A tracking control method and system for a cascade continuous stirred reactor time-delay system under state-dependent constraints

Abstract

The present application discloses a tracking control method and system for a cascade continuous stirred reactor time-delay system under state-dependent constraints, belonging to the field of automatic control technology. The present application obtains a nonlinear time-delay system model of the cascade continuous stirred reactor time-delay system subject to state-dependent constraints based on a mechanism model of the cascade continuous stirred reactor time-delay system; obtains a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor based on the nonlinear time-delay system model; obtains a controller of the continuous stirred reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, and tracks and controls the cascade continuous stirred reactor through the controller. The present application is based on a nonlinear time-delay system model subject to state-dependent constraints, and combines the time-varying asymmetric Lyapunov function with the Lyapunov-Krasovsky function to eliminate the effects of time-varying delay and state-dependent constraints on system performance.

Description

Technical Field

[0001] The present application belongs to the field of automatic control technology, and specifically relates to a tracking control method and system for a cascade continuous stirred reactor time-delay system under state-dependent constraints. Background Art

[0002] The continuous stirred tank reactor (CSTR) is one of the most critical pieces of equipment in continuous chemical production. With the rapid development of new technologies such as artificial intelligence, my country's automatic control technology has matured. In actual operation, CSTRs often exhibit strong nonlinearity, uncertainty, and complex dynamic mechanisms, making their modeling and control a hot research topic in process control. The goal of controlling CSTR systems is to optimize operating conditions and modes while ensuring system stability, thereby increasing yield and improving product quality. In actual chemical reactions, parameters such as reactant concentration, reactor temperature, and pressure must be rationally controlled to maintain stable ranges. Simultaneously, with the rapid development of modern industrial production, industrial processes are often scaled and integrated, resulting in the widespread use of large, interconnected systems with complex structures. CSTRs are often implemented in the form of multiple cascade continuous stirred tank reactors (CCSTRs). In practical engineering systems, time delay is often a key factor contributing to system instability. Therefore, time delay must be considered in system modeling. How to make the system achieve better tracking performance without violating state-dependent constraints and design a high-performance and highly stable control method for the CCSTR system is a key and challenging problem that needs to be solved urgently. Summary of the Invention

[0003] In response to the above-mentioned shortcomings of the prior art, the present application has developed a tracking control method and system for a cascade continuous stirred reactor time-delay system under state-dependent constraints, comprising:

[0004] Technical Solution: In a first aspect, an embodiment of the present application provides a tracking control method for a cascade continuous stirred reactor time-delay system under a state-dependent constraint, which is applied to a cascade continuous stirred reactor, comprising:

[0005] Obtaining a mechanism model of the cascade type continuous stirred tank time-delay system, and obtaining a state-dependent nonlinear time-delay system model of the cascade type continuous stirred tank time-delay system based on the mechanism model;

[0006] Based on the nonlinear time-delay system model, obtaining a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor;

[0007] Obtaining a controller for the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function;

[0008] The cascade type continuous stirred reactor is tracked and controlled based on the controller.

[0009] In some embodiments, the step of obtaining the time-varying asymmetric Lyapunov function of the cascade continuous stirred tank reactor includes:

[0010] Determining a dependency constraint boundary function based on the nonlinear time-delay system model, and acquiring an error constraint boundary function based on the dependency constraint boundary function;

[0011] Obtaining a tracking error based on the nonlinear time-delay system model and the desired trajectory signal;

[0012] A time-varying asymmetric Lyapunov function of the continuous stirred reactor is determined based on the error constraint boundary function, the tracking error, and the expected trajectory signal.

[0013] In some embodiments, the cascaded continuous stirred reactor includes a first reactor and a second reactor in the cascade. In the first reactor or the second reactor, the characterization formula of the time-varying asymmetric Lyapunov function includes:

[0014]

[0015] Among them, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is the first symbolic discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z1 represents the first independent variable of the input, when z1>0, s1(z1)=1, when z1<0, s1(z1)=0; P is the feed rate; and is the first error constraint boundary function, is the first error constraint lower bound function, is the upper bound function of the first error constraint; y d is the expected trajectory signal; e1 is the first tracking error; t is time; is the first estimation error; Γ1 is the first constant gain matrix; T is the transpose operation.

[0016] In some embodiments, the step of obtaining the controller of the continuous stirred tank reactor includes:

[0017] Obtaining a first parameter based on a first-order derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function;

[0018] Based on the approximation of the neural network, approximate the first parameter to obtain a second parameter;

[0019] Based on the second parameter and the first-order derivative function of the time-varying asymmetric Lyapunov function, a virtual controller and a first adaptive law are obtained.

[0020] In some embodiments, the cascade continuous stirred tank reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the nonlinear time-delay system model includes:

[0021]

[0022] in, and Represents R A and R B The expected value of, A is the first reactor; B is the second reactor; R is the reactant concentration of the reactor; y is the output; i = 1, 2; represents an unknown smooth nonlinear function, represents the unknown control gain function, is the unknown nonlinear time-delay function, τ i (t) is the time lag term, τ i The upper bound of (t) is a positive constant τ max , there exists a positive constant satisfy V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the circulation flow rate; P is the feed rate; Q is a nonlinear function with uncertainty and external interference; t is time; d is the time lag term, which is used to characterize the time lag between the first reactor and the second reactor during the reaction process.

