Tracking control method and system for cascade type continuous stirring reaction kettle time delay system under state dependence constraint

Abstract

The invention discloses a tracking control method and system for a cascade type continuous stirring reaction kettle time delay system under a state dependence constraint, and belongs to the technical field of automatic control. The method comprises the following steps: acquiring a nonlinear time-lag system model, constrained by state dependence, of a cascade type continuous stirred tank time-lag system based on a mechanism model of the cascade type continuous stirred tank time-lag system; obtaining a time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor based on the nonlinear time-delay system model; and based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovski function, obtaining a controller of the continuous stirred tank reactor, and performing tracking control on the cascaded continuous stirred tank reactor through the controller. According to the method, a nonlinear time delay system model subjected to state dependence constraint is taken as a basis, and the influence of time-varying time delay and state dependence constraint on the system performance is eliminated by combining a time-varying asymmetric Lyapunov function and a Lyapunov-Crasovski function.

Description

Technical Field

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

[0002] A continuous stirred tank reactor (CSTR) is one of the most critical production equipments in the continuous chemical production process. With the rapid development of new technologies such as artificial intelligence, the level of China's automatic control technology has become mature. CSTR often exhibits characteristics such as strong nonlinearity, uncertainty, and complex dynamic mechanisms during actual operation, and its modeling and control have become hot research issues in the field of process control. The purpose of controlling the CSTR system is to optimize the operating conditions and operating modes on the premise of ensuring the stability of the system, so as to increase production and improve product quality. In the actual chemical reaction process, it is necessary to reasonably control parameters such as the concentration of reactants, the temperature and pressure in the reaction kettle, and control them within a stable range. At the same time, modern industrial production develops rapidly, and industrial production processes often tend to be large-scale and integrated. Large interconnected systems with complex structures are widely used in actual production, and CSTR often exists in the form of multi-tank cascade (cascade continuous stirred tank reactor, CCSTR) in actual production. In actual engineering systems, the time-delay phenomenon is often one of the important factors causing system instability. Therefore, the time-delay problem should be inevitably considered in system modeling. How to make the system obtain better tracking performance without violating state-dependent constraints and design a high-performance and strongly 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 view of the above-mentioned shortcomings of the prior art, this application has developed a tracking control method and system for a time-delay system of a cascaded continuous stirred tank reactor with state-dependent constraints, including:

[0004] Technical Solution: In a first aspect, an embodiment of this application provides a tracking control method for a time-delay system of a cascaded continuous stirred tank reactor with state-dependent constraints, which is applied to a cascaded continuous stirred tank reactor and includes:

[0005] Obtain the mechanism model of the time-delay system of the cascaded continuous stirred tank reactor, and based on the mechanism model, obtain the non-linear time-delay system model of the time-delay system of the cascaded continuous stirred tank reactor subject to state-dependent constraints;

[0006] Based on the non-linear time-delay system model, obtain the time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor;

[0007] Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain the controller of the continuous stirred tank reactor;

[0008] Based on the controller, perform tracking control on the cascaded continuous stirred tank reactor.

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

[0010] Based on the non-linear time-delay system model, determine the dependent constraint boundary function, and based on the dependent constraint boundary function, obtain the error constraint boundary function;

[0011] Based on the non-linear time-delay system model and the desired trajectory signal, obtain the tracking error;

[0012] Based on the error constraint boundary function, the tracking error, and the desired trajectory signal, determine the time-varying asymmetric Lyapunov function of the continuous stirred tank reactor.

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

[0014]

[0015] where, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is the first sign discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z1 represents the first independent variable input. When z1 > 0, s1(z1) = 1, and 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 first error constraint upper bound function; y d is the desired trajectory signal; e1 is the first tracking error; t is the 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] Based on the first derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain the first parameter;

[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 derivative function of the time-varying asymmetric Lyapunov function, obtain a virtual controller and a first adaptation law.

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

[0021]

[0022] Wherein, and respectively represent the expected values of R A and R B ; 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 non-linear function, represents an unknown control gain function, is an unknown non-linear time-delay function, τ i (t) is the time-delay term, and the upper bound of τ i (t) is a positive constant τ max , and there exists a positive constant satisfying V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the recycle flow rate; P is the feed rate; Q is a non-linear function with uncertainty and external disturbance; t is the time; d is the time-delay term, which is used to characterize the time-delay in the reaction process between the first reactor and the second reactor.

