A residual driving urban sewage treatment process executor fault active fault-tolerant control method

Abstract

The application provides a residual driving municipal sewage treatment process executor fault active fault-tolerant control method, which realizes stable control of dissolved oxygen concentration and nitrate nitrogen concentration of a municipal sewage treatment process under the influence of an executor fault. First, a residual trigger adaptive threshold fault detection mechanism is designed to accurately detect the executor fault. Second, a radial basis function neural network model is constructed to estimate unknown nonlinear functions and executor multiplicative fault coefficients, so as to accurately represent the time-varying nonlinear dynamic characteristics of the sewage treatment process. Then, a fault-tolerant controller based on a zero-sum game is designed to solve the approximate optimal control law, so as to solve the problem that the dissolved oxygen concentration and the nitrate nitrogen concentration deviate from the set value due to the executor fault. Experimental results show that the method can realize fault-tolerant control of the dissolved oxygen concentration and the nitrate nitrogen concentration, and ensure stable operation of the municipal sewage treatment process.

Description

Technical Field

[0001] This invention proposes a residual-driven active 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 failure. 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 (such as aeration pumps and internal return flow regulating valves) are prone to failure, causing 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 data-driven optimal control technique, adaptive dynamic programming has been widely applied in fault-tolerant control of nonlinear systems, effectively balancing system stability and optimal control performance. However, existing fault-tolerant control methods face two major challenges when applied to urban wastewater treatment processes: First, wastewater treatment processes are highly nonlinear, time-varying, and uncertain, and actuators often face the combined effects of multiplicative and additive faults during actual operation. Existing control methods based on single fault assumptions or known system dynamics struggle to address such complex adversarial optimization problems. Second, effective fault-tolerant control heavily relies on timely and accurate fault detection, but most existing fault-tolerant control methods lack dynamic fault detection mechanisms, easily leading to overlooked faults and subsequent system instability. Therefore, there is an urgent need to develop a novel active fault-tolerant control method combining dynamic fault detection and zero-sum game theory. This method can achieve timely fault warnings through residual triggering mechanisms and utilize a zero-sum game framework to handle the adversarial relationship between the controller and compound faults, thereby achieving the goal of maintaining stable tracking control even when affected by actuator faults in urban wastewater treatment processes.

[0004] This invention designs a residual-driven active fault-tolerant control method for actuator faults in urban wastewater treatment processes. By designing an adaptive threshold fault detection mechanism triggered by residuals, the method accurately detects actuator faults, constructs a radial basis function neural network model to estimate unknown nonlinear functions and actuator multiplicative fault coefficients, and designs a fault-tolerant controller based on zero-sum game theory to achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration. Summary of the Invention

[0005] This invention designs a residual-driven active fault-tolerant control method for actuator failure in urban wastewater treatment processes. It designs an adaptive threshold mechanism triggered by residuals to dynamically detect actuator failures, constructs a radial basis function neural network to estimate unknown nonlinear functions and multiplicative failure coefficients, and designs an active fault-tolerant controller based on zero-sum game and single-evaluation network adaptive dynamic programming. This solves the problem of stable tracking control of urban wastewater treatment processes under the influence of actuator failures.

[0006] The present invention adopts the following technical solution and implementation steps:

[0007] A residual-driven active fault-tolerant control method for actuator failures in urban wastewater treatment processes is characterized by: constructing a control system for urban wastewater treatment processes with actuator failures; designing an adaptive threshold fault detection mechanism triggered by residuals; constructing a radial basis function network model to estimate unknown nonlinear functions and actuator multiplicative fault coefficients; and designing a fault-tolerant controller based on zero-sum game theory to achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration; including the following steps:

[0008] (1) Constructing a process control system for urban wastewater treatment with actuator failure

[0009] First, consider a municipal wastewater treatment process control system without actuator failures, its expression is:

[0010]

[0011] 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)) and f2(x(k)) represent the unknown nonlinear functions at time k, respectively, and let f1(x(k)) and f2(x(k)) represent the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k; 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 A diagonal matrix of (k), where the diagonal elements are 8-S O5 (k) and 0.00537S NO2 (k) is in mg / L; u f (k) represents the actual control quantity output by the actuator at time k;

[0012] Considering the simultaneous existence of multiplicative and additive actuator faults, the actuator fault model is established as follows:

[0013]

[0014] Where, v(k)=[K L a5(k), Q a (k)] T Let K represent the ideal 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, σ(k) represents the multiplicative fault coefficient of the actuator at time k, which represents the unknown efficiency decay of the actuator in actual operation, and its value range is 0 < σ(k) ≤ 1; w(k) represents the additive fault value of the actuator at time k, which represents the unknown fixed deviation of the actuator in actual operation, and it is a bounded unknown variable.

