripts (1), (2) and (3) represent the input layer, the hidden layer and the output layer respectively, and f(·) represents a hidden layer activation function;

[0144] The input and output of the output layer node O (3) of the network are:

[0145]

[0146] where w ij (2) is a hidden layer weighting coefficient, and g(·) represents an output layer activation function, and the output of the output layer node corresponds to three adjustable parameters K p ,K i ,K d ; l = 1, 2, 3. The function of the PID controller can be realized by connecting the above-mentioned three-layer BP neural network with the incremental PID algorithm. The control effect of the PID controller is determined by the parameters K p ,K i ,K d , and in the three-layer BP neural network, the values of K p ,K i ,K d depend on the network weighting coefficients w ij (2) and w ij (2)Therefore, in order to realize the best control law, it is necessary to establish an optimization method of network weight coefficients, and the method used in the application is as follows:

[0147] Taking the performance index as:

[0148]

[0149] Wherein, J represents the performance index, r(k+1) represents the given target value at k+1 time, and y represents the actual output of the control system at k+1 time.

[0150] According to the steepest descent method, the adjustment formula of the output layer weight coefficient of the network is obtained:

[0151]

[0152] In the formula, η is a learning rate, representing the speed of network correction, η>0, and α is an inertia coefficient, 0<α<1.

[0153] The adjustment formula of the hidden layer weight coefficient is:

[0154]

[0155] Step (4) performs time-varying reliability analysis of the neural network PID controller.

[0156] m i The uncertainty of m is represented as Wherein represents the nominal value of the mass block, represents the dispersion degree of m i , and satisfies The c i = c i (1+p ci ) is established, Wherein, represents the nominal value of the damper, represents the dispersion degree of c i , and satisfies represents the nominal value of the spring stiffness, represents the dispersion degree of k i , and satisfies

[0157] The uncertainty of the system parameters will eventually lead to the uncertainty of the vibration response, and in the case of a determined input excitation, taking the uncertainty of the mass block m1 as an example, the uncertainty of the closed-loop system response is represented by Taylor expansion as:

[0158]

[0159] In formula (22), a small perturbation of the representative parameter m1;

[0160] To measure the reliability of the control system, it is assumed that the system response only once can exceed the threshold value in the time [t1, t2], the following limit state equation is defined:

[0161]

[0162] Where, R cr The maximum response allowed by the closed-loop control system, the maximum value of the actual response of the system in the time [t1, t2], in order to express simply, define When g>0, the system is safe; when g<0, the system fails; when g=0, the system is in a critical state; therefore, the reliability of the system is defined as:

[0163]

[0164] Where, Prob(·) represents the probability of system failure, when the system response has multiple times that can exceed the threshold value in the time [t1, t5], the reliability of the time [t1, t5] is defined as:

[0165] P s =P s1 ·P s2 ·…·P sp (25)

[0166] Where, p is the number of times that the system response can exceed the given value in [t1, t5], P si (i=1, 2, …, p) represents the probability of system response exceeding the given value in the time [t i-1 , t i ], divided into [t1, t2], [t2, t3], [t3, t4], [t4, t5] four parts, at this time, the reliability of the control system can be obtained by interval method, that is:

[0167]

[0168] In the formula,

[0169] Embodiment:

[0170] In order to more fully understand the characteristics of the present application and its applicability to engineering practice, the present application is aimed at the low frequency vibration problem of two degree of freedom spring mass damping system, and the PID control method with fixed parameters and the neural network PID control method are compared and verified. Then, in order to verify the reliability analysis method of the control system, the present application further analyzes and verifies the reliability of the aforementioned spring mass damping system.

[0171] In the control method validity verification, the system parameters used are m1=1.2 kg, m2=1 kg, k1=350 Nm, k2=300 Nm, c1=4 Nsm, c2=3 Nsm - , - , - , - , Thus, the transfer function obtained is:

[0172]

[0173]

[0174] Wherein:

[0175] D(s)=1.2s 4 +10.6s 3 +1022s 2 +2250s+105000

[0176] The Simulink is used to build the fixed parameter PID feedback control framework as shown in Figure 6 , the neural network PID feedback control framework as shown in Figure 7 , and the disturbance signal is set to be a plurality of sine wave signals superimposed, and the specific expression is:

[0177] f2(t)=1500*sin(3.14t)+1750*sin(2.64t)+1125*sin(4.52t)

[0178] The initial parameters of the fixed parameter PID controller and the neural network PID controller are K p =100, K i =40, K d =10, and the control effect comparison is obtained as shown in Figure 8 It can be seen that the neural network PID control effect is better than the fixed parameter PID controller. The neural network PID parameter setting process is shown in Figure 9a , Figure 9b , Figure 9c (the parameter fluctuation process is shown in the figure), and the results show that the neural network PID parameter setting method proposed in the application can ensure the rapid convergence of the PID parameters.

[0179] In the reliability verification embodiment of the active control system, the application assumes that the six system parameters have a dispersion degree of 20%, that is The final response distribution of the neural network PID control system is shown in Figure 10 , and the final response distribution of the constant PID control system is shown in Figure 11 The design requirement is R crAccording to the reliability calculation method of the application, the reliability of the neural network PID control system is 0.7534, and the reliability of the fixed parameter PID control system is 0.4375, that is, the neural network PID control is more effective in dealing with the vibration suppression problem of system parameter uncertainty.