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

[0024] Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain a Lyapunov function:

[0025] V = V B2 + V L ;

[0026]

[0027]

[0028] where V is the Lyapunov function; V B2 is the time-varying asymmetric Lyapunov function of the cascaded first reactor and the second reactor; V L is the Lyapunov-Krasovskii function; V B1 is the time-varying asymmetric Lyapunov function of the cascaded first reactor or the second reactor; s2(z2) is the second sign discriminant function for meeting the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the input second independent variable, when z2 > 0, s2(z2) = 1, when z2 < 0, s2(z2) = 0; and is the second error constraint boundary function, is the second error constraint lower bound function, is the second error constraint upper bound function; 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 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 delay term; d is the integral operation; is the system state variable;

[0030] Based on the first derivative function of the Lyapunov function, obtain the actual controller and the second adaptation law:

[0031]

[0032] where u is the actual controller; is the second adaptation law; g 2 is a positive constant for characterizing the lower bound of, that is 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 steps of obtaining the nonlinear time-delay system model subject to state-dependent constraints for the cascaded continuous stirred tank reactor time-delay system based on the mechanism model include:

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

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

[0037] In some embodiments, the cascaded 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] where A is the first reactor; B is the second reactor; R is the reactant concentration of the reactor; V is the volume of the reactor; K is the reaction constant; D is the reaction residence time; F is the recycle flow rate; P is the feed rate; q is a nonlinear function with complex behavior; Q is a nonlinear function with uncertainty and external disturbance; t is time; s is the time-delay term, which is used to characterize the time-delay between the first reactor and the second reactor during the reaction process.

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

[0041] q A =R A +R A (t - d A );

[0042]

[0043] Q A =0;

[0044] Q B =0.

[0045] Second, an embodiment of the present application further provides a tracking control system for a cascaded continuous stirred tank reactor time-delay system subject to state-dependent constraints, which is applied to a cascaded continuous stirred tank reactor. The tracking control system includes:

[0046] A time-delay system acquisition module, which is used to acquire the mechanism model of the cascaded continuous stirred tank reactor time-delay system and obtain the nonlinear time-delay system model subject to state-dependent constraints for the cascaded continuous stirred tank reactor time-delay system based on the mechanism model;

[0047] Function acquisition module, which is used to obtain the time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor based on the non-linear time-delay system model;

[0048] Controller acquisition module, which is used to obtain the controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function;

[0049] Tracking control module, which is used to perform tracking control on the continuous stirred tank reactor based on the controller.

[0050] Beneficial effects: Compared with the prior art, the tracking control method for the time-delay system of the cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiment of the present application obtains a non-linear time-delay system model with state-dependent constraints of the time-delay system of the cascaded continuous stirred tank reactor based on the mechanism model of the cascaded continuous stirred tank time-delay system; obtains the time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor based on the non-linear time-delay system model; obtains the controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, and performs tracking control on the cascaded continuous stirred tank reactor through the controller. The present application is based on the non-linear time-delay system model with state-dependent constraints converted from the time-delay system of the cascaded continuous stirred tank reactor, and combines the time-varying asymmetric Lyapunov function with the Lyapunov-Krasovskii function to eliminate the influence of time-varying time delay and state-dependent constraints on the system performance. Description of the drawings

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

[0052] Figure 1 It is a step flow chart of the tracking control method for the time-delay system of the cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiment of the present application;

[0053] Figure 2 It is a specific flow chart of step 100 in the tracking control method for the time-delay system of the cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiment of the present application;

[0054] Figure 3 It is a specific flow chart of step 200 in the tracking control method for the time-delay system of the cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiment of the present application;

[0055] Figure 4 It is the specific flowchart of step 300 in the tracking control method for the cascaded continuous stirred tank reactor time-delay system with state-dependent constraints provided by the embodiments of the present application;

[0056] Figure 5 It is the module connection diagram of the tracking control system for the cascaded continuous stirred tank reactor time-delay system with state-dependent constraints provided by the embodiments of the present application;

[0057] Figure 6 It is the mechanism diagram of the cascaded continuous stirred tank reactor provided by the embodiments of the present application;

[0058] Figure 7 It shows the reaction concentration x1 of the continuous stirred tank reactor A in the cascaded continuous stirred tank reactor time-delay system and the desired trajectory y d tracking performance diagram;

[0059] Figure 8 It shows the trajectory diagram of the reaction concentration x2 of the continuous stirred tank reactor B in the cascaded continuous stirred tank reactor time-delay system;

[0060] Figure 9 It shows the trajectory diagrams of the errors e1 and e2 in the cascaded continuous stirred tank reactor time-delay system;

[0061] Figure 10 It shows the trajectory diagram of the controller u in the cascaded continuous stirred tank reactor time-delay system;

[0062] Figure 11 It shows the adaptive parameters of the cascaded continuous stirred tank reactor time-delay system and variation curve diagram;

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

[0064] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without making creative efforts belong to the scope of protection of the present application.