[0015] Substituting the actuator failure model (2) into the ideal system model (1), the expression for the urban wastewater treatment process control system with actuator failure is obtained as follows:

[0016]

[0017] (2) Design an adaptive threshold fault detection mechanism triggered by residuals.

[0018] Establish a state observer to estimate the system's state values ​​when there are no faults, specifically as follows:

[0019]

[0020] in, This represents the state estimate of the system at time k+1 under fault-free conditions. Let f̂(x(k)) represent the state estimate of the system at time k under fault-free conditions, f̂(x(k)) represent the estimate of the unknown nonlinear function at time k, and L represent the preset observer gain matrix.

[0021] Design detection and evaluation functions for:

[0022]

[0023] Among them, e r (k)= Represents the residual vector;

[0024] Design an adaptive dynamic threshold J th (k), represented as:

[0025]

[0026] Where ∘ represents the Hadama product, Denotes the 1-norm, e d (k) is the decision variable;

[0027] Define F(k) = 0 to indicate that the system has not detected a fault, and F(k) = 1 to indicate that a fault has been detected. >J th When (k) is reached, let F(k) = 1 and trigger fault-tolerant control. Design the following fault detection mechanism:

[0028]

[0029] Furthermore, the decision variable e at time k+1 d The update for (k+1) is as follows:

[0030]

[0031] Where, k h e represents the nearest fault-free time before time k. r (k h ) represents the residual error vector recorded at a fault-free time.

[0032] (3) Construct a radial basis function neural network model to estimate unknown nonlinear functions and actuator efficiency.

[0033] 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:

[0034]

[0035] 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 l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k. Let ω represent 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, r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model, and ω r (k), c lr (k), Randomly assign values ​​within the range [0, 1].

[0036] The unknown nonlinear function f(x(k)) is estimated using a radial basis function neural network (9). The number of neurons in the hidden layer of the radial basis function neural network model is set to a. f =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k)=f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T The weight parameter update strategy is designed as follows:

[0037]

[0038] Among them, f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T Let f(x(k)) represent the estimated value of the unknown nonlinear function f(x(k)) at time k, and f̂1(x(k)) and f̂2(x(k)) represent the estimated values ​​of the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k, respectively; ω f(k+1) represents the connection weights between the r-th hidden layer neuron and the output layer neuron of the neural network used to estimate the unknown nonlinear function at time k+1. f η represents the learning rate of the neural network. f (k) represents the hidden layer output vector of the neural network at time k. This represents the predicted state of the system;

[0039] The multiplicative failure coefficients of the actuator are estimated using a radial basis function neural network (9). Set the number of neurons 'a' in the hidden layer of the radial basis function neural network model. σ =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k) = The weight parameter update strategy is designed as follows:

[0040]

[0041] in, Indicates the multiplicative failure coefficient of the unknown actuator at time k. The estimated value, ω represents the estimated multiplicative fault coefficient of the oxygen transfer coefficient and the internal return flow rate at time k, respectively; σ (k+1) represents the connection weights of the r-th hidden layer neuron and the output layer neuron in the neural network used to estimate the multiplicative fault coefficients of the actuator at time k+1. σ η represents the learning rate of the neural network. σ (k) represents the hidden layer output vector of the neural network at time k;

[0042] Design a state identifier to predict the state at time k+1:

[0043]

[0044] in, This represents the predicted state of the faulty system at time k+1. denoted by k, where k represents the state estimate of the faulty system at time k, and K represents the preset state identifier gain matrix.