[0180] The part of the application not described in detail belongs to the known technology of those skilled in the art.

[0181] The above is only the specific steps of the application, and does not constitute any limitation on the protection scope of the application; it can be extended to the vibration active control system design and reliability analysis of general objects, and any technical solution formed by equivalent transformation or equivalent replacement falls within the protection scope of the application.

Claims

1. A neural network-based PID vibration active control and reliability analysis method, characterized in that, The method comprises the following steps: Step 1: establishing the dynamic differential equation of a two-degree-of-freedom spring mass damping system; Step 2: constructing a PID feedback control framework based on a classical active control theory; Step 3: designing a neural network PID controller based on a BP neural network algorithm, comprising: such that a representative PID controller is denoted as: （7） wherein , , respectively represent the coefficients of the gain, the derivative, the integral element. Or the control law is expressed as: （8） wherein represents a control force output by the control system, is a proportional coefficient, is an integral time constant, is a differential time constant, represents a control deviation constituted by a given value and an actual output value; The integral is replaced by summation, and the differential is replaced by difference quotient, and the following approximate transformation is performed on formula (8): （9） wherein is a sample number, , is a sample period, is a sample time, represents a discrete time point, i.e. ; represents a discrete time point corresponds to an error signal; Thus, the position formula PID algorithm is as follows: （10） Or （11） wherein , , and are the deviation signals obtained at the and the instants, respectively, is the computer output value at the sampling instant; In order to prevent the error from accumulating in the calculation process, according to the recursive principle, the following is obtained: （12） Subtracting formula (9) from formula (10), the incremental PID algorithm is obtained: （13） The above formula is expressed as: （14） In the formula, is a nonlinear function, so the optimal control law can be obtained by training the neural network with the nonlinear fitting ability of the BP neural network. A three-layer BP neural network is designed, including 4 input nodes, hidden layer nodes and 3 output nodes; BP neural network input layer nodes The input is: （15） The output of the network input layer node is: （16） Network hidden layer nodes The input and output of the network are: （17） wherein are the weights of the hidden layer, the superscript represent the input layer, the hidden layer and the output layer, respectively, represents the activation function of the hidden layer; Network output layer node The input and output are: （18） In the formula, is the output layer weighting coefficient, represents the output layer activation function, and the output of the output layer node is three adjustable parameters ; The function of the PID controller is realized by connecting the above-mentioned three-layer BP neural network with the incremental PID algorithm; the control effect of the PID controller is determined by the parameters In the three-layer BP neural network, the values of the parameters depend on the weight coefficients and of the network; The performance index is taken as: （19） wherein representative performance indicators, represent a target value at a time instant, represent an actual output of the control system at a time instant; According to the steepest descent method, the weight function of the network is corrected, and the adjustment formula of the output layer weight coefficient is obtained as: （20） In the formula, is a learning rate, and represents the speed of network correction, is an inertia coefficient, ;​ The adjustment formula of the weight coefficient of the hidden layer is: (21) Step 4: time-varying reliability analysis of the neural network PID controller.

2. The neural network-based PID vibration active control and reliability analysis method according to claim 1, characterized in that, the first step comprises: The mass Force analysis is performed to obtain the dynamic differential equation as follows: （1） By analogy, we obtain The kinetic differential equation is: （2） Assembling to establish the dynamic differential equation of a two-degree-of-freedom spring mass damping system is: （3） wherein, represents the mass, represents the spring stiffness, represents the damper damping, i = 1, 2; represents the control force applied to the mass , represents the disturbance force to the mass ; are the global mass matrix, the global damping matrix and the global stiffness matrix, respectively, represent the acceleration, velocity and displacement vectors of the controlled structure, respectively, and represent the corresponding displacement and velocity vectors at the initial time , and are the given input conditions of the initial displacement and initial velocity, respectively, is the external disturbance force vector, denotes the control force vector. 3.The PID vibration active control and reliability analysis method based on neural network according to claim 2, wherein, The second step comprises; According to the dynamic differential equation, the Laplace transform is performed to obtain: （4） That is: （5） （6） wherein denotes the displacement vector of the mass ; denotes the target signal for vibration suppression, i.e. the zero vector; denotes the control force vector; denotes the external disturbance force vector.

4. The neural network-based PID vibration active control and reliability analysis method according to claim 3, characterized in that, the fourth step comprises; The uncertainty is expressed as ,in The nominal value representing the mass block. represent The dispersion satisfies ;Establish , ,in, This represents the nominal value of the damper. represent The dispersion satisfies , The nominal value representing the spring stiffness. represent The dispersion satisfies ; For the mass There is uncertainty, the uncertainty of the closed loop system response is expressed by Taylor expansion as: （22） In formula (22), representative parameters a small perturbation of To measure the reliability of the control system, it is assumed that the system response exceeds the threshold value only once in time and the following limit state equation is defined: （23） where, is the maximum response allowed for the closed loop control system, is time is the maximum value of the inner system true response, for simplicity of presentation, define ; when , the system is safe; when , the system fails; when , the system is in a critical state; therefore, the reliability of the system is defined as: （24） wherein, the probability that the system does not fail when, within a time the system's response exceeds the threshold value a number of times, the time reliability is defined as: （25） where, the number of times the system response exceeds a given value within , the probability that the system response exceeds a given value within a time , is divided into four parts, in which case the reliability of the control system can be obtained by the interval method, i.e.: （26） In the formulae, .

Images