Fault-tolerant control method for actuator fault in urban sewage treatment process

Abstract

According to the fault-tolerant control method for the actuator fault in the urban sewage treatment process, stable control over the dissolved oxygen concentration and the nitrate nitrogen concentration in the urban sewage treatment process under the influence of the actuator fault is achieved. Firstly, a nonlinear observer is designed to accurately estimate an actuator fault; secondly, constructing a radial basis function neural network to accurately characterize the time-varying nonlinear dynamic characteristics of the sewage treatment process; a fault-tolerant controller based on self-adaptive dynamic programming is designed, and the problem that the dissolved oxygen concentration and the nitrate nitrogen concentration deviate from set values due to actuator faults is solved jointly by constructing an evaluation network and an execution network, training a neural network by adopting a strategy iterative algorithm, solving an approximate optimal control law and fusing a prior classical controller. Experimental results show that the method can realize fault-tolerant control of the dissolved oxygen concentration and the nitrate nitrogen concentration, and ensures stable operation of the urban sewage treatment process.

Description

Technical Field

[0001] This invention proposes a fault-tolerant control method for actuator failures in urban wastewater treatment processes, enabling stable control of dissolved oxygen and nitrate nitrogen concentrations even in the event of actuator malfunction. As key process variables in urban wastewater treatment, the stable control of dissolved oxygen and nitrate nitrogen concentrations has a significant impact on the real-time attainment of effluent quality standards. This invention belongs to both the research fields of intelligent control and water treatment. Background Technology

[0002] With the acceleration of urbanization, water scarcity and water pollution problems are becoming increasingly serious. Urban wastewater treatment processes, primarily based on activated sludge, play a positive role in addressing water pollution and are of great significance in alleviating water shortages. In wastewater treatment, dissolved oxygen and nitrate nitrogen concentrations are key control parameters, closely related to the biochemical reactions of activated sludge microorganisms. Their stable control directly affects effluent quality compliance and system operating efficiency. However, due to the complex and variable environment of urban wastewater treatment processes, and the long-term operation of equipment in harsh conditions, actuators (aeration pumps and internal return flow regulating valves) are prone to failure, leading to insufficient oxygen supply and internal return flow. This causes dissolved oxygen and nitrate nitrogen concentrations to deviate from set values, making precise tracking and control difficult. Therefore, developing a fault-tolerant control strategy to address actuator failures in urban wastewater treatment processes is of great importance.

[0003] As an advanced intelligent control technology for industrial processes, adaptive dynamic programming (ACT) has been widely applied in fault-tolerant control of nonlinear systems. However, wastewater treatment processes involve complex biological, chemical, and physical reactions, exhibiting more complex dynamic characteristics and significant time-varying and uncertainties. This makes existing fault-tolerant control methods based on ACT difficult to directly address these challenges. Furthermore, classical controllers, proven feasible in actual urban wastewater treatment processes, exist. Utilizing the prior knowledge of existing classical controllers can reduce the design difficulty and learning cycle of ACT based controllers. However, traditional classical controllers generally suffer from insufficient fault tolerance, failing to maintain system performance when actuators fail. Therefore, there is an urgent need to develop a novel fault-tolerant control method that combines classical controllers with ACT. This method can integrate proven classical controller knowledge to construct a more robust control framework, thereby achieving the goal of maintaining stable tracking control even when affected by actuator failures in urban wastewater treatment processes.

[0004] This invention designs a fault-tolerant control method for actuator failures in urban wastewater treatment processes. By establishing a nonlinear observer to estimate actuator failures, constructing a radial basis function neural network model to estimate unknown nonlinear functions, and designing a robust fault-tolerant controller based on adaptive dynamic programming, fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration is achieved. Summary of the Invention

[0005] This invention proposes a fault-tolerant control method for actuator failures in urban sewage treatment processes. It establishes a nonlinear observer to estimate actuator failures, constructs a radial basis function neural network model to estimate unknown nonlinear functions, and designs a fault-tolerant controller based on adaptive dynamic programming. This solves the problem of stable tracking control of urban sewage treatment processes under the influence of actuator failures.