[0045] (4) Design a fault-tolerant controller based on zero-sum game theory, specifically:

[0046] ① Design a fault-tolerant controller based on adaptive dynamic programming, specifically:

[0047] For the aforementioned fault-tolerant control problem, the objective is to solve for the optimal control law and the worst-case additive fault value, such that when the worst-case additive fault occurs within the allowable range, the designed control law minimizes the cost function, which is expressed as follows:

[0048]

[0049] Where, e(k) = x(k) - x d (k) represents the system error state vector at time k, x d (k) represents the desired state vector of the system at time k, τ=k, k+1, k+2,… represent any time after k, and r(e(k), v(k), w(k))=e T (k)e(k)+ 0.05v T (k)v(k)+ 0.05w T (k)w(k) is the utility function at time k;

[0050] According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation:

[0051]

[0052] Among them, V * (e(k)) represents the optimal cost function at time k, V * (e(k+1)) represents the optimal cost function at time k+1. This represents a minimax operation based on a zero-sum game, which means finding the optimal control input v(k) to minimize the cost function, given that the worst additive fault w(k) maximizes the cost function.

[0053] The optimal control input and worst-case additive fault can be solved using the following formula:

[0054] in, V represents * Take the partial derivative of (e(k+1)) with respect to e(k+1);

[0055] A policy iterative adaptive dynamic programming framework is constructed to solve for the optimal control input and the allowable worst-case additive fault, 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, v0(k) = [0, 0]. T and w0(k)=[0, 0] TBegin the iterative process and solve for the iterative cost function:

[0056]

[0057] Update iterative control inputs and additive faults:

[0058]

[0059]

[0060] The above iterative process is expressed as:

[0061]

[0062] When ǁV i+1 (e(k))-V i (e(k))ǁ<10 -3 Or i reaches the maximum number of iterations i max When the iteration stops and the approximate optimal control input v is output, the iteration stops. * (k) and the worst-case additive fault approximation w*(k); otherwise, let the iteration step number i=i+1, and continue the iteration process shown in formulas (17)-(19);

[0063] ② Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows:

[0064] Since the cost function (13) cannot be directly obtained, an evaluation network is constructed using a radial basis function neural network (9) for estimation. The number of neurons in the hidden layer of the radial basis function neural network model is a. c =14, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[e1(k), e2(k))] T Where e1(k) represents the error of dissolved oxygen concentration at time k, e2(k) represents the error of nitrate nitrogen concentration at time k, and the output is h(k) = The parameters of the neural network are updated using a parameter update strategy, as follows:

[0065]

[0066] Among them, ο c This represents the learning rate of the neural network, and the loss function is E(k) = 1 / 2e c(i) 2 (k), e c(i)(k) represents the estimation error, expressed as follows:

[0067]

[0068] Then, the approximate control input and the approximate additive fault are calculated as follows:

[0069]

[0070]

[0071] Where I2 represents a 2×2 identity matrix;

[0072] When ǁ ǁ<10 -3 Alternatively, when i reaches the maximum number of iterations (20), the iteration stops, at which point we have and This allows us to obtain the approximate optimal control input. And the worst-case additive fault allowed by approximation ;

[0073] (5) Control Law v * (k) includes the oxygen transfer coefficient and internal return flow required to handle actuator failure at time k. 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.

[0074] The inventiveness of this invention is mainly reflected in:

[0075] (1) This invention designs an adaptive threshold fault detection mechanism triggered by residuals, which can dynamically update the detection threshold online based on historical residual statistics, thus realizing timely and accurate identification of actuator faults in urban sewage treatment process; at the same time, it constructs a radial basis function neural network to accurately estimate unknown nonlinear functions and actuator multiplicative fault coefficients, providing complete and accurate system dynamic information for the subsequent design of fault-tolerant controllers based on zero-sum game.

[0076] (2) To address the problem of dissolved oxygen and nitrate nitrogen concentrations deviating from setpoints due to actuator failure, this invention establishes an adversarial zero-sum game framework, treating the controller and the additive actuator failure as opposing sides in the game. Furthermore, a simplified adaptive dynamic programming method containing only a single evaluation network is designed, which significantly reduces computational complexity while learning the optimal fault-tolerant control law online, achieving stable tracking control of dissolved oxygen and nitrate nitrogen concentrations. Attached Figure Description

[0077] Figure 1 This is a diagram showing the fault detection results of the present invention;

[0078] Figure 2 This is a graph showing the control results of dissolved oxygen concentration according to the present invention;

[0079] Figure 3 This is an error graph showing the control results of dissolved oxygen concentration according to the present invention;

