Nonlinear Compensation Fault-Tolerant Control Method for Water Tank Level System with Time Delay Characteristics

Abstract

A nonlinear compensation fault-tolerant control method for a water tank level system with time-delay characteristics belongs to the field of advanced industrial process control. The method includes the following steps: Step 1: Establishing a state-space model of the water tank level control system with uncertainties, time-varying time delays, partial actuator failures, and high-order nonlinear components; Step 2: Designing a robust fault-tolerant controller for the time-delay system; Step 3: Designing a dynamic compensation fault-tolerant controller for the previous time step that was not modeled; Step 4: Designing a dynamic incremental compensation fault-tolerant controller that was not modeled. This method proposes a novel solution for water tank level control with time-varying time delays, parameter uncertainties, partial actuator failures, and strong nonlinearity. It integrates data-driven approaches, signal compensation, and the one-step optimal principle to minimize system tracking errors and enable the output to better track the setpoint.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of advanced control of industrial processes, and relates to a nonlinear compensation fault-tolerant control method for a water tank liquid level system with time delay characteristics. BACKGROUND

[0002] In recent years, with the continuous advancement of industrialization, the requirements of modern society for industrial production processes are also increasingly high. As a major field in actual industrial production processes, liquid level control systems are widely used in process industries. Such systems usually have characteristics such as uncertainty, time-varying time delay, partial actuator failure, and strong nonlinearity, which makes it difficult to effectively control such complex industrial processes.

[0003] The present patent is directed to the study of a water tank liquid level control system with time delay characteristics. On the one hand, it is difficult to establish an accurate liquid level control system model, and the control system will change with the change of external conditions in the control process, so the liquid level control system has certain uncertainty and lag characteristics, which may lead to deterioration of the control performance of the system if not handled properly. On the other hand, the water tank liquid level control system often has nonlinear characteristics, and when modeling such a complex system, simplification is usually performed, so there will inevitably be unmodeled elements, and the dynamic characteristics of these unmodeled elements may have a great impact on the system. In the control model established in the present patent, these unmodeled dynamics are referred to as unmodeled dynamics. When dealing with these unmodeled dynamics, if they are simply ignored, the actual control may not achieve the desired effect in a relatively ideal situation. In addition, actuators are the most prone to failure part of the control system, in order to ensure the safe and stable operation of the liquid level control system, it is necessary to design an advanced control method for such problems, which has become an inevitable requirement for actual industrial process control.

[0004] Currently, the mainstream control method for industrial liquid level control systems is PID control. However, with the increasing complexity of industrial process control objects and the increasingly high requirements for control indicators, for process control objects with uncertainty, time-varying time delay, partial actuator failure, and strong nonlinearity, traditional PID control has been difficult to achieve ideal control effect. Therefore, it is of great significance to carry out research on experimental devices with such industrial characteristics and integrate advanced control algorithms into them for the study of industrial process control systems. SUMMARY

[0005] In order to solve the above technical problems, the patent proposes a nonlinear compensation fault-tolerant control method for a liquid level system with time delay characteristics. This method combines robust predictive fault-tolerant control and nonlinear compensation control for time delay systems, considering the uncertainty, time-varying time delay, partial actuator failure and strong nonlinearity of the water tank liquid level control system, to better control the water tank liquid level within the target range.

[0006] To make the controller better applied to the actual system, the uncertainty, time-varying time delay, partial actuator failure and strong nonlinearity of the system need to be considered and handled during the design of the controller, which brings great challenges to the design of the controller. This method is developed to solve this problem. First, the state space model of the water tank liquid level system, i.e. the controlled object simulation model, is established. On this basis, the controller is designed: based on the linear part of the state space model, the robust predictive fault-tolerant controller for time delay systems is designed, considering the uncertainty, time-varying time delay and partial actuator failure of the system at the beginning of the design, which enables the entire liquid level control system to perform well in the face of various actual influencing factors. Then, based on the signal compensation method, the previous time unmodeled dynamic compensation fault-tolerant controller is designed to eliminate the influence of the previous time unmodeled dynamic data information on the closed-loop system. Finally, the unmodeled dynamic increment compensation fault-tolerant controller is designed to eliminate the influence of unmodeled dynamic increment on the system performance.