[0023] In some embodiments, the cascaded continuous stirred reactor includes a first reactor and a second reactor in the cascade. In the first reactor and the second reactor in the cascade, the step of obtaining a controller for the continuous stirred reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function further includes:

[0024] Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, a Lyapunov function is obtained:

[0025] V=V B2 +V L ;

[0026]

[0027]

[0028] Wherein, V is the Lyapunov function; V B2 is the time-varying asymmetric Lyapunov function of the first reactor and the second reactor in cascade; V L is the Lyapunov-Krasovsky function; V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor of the cascade; s2(z2) is a second symbolic discriminant function used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the second independent variable of the input, when z2>0, s2(z2)=1, when z2<0, s2(z2)=0; and is the second error constraint boundary function, is the lower bound function of the second error constraint, is the upper bound function of the second error constraint; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant that satisfies 2p≥n+2, where n is the system state dimension;

[0029] Γ2 is the second constant gain matrix; i is the current iteration index; j is the index related to the parameter state; τ is the time lag term; d is the integration operation; is the system state variable;

[0030] Based on the first-order derivative function of the Lyapunov function, the actual controller and the second adaptive law are obtained:

[0031]

[0032] Among them, u is the actual controller; is the second adaptive law; g 2 is a positive constant, used to represent The lower bound of λ2, a2 ​​and μ are positive constants; e2 is the second tracking error; is the estimated value of W2, where W2 is the second optimal neural network weight vector; G2(Z2) is the Gaussian function vector; γ2 is a positive constant,

[0033] Γ2 is the second constant gain matrix; k2 is a positive constant.

[0034] In some embodiments, the step of obtaining a state-dependent nonlinear time-delay system model of the cascade continuous stirred reactor time-delay system based on the mechanism model includes:

[0035] When the cascade continuous stirred tank reactor is in an equilibrium state, obtaining an expected value of a reactant concentration in the reactor based on the mechanism model;

[0036] The reaction concentration of the reactor is obtained based on the expected value, and the nonlinear time-delay system model is obtained based on the reaction concentration.

[0037] In some embodiments, the cascade continuous stirred tank reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the mechanism model includes:

[0038]

[0039] Wherein, A is the first reactor; B is the second reactor; R is the reactant concentration of the reactor; V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the circulation flow rate; P is the feed rate; q is a nonlinear function with complex behavior; Q is a nonlinear function with uncertainty and external interference; t is time; s is the time lag term, which is used to characterize the time lag between the first reactor and the second reactor during the reaction process.

[0040] In some embodiments, when the continuous stirred tank reactor is in equilibrium:

[0041] q A =R A +R A (td A );

[0042]

[0043] Q A =0;

[0044] Q B =0.

[0045] In a second aspect, an embodiment of the present application further provides a tracking control system for a cascade continuous stirred reactor time-delay system under a state-dependent constraint, which is applied to a cascade continuous stirred reactor, and the tracking control system includes:

[0046] A time-delay system acquisition module, the time-delay system acquisition module is used to obtain a mechanism model of the cascade type continuous stirred tank time-delay system, and obtain a state-dependent nonlinear time-delay system model of the cascade type continuous stirred tank time-delay system based on the mechanism model;

[0047] A function acquisition module, configured to acquire a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor based on the nonlinear time-delay system model;

[0048] A controller acquisition module, configured to acquire a controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function;

[0049] A tracking control module is used to track and control the continuous stirred reactor based on the controller.

[0050] Beneficial effects: Compared with the prior art, the embodiment of the present application provides a tracking control method for a cascade continuous stirred reactor time-delay system under a state-dependent constraint. Based on the mechanism model of the cascade continuous stirred reactor time-delay system, a nonlinear time-delay system model of the cascade continuous stirred reactor time-delay system subject to state-dependent constraints is obtained; based on the nonlinear time-delay system model, a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor is obtained; based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, a controller of the continuous stirred reactor is obtained, and the cascade continuous stirred reactor is tracked and controlled by the controller. The present application is based on the nonlinear time-delay system model of the cascade continuous stirred reactor time-delay system subject to state-dependent constraints after conversion, and combines the time-varying asymmetric Lyapunov function with the Lyapunov-Krasovsky function to eliminate the influence of time-varying delay and state-dependent constraints on system performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For those skilled in the art, other drawings can be obtained based on these drawings without creative work.