[0080] Figure 4 This is a graph showing the control results of nitrate nitrogen concentration according to the present invention;

[0081] Figure 5 This is an error graph showing the control results of nitrate nitrogen concentration according to the present invention; Detailed Implementation

[0082] A residual-driven active fault-tolerant control method for actuator failures in urban wastewater treatment processes is characterized by: constructing a control system for urban wastewater treatment processes with actuator failures; designing an adaptive threshold fault detection mechanism triggered by residuals; constructing a radial basis function network model to estimate unknown nonlinear functions and actuator multiplicative fault coefficients; and designing a fault-tolerant controller based on zero-sum game theory to achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration; including the following steps:

[0083] (1) Constructing a process control system for urban wastewater treatment with actuator failure

[0084] First, consider a municipal wastewater treatment process control system without actuator failures, its expression is:

[0085]

[0086] 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)) and f2(x(k)) represent the unknown nonlinear functions at time k, respectively, and let f1(x(k)) and f2(x(k)) represent the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k; 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 A diagonal matrix of (k), where the diagonal elements are 8-S O5 (k) and 0.00537S NO2 (k) is in mg / L; u f (k) represents the actual control quantity output by the actuator at time k;

[0087] Considering the simultaneous existence of multiplicative and additive actuator faults, the actuator fault model is established as follows:

[0088]

[0089] Where, v(k)=[K L a5(k), Q a (k)] T Let K represent the ideal 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, σ(k) represents the multiplicative fault coefficient of the actuator at time k, which represents the unknown efficiency decay of the actuator in actual operation, and its value range is 0 < σ(k) ≤ 1; w(k) represents the additive fault value of the actuator at time k, which represents the unknown fixed deviation of the actuator in actual operation, and it is a bounded unknown variable.

[0090] Substituting the actuator failure model (2) into the ideal system model (1), the expression for the urban wastewater treatment process control system with actuator failure is obtained as follows:

[0091]

[0092] (2) Design an adaptive threshold fault detection mechanism triggered by residuals.

[0093] Establish a state observer to estimate the system's state values ​​when there are no faults, specifically as follows:

[0094]

[0095] in, This represents the state estimate of the system at time k+1 under fault-free conditions. Let f̂(x(k)) represent the state estimate of the system at time k under fault-free conditions, f̂(x(k)) represent the estimate of the unknown nonlinear function at time k, and L represent the preset observer gain matrix.

[0096] Design detection and evaluation functions for:

[0097]

[0098] Among them, e r (k)= Represents the residual vector;

[0099] Design an adaptive dynamic threshold J th (k), represented as:

[0100]

[0101] Where ∘ represents the Hadama product, Denotes the 1-norm, e d (k) is the decision variable;

[0102] Define F(k) = 0 to indicate that the system has not detected a fault, and F(k) = 1 to indicate that a fault has been detected. >J th When (k) is reached, let F(k) = 1 and trigger fault-tolerant control. Design the following fault detection mechanism:

[0103]

[0104] Furthermore, the decision variable e at time k+1 d The update for (k+1) is as follows:

[0105]

[0106] Where, k h e represents the nearest fault-free time before time k. r (k h ) represents the residual error vector recorded at a fault-free time.

[0107] (3) Construct a radial basis function neural network model to estimate unknown nonlinear functions and actuator efficiency.

[0108] 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:

[0109]

[0110] 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 l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k. Let ω represent 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, r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model, and ω r (k), c lr (k), Randomly assign values ​​within the range [0, 1].

[0111] The unknown nonlinear function f(x(k)) is estimated using a radial basis function neural network (9). The number of neurons in the hidden layer of the radial basis function neural network model is set to a. f =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k)=f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T The weight parameter update strategy is designed as follows:

[0112]

[0113] Among them, f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T Let f(x(k)) represent the estimated value of the unknown nonlinear function f(x(k)) at time k, and f̂1(x(k)) and f̂2(x(k)) represent the estimated values ​​of the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k, respectively; ω f (k+1) represents the connection weights between the r-th hidden layer neuron and the output layer neuron of the neural network used to estimate the unknown nonlinear function at time k+1.f η represents the learning rate of the neural network. f (k) represents the hidden layer output vector of the neural network at time k. This represents the predicted state of the system;

[0114] The multiplicative failure coefficients of the actuator are estimated using a radial basis function neural network (9). Set the number of neurons 'a' in the hidden layer of the radial basis function neural network model. σ =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k) = The weight parameter update strategy is designed as follows:

[0115]

[0116] in, Indicates the multiplicative failure coefficient of the unknown actuator at time k. The estimated value, ω represents the estimated multiplicative fault coefficient of the oxygen transfer coefficient and the internal return flow rate at time k, respectively; σ (k+1) represents the connection weights of the r-th hidden layer neuron and the output layer neuron in the neural network used to estimate the multiplicative fault coefficients of the actuator at time k+1. σ η represents the learning rate of the neural network. σ (k) represents the hidden layer output vector of the neural network at time k;

[0117] Design a state identifier to predict the state at time k+1:

[0118]

[0119] in, This represents the predicted state of the faulty system at time k+1. denoted by k, where k represents the state estimate of the faulty system at time k, and K represents the preset state identifier gain matrix.

[0120] (4) Design a fault-tolerant controller based on zero-sum game theory, specifically:

[0121] ③ Design a fault-tolerant controller based on adaptive dynamic programming, specifically:

[0122] For the aforementioned fault-tolerant control problem, the objective is to solve for the optimal control law and the worst-case additive fault value, such that when the worst-case additive fault occurs within the allowable range, the designed control law minimizes the cost function, which is expressed as follows:

[0123]

[0124] Where, e(k) = x(k) - x d (k) represents the system error state vector at time k, x d (k) represents the desired state vector of the system at time k, τ=k, k+1, k+2,… represent any time after k, and r(e(k), v(k), w(k))=e T (k)e(k)+ 0.05v T (k)v(k)+ 0.05w T (k)w(k) is the utility function at time k;

[0125] According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation:

[0126]

[0127] Among them, V * (e(k)) represents the optimal cost function at time k, V * (e(k+1)) represents the optimal cost function at time k+1. This represents a minimax operation based on a zero-sum game, which means finding the optimal control input v(k) to minimize the cost function, given that the worst additive fault w(k) maximizes the cost function.

[0128] The optimal control input and worst-case additive fault can be solved using the following formula:

[0129]

[0130]

[0131] in, V represents * Take the partial derivative of (e(k+1)) with respect to e(k+1);

[0132] A policy iterative adaptive dynamic programming framework is constructed to solve for the optimal control input and the allowable worst-case additive fault, where i = 0, 1, 2, ... i max Indicates the iteration step number, i maxThis indicates that, given the maximum number of iterations, at each time step, v0(k) = [0, 0]. T and w0(k)=[0, 0] T Begin the iterative process and solve for the iterative cost function:

[0133]

[0134] Update iterative control inputs and additive faults:

[0135]

[0136]

[0137] The above iterative process is expressed as:

[0138]

[0139] When ǁV i+1 (e(k))-V i (e(k))ǁ<10 -3 Or i reaches the maximum number of iterations i max When the iteration stops and the approximate optimal control input v is output, the iteration stops. * (k) and the worst-case additive fault approximation w*(k); otherwise, let the iteration step number i=i+1, and continue the iteration process shown in formulas (17)-(19);

[0140] ④ Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows:

[0141] Since the cost function (13) cannot be directly obtained, an evaluation network is constructed using a radial basis function neural network (9) for estimation. The number of neurons in the hidden layer of the radial basis function neural network model is a. c =14, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[e1(k), e2(k))] T Where e1(k) represents the error of dissolved oxygen concentration at time k, e2(k) represents the error of nitrate nitrogen concentration at time k, and the output is h(k) = The parameters of the neural network are updated using a parameter update strategy, as follows:

[0142]

[0143] Among them, ο cThis represents the learning rate of the neural network, and the loss function is E(k) = 1 / 2e c(i) 2 (k), e c(i) (k) represents the estimation error, expressed as follows:

[0144]

[0145] Then, the approximate control input and the approximate additive fault are calculated as follows:

[0146]

[0147]

[0148] Where I2 represents a 2×2 identity matrix;

[0149] When ǁ ǁ<10 -3 Alternatively, when i reaches the maximum number of iterations (20), the iteration stops, at which point we have and This allows us to obtain the approximate optimal control input. And the worst-case additive fault allowed by approximation ;

[0150] (5) Achieve fault-tolerant control of dissolved oxygen concentration and nitrate nitrogen concentration.