[0007] The method is implemented through the following technical solutions:

[0008] Step 1: Establish the state space model of the water tank liquid level control system with uncertainty, time-varying time delay, partial actuator failure and high-order nonlinear part;

[0009]

[0010] In the formula, x(k+1) represents the system state at discrete k+1, x(k) is the system state at discrete k, u(k) is the system input at discrete k, y(k) represents the system output at discrete k, v(k) represents the high-order nonlinear part between the discretized model at discrete k and the original nonlinear model, d(k) represents the time delay step at discrete k, x(k-d(k)) represents the system state considering the time delay step at discrete k. A(k) = A + Δ a (k) is the state matrix of the system at discrete k, A, B are model parameters to be identified, A d is the time delay matrix, C is the output matrix, Δ a (k) = NΔ(k)H is the uncertainty perturbation at discrete k, N, H are constant matrices of corresponding dimensions, Δ(k) is the uncertainty perturbation dependent on discrete time k, α represents the actuator failure coefficient, which satisfies α ≤1 is the lower bound of the failure coefficient. This represents the upper bound of the failure coefficient.

[0011] To simplify the design, the fault coefficient is defined to satisfy α=(I+α0)β, where α0 is a fault coefficient factor and is a random number that satisfies |α0|≤β0≤I, β0 is the ratio of the sum and difference of the upper and lower bounds of the fault coefficient, β is the median of the upper and lower bounds of the fault coefficient, and I is an identity matrix of appropriate dimension.

[0012] Based on Δx(k+1)=x(k+1)-x(k), the incremental state-space model is designed as follows:

[0013]

[0014] In the formula, Δx(k+1) is the state increment of the system at discrete time k+1, Δx(k)=x(k)-x(k-1) is the state increment of the system at discrete time k, x(k-1) is the system state at discrete time k-1, Δu(k)=u(k)-u(k-1) is the control input increment of the system at discrete time k, u(k-1) is the control input at discrete time k-1, Δy(k)=y(k)-y(k-1) is the system output increment of the system at discrete time k, y(k-1) is the system output at discrete time k-1. The lumped dynamic nonlinear term of the system at discrete time k is: Where Δ a (k-1) represents the uncertain perturbation of the state matrix at discrete time k-1.

[0015] Split the unmodeled parts And compensation controllers were designed for them respectively. Among them, This represents the unmodeled dynamics between the discretized model and the original nonlinear model at time k-1. This represents the unmodeled dynamic increment of the system at discrete time k.

[0016] Step 2: Design a robust fault-tolerant controller for the time-delay system;

[0017] The control law of a robust fault-tolerant controller for a time-delay system can be expressed as:

[0018] Δu1(k)=K1(k)Δx(k) (3)

[0019] In the formula, K1(k) is the control law gain of Δu1(k) at discrete time k.

[0020] The following performance metrics are introduced:

[0021]

[0022] In the formula, Δx(k+i|k) represents the state prediction value at time k+i based on the incremental model at time k, Δu(k+i|k) represents the prediction value at time k+i of the input sequence that makes the performance index hold at time k, and Q and R are the controller weighting matrices.

[0023] According to Lyapunov's theorem and Schul's complement lemma, if there exists a known scalar θ > 0, 0 ≤ d m ≤d M Unknown symmetric positive matrix Unknown matrix Given unknown scalars ε1, ε2 > 0, such that the following LMI condition holds, the controlled system is asymptotically stable.

[0024]

[0025] In the formula, φ1=-L1+M3+D1S2+S2-X2, D1=(d M -d m +1)I,D2=(d M ) 2 I, * denote the transpose of the symmetric element of the matrix. The elements in LMI are shown below:

[0026]

[0027]

[0028]

[0029] Under the condition of LMI mentioned above, the control law gain K1(k) = Y1L1 of the robust fault-tolerant controller of the time-delay system can be obtained. -1 .