[0052] Figure 1 A flowchart of the steps of a tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application;

[0053] Figure 2 A specific flow chart of step 100 in the tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application;

[0054] Figure 3 A specific flow chart of step 200 in the tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application;

[0055] Figure 4 A specific flow chart of step 300 in the tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application;

[0056] Figure 5 A module connection diagram of a tracking control system for a cascaded continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application;

[0057] Figure 6 This is a mechanism diagram of the cascade continuous stirred reactor provided in the examples of the present application;

[0058] Figure 7 The reaction concentration x1 and the expected trajectory y of the continuous stirred reactor A in the cascade continuous stirred reactor time-delay system are shown. d Tracking performance graph;

[0059] Figure 8 The x2 trajectory of the reaction concentration in the continuous stirred reactor B in the cascade continuous stirred reactor time-delay system is shown;

[0060] Figure 9 The error e1 and e2 trajectory diagrams of the cascade continuous stirred tank time-delay system are shown;

[0061] Figure 10 The trajectory diagram of the controller u for the cascade continuous stirred reactor time-delay system is shown;

[0062] Figure 11 The adaptive parameters of the cascade continuous stirred reactor time-delay system are shown. and The change curve diagram of

[0063] Reference numerals: 10, time-delay system acquisition module; 20, function acquisition module; 30, controller acquisition module; 40, tracking control module. DETAILED DESCRIPTION

[0064] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the embodiments described are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without making creative efforts are within the scope of protection of this application.

[0065] The continuous stirred tank reactor (CSTR) is one of the most critical pieces of equipment in continuous chemical production. With the rapid development of new technologies such as artificial intelligence, my country's automatic control technology has matured. In actual operation, CSTRs often exhibit strong nonlinearity, uncertainty, and complex dynamic mechanisms, making their modeling and control a hot research topic in process control. The goal of controlling CSTR systems is to optimize operating conditions and modes while ensuring system stability, thereby increasing yield and improving product quality. In actual chemical reactions, parameters such as reactant concentration, reactor temperature, and pressure must be rationally controlled to maintain stable ranges. Simultaneously, with the rapid development of modern industrial production, industrial processes are often scaled and integrated, resulting in the widespread use of large, interconnected systems with complex structures. CSTRs are often implemented in the form of multiple cascade continuous stirred tank reactors (CCSTRs). In practical engineering systems, time delay is often a key factor contributing to system instability. Therefore, time delay must be considered in system modeling. How to make the system achieve better tracking performance without violating state-dependent constraints and design a high-performance and highly stable control method for the CCSTR system is a key and challenging problem that needs to be solved urgently.

[0066] In view of this, an embodiment of the present application provides a tracking control method for a cascade type continuous stirred reactor time-delay system under a state-dependent constraint, which is specifically applied to a cascade type continuous stirred reactor time-delay system. It mainly realizes that the system state does not violate the state-dependent constraint while eliminating the influence of the time-delay problem on the system, greatly improving the safety and stability of the chemical production process. Specifically, the present application obtains a nonlinear time-delay system model of the cascade type continuous stirred reactor time-delay system subject to state-dependent constraints based on the mechanism model of the cascade type continuous stirred reactor time-delay system; obtains a time-varying asymmetric Lyapunov function of the cascade type continuous stirred reactor based on the nonlinear time-delay system model; obtains a controller of the continuous stirred reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, and tracks and controls the cascade type continuous stirred reactor through the controller. This application is based on a nonlinear time-delay system model that is converted from a cascaded continuous stirred reactor time-delay system and subject to state-dependent constraints. By combining a time-varying asymmetric Lyapunov function with a Lyapunov-Krasovsky function, the effects of time-varying delay and state-dependent constraints on system performance are eliminated. A neural network is also used to approximate unknown dynamics, ensuring good tracking performance without violating state-dependent constraints.

[0067] In some embodiments, see Figure 1 and Figure 6 , Figure 1 This is a flowchart of the tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application. Figure 6 This is a mechanism diagram of a cascade continuous stirred reactor provided in an embodiment of the present application. Specifically, the tracking control method of a cascade continuous stirred reactor time-delay system under a state-dependent constraint provided in an embodiment of the present application is applied to the cascade continuous stirred reactor time-delay system, specifically completed through steps 100 to 400:

[0068] Step 100: Obtain a mechanism model of a cascade continuous stirred tank time-delay system, and obtain a state-dependent nonlinear time-delay system model of the cascade continuous stirred tank time-delay system based on the mechanism model.

[0069] In some embodiments, the cascade continuous stirred tank reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the mechanism model includes:

[0070]

[0071] The physical meaning of each symbol is as follows:

[0072]

[0073]

[0074] In some embodiments, see Figure 2 , Figure 2 This is a specific flow chart of step 100 in the tracking control method for a cascaded continuous stirred reactor time-delay system under a state-dependent constraint provided in an embodiment of the present application. The method for obtaining a nonlinear time-delay system model of a cascaded continuous stirred reactor time-delay system under a state-dependent constraint based on a mechanism model is specifically implemented through steps 110 to 120:

[0075] Step 110: When the cascade continuous stirred tank reactor is in an equilibrium state, an expected value of the reactant concentration of the reactor is obtained based on the mechanism model.

[0076] In some embodiments, when the continuous stirred tank reactor is in equilibrium:

[0077] q A =R A +R A (td A );

[0078]

[0079] QA =0;

[0080] Q B =0;

[0081] The equilibrium point of the cascade continuous stirred reactor time-delay system can be obtained, which satisfies the following form:

[0082]

[0083] in and Represents R A and R B expected value.