[0006] This invention provides a fault-tolerant control method for actuator failures in urban wastewater treatment processes. The method is characterized by: constructing a control system for an urban wastewater treatment process with actuator failures; establishing a nonlinear observer to estimate the actuator failures; constructing a radial basis function network model to estimate unknown nonlinear functions; and designing a fault-tolerant controller based on adaptive dynamic programming to achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration. The method includes the following steps:

[0007] (1) Constructing a process control system for urban wastewater treatment with actuator failure

[0008] The expression for a municipal wastewater treatment process control system with actuator malfunction is:

[0009] x(k+1)=f(x(k))+g(x(k))(u(k)-u f (k)) (1)

[0010] Where, x(k+1)=[S O5 (k+1),S NO2 (k+1)] T S represents the system state at time k+1. O5 (k+1) represents the dissolved oxygen concentration at time k+1, S NO2 (k+1) represents the nitrate nitrogen concentration at time k+1, T represents the transpose of the vector, and x(k) = [S O5 (k),S NO2 (k)] T S represents the system state at time k. O5 (k) represents the dissolved oxygen concentration at time k, S NO2 (k) represents the nitrate nitrogen concentration at time k, f(x(k)) = [f1(x(k)), f2(x(k))] TLet f1(x(k)) represent the unknown nonlinear function at time k, f2(x(k)) represent the unknown nonlinear function of the dissolved oxygen concentration control loop at time k, and g(x(k)) = diag(8-S) O5 (k), 0.00537S NO2 (k)) represents the coefficient of the control input at time k, and diag represents the diagonal matrix symbol, used to indicate that g(x(k)) is a matrix with 8-S diagonal elements. O5 (k) and 0.00537S NO2 The diagonal matrix of (k), u(k)=[K L a5(k),Q a (k)] T K represents the control input at time k. L a5(k) represents the oxygen transfer coefficient at time k, Q a (k) represents the internal return flow at time k, u f (k) indicates that the actuator fails at time k;

[0011] (2) Establish a nonlinear observer to estimate actuator faults

[0012] Constructing a nonlinear observer for actuator fault u f (k) is estimated, specifically as follows:

[0013]

[0014] in, Let x(k+1) represent the estimated value at time k+1. Let x(k) represent the estimated value at time k. This represents the estimated value of the unknown nonlinear function at time k. U represents time k f The estimated value of (k), κ1 = 0.5I2 represents a positive definite observer coefficient, and I2 represents a 2×2 identity matrix. This represents the estimation error of x(k) at time k;

[0015] design The adaptive update law is:

[0016]

[0017] Where κ2=0.05I2 represents a positive definite observer coefficient;

[0018] (3) Constructing a radial basis function neural network model to estimate unknown nonlinear functions

[0019] The input to the radial basis function neural network model is [α1(k), α2(k), ..., α b (k)] T The output expression of the radial basis function neural network model is:

[0020]

[0021] Where, α l (k) represents the l-th input of the radial basis function neural network model at time k, ω r (k) represents the connection weights between the r-th hidden layer neuron and the output layer neuron in the radial basis function neural network model at time k, c lr (k) represents the center value of the r-th hidden layer neuron corresponding to the l-th input layer neuron in the radial basis function neural network model at time k, σ lr (k) represents the width of the l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k, where l = 1, 2, ..., b represents the number of input layer neurons in the radial basis function neural network model, and r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model. ω r (k),c lr (k),σ lr (k) is randomly assigned a value in [0,1];

[0022] The parameter update strategy for the radial basis function neural network model is designed as follows:

[0023]

[0024] Where E(k) represents the loss function of the radial basis function neural network at time k;

[0025] The unknowns in the nonlinear observer (2) are estimated using a radial basis function neural network (4). Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 20, and the number of neurons in the input layer be b = 12, with the input being [α1(k), α2(k), ..., α6(k)]. T Each element is defined as α z (k) = x(k-(z-1)) (z = 1, 2, ..., 6), representing the system state at time k-(z-1), and the output is... The loss function is in To estimate the error;

[0026] (4) Design a fault-tolerant controller based on adaptive dynamic programming, specifically:

[0027] For the control system of urban wastewater treatment process, there is an original incremental PID controller that can track and control the dissolved oxygen concentration and nitrate nitrogen concentration. However, the controller will fail when the actuator fails. Therefore, an auxiliary controller based on adaptive dynamic programming is designed to achieve fault-tolerant control.

[0028] The fault-tolerant controller consists of two parts: an incremental PID controller and an auxiliary controller based on adaptive dynamic programming.

[0029] ① Establish an incremental PID controller, specifically as follows:

[0030] u P (k)=K p (e(k)-e(k-1))+K i (e(k))+K d (e(k)-2e(k-1)+e(k-2)) (8)

[0031] Among them, u P (k)=[K L a 5(P) (k),Q a(P) (k)] T K represents the control increment given by the incremental PID controller at time k. L a 5(o) (k) represents the increment of the oxygen transfer coefficient given by the incremental PID controller at time k, Q a(o) (k) represents the increment of the internal return flow rate given by the incremental PID controller at time k, K p =[200,50000] T K represents the proportionality coefficient. i =[15,5000] T K represents the integral coefficient. d =[2,400] T Denotes the differential coefficient, e(k) = x(k) - x d (k) represents the tracking error of the system state at time k, x d (k) represents the setpoint of the system state at time k, e(k) = x(k-1) - x d (k-1) represents the tracking error of the system state at time k-1, x d (k-1) represents the setpoint of the system state at time k-1, e(k-2) = x(k-2) - x d (k-2) represents the tracking error of the system state at time k, x d (k-2) represents the setpoint of the system state at time k-2, x(k), x(k-1), and x(k-2) are the system states at times k, k-1, and k-2, respectively. d (k), xd (k-1) and x d (k-2) represents the setpoints for the system state at times k, k-1, and k-2;

[0032] ② Design an auxiliary controller based on adaptive dynamic programming, specifically:

[0033] For the aforementioned fault-tolerant control problem, the goal is to find a suitable control input that minimizes the cost function, the expression of which is as follows:

[0034]

[0035] Where τ = k, k+1, k+2, ... represents any time from k onwards, e(τ) represents the tracking error of the system state at time τ, u(τ) represents the control input at time τ, and U(e(k), u(k)) = e T (k)I2e(k)+u T (k)(0.05I2)u(k) is the utility function at time k, and ρ represents the weighting coefficient of the estimated value of actuator failure. This represents the estimated value of the actuator fault at time k obtained by the nonlinear observer (2)-(3);

[0036] According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation:

[0037]

[0038] Among them, Q * (e(k),u(k)) and Q * (e(k+1),u(k+1)) represent the optimal cost functions at times k and k+1, respectively;

[0039] The optimal control input is solved using the following formula:

[0040]

[0041] Among them, u * (k) represent the optimal control input at time k. This means finding the solution that makes the optimal cost function Q * (e(k),u(k)) represents the minimum control input u(k);

[0042] Construct an iterative adaptive dynamic programming framework to solve for the optimal control input, where i = 0, 1, 2, ... i max Indicates the iteration step number, i max This indicates that, given the maximum number of iterations, at each time step, u0(k) = [0,0] T Begin the iterative process and solve for the iterative cost function:

[0043]

[0044] Among them, Q i (e(k),u(k)) and Q i (e(k+1),u(k+1)) represent the cost functions of the i-th iteration at times k and k+1, respectively;

[0045] Update iterative control input:

[0046]

[0047] Among them, u i+1 (k) represents the control input for the (i+1)th iteration at time k;

[0048] The above iterative process is expressed as:

[0049] u0(k),→Q0(e(k),u(k))→u1(k)→...→u i (k)→Q i+1 (e(k),u(k))→... (14)

[0050] Where Q0(e(k),u(k)) and Q i+1 (e(k), u(k)) represent the cost functions for the 0th and (i+1)th iterations at time k, respectively, where u0(k), u1(k), and u... i+1 (k) represents the control input at time k for the 0th, 1st, and (i+1)th iterations;

[0051] When ||Q i+1 (e(k),u(k))-Q i (e(k),u(k))||<10 -3 Or i reaches the maximum number of iterations i max When the iteration stops and the approximate optimal control input u is output, the iteration is stopped. * (k); otherwise, let the iteration step number i = i + 1, and continue the iteration process shown in formulas (12)-(13);

[0052] The control input at time k+1 is calculated as follows:

[0053]

[0054] ③ Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows:

[0055] First, since the cost function cannot be directly obtained, an evaluation network is constructed based on a radial basis function neural network (4) to estimate the cost function (9). The number of neurons in the hidden layer of the radial basis function neural network model is a = 14, and the number of neurons in the input layer of the radial basis function neural network model is b = 4. The input is [α1(k), α2(k)]. T =[e(k),u i (k)] T , where u i (k) represents the control input at time k and the i-th iteration, and the output is: The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function E(k) = 1 / 2e c(i) 2 (k), where e c(i) (k) represents the estimation error, expressed as follows:

[0056]

[0057] Then, based on the radial basis function neural network (4), an execution network is constructed to estimate the control input. Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 10, the number of neurons in the input layer of the radial basis function neural network model be b = 2, and the input be α1(k) = e(k). The output of each iteration step is... The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function being E(k) = 1 / 2e a(i) 2 (k), where To estimate the error;

[0058] In the neural network implementation of the iterative adaptive dynamic programming algorithm, when the iterative algorithm satisfies the convergence requirement, that is... Or i reaches the maximum number of iterations i max Sometimes, This allows us to obtain approximately optimal control input.