[0030] Step 3: Design a dynamic compensation fault-tolerant controller that was not modeled in the previous moment;

[0031] The control law of the unmodeled dynamic compensation fault-tolerant controller in the previous moment can be expressed as:

[0032]

[0033] In the formula, K2(k) is the control law gain of Δu2(k) at discrete time k.

[0034] Based on Δu1(k), to eliminate To design an incremental dynamic compensation controller for the previous time step that was not modeled, the following performance metrics are introduced:

[0035]

[0036] Define generalized output:

[0037]

[0038] Generalized ideal output:

[0039]

[0040] In the formula P(z) -1 )=A(z -1 )-z -1 K1(k), A(z) -1 )=1-z -1 A(k)-z -1-d(k) A d ,z -1 For the shift operator, d(k) represents the time delay step size at discrete time k, and Q(z) -1 ) is the coefficient polynomial of Δu(k). yes coefficient polynomial:

[0041]

[0042] When the generalized output approaches the generalized ideal output infinitely, the following condition is met:

[0043]

[0044] Therefore, the control law gain K2(k) = B at discrete time k is obtained. -1 (β+β0β) -1 .

[0045] Step 4: Design an unmodeled dynamic incremental compensation fault-tolerant controller;

[0046] Finally, due to The tracking error is unknown, but the resulting error is known. Therefore, the incremental unmodeled dynamic incremental compensator control law is designed to minimize the tracking error. Based on the incremental state equation and output equation of the closed-loop system, the relationship between the tracking error e(k+1) and Δu3(k) at the previous time step is determined:

[0047]

[0048] In the formula, H(z) -1 )=1-z -1 For z -1 The incremental polynomial.

[0049] Based on the one-step optimal feedforward principle:

[0050]

[0051] In the formula, Let z be the unmodeled dynamic increment of the system at discrete time k-1. -2 This represents a shift operator with a step size of 2. To minimize the tracking error, the control rate of the unmodeled dynamic incremental compensation fault-tolerant controller is:

[0052]

[0053] The proposed nonlinear compensation fault-tolerant controller can be expressed as:

[0054]

[0055] Compared with existing technologies, this method offers the following advantages: It proposes a nonlinear compensated fault-tolerant control method for water tank level control systems with time-delay characteristics. Firstly, based on robust predictive fault-tolerant control of time-delay systems, this method integrates adaptive compensation principles and data-driven theory. For unmodeled dynamic components in the model, a compensated fault-tolerant controller is designed, and combined with the properties of the closed-loop control system, continuous rolling optimization significantly improves the control performance of the nonlinear dynamic system. Secondly, unlike traditional nonlinear control methods, considering the differences between nonlinear and linear models, and incorporating the one-step optimal control concept, this method effectively utilizes the system's tracking error information to design a compensated controller that eliminates error signals. This results in a nonlinear compensated fault-tolerant control method with low conservatism and good tracking performance.

[0056] Instruction manual illustrations

[0057] Figure 1 This is a random fault signal;

[0058] Figure 2 It is a time-varying, time-delay signal;

[0059] Figure 3 A simulation comparison graph showing the output response and control input of the control system;

[0060] Figure 4 A comparison chart of the probability distribution of tracking errors in a control system;

[0061] Figure 5 This is a diagram illustrating the actual application effect of the method described in this patent.

[0062] Figure 6 This is a flowchart of the steps in this patent. Detailed Implementation

[0063] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0064] Example:

[0065] This patented method is applied to the TTS20 three-tank water level control system. The TTS20 three-tank water system has a height of H = 0.6m and a cross-sectional area of ​​S = 0.154m². 2 It consists of three resin glass containers, T1, T3, and T2, which are connected by a cross-sectional area of ​​S. n =5×10 -5 m 2 The cylindrical pipes are interconnected, each with a connecting valve. The flow rates of pumps 1 and 2 are represented by Q1 and Q2, respectively, and controlled by a digital controller. Pressure sensors are installed in each of the three tanks to measure the liquid level. Using the flow rate of pump 1 as the control input and the liquid levels of tanks T1 and T3 as the controlled outputs, a single-input single-output second-order model of the three tanks is established:

[0066]

[0067] In the formula, h1 and h3 are the liquid level heights of water tank T1 and water tank T3, respectively. Let T1 be the flow rate of liquid flowing from tank T1 to tank T3. Q is the flow rate of liquid flowing from the bottom of tank T3. in Let y be the flow rate of pump 1, y be the defined controlled liquid level output of the water tank, g be the local gravitational acceleration, h2 be the liquid level height of tank T2, az1 = 0.48, az2 = 0.58 be the liquid flow coefficients of tanks T1 and T2, and S1 = S n =5×10 - 5 m 2 H is the cross-sectional area of ​​the connecting pipe. max =0.6m is the upper limit of the liquid level height, sgn(·) is the sign function, then the state variables and state inputs are respectively:

[0068]

[0069] u(k)=Q in (k) (18)

[0070] In the formula, x(k) represents the system state at discrete time k, x1(k) represents system state 1 at discrete time k, x2(k) represents system state 2 at discrete time k, u(k) represents the control input at discrete time k, h1 and h3 represent the liquid level heights of tanks T1 and T3 at discrete time k, and Q in (k) represents the flow rate of pump 1 at discrete time k.

[0071] The above model is applied at the operation point 0.33H. maxLocal linearization is performed at the point of origin, resulting in a discrete state-space model with parameter uncertainties and unmodeled dynamics. The model parameters are then obtained using an alternating method of least squares and neural network identification. Assuming time delay matrix Uncertainty matrix The system output matrix C = [1 0]. During the simulation, random noise is superimposed as the fault signal and time-delay signal of the controlled system, respectively, such as... Figure 1 , 2 As shown.

[0072] When the control target is y r (k) = 45cm. Since the controller parameters designed by this patent method will be continuously optimized as the system runs, and the controller parameters will only fluctuate within a small range, only the controller parameters when k = 500s are given here for reference, and the values ​​are as follows:

[0073]

[0074] Therefore, at k = 500s, the nonlinear compensation fault-tolerant controller u(k) = 0.0193.

[0075] A simulation comparison was conducted between the traditional robust fault-tolerant control method and the method of this patent, as follows: Figure 3 As shown, under the same external noise influence, the proposed method can better control the water tank level to near the set value when the input signal fluctuation is small, while the traditional robust fault-tolerant control method has several cases of large fluctuations, which greatly reduces the accuracy of the water tank level controller.

[0076] Depend on Figure 4 It is known that the absolute value of the tracking error in traditional robust fault-tolerant control methods sometimes exceeds the specified error upper limit, and the time exceeding the upper limit accounts for 2.4% of the total simulation time. In contrast, the proposed method controls the water tank level within the required range, and the time exceeding the upper limit accounts for 0% of the total simulation time. This improves the tracking performance of the system.

[0077] In addition, the performance evaluation indices mean squared error (MSE) and integrated absolute error (IAE) shown in Equations (20) and (21) were used to compare the control effect, as shown in Table 1, where N represents the running time.

[0078]

[0079]

[0080] Table 1 Comparison of Control Performance

[0081]

[0082] As shown in Table 1, the MSE of the water tank level is 7.1239 when using the traditional robust pre-fault-tolerant control method, while the MSE of the water tank level is 8.7768 when using the method of this patent, a reduction of 18.83%. The IAE of the water tank level is 1.0824 × 10⁻⁶ when using the traditional robust fault-tolerant control method. 3 When using the method of this patent, the IAE of the water tank level is 1.1434 × 10⁻⁶. 3 This represents a decrease of 5.33%.

[0083] The simulation results above show that the method of this patent is superior to the traditional robust fault-tolerant control method.

[0084] A practical application experiment was conducted using a TTS20 three-tank system to verify the effectiveness and practicality of this method. The control effect of the nonlinear compensation fault-tolerant control method is as follows: Figure 5 As shown in the figure, this method has a good control effect, making the water tank level fluctuate less around the set value, and has good control performance.

[0085] In summary, the nonlinear compensation fault-tolerant control method can be effectively applied to the TTS20 three-tank water level control system. Through simulation experiments and analysis of actual operational results, this method demonstrates superior control performance compared to traditional robust fault-tolerant control methods. Therefore, the nonlinear compensation fault-tolerant control method better meets practical control requirements, offering a novel design solution for industrial control systems exhibiting uncertainty, time-varying delays, partial actuator failures, and strong nonlinearity. It possesses significant application value and promising prospects, and is of great importance to achieving my country's ultimate goal of leading the global technological system.

Claims

1. A nonlinear compensation fault-tolerant control method for a water tank level system with time delay characteristics, characterized in that : Firstly, the robust fault-tolerant controller of the time-delay system is designed, and the following performance index is introduced: (1) wherein, denotes the state prediction value at time k+i based on the incremental model at time k, denotes the prediction value at time k+i of the input sequence that makes the performance index true at time k, Q and R are controller weighting matrices, is a uniform matrix of system uncertainty, is a state matrix of system uncertainty, is an uncertainty perturbation depending on the discrete time k; According to Lyapunov theorem and Schur complement lemma, if there exist known scalar unknown symmetric positive definite matrix unknown matrix and unknown scalar such that the following LMI condition holds, the controlled system is asymptotically stable; (2) where , , , denotes the transpose of the matrix symmetric position elements, , the elements in the LMI are as follows: ， ， ， ， ， , and B are model parameters to be identified; for simplifying design, define the failure coefficient to satisfy , is a failure coefficient factor, and satisfies a random number, is the ratio of the difference between the upper and lower bounds of the failure coefficient, is the median of the upper and lower bounds of the failure coefficient, and I is an identity matrix of appropriate dimension. Robust fault-tolerant controller for time-delay system , is the state increment of the system at discrete time k; the control law gain of the robust fault-tolerant controller for the time-delay system can be solved under the condition of satisfying the above LMI ; Secondly, the dynamic compensation controller of the former time is designed to eliminate the influence of the unmodeled dynamics. The incremental dynamic compensation controller of the former time is designed to eliminate the influence of the unmodeled dynamics.​ (3) wherein , , is a time delay matrix, is a state matrix of the system at discrete k time instant, is a backward shift operator, denotes a time delay step at discrete k time instant, is a state increment of the system at discrete k+1 time instant; Generalized ideal output: (4) wherein is the control input increment of the system at discrete k time instant, is the un-modeled dynamics between the discretized model of the system at discrete k-1 time instant and the original nonlinear model, is the coefficient polynomial of is the coefficient polynomial of ​​ (5) When the generalized output is infinitely close to the generalized ideal output, the following equation is satisfied: (6) Thus, the control law of the preceding moment unmodeled dynamics compensating fault-tolerant controller , the control law gain at discrete k moment may be obtained Finally, the design of the unmodeled dynamics incremental compensation fault-tolerant controller, because Unknown, but the error caused by the known, so as to minimize tracking error unmodeled dynamics incremental compensation fault-tolerant controller control rate design; according to the incremental state equation and output equation of closed-loop system, determine the previous time tracking error And unmodeled dynamics incremental compensation fault-tolerant controller control rate The relationship: (7) where C is the system output matrix, is the incremental polynomial with respect to is the incremental polynomial with respect to is the unmodeled dynamics of the system at discrete time k. Based on the one-step optimal feedforward principle: (8) wherein is the unmodeled dynamics increment of the system at discrete time k-1, represents the backward shift operator with step 2; the minimum tracking error can be obtained, and the control rate of the unmodeled dynamics increment compensation fault-tolerant controller is: (9) The proposed nonlinear compensation fault-tolerant controller can be expressed as: (10) In summary, the nonlinear compensation fault-tolerant control system of the water tank level system with time-delay characteristics is composed of the robust fault-tolerant controller of the time-delay system, the fault-tolerant controller of the unmodeled dynamics at the previous time, and the fault-tolerant controller of the unmodeled dynamics increment.

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