[0084] Step 120: Obtain the reaction concentration of the reactor based on the expected value, and obtain a nonlinear time-delay system model based on the reaction concentration.

[0085] In some embodiments, a cascade continuous stirred tank reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the nonlinear time-delay system model includes:

[0086]

[0087] x1 represents the reaction concentration of continuous stirred reactor A, x2 represents the reaction concentration of continuous stirred reactor B, and Represents R A and R B where A is the first reactor; B is the second reactor; R is the reactant concentration in the reactor; y is the output; i = 1, 2; V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the circulation flow rate; P is the feed rate; Q is a nonlinear function with uncertainty and external interference; t is time; and d is the lag term, which is used to characterize the lag between the first reactor and the second reactor during the reaction process.

[0088] Further, let Where x1 represents the reaction concentration of continuous stirred reactor A, and x2 represents the reaction concentration of continuous stirred reactor B. The mechanism model can be described as:

[0089]

[0090] Among them, for i=1,2, Denotes an unknown smooth nonlinear function, let as well as Represents the unknown control gain function, let as well as is defined as an unknown nonlinear time-delay function. It is worth noting that τ i (t) represents the time lag term, and defines a positive constant τ max As its upper bound, and there exists a positive constant satisfy make as well as

[0091] Step 200: Based on the nonlinear time-delay system model, a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor is obtained.

[0092] In some embodiments, see Figure 3 , Figure 3 This is a specific flow chart of step 200 in the tracking control method for a cascaded continuous stirred reactor time-delay system under a state-dependent constraint provided in an embodiment of the present application. Obtaining the time-varying asymmetric Lyapunov function of the cascaded continuous stirred reactor is specifically implemented through steps 210 to 230:

[0093] Step 210: Determine a dependency constraint boundary function based on the nonlinear time-delay system model, and obtain an error constraint boundary function based on the dependency constraint boundary function.

[0094] In some embodiments, the present application introduces a state-dependent constraint boundary function, requiring that all system states x1(t) and x2(t) in the model satisfy the following form:

[0095]

[0096] It is understandable that the constraints considered in this application are state-dependent constraints, that is, functions related to both state variables and time. and Represents the state-dependent constraint boundary function and satisfies Among them, the state variables should satisfy χ i =[y d ,x1] T ,i=1,2.

[0097] Introducing error constraint boundary function and as follows:

[0098]

[0099] Step 220: Obtain a tracking error based on the nonlinear time-delay system model and the desired trajectory signal.

[0100] In some embodiments, the tracking error is e1=x1-y d, where y d represents the expected trajectory signal.

[0101] Step 230: Determine the time-varying asymmetric Lyapunov function of the continuous stirred tank reactor based on the error constraint boundary function, the tracking error, and the expected trajectory signal.

[0102] In some embodiments, a cascade continuous stirred tank reactor includes a first reactor and a second reactor in the cascade, and in the first reactor or the second reactor, a characterization formula of the time-varying asymmetric Lyapunov function includes:

[0103]

[0104] Among them, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is a piecewise function that serves as the first symbolic discriminant function to meet the characteristic requirements of the time-varying asymmetric Lyapunov function. Where z1 represents the input independent variable, when z i >0, s1(z1)=1, when z1<0, s1(z1)=0; P is the feed rate; and is the error constraint boundary function, is the error constraint lower bound function, is the upper bound function of the error constraint; y d is the expected trajectory signal; e1 is the first tracking error; t is time; is the first estimation error, W1 is the first optimal neural network weight vector, Used to characterize the estimated value of W1; Γ1 is the first constant gain matrix, T is the transpose operation.

[0105] Step 300: Obtain a controller for a continuous stirred tank reactor based on a time-varying asymmetric Lyapunov function and a Lyapunov-Krasovsky function.

[0106] In some embodiments, see Figure 4 , Figure 4 This is a specific flow chart of step 300 in the tracking control method for a cascaded continuous stirred reactor time-delay system under a state-dependent constraint provided in an embodiment of the present application. The method for obtaining a controller for a continuous stirred reactor based on a time-varying asymmetric Lyapunov function and a Lyapunov-Krasovsky function is implemented through steps 310 to 350:

[0107] Step 310: Obtain a first parameter based on the first-order derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function.

[0108] In some embodiments, V B1 Taking the derivative we can get:

[0109]

[0110] in, definition as well as definition

[0111] Furthermore, the characterization formula of the first parameter includes:

[0112]

[0113] Among them, U1(Z1) is the first parameter, f1(x1(t)) represents the unknown smooth nonlinear function; and the Lyapunov-Krasovskii function is Its function is to deal with the time lag term make in represents a positive constant that satisfies

[0114] Step 320: Based on the approximation of the neural network, perform approximation processing on the first parameter to obtain the second parameter.

[0115] According to the approximation of the neural network, the first parameter U1(Z1) can be approximated as the second parameter. The characterization formula of the second parameter includes:

[0116]

[0117] Where W1 is the first optimal neural network weight vector, σ1(Z1) represents the approximation error and satisfies Represents a positive constant; G1(Z1)∈R l Represents the Gaussian function vector, and the number of neural network nodes is l.

[0118] Step 330: Obtain a virtual controller and a first adaptive law based on the second parameter and the first derivative function of the time-varying asymmetric Lyapunov function.

[0119] In some embodiments, the virtual controller α1 and the first adaptive law The characterization formulas include:

[0120]

[0121] Among them, a1 is a positive constant; g1 is a positive constant that satisfies the lower bound of the nonlinear smooth function g1(x1(t)), that is, g 1≤g1(x1(t)); λ1 is a positive constant; a1 is a positive constant; μ is a positive constant; e1 is the first tracking error; is the estimated value of W1, where W1 represents the first optimal neural network weight vector; G1(Z1) is the Gaussian function vector; Represents a function, specifically γ1 represents a positive constant; κ1 represents a function, specifically Represents a function, specifically k1 is a positive constant; Γ1 is the first constant gain matrix.

[0122] The actual system model may cause the expression of Lyapunov function and its derivative to be very complicated and difficult to analyze directly. as well as By appropriately scaling each term in , we can transform it into a simpler and easier-to-handle form so that it can be combined with the judgment conditions of positive definite functions. This leads to a conclusion on the stability of the system. Based on Young's inequality, we can obtain:

[0123]

[0124] Where σ1(Z1) represents the approximation error and satisfies And a1 represents a positive constant.

[0125] Substituting (9)-(14) into (8), we can obtain:

[0126]

[0127] in, and It will be offset in the next step.

[0128] Step 340: Obtain a Lyapunov function based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function.

[0129] In some embodiments, the characterization formula of the Lyapunov function includes:

[0130] V=V B2 +V L ; (15)

[0131]

[0132] Where V is the Lyapunov function; V B2is the time-varying asymmetric Lyapunov function of the first and second reactors in the cascade; V L is the Lyapunov-Krasovsky function; V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor of the cascade; s2(z2) is the second symbolic discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the second independent variable of the input, when z2>0, s2(z2)=1, when z2<0, s2(z2)=0. and is the second error constraint boundary function, is the lower bound function of the second error constraint, is the upper bound function of the second error constraint; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant satisfying 2p≥n+2, n is the system state dimension; Γ2 is the second constant gain matrix; i is the current iteration index; j is the index related to the parameter state; τ is the time lag term; d is the integration operation; is the system state variable.

[0133] According to Lyapunov's stability theorem, we first take the derivative of V to determine the stability of the cascade continuous stirred reactor time-delay system. Taking the derivative of V, we can get:

[0134]

[0135] V B2 Taking the derivative we can get:

[0136]

[0137] V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor, V B2 is the time-varying asymmetric Lyapunov function of the first and second reactors in the cascade; V B2 With V B1 This is a cumulative relationship, combined with backstepping techniques. Its primary purpose is to ensure system stability by constructing appropriate Lyapunov functions and control laws based on Lyapunov stability theory, combined with neural network control. It enables the system state to gradually converge to the desired equilibrium point. That is, over time, the system output can approach a given reference signal, thereby achieving high-precision control of cascaded continuous stirred reactor time-delay systems.

[0138] For the cascade continuous stirred reactor time-delay system, a stable controller is designed to adjust the system input so that the system state can converge to the desired equilibrium point, prevent the system from becoming unstable, and ensure the normal operation of the system. The adaptive law continuously adjusts the parameters or structure of the controller so that the system can maintain good stability and performance under different working conditions and environments. Even when the system parameters change significantly or are subject to strong interference, it can still operate normally, reducing the system's dependence on accurate models and improving the system's reliability and adaptability. Design the actual controller u and the second adaptive law as follows:

[0139]

[0140] Among them, u is the actual controller; is the second adaptive law. g 2 is a positive constant that satisfies the nonlinear smooth function The lower bound of λ2 represents a positive constant; a2 represents a positive constant; μ represents a positive constant; e2 represents the second tracking error; Represents the estimated value of W2, where W2 represents the second optimal neural network weight vector; G2(Z2) represents the Stochastic function vector; Represents a function, specifically γ2 represents a positive constant; κ2 represents a function, specifically Represents a function, specifically κ2 represents a positive constant; Γ2 represents a constant gain matrix.

[0141] The role of Young's inequality is to convert V B2 as well as By appropriately scaling each term in , we can transform it into a simpler and easier-to-handle form, and then draw conclusions about the stability of the system. Based on Young's inequality, we can get:

[0142]

[0143] Among them, a2 represents a positive constant, we can get in and

[0144] Furthermore, in order to simplify V B2 Expression, substitute (21)-(24) into (20), can be transformed into:

[0145]

[0146] Next we deal with the time lag function. First, we need to calculate V L Taking the derivative, we can get:

[0147]

[0148] in, as well as

[0149] Time lag term τ j (t) Satisfaction Furthermore, by combining μ = 1-τ, we can obtain therefore, It can be expressed as:

[0150]

[0151] Step 1.9: After the Young's inequality scaling process, by combining (26) and (27), It can be expressed as:

[0152]

[0153] in, And Θ < 0. Θ is just a symbol that groups the time-delay components together, and after sorting, the equation is less than zero, so it can be omitted.