[0059] Finally, according to formula (15), the optimal fault-tolerant control law u(k+1) at time k+1 is calculated:

[0060]

[0061] (5) The control law u(k+1) includes the oxygen transfer coefficient and internal return flow required to handle the actuator failure at time k+1. The programmable logic controller controls the frequency of the frequency converter according to the calculated oxygen transfer coefficient. The frequency converter controls the aeration volume by adjusting the speed of the blower. The electric regulating valve controls the internal return flow by adjusting the valve opening according to the calculated internal return flow. Finally, the fault-tolerant control of the urban sewage treatment process is achieved by regulating the dissolved oxygen concentration and nitrate nitrogen concentration.

[0062] The inventiveness of this invention is mainly reflected in:

[0063] (1) This invention designs a nonlinear observer to estimate actuator faults and constructs a radial basis function neural network to estimate unknown nonlinear functions, providing necessary information for the design of cost functions of fault-tolerant controllers based on adaptive dynamic programming;

[0064] (2) In view of the problem that the dissolved oxygen concentration and nitrate nitrogen concentration deviate from the set value due to actuator failure, the present invention designs a fault-tolerant control method based on adaptive dynamic programming. By constructing and training the evaluation network and the execution network, the cost function is minimized, and then the optimal control law is obtained. This makes up for the lack of fault tolerance of the classical controller and works with the classical controller to achieve stable control of dissolved oxygen concentration and nitrate nitrogen concentration. Attached Figure Description

[0065] Figure 1 This is a graph showing the control results of dissolved oxygen concentration according to the present invention;

[0066] Figure 2 This is an error graph showing the control results of dissolved oxygen concentration according to the present invention;

[0067] Figure 3 This is a graph showing the control results of nitrate nitrogen concentration according to the present invention;

[0068] Figure 4 This is an error graph showing the control results of nitrate nitrogen concentration according to the present invention. Detailed Implementation

[0069] A fault-tolerant control method for actuator failure in urban wastewater treatment processes is characterized by: constructing a control system for urban wastewater treatment processes with actuator failures, establishing a nonlinear observer to estimate actuator failures, constructing a radial basis function network model to estimate unknown nonlinear functions, and designing a fault-tolerant controller based on adaptive dynamic programming to achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration; including the following steps:

[0070] (1) Constructing a process control system for urban wastewater treatment with actuator failure

[0071] The expression for a municipal wastewater treatment process control system with actuator malfunction is:

[0072] x(k+1)=f(x(k))+g(x(k))(u(k)-u f (k)) (18)

[0073] Where, x(k+1)=[S O5 (k+1),S NO2 (k+1)] TS represents the system state at time k+1. O5 (k+1) represents the dissolved oxygen concentration at time k+1, S NO2 (k+1) represents the nitrate nitrogen concentration at time k+1, T represents the transpose of the vector, and x(k) = [S O5 (k),S NO2 (k)] T S represents the system state at time k. O5 (k) represents the dissolved oxygen concentration at time k, S NO2 (k) represents the nitrate nitrogen concentration at time k, f(x(k)) = [f1(x(k)), f2(x(k))] T Let f1(x(k)) represent the unknown nonlinear function at time k, f2(x(k)) represent the unknown nonlinear function of the dissolved oxygen concentration control loop at time k, and g(x(k)) = diag(8-S) O5 (k), 0.00537S NO2 (k)) represents the coefficient of the control input at time k, and diag represents the diagonal matrix symbol, used to indicate that g(x(k)) is a matrix with 8-S diagonal elements. O5 (k) and 0.00537S NO2 The diagonal matrix of (k), u(k) = [K L a5(k),Q a (k)] T K represents the control input at time k. L a5(k) represents the oxygen transfer coefficient at time k, Q a (k) represents the internal return flow at time k, u f (k) indicates that the actuator fails at time k;

[0074] (2) Establish a nonlinear observer to estimate actuator faults

[0075] Constructing a nonlinear observer for actuator fault u f (k) is estimated, specifically as follows:

[0076]

[0077] in, Let x(k+1) represent the estimated value at time k+1. Let x(k) represent the estimated value at time k. This represents the estimated value of the unknown nonlinear function at time k. U represents time k f The estimated value of (k), κ1 = 0.5I2 represents a positive definite observer coefficient, and I2 represents a 2×2 identity matrix. This represents the estimation error of x(k) at time k;

[0078] design The adaptive update law is:

[0079]

[0080] Where κ2=0.05I2 represents a positive definite observer coefficient;

[0081] (3) Constructing a radial basis function neural network model to estimate unknown nonlinear functions

[0082] The input to the radial basis function neural network model is [α1(k), α2(k), ..., α b (k)] T The output expression of the radial basis function neural network model is:

[0083]

[0084] Where, α l (k) represents the l-th input of the radial basis function neural network model at time k, ω r (k) represents the connection weights between the r-th hidden layer neuron and the output layer neuron in the radial basis function neural network model at time k, c lr (k) represents the center value of the r-th hidden layer neuron corresponding to the l-th input layer neuron in the radial basis function neural network model at time k, σ lr (k) represents the width of the l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k, where l = 1, 2, ..., b represents the number of input layer neurons in the radial basis function neural network model, and r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model. ω r (k),c lr (k),σ lr (k) is randomly assigned a value in [0,1];

[0085] The parameter update strategy for the radial basis function neural network model is designed as follows:

[0086]

[0087] Where E(k) represents the loss function of the radial basis function neural network at time k;

[0088] The unknowns in the nonlinear observer (2) are estimated using a radial basis function neural network (4). Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 20, and the number of neurons in the input layer be b = 12, with the input being [α1(k), α2(k), ..., α6(k)]. T Each element is defined as α z (k) = x(k-(z-1)) (z = 1, 2, ..., 6), representing the system state at time k-(z-1), and the output is... The loss function is in To estimate the error;

[0089] (4) Design a fault-tolerant controller based on adaptive dynamic programming, specifically:

[0090] For the control system of urban wastewater treatment process, there is an original incremental PID controller that can track and control the dissolved oxygen concentration and nitrate nitrogen concentration. However, the controller will fail when the actuator fails. Therefore, an auxiliary controller based on adaptive dynamic programming is designed to achieve fault-tolerant control.

[0091] The fault-tolerant controller consists of two parts: an incremental PID controller and an auxiliary controller based on adaptive dynamic programming.

[0092] ① Establish an incremental PID controller, specifically as follows:

[0093] u P (k)=K p (e(k)-e(k-1))+K i (e(k))+K d (e(k)-2e(k-1)+e(k-2)) (25)

[0094] Among them, u P (k)=[K L a 5(P) (k),Q a(P) (k)] T K represents the control increment given by the incremental PID controller at time k. L a 5(o) (k) represents the increment of the oxygen transfer coefficient given by the incremental PID controller at time k, Q a(o) (k) represents the increment of the internal return flow rate given by the incremental PID controller at time k, K p =[200,50000] T K represents the proportionality coefficient. i =[15,5000] T K represents the integral coefficient. d =[2,400] TDenotes the differential coefficient, e(k) = x(k) - x d (k) represents the tracking error of the system state at time k, x d (k) represents the setpoint of the system state at time k, e(k) = x(k-1) - x d (k-1) represents the tracking error of the system state at time k-1, x d (k-1) represents the setpoint of the system state at time k-1, e(k-2) = x(k-2) - x d (k-2) represents the tracking error of the system state at time k, x d (k-2) represents the setpoint of the system state at time k-2, x(k), x(k-1), and x(k-2) are the system states at times k, k-1, and k-2, respectively. d (k), x d (k-1) and x d (k-2) represents the setpoints for the system state at times k, k-1, and k-2;

[0095] ② Design an auxiliary controller based on adaptive dynamic programming, specifically:

[0096] For the aforementioned fault-tolerant control problem, the goal is to find a suitable control input that minimizes the cost function, the expression of which is as follows:

[0097]

[0098] Where τ = k, k+1, k+2, ... represents any time from k onwards, e(τ) represents the tracking error of the system state at time τ, u(τ) represents the control input at time τ, and U(e(k), u(k)) = e T (k)I2e(k)+u T (k)(0.05I2)u(k) is the utility function at time k, and ρ represents the weighting coefficient of the estimated value of the actuator fault. In the example, it is taken as 2, but it is not limited to this. This represents the estimated value of the actuator fault at time k obtained by the nonlinear observer (2)-(3);

[0099] According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation:

[0100]

[0101] Among them, Q * (e(k),u(k)) and Q * (e(k+1),u(k+1)) represent the optimal cost functions at times k and k+1, respectively;

[0102] The optimal control input is solved using the following formula:

[0103]

[0104] Among them, u * (k) represent the optimal control input at time k. This means finding the solution that makes the optimal cost function Q * (e(k),u(k)) represents the minimum control input u(k);

[0105] Construct an iterative adaptive dynamic programming framework to solve for the optimal control input, where i = 0, 1, 2, ... i max Indicates the iteration step number, i max This indicates that, given the maximum number of iterations, at each time step, u0(k) = [0,0] T Begin the iterative process and solve for the iterative cost function:

[0106]

[0107] Among them, Q i (e(k),u(k)) and Q i (e(k+1),u(k+1)) represent the cost functions of the i-th iteration at times k and k+1, respectively;

[0108] Update iterative control input:

[0109]

[0110] Among them, u i+1 (k) represents the control input for the (i+1)th iteration at time k;

[0111] The above iterative process is expressed as:

[0112] u0(k),→Q0(e(k),u(k))→u1(k)→...→u i (k)→Q i+1 (e(k),u(k))→... (31)

[0113] Where Q0(e(k),u(k)) and Q i+1 (e(k), u(k)) represent the cost functions for the 0th and (i+1)th iterations at time k, respectively, where u0(k), u1(k), and u... i+1 (k) represents the control input at time k for the 0th, 1st, and (i+1)th iterations;

[0114] When ||Q i+1 (e(k),u(k))-Q i (e(k),u(k))||<10 -3 Or i reaches the maximum number of iterations imax When the iteration stops and the approximate optimal control input u is output, the iteration is stopped. * (k); otherwise, let the iteration step number i = i + 1, and continue the iteration process shown in formulas (12)-(13);

[0115] The control input at time k+1 is calculated as follows:

[0116]

[0117] ③ Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows:

[0118] First, since the cost function cannot be directly obtained, an evaluation network is constructed based on a radial basis function neural network (4) to estimate the cost function (9). The number of neurons in the hidden layer of the radial basis function neural network model is a = 14, and the number of neurons in the input layer of the radial basis function neural network model is b = 4. The input is [α1(k), α2(k)]. T =[e(k),u i (k)] T , where u i (k) represents the control input at time k and the i-th iteration, and the output is: The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function E(k) = 1 / 2e c(i) 2 (k), where e c(i) (k) represents the estimation error, expressed as follows:

[0119]

[0120] Then, based on the radial basis function neural network (4), an execution network is constructed to estimate the control input. Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 10, the number of neurons in the input layer of the radial basis function neural network model be b = 2, and the input be α1(k) = e(k). The output of each iteration step is... The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function being E(k) = 1 / 2e a(i) 2 (k), where To estimate the error;

[0121] In the neural network implementation of the iterative adaptive dynamic programming algorithm, when the iterative algorithm satisfies the convergence requirement, that is... Or i reaches the maximum number of iterations i max Sometimes, This allows us to obtain approximately optimal control input. Maximum number of iterations i maxIn this example, we take 20; however, we are not limited to this. Finally, according to formula (15), we calculate the optimal fault-tolerant control law u(k+1) at time k+1:

[0122]

[0123] (5) Achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration.

[0124] The control law u(k+1) includes the oxygen transfer coefficient and internal return flow required to handle actuator failure at time k+1. The programmable logic controller controls the frequency of the frequency converter according to the calculated oxygen transfer coefficient. The frequency converter controls the aeration volume by adjusting the speed of the blower. The electric regulating valve controls the internal return flow by adjusting the valve opening according to the calculated internal return flow. Finally, the fault-tolerant control of the urban sewage treatment process is achieved by regulating the dissolved oxygen concentration and nitrate nitrogen concentration. Figure 1 The display shows the tracking and control results of dissolved oxygen concentration. X-axis: time, in days; Y-axis: dissolved oxygen concentration value, in milligrams per liter. Figure 2 Displays the error between the actual dissolved oxygen concentration and the set dissolved oxygen concentration. X-axis: time, in days; Y-axis: dissolved oxygen concentration error value, in milligrams per liter. Figure 3 This displays the tracking and control results of nitrate nitrogen concentration. X-axis: time (days); Y-axis: nitrate nitrogen concentration (mg / L). Figure 4 The error between the actual nitrate nitrogen concentration and the set dissolved oxygen concentration is displayed. X-axis: time, in days; Y-axis: nitrate nitrogen concentration error value, in milligrams per liter. The results demonstrate the effectiveness of the method.

Claims

1. A fault-tolerant control method for actuator failure in urban wastewater treatment processes, characterized in that: A process control system for urban wastewater treatment with actuator failures is constructed. A nonlinear observer is established to estimate the actuator failures. A radial basis function network model is built to estimate unknown nonlinear functions. A fault-tolerant controller based on adaptive dynamic programming is designed to achieve fault-tolerant control of dissolved oxygen and nitrate nitrogen concentrations. The process includes the following steps: (1) Constructing a process control system for urban wastewater treatment with actuator failure The expression for a municipal wastewater treatment process control system with actuator malfunction is: x(k+1)=f(x(k))+g(x(k))(u(k)-u f (k)) (1) Where, x(k+1)=[S O5 (k+1),S NO2 (k+1)] T S represents the system state at time k+1. O5 (k+1) represents the dissolved oxygen concentration at time k+1, S NO2 (k+1) represents the nitrate nitrogen concentration at time k+1, T represents the transpose of the vector, and x(k) = [S O5 (k),S NO2 (k)] T S represents the system state at time k. O5 (k) represents the dissolved oxygen concentration at time k, S NO2 (k) represents the nitrate nitrogen concentration at time k, f(x(k)) = [f1(x(k)), f2(x(k))] T Let f1(x(k)) represent the unknown nonlinear function at time k, f2(x(k)) represent the unknown nonlinear function of the dissolved oxygen concentration control loop at time k, and g(x(k)) = diag(8-S) O5 (k), 0.00537S NO2 (k)) represents the coefficient of the control input at time k, and diag represents the diagonal matrix symbol, used to indicate that g(x(k)) is a matrix with 8-S diagonal elements. O5 (k) and 0.00537S NO2 The diagonal matrix of (k), u(k)=[K L a5(k),Q a (k)] T K represents the control input at time k. L a5(k) represents the oxygen transfer coefficient at time k, Q a (k) represents the internal return flow at time k, u f (k) indicates that the actuator fails at time k; (2) Establish a nonlinear observer to estimate actuator faults Constructing a nonlinear observer for actuator fault u f (k) is estimated, specifically as follows: in, Let x(k+1) represent the estimated value at time k+1. Let x(k) represent the estimated value at time k. This represents the estimated value of the unknown nonlinear function at time k. U represents time k f The estimated value of (k), κ1 = 0.5I2 represents a positive definite observer coefficient, and I2 represents a 2×2 identity matrix. This represents the estimation error of x(k) at time k; design The adaptive update law is: Where κ2=0.05I2 represents a positive definite observer coefficient; (3) Constructing a radial basis function neural network model to estimate unknown nonlinear functions The input to the radial basis function neural network model is [α1(k), α2(k), ..., α b (k)] T The output expression of the radial basis function neural network model is: Where, α l (k) represents the l-th input of the radial basis function neural network model at time k, ω r (k) represents the connection weights between the r-th hidden layer neuron and the output layer neuron in the radial basis function neural network model at time k, c lr (k) represents the center value of the r-th hidden layer neuron corresponding to the l-th input layer neuron in the radial basis function neural network model at time k, σ lr (k) represents the width of the l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k, where l = 1, 2, ..., b represents the number of input layer neurons in the radial basis function neural network model, and r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model. ω r (k),c lr (k),σ lr (k) is randomly assigned a value in [0,1]; The parameter update strategy for the radial basis function neural network model is designed as follows: Where E(k) represents the loss function of the radial basis function neural network at time k; The unknowns in the nonlinear observer (2) are estimated using a radial basis function neural network (4). Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 20, and the number of neurons in the input layer be b = 12, with the input being [α1(k), α2(k), ..., α6(k)]. T Each element is defined as α z (k) = x(k-(z-1)) (z = 1, 2, ..., 6), representing the system state at time k-(z-1), and the output is... The loss function is in To estimate the error; (4) Design a fault-tolerant controller based on adaptive dynamic programming, specifically: For the control system of urban wastewater treatment process, there is an original incremental PID controller that can track and control the dissolved oxygen concentration and nitrate nitrogen concentration. However, the controller will fail when the actuator fails. Therefore, an auxiliary controller based on adaptive dynamic programming is designed to achieve fault-tolerant control. The fault-tolerant controller consists of two parts: an incremental PID controller and an auxiliary controller based on adaptive dynamic programming. ① Establish an incremental PID controller, specifically as follows: u P (k)=K p (e(k)-e(k-1))+K i (e(k))+K d (e(k)-2e(k-1)+e(k-2)) (8) Among them, u P (k)=[K L a 5(P) (k),Q a(P) (k)] T K represents the control increment given by the incremental PID controller at time k. L a 5(o) (k) represents the increment of the oxygen transfer coefficient given by the incremental PID controller at time k, Q a(o) (k) represents the increment of the internal return flow rate given by the incremental PID controller at time k, K p =[200,50000] T K represents the proportionality coefficient. i =[15,5000] T K represents the integral coefficient. d =[2,400] T Denotes the differential coefficient, e(k) = x(k) - x d (k) represents the tracking error of the system state at time k, x d (k) represents the setpoint of the system state at time k, e(k) = x(k-1) - x d (k-1) represents the tracking error of the system state at time k-1, x d (k-1) represents the setpoint of the system state at time k-1, e(k-2) = x(k-2) - x d (k-2) represents the tracking error of the system state at time k, x d (k-2) represents the setpoint of the system state at time k-2, x(k), x(k-1), and x(k-2) are the system states at times k, k-1, and k-2, respectively. d (k), x d (k-1) and x d (k-2) represents the setpoints for the system state at times k, k-1, and k-2; ② Design an auxiliary controller based on adaptive dynamic programming, specifically: For the aforementioned fault-tolerant control problem, the goal is to find a suitable control input that minimizes the cost function, the expression of which is as follows: Where τ = k, k+1, k+2, ... represents any time from k onwards, e(τ) represents the tracking error of the system state at time τ, u(τ) represents the control input at time τ, and U(e(k), u(k)) = e T (k)I2e(k)+u T (k)(0.05I2)u(k) is the utility function at time k, and ρ represents the weighting coefficient of the estimated value of actuator failure. This represents the estimated value of the actuator fault at time k obtained by the nonlinear observer (2)-(3); According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation: Among them, Q * (e(k),u(k)) and Q * (e(k+1),u(k+1)) represent the optimal cost functions at times k and k+1, respectively; The optimal control input is solved using the following formula: Among them, u * (k) represent the optimal control input at time k. This means finding the solution that makes the optimal cost function Q * (e(k),u(k)) represents the minimum control input u(k); Construct an iterative adaptive dynamic programming framework to solve for the optimal control input, where i = 0, 1, 2, ... i max Indicates the iteration step number, i max This indicates that, given the maximum number of iterations, at each time step, u0(k) = [0,0] T Begin the iterative process and solve for the iterative cost function: Among them, Q i (e(k),u(k)) and Q i (e(k+1),u(k+1)) represent the cost functions of the i-th iteration at times k and k+1, respectively; Update iterative control input: Among them, u i+1 (k) represents the control input for the (i+1)th iteration at time k; The above iterative process is expressed as: u0(k),→Q0(e(k),u(k))→u1(k)→...→u i (k)→Q i+1 (e(k),u(k))→... (14) Where Q0(e(k),u(k)) and Q i+1 (e(k), u(k)) represent the cost functions for the 0th and (i+1)th iterations at time k, respectively, where u0(k), u1(k), and u... i+1 (k) represents the control input at time k for the 0th, 1st, and (i+1)th iterations; When ||Q i+1 (e(k),u(k))-Q i (e(k),u(k))||<10 -3 Or i reaches the maximum number of iterations i max When the iteration stops and the approximate optimal control input u is output, the iteration is stopped. * (k); otherwise, let the iteration step number i = i + 1, and continue the iteration process shown in formulas (12)-(13); The control input at time k+1 is calculated as follows: ③ Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows: First, since the cost function cannot be directly obtained, an evaluation network is constructed based on a radial basis function neural network (4) to estimate the cost function (9). The number of neurons in the hidden layer of the radial basis function neural network model is a = 14, and the number of neurons in the input layer of the radial basis function neural network model is b = 4. The input is [α1(k), α2(k)]. T =[e(k),u i (k)] T , where u i (k) represents the control input at time k and the i-th iteration, and the output is: The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function E(k) = 1 / 2e c(i) 2 (k), where e c(i) (k) represents the estimation error, expressed as follows: Then, based on the radial basis function neural network (4), an execution network is constructed to estimate the control input. Let the number of neurons in the hidden layer of the radial basis function neural network model be a = 10, the number of neurons in the input layer of the radial basis function neural network model be b = 2, and the input be α1(k) = e(k). The output of each iteration step is... The neural network parameters are updated using parameter update strategies (5)-(7), with the loss function being E(k) = 1 / 2e a(i) 2 (k), where To estimate the error; In the neural network implementation of the iterative adaptive dynamic programming algorithm, when the iterative algorithm satisfies the convergence requirement, that is... Or i reaches the maximum number of iterations i max Sometimes, This allows us to obtain approximately optimal control input. Finally, according to formula (15), the optimal fault-tolerant control law u(k+1) at time k+1 is calculated: (5) The control law u(k+1) includes the oxygen transfer coefficient and internal return flow required to handle the actuator failure at time k+1. The programmable logic controller controls the frequency of the frequency converter according to the calculated oxygen transfer coefficient. The frequency converter controls the aeration volume by adjusting the speed of the blower. The electric regulating valve controls the internal return flow by adjusting the valve opening according to the calculated internal return flow. Finally, the fault-tolerant control of the urban sewage treatment process is achieved by regulating the dissolved oxygen concentration and nitrate nitrogen concentration.

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