[0065] The Continuous Stirred Tank Reactor (CSTR) is one of the most critical production equipment in the continuous chemical production process. With the rapid development of new technologies such as artificial intelligence, the level of automatic control technology in China has become mature. The CSTR often exhibits characteristics such as strong nonlinearity, uncertainty, and complex dynamic mechanisms during actual operation, and its modeling and control have become hot research issues in the field of process control. The purpose of controlling the CSTR system is to optimize the operating conditions and modes on the premise of ensuring system stability, so as to increase production and improve product quality. In the actual chemical reaction process, it is necessary to reasonably control parameters such as the concentration of reactants, the temperature and pressure in the reaction kettle, and control them within a stable range. At the same time, modern industrial production has developed rapidly, and industrial production processes often tend to be large-scale and integrated. Large interconnected systems with complex structures are widely used in actual production, and the CSTR often exists in the form of a cascade continuous stirred tank reactor (CCSTR) in actual production. In actual engineering systems, the time-delay phenomenon is often one of the important factors causing system instability. Therefore, the time-delay problem should inevitably be considered in system modeling. How to make the system obtain better tracking performance without violating the state-dependent constraints and design a high-performance and strongly stable control method for the CCSTR system is a key and challenging problem that needs to be solved urgently at present.

[0066] In view of this, the tracking control method for the time-delay system of the cascade continuous stirred tank reactor under state-dependent constraints provided by the embodiments of the present application is specifically applied to the time-delay system of the cascade continuous stirred tank reactor, and mainly realizes that the system state eliminates the influence of the time-delay problem on the system while not violating the state-dependent constraints, 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 continuous stirred tank reactor time-delay system subject to state-dependent constraints based on the mechanism model of the cascade continuous stirred tank time-delay system; obtains a time-varying asymmetric Lyapunov function of the cascade continuous stirred tank reactor based on the nonlinear time-delay system model; obtains a controller for the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, and performs tracking control on the cascade continuous stirred tank reactor through the controller. The present application is based on the nonlinear time-delay system model of the cascade continuous stirred tank reactor time-delay system subject to state-dependent constraints after conversion, and combines the time-varying asymmetric Lyapunov function with the Lyapunov-Krasovskii function to eliminate the influence of time-varying time delay and state-dependent constraints on the system performance. At the same time, a neural network is used to approximate the unknown dynamics, so as to ensure that the system has good tracking performance without violating the state-dependent constraints.

[0067] In some embodiments, please refer to Figure 1 and Figure 6 , Figure 1 which is a flowchart of the steps of the tracking control method for the state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application. Figure 6 is a mechanism diagram of the cascaded continuous stirred tank reactor provided by the embodiments of the present application. Specifically, the tracking control method for the state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application is applied to the cascaded continuous stirred tank reactor time-delay system and is specifically completed through steps 100 to 400:

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

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

[0070]

[0071] wherein, the physical meanings of the symbols are as follows:

[0072]

[0073]

[0074] In some embodiments, please refer to Figure 2 , Figure 2 which is a specific flowchart of step 100 in the tracking control method for the state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application. The method for obtaining the state-dependent constraint nonlinear time-delay system model of the cascaded continuous stirred tank reactor time-delay system based on the mechanism model is specifically implemented through steps 110 to 120:

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

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

[0077] q A = R A + R A (t - d A );

[0078]

[0079] QA = 0;

[0080] Q B = 0;

[0081] The equilibrium point of the cascaded continuous stirred tank reactor time-delay system can be obtained and satisfies the following form:

[0082]

[0083] where and respectively represent the expected values of R A and R B .

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

[0085] In some embodiments, the cascaded 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 the continuous stirred tank reactor A, x2 represents the reaction concentration of the continuous stirred tank reactor B, and respectively represent the expected values of R A and R B . 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; V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the recycle flow rate; P is the feed rate; Q is a nonlinear function with uncertainty and external disturbance; t is the time; d is the time-delay term, which is used to characterize the time-delay between the first reactor and the second reactor during the reaction process.