[0154] according to as well as Therefore, the following inequality can be obtained:

[0155]

[0156] Combining (7), (16) and (17), V can be converted to:

[0157]

[0158] Finally, we can get:

[0159]

[0160] Among them, c and ρ represent the set of constant parts, specifically ρ=min{2λ i ,k i o(Γ i ) min{}}. After the derivation of the above formula, it can finally be transformed into the Lyapunov stability conclusion. The purpose of this step is to determine whether the cascade continuous stirred reactor time-delay system is globally asymptotically stable.

[0161] Step 350: Based on the first-order derivative function of the Lyapunov function, obtain the actual controller and the second adaptive law.

[0162] In some embodiments, the characterization formula of the actual controller and the second adaptive law includes:

[0163]

[0164] Among them, u is the actual controller; is the second adaptive law. g 2 is a positive constant that satisfies the nonlinear smooth function The lower bound of λ2 represents a positive constant; a2 represents a positive constant; μ represents a positive constant; e2 represents the second tracking error; Represents the estimated value of W2, where W2 represents the second optimal neural network weight vector; G2(Z2) represents the Stochastic function vector; Represents a function, specifically γ2 represents a positive constant; κ2 represents a function, specifically Represents a function, specifically κ2 represents a positive constant; Γ2 represents a constant gain matrix.

[0165] Step 400: Tracking and controlling the cascade continuous stirred tank reactor based on a controller.

[0166] Specifically, in the embodiment of the present invention, the basic physical parameters of the cascade type continuous stirred reactor time-delay system mechanism model are P=0.5, K A =K B =0.5, D A =D B =0.5, R A =R B =0.5, F A =F B =2 and Expected trajectory y d =0.6sin(πt)e -2t +0.6, the lag function is designed to be Where τ1(t)=1-sin(0.5t)cos(2t), τ2=1.55-0.5sin(0.5t). The initial value of the system state and the adaptive parameters are selected as follows: In addition, the system state x i ,i=1,2 and e i ,i=1,2 must satisfy the state dependency constraint boundary conditions:

[0167]

[0168] where χ1 = y d And X2=x1, the state-dependent constraint boundary function is designed as follows:

[0169]

[0170] Under this initial condition, simulation verification is carried out and the simulation results are shown in the figure below. Figure 7-11 shown. Figure 7 The reaction concentration x1 and the expected trajectory y of the continuous stirred reactor A in the cascade continuous stirred reactor time-delay system are shown. d Tracking performance graph; Figure 8 The x2 trajectory of the reaction concentration in the continuous stirred reactor B in the cascade continuous stirred reactor time-delay system is shown; Figure 9 The error e1 and e2 trajectory diagrams of the cascade continuous stirred tank time-delay system are shown; Figure 10 The trajectory diagram of the controller u for the cascade continuous stirred reactor time-delay system is shown; Figure 11 The adaptive parameters of the cascade continuous stirred reactor time-delay system are shown. and 's change curve.

[0171] It can be understood that the tracking control method for the cascade type continuous stirred reactor time-delay system under the state-dependent constraint provided in the embodiment of the present application is specifically applied to the cascade type continuous stirred reactor time-delay system, which mainly realizes that the system state does not violate the state-dependent constraint while eliminating the influence of the time-delay problem on the system, greatly improving the safety and stability of the chemical production process. Specifically, the present application obtains a nonlinear time-delay system model of the cascade type continuous stirred reactor time-delay system subject to state-dependent constraints based on the mechanism model of the cascade type continuous stirred reactor time-delay system; obtains the time-varying asymmetric Lyapunov function of the cascade type continuous stirred reactor based on the nonlinear time-delay system model; obtains the controller of the continuous stirred reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, and tracks and controls the cascade type continuous stirred reactor through the controller. This application is based on a nonlinear time-delay system model that is converted from a cascaded continuous stirred reactor time-delay system and subject to state-dependent constraints. By combining a time-varying asymmetric Lyapunov function with a Lyapunov-Krasovsky function, the effects of time-varying delay and state-dependent constraints on system performance are eliminated. A neural network is also used to approximate unknown dynamics, ensuring good tracking performance without violating state-dependent constraints.

[0172] Accordingly, see Figure 5 , Figure 5 This is a module connection diagram of a tracking control system for a cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application. The cascade type continuous stirred reactor time-delay system under state-dependent constraints provided in an embodiment of the present application includes:

[0173] The time-delay system acquisition module 10 is used to obtain a mechanism model of the cascade type continuous stirred reactor time-delay system, and obtain a state-dependent nonlinear time-delay system model of the cascade type continuous stirred reactor time-delay system based on the mechanism model;

[0174] The function acquisition module 20 is used to obtain the time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor based on the nonlinear time-delay system model;

[0175] A controller acquisition module 30 is used to acquire a controller of a continuous stirred tank reactor based on a time-varying asymmetric Lyapunov function and a Lyapunov-Krasovsky function;

[0176] The tracking control module 40 is used to track and control the continuous stirred reactor based on the controller.

[0177] In some embodiments, the function acquisition module 20 is specifically configured to:

[0178] Determine a dependency constraint boundary function based on a nonlinear time-delay system model, and obtain an error constraint boundary function based on the dependency constraint boundary function;

[0179] Obtain tracking error based on nonlinear time-delay system model and desired trajectory signal;

[0180] The time-varying asymmetric Lyapunov function of a continuously stirred reactor is determined based on the error constraint boundary function, tracking error, and expected trajectory signal.

[0181] In some embodiments, the controller acquisition module 30 is specifically configured to:

[0182] Obtaining a first parameter based on the first derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function;

[0183] Based on the approximation of the neural network, the first parameter is approximated to obtain the second parameter;

[0184] Based on the second parameter and the first derivative function of the time-varying asymmetric Lyapunov function, a virtual controller and a first adaptive law are obtained.

[0185] In some embodiments, the controller acquisition module 30 is specifically configured to:

[0186] Obtain the Lyapunov function based on the time-varying asymmetric Lyapunov function and the Lyapunov–Krasovsky function:

[0187] V=V B2 +V L ;

[0188]

[0189]

[0190] Where V is the Lyapunov function; V B2 is the time-varying asymmetric Lyapunov function of the first and second reactors in the cascade; V L is the Lyapunov-Krasovsky function; V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor of the cascade; s2(z2) is the second symbolic discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the second independent variable of the input, when z2>0, s2(z2)=1, when z2<0, s2(z2)=0; and is the second error constraint boundary function, is the lower bound function of the second error constraint, is the upper bound function of the second error constraint; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant satisfying 2p≥n+2, where n is the dimension of the system state; Γ2 is the second constant gain matrix; i is the current iteration index; j is the index related to the parameter state; τ is the time lag term; d is the integration operation; is the system state variable;

[0191] Based on the first-order derivative of the Lyapunov function, the actual controller and the second adaptive law are obtained:

[0192]

[0193] Among them, u is the actual controller; is the second adaptive law; g2 is a positive constant used to characterize The lower bound of λ2, a2 ​​and μ are positive constants; e2 is the second tracking error; is the estimated value of W2, where W2 is the second optimal neural network weight vector; G2(Z2) is the Gaussian function vector; γ2 is a positive constant, Γ2 is the second constant gain matrix; k2 is a positive constant.

[0194] In some embodiments, the time-delay system acquisition module 10 is specifically configured to:

[0195] When the cascade continuous stirred tank reactor is in equilibrium, the expected value of the reactant concentration in the reactor is obtained based on the mechanism model;

[0196] The reaction concentration of the reactor is obtained based on the expected value, and the nonlinear time-delay system model is obtained based on the reaction concentration.

[0197] The above application provides a detailed introduction to the tracking control method and system of a cascade continuous stirred reactor time-delay system under a state-dependent constraint provided by the embodiment of the present application. Specific examples are used herein to illustrate the principles and implementation methods of the present application. The description of the above embodiments is only used to help understand the method of the present application and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present application, there may be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation to the present application.

Claims

1. A tracking control method for a cascade continuous stirred reactor time-delay system under state-dependent constraints, applied to a cascade continuous stirred reactor comprising a first reactor and a second reactor connected in cascade. The tracking control method comprises: Obtaining a mechanism model of the cascade type continuous stirred tank time-delay system, and obtaining a state-dependent nonlinear time-delay system model of the cascade type continuous stirred tank time-delay system based on the mechanism model; Based on the nonlinear time-delay system model, obtaining a time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor; In the first reactor or the second reactor, the characterization formula of the time-varying asymmetric Lyapunov function includes: Among them, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is the first symbolic discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z1 represents the first independent variable of the input, when z1>0, s1(z1)=1, when z1<0, s1(z1)=0; P is the feed rate; and is the first error constraint boundary function, is the first error constraint lower bound function, is the upper bound function of the first error constraint; y d is the expected trajectory signal; e1 is the first tracking error; t is time; is the first estimation error; Γ1 is the first constant gain matrix; T is the transpose operation; Obtaining a controller for the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function; The cascade type continuous stirred reactor is tracked and controlled based on the controller.

2. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 1, characterized in that: The step of obtaining the time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor comprises: Determining a dependency constraint boundary function based on the nonlinear time-delay system model, and acquiring an error constraint boundary function based on the dependency constraint boundary function; Obtaining a tracking error based on the nonlinear time-delay system model and the desired trajectory signal; A time-varying asymmetric Lyapunov function of the continuous stirred reactor is determined based on the error constraint boundary function, the tracking error, and the expected trajectory signal.

3. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 1, characterized in that: The step of obtaining the controller of the continuous stirred reactor comprises: Obtaining a first parameter based on a first-order derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function; Based on the approximation of the neural network, approximate the first parameter to obtain a second parameter; Based on the second parameter and the first-order derivative function of the time-varying asymmetric Lyapunov function, a virtual controller and a first adaptive law are obtained.

4. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 1, characterized in that: The cascade type continuous stirred reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the nonlinear time-delay system model includes: in, and Represents R A and R B The expected value of, A is the first reactor; B is the second reactor; R is the reactant concentration of the reactor; y is the output; i = 1, 2; represents an unknown smooth nonlinear function, represents the unknown control gain function, is the unknown nonlinear time-delay function, τ i (t) is the time lag term, τ i The upper bound of (t) is a positive constant τ max , there exists a positive constant satisfy V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the circulation flow rate; P is the feed rate; Q is a nonlinear function with uncertainty and external interference; t is time; d is the time lag term, which is used to characterize the time lag between the first reactor and the second reactor during the reaction process.

5. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 4, characterized in that: The cascade type continuous stirred reactor includes a first reactor and a second reactor in the cascade. In the first reactor and the second reactor in the cascade, the step of obtaining a controller for the continuous stirred reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function further includes: Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function, a Lyapunov function is obtained: V=V B2 +V L ; Wherein, V is the Lyapunov function; V B2 is the time-varying asymmetric Lyapunov function of the first reactor and the second reactor in cascade; V L is the Lyapunov-Krasovsky function; V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor of the cascade; s2(z2) is a second symbolic discriminant function used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the second independent variable of the input, when z2>0, s2(z2)=1, when z2<0, s2(z2)=0; and is the second error constraint boundary function, is the lower bound function of the second error constraint, is the upper bound function of the second error constraint; χ2 is the state variable; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant satisfying 2p≥n+2, where n is the dimension of the system state; Γ2 is the second constant gain matrix; i is the current iteration index; j is the index related to the parameter state; τ is the time lag term; d is the integration operation; is the system state variable; Based on the first-order derivative function of the Lyapunov function, the actual controller and the second adaptive law are obtained: Among them, u is the actual controller; is the second adaptive law; g 2 is a positive constant, used to represent The lower bound of λ2, a2 ​​and μ are positive constants; e2 is the second tracking error; is the estimated value of W2, where W2 is the second optimal neural network weight vector; G2(Z2) is the Gaussian function vector; γ2 is a positive constant, Γ2 is the second constant gain matrix; k2 is a positive constant.

6. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 1, characterized in that: The step of obtaining a state-dependent nonlinear time-delay system model of the cascade continuous stirred reactor time-delay system based on the mechanism model comprises: When the cascade continuous stirred tank reactor is in an equilibrium state, obtaining an expected value of a reactant concentration in the reactor based on the mechanism model; The reaction concentration of the reactor is obtained based on the expected value, and the nonlinear time-delay system model is obtained based on the reaction concentration.

7. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 1, characterized in that: The cascade type continuous stirred reactor includes a first reactor and a second reactor arranged in cascade, and the characterization formula of the mechanism model includes: Among them, A is the first reactor; B is the second reactor; R is the reactant concentration of the reactor; V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the circulation flow rate; P is the feed rate; q is a nonlinear function with complex behavior; Q is a nonlinear function with uncertainty and external interference; u is the actual controller; t is time; d is the time lag term, which is used to characterize the time lag between the first reactor and the second reactor during the reaction process.

8. The tracking control method for a cascade type continuous stirred reactor time-delay system under state-dependent constraints according to claim 7, characterized in that: When the continuous stirred reactor is in equilibrium: q A =R A +R A (t-d A ); Q A ＝0； Q B ＝0。 9. A tracking control system for a cascade type continuous stirred reactor time-delay system under state-dependent constraints, characterized in that: Applicable to a cascade type continuous stirred reactor, the cascade type continuous stirred reactor comprising a first reactor and a second reactor in cascade, the tracking control system comprising: A time-delay system acquisition module (10), the time-delay system acquisition module (10) is used to acquire a mechanism model of the cascade type continuous stirred tank time-delay system, and based on the mechanism model, acquire a state-dependent nonlinear time-delay system model of the cascade type continuous stirred tank time-delay system; A function acquisition module (20) is used to acquire the time-varying asymmetric Lyapunov function of the cascade continuous stirred reactor based on the nonlinear time-delay system model; in the first reactor or the second reactor, the characterization formula of the time-varying asymmetric Lyapunov function includes: Among them, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is the first symbolic discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z1 represents the first independent variable of the input, when z1>0, s1(z1)=1, when z1<0, s1(z1)=0; P is the feed rate; and is the first error constraint boundary function, is the first error constraint lower bound function, is the upper bound function of the first error constraint; y d is the expected trajectory signal; e1 is the first tracking error; t is time; is the first estimation error; Γ1 is the first constant gain matrix; T is the transpose operation; A controller acquisition module (30), the controller acquisition module (30) is used to acquire a controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovsky function; A tracking control module (40) is used to track and control the continuous stirred reactor based on the controller.

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