[0151] Control law v * (k) includes the oxygen transfer coefficient and internal return flow required to handle actuator failure at time k. 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 Displays fault detection results, X-axis: time, in days, Y-axis: fault detection signal value, usually dimensionless; Figure 2 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 3 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 4 This displays the tracking and control results of nitrate nitrogen concentration. X-axis: time (days); Y-axis: nitrate nitrogen concentration (mg / L). Figure 5The 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 residual-driven active fault-tolerant control method for actuator faults in urban wastewater treatment processes, characterized in that: Includes the following steps: (1) Constructing a process control system for urban wastewater treatment with actuator failure First, consider a municipal wastewater treatment process control system without actuator failures, its expression is: ； 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)) and f2(x(k)) represent the unknown nonlinear functions at time k, respectively, and let f1(x(k)) and f2(x(k)) represent the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k; 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 A diagonal matrix of (k), where the diagonal elements are 8-S O5 (k) and 0.00537S NO2 (k) is in mg / L; u f (k) represents the actual control quantity output by the actuator at time k; Considering the simultaneous existence of multiplicative and additive actuator faults, the actuator fault model is established as follows: ； Where, v(k)=[K L a5(k), Q a (k)] T Let K represent the ideal 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, σ(k) represents the multiplicative fault coefficient of the actuator at time k, which represents the unknown efficiency decay of the actuator in actual operation, and its value range is 0 < σ(k) ≤ 1; w(k) represents the additive fault value of the actuator at time k, which represents the unknown fixed deviation of the actuator in actual operation, and it is a bounded unknown variable. Substituting the actuator failure model (2) into the ideal system model (1), the expression for the urban wastewater treatment process control system with actuator failure is obtained as follows: ； (2) Design an adaptive threshold fault detection mechanism triggered by residuals. Establish a state observer to estimate the system's state values ​​when there are no faults, specifically as follows: ； in, This represents the state estimate of the system at time k+1 under fault-free conditions. Let f̂(x(k)) represent the state estimate of the system at time k under fault-free conditions, f̂(x(k)) represent the estimate of the unknown nonlinear function at time k, and L represent the preset observer gain matrix. Design detection and evaluation functions for: ； Among them, e r (k)= Represents the residual vector; Design an adaptive dynamic threshold J th (k), represented as: ； Where ∘ represents the Hadama product, Denotes the 1-norm, e d (k) is the decision variable; Define F(k) = 0 to indicate that the system has not detected a fault, and F(k) = 1 to indicate that a fault has been detected. >J th When (k) is reached, let F(k) = 1 and trigger fault-tolerant control. Design the following fault detection mechanism: ； Furthermore, the decision variable e at time k+1 d The update for (k+1) is as follows: ； Where, k h e represents the nearest fault-free time before time k. r (k h ) represents the residual error vector recorded at a fault-free time. (3) Construct a radial basis function neural network model to estimate unknown nonlinear functions and actuator efficiency. 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 l-th input layer neuron corresponding to the r-th hidden layer neuron in the radial basis function neural network model at time k. Let ω represent 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, r = 1, 2, ..., a represents the number of hidden layer neurons in the radial basis function neural network model, and ω represents the width of the l-th input layer neurons in the radial basis function neural network model. r (k), c lr (k), Randomly assign values ​​within the range [0, 1]. The unknown nonlinear function f(x(k)) is estimated using a radial basis function neural network (9). The number of neurons in the hidden layer of the radial basis function neural network model is set to a. f =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k)=f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T The weight parameter update strategy is designed as follows: ； Among them, f̂(x(k))=[f̂1(x(k)), f̂2(x(k))] T ω represents the estimated value of the unknown nonlinear function f(x(k)) at time k, and f̂1(x(k)) and f̂2(x(k)) represent the estimated values ​​of the unknown nonlinear function components that are dynamically related to dissolved oxygen concentration and nitrate nitrogen concentration at time k, respectively; f (k+1) represents the connection weights between the r-th hidden layer neuron and the output layer neuron of the neural network used to estimate the unknown nonlinear function at time k+1. f η represents the learning rate of the neural network. f (k) represents the hidden layer output vector of the neural network at time k. This represents the predicted state of the system; The multiplicative failure coefficients of the actuator are estimated using a radial basis function neural network (9). Set the number of neurons 'a' in the hidden layer of the radial basis function neural network model. σ =20, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[S O5 (k), S NO2 (k)] T The output is h(k) = The weight parameter update strategy is designed as follows: ； in, Indicates the multiplicative failure coefficient of the unknown actuator at time k. The estimated value, ω represents the estimated multiplicative fault coefficient of the oxygen transfer coefficient and the internal return flow rate at time k, respectively; σ (k+1) represents the connection weights of the r-th hidden layer neuron and the output layer neuron in the neural network used to estimate the multiplicative fault coefficients of the actuator at time k+1. σ η represents the learning rate of the neural network. σ (k) represents the hidden layer output vector of the neural network at time k; Design a state identifier to predict the state at time k+1: ； in, This represents the predicted state of the faulty system at time k+1. denoted by k, where k represents the state estimate of the faulty system at time k, and K represents the preset state identifier gain matrix. (4) Design a fault-tolerant controller based on zero-sum game theory, specifically: ① Design a fault-tolerant controller based on adaptive dynamic programming, specifically: For the aforementioned fault-tolerant control problem, the objective is to solve for the optimal control law and the worst-case additive fault value, such that when the worst-case additive fault occurs within the allowable range, the designed control law minimizes the cost function, which is expressed as follows: ； Where, e(k) = x(k) - x d (k) represents the system error state vector at time k, x d (k) represents the desired state vector of the system at time k, τ=k, k+1, k+2,… represent any time after k, and r(e(k), v(k), w(k))=e T (k)e(k)+0.05v T (k)v(k)+ 0.05w T (k)w(k) is the utility function at time k; According to the Bellman optimality principle, the optimal cost function satisfies the following discrete-time HJB equation: ； Among them, V * (e(k)) represents the optimal cost function at time k, V * (e(k+1)) represents the optimal cost function at time k+1. This represents a minimax operation based on a zero-sum game, which means finding the optimal control input v(k) to minimize the cost function, given that the worst additive fault w(k) maximizes the cost function. The optimal control input and worst-case additive fault can be solved using the following formula: ； ； in, V represents * Take the partial derivative of (e(k+1)) with respect to e(k+1); A policy iterative adaptive dynamic programming framework is constructed to solve for the optimal control input and the allowable worst-case additive fault, 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 v0(k) = [0, 0] T and w0(k)=[0, 0] T Begin the iterative process and solve for the iterative cost function: ； Update iterative control inputs and additive faults: ； ； The above iterative process is expressed as: ； When ǁV i+1 (e(k))-V i (e(k))ǁ<10 -3 Or i reaches the maximum number of iterations i max When the iteration stops and the approximate optimal control input v is output, the iteration stops. * (k) and the worst-case additive fault approximation w*(k); otherwise, let the iteration step number i=i+1, and continue the iteration process shown in formulas (17)-(19); ② Adaptive dynamic programming is implemented using a radial basis function neural network model, specifically as follows: Since the cost function (13) cannot be directly obtained, an evaluation network is constructed using a radial basis function neural network (9) for estimation. The number of neurons in the hidden layer of the radial basis function neural network model is a. c =14, the number of neurons in the input layer of the radial basis function neural network model is b=2, and the input is [α1(k), α2(k)]. T =[e1(k), e2(k))] T Where e1(k) represents the error of dissolved oxygen concentration at time k, e2(k) represents the error of nitrate nitrogen concentration at time k, and the output is h(k) = The parameters of the neural network are updated using a parameter update strategy, as follows: ； Among them, ο c This represents the learning rate of the neural network, and the loss function is E(k) = 1 / 2e c(i) 2 (k), e c(i) (k) represents the estimation error, expressed as follows: ； Then, the approximate control input and the approximate additive fault are calculated as follows: ； ； Where I2 represents a 2×2 identity matrix; When ǁ ǁ<10 -3 Alternatively, when i reaches the maximum number of iterations (20), the iteration stops, at which point we have and This allows us to obtain the approximate optimal control input. And the worst-case additive fault allowed by approximation ; (5) Control Law v * (k) includes the oxygen transfer coefficient and internal return flow required to handle actuator failure at time k. 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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