[0088] Furthermore, let where x1 represents the reaction concentration of the continuous stirred tank reactor A, x2 represents the reaction concentration of the continuous stirred tank reactor B, and the mechanism model can be described as:

[0089]

[0090] where, for i = 1, 2, represents an unknown smooth nonlinear function. Let and represent unknown control gain functions. Let and is defined as an unknown non - linear time - delay function. It should be noted that τ i (t) represents the time - delay term. Define a positive constant τ max as its upper bound, and there exists a positive constant satisfying Let and

[0091] Step 200: Based on the non - linear time - delay system model, obtain the time - varying asymmetric Lyapunov function of the cascaded continuous stirred - tank reactor.

[0092] In some embodiments, refer to Figure 3 , Figure 3 which is the specific flowchart of Step 200 in the tracking control method for the time - delay system of the cascaded continuous stirred - tank reactor under state - dependent constraints provided by the embodiments of the present application. The time - varying asymmetric Lyapunov function of the cascaded continuous stirred - tank reactor is specifically obtained through Steps 210 to 230:

[0093] Step 210: Determine the dependence - constraint boundary function based on the non - linear time - delay system model, and obtain the error - constraint boundary function based on the dependence - 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 can be understood that the constraints considered in the present application are state - dependent constraints, that is, functions related to both state variables and time. Therefore, and represent the state - dependent constraint boundary functions, and satisfy where the state variables should satisfy χ i =[y d ,x1] T , i = 1, 2.

[0097] Introduce the error - constraint boundary functions and as follows:

[0098]

[0099] Step 220: Based on the non - linear time - delay system model and the desired trajectory signal, obtain the tracking error.

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

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

[0103]

[0104] where, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is a piecewise function, serving as the first sign discriminant function, which is used 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 are the error constraint boundary functions, is the error constraint lower bound function, is the error constraint upper bound function; y d is the desired 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, which is used to represent the estimated value of W1; Γ1 is the first constant gain matrix, T represents the transpose operation.

[0105] Step 300: Obtain the controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function.

[0106] In some embodiments, please refer to Figure 4 , Figure 4 is the specific flowchart of Step 300 in the tracking control method for the time-delay system of the cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiments of the present application. The method for obtaining the controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function is implemented through Steps 310 to 350:

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

[0108] In some embodiments, for V B1 Taking the derivative gives:

[0109]

[0110] where, Define and Define

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

[0112]

[0113] where U1(Z1) is the first parameter, and f1(x1(t)) represents an unknown smooth nonlinear function; And the Lyapunov-Krasovskii function is Its role is to handle the time-delay term Let where represents a positive constant, satisfying

[0114] Step 320: Based on the approximation of the neural network, approximate 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 representation 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 a Gaussian function vector, and the number of neural network nodes is l.

[0118] Step 330: Based on the second parameter and the first derivative function of the time-varying asymmetric Lyapunov function, obtain the virtual controller and the first adaptation law.

[0119] In some embodiments, the representation formulas of the virtual controller α1 and the first adaptation law include:

[0120]

[0121] where, a1 is a positive constant; g1 is a positive constant that satisfies the lower bound of the non - linear smooth function g1(x1(t)), i.e., 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 a 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 lead to very complex expressions of the Lyapunov function and its derivative, making it difficult to directly analyze. The role of Young's inequality is to as well as scale the terms in

[0123]

[0124] appropriately, transform them into a simpler and more tractable form, and enable them to be combined with the determination conditions of positive definite functions. Furthermore, a conclusion about the system stability can be obtained. Based on Young's inequality, we can get: where, σ1(Z1) represents the approximation error and satisfies

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

[0126]

[0127] where, and will be cancelled out in the next step.

[0128] Step 340: Based on the time - varying asymmetric Lyapunov function and the Lyapunov - Krasovskii function, obtain the Lyapunov function.

[0129] In some embodiments, the representation 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 cascaded first and second reactors; V L is the Lyapunov-Krasovskii function; V B1 is the time-varying asymmetric Lyapunov function of the cascaded first or second reactor; s2(z2) is the second sign 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 second error constraint lower bound function, is the second error constraint upper bound function; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant and satisfies 2p ≥ n + 2, where 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 delay term; d is the integral operation; is the system state variable.

[0133] According to the Lyapunov stability theorem, first, the derivative of V is calculated to facilitate the judgment of the stability of the cascaded continuous stirred tank reactor time-delay system. The derivative of V can be obtained as follows:

[0134]

[0135] For V B2 the derivative can be obtained as follows:

[0136]

[0137] V B1 is the time-varying asymmetric Lyapunov function of the first or second reactor, V B2 is the time-varying asymmetric Lyapunov function of the cascaded first and second reactors; V B2 and V B1 have an accumulative relationship. Combining the backstepping technique, its main role is to ensure the stability of the system based on the Lyapunov stability theory by constructing appropriate Lyapunov functions and control laws, and simultaneously combining neural network control. It can make the system state asymptotically converge to the desired equilibrium point, that is, as time goes by, the output of the system can infinitely approach the given reference signal, thereby realizing the control of the high-precision cascaded continuous stirred tank reactor time-delay system.

[0138] For the time-delay system of cascaded continuous stirred tank reactors, by designing a stable controller to adjust the input of the system, 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 adaptation law continuously adjusts the parameters or structure of the controller, enabling the system to maintain good stability and performance under different working conditions and environments. Even when the system parameters change significantly or are subject to strong disturbances, the system can still operate normally, reducing the dependence on an accurate model and improving the reliability and adaptability of the system. Design the actual controller u and the second adaptation law as follows:

[0139]

[0140] where u is the actual controller; is the second adaptation law. g 2 is a positive constant that satisfies the lower bound of the nonlinear smooth function i.e., 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 sigmoid 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 appropriately scale the terms in V B2 and transforming them into a simpler and more tractable form, and then drawing conclusions about the system stability. Based on Young's inequality, we can obtain:

[0142]

[0143] where a2 represents a positive constant, and we can obtain where and

[0144] Furthermore, to simplify the expression of V B2 substitute (21)-(24) into (20), which can be transformed into:

[0145]

[0146] Next, deal with the time-delay function. First, take the derivative of V L to obtain:

[0147]

[0148] wherein, and

[0149] the time-delay term τ j (t) satisfies In addition, by combining μ = 1 - τ, we can obtain Therefore,[[]] can be expressed as:

[0150]

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

[0152]

[0153] wherein, and it satisfies Θ < 0. Θ only represents the symbol for grouping the time-delay part together, and after arrangement, it can be obtained that this formula is less than zero, so it can be omitted.[[]]

[0154] According to and Therefore, the following inequality can be obtained:

[0155]

[0156] Combining (7), (16), and (17), V can be transformed into:

[0157]

[0158] Finally, it can be obtained that:

[0159]

[0160] wherein, 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 reduced to the Lyapunov stability conclusion. This step is used to determine that the time-delay system of the cascaded continuous stirred tank reactor is globally asymptotically stable.

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

[0162] In some embodiments, the representation formulas of the actual controller and the second adaptation law include:

[0163]

[0164] where u is the actual controller; is the second adaptation law. g 2 is a positive constant that satisfies the lower bound of the non-linear smooth function i.e., 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 sigmoid 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: Based on the controller, perform tracking control on the cascaded continuous stirred tank reactor.

[0166] Specifically, in the embodiments of the present invention, the basic physical parameters of the mechanism model of the time-delay system of the cascaded continuous stirred tank reactor 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 the desired trajectory y d = 0.6sin(πt)e -2t + 0.6, and the time-delay function is designed as where τ1(t) = 1 - sin(0.5t)cos(2t), τ2 = 1.55 - 0.5sin(0.5t). The initial values of the system state and the adaptation parameters are selected as In addition, the system state x i , i = 1, 2 and e i , i = 1, 2 must satisfy the state-dependent constraint boundary conditions:

[0167]

[0168] where χ1 = y d and X2 = x1, and the state-dependent constraint boundary function is designed in the following form:

[0169]

[0170] Under this initial condition, simulation verification is carried out, and the simulation result graph is as Figures 7 - 11 shown. Figure 7 It shows the tracking performance graph of the reaction concentration x1 of the continuous stirred tank reactor A in the cascaded continuous stirred tank reactor time-delay system and the desired trajectory y d ; Figure 8 It shows the trajectory graph of the reaction concentration x2 of the continuous stirred tank reactor B in the cascaded continuous stirred tank reactor time-delay system; Figure 9 It shows the trajectory graphs of the errors e1 and e2 in the cascaded continuous stirred tank reactor time-delay system; Figure 10 It shows the trajectory graph of the controller u in the cascaded continuous stirred tank reactor time-delay system; Figure 11 It shows the adaptive parameters of the cascaded continuous stirred tank reactor time-delay system and variation curve graph.

[0171] Understandably, the tracking control method for the state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application is specifically applied to the cascaded continuous stirred tank reactor time-delay system, which mainly realizes that the system state eliminates the influence of the time-delay problem on the system without violating the state-dependent constraints, greatly improving the safety and stability of the chemical production process. Specifically, the present application obtains a non-linear time-delay system model of the cascaded continuous stirred tank reactor time-delay system subject to state-dependent constraints based on the mechanism model of the cascaded continuous stirred tank reactor time-delay system; obtains a time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor based on the non-linear time-delay system model; obtains a controller for the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, and performs tracking control on the cascaded continuous stirred tank reactor through the controller. Based on the non-linear time-delay system model of the cascaded continuous stirred tank reactor time-delay system converted under state-dependent constraints, the present application combines the time-varying asymmetric Lyapunov function with the Lyapunov-Krasovskii function to eliminate the influence of time-varying time-delay and state-dependent constraints on the system performance. At the same time, a neural network is used to approximate the unknown dynamics, so as to ensure that the system has good tracking performance without violating the state-dependent constraints.

[0172] Correspondingly, please refer to Figure 5 , Figure 5 which is the module connection diagram of the tracking control system for the state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application. The state-dependent constraint lower cascaded continuous stirred tank reactor time-delay system provided by the embodiments of the present application includes:

[0173] A time-delay system acquisition module 10, which is used to acquire the mechanism model of the cascaded continuous stirred tank reactor time-delay system and obtain a non-linear time-delay system model of the cascaded continuous stirred tank reactor time-delay system subject to state-dependent constraints based on the mechanism model;

[0174] A function acquisition module 20, which is used to obtain a time-varying asymmetric Lyapunov function of the cascaded continuous stirred tank reactor based on the non-linear time-delay system model;

[0175] A controller acquisition module 30, which is used to obtain a controller for the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function;

[0176] A tracking control module 40, which is used to perform tracking control on the continuous stirred tank reactor based on the controller.

[0177] In some embodiments, the function acquisition module 20 is specifically used for:

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

[0179] Obtain a tracking error based on the non - linear time - delay system model and the desired trajectory signal;

[0180] Determine a time - varying asymmetric Lyapunov function of the continuous stirred tank reactor based on the error constraint boundary function, the tracking error, and the desired trajectory signal.

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

[0182] Obtain a first parameter based on the first - order derivative function of the time - varying asymmetric Lyapunov function and the Lyapunov - Krasovskii function;

[0183] Perform an approximation process on the first parameter based on the approximation of the neural network to obtain a second parameter;

[0184] Obtain a virtual controller and a first adaptation law based on the second parameter and the first - order derivative function of the time - varying asymmetric Lyapunov function.

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

[0186] Obtain a Lyapunov function based on the time - varying asymmetric Lyapunov function and the Lyapunov - Krasovskii 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 cascaded first reactor and second reactor; V L is the Lyapunov - Krasovskii function; V B1 is the time - varying asymmetric Lyapunov function of the cascaded first reactor or second reactor; s2(z2) is the second sign discriminant function, which is used to meet the characteristic requirements of the time - varying asymmetric Lyapunov function, where z2 represents the input second independent variable. When z2 > 0, s2(z2)=1; when z2 < 0, s2(z2)=0; and is the second error constraint boundary function, is the second error constraint lower - bound function, is the second error constraint upper - bound function; 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-delay term; d is the integral operation; is the system state variable;

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

[0192]

[0193] where u is the actual controller; is the second adaptation law; g2 is a positive constant used to characterize the lower bound of, that is 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 cascaded continuous stirred tank reactor is in an equilibrium state, the expected value of the reactant concentration of the reactor is obtained based on the mechanism model;

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

[0197] The above detailedly introduces a tracking control method and system for a time-delay system of a cascaded continuous stirred tank reactor under state-dependent constraints provided by the embodiments of the present application. Specific examples are used in this article to elaborate on the principle and implementation manner of the present application. The description of the above embodiments is only used to help understand the method and its core idea of the present application; at the same time, for those skilled in the art, according to the idea of the present application, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation to the present application.

Claims

1. A tracking control method for a time-delay system of a cascade continuous stirred tank reactor with state-dependent constraints, which is applied to a cascade continuous stirred tank reactor. The tracking control method includes: Obtain the mechanism model of the time-delay system of the cascade continuous stirred tank reactor, and based on the mechanism model, obtain the non-linear time-delay system model of the cascade continuous stirred tank reactor subject to state-dependent constraints; Based on the non-linear time-delay system model, obtain the time-varying asymmetric Lyapunov function of the cascade continuous stirred tank reactor; Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain the controller of the continuous stirred tank reactor; Perform tracking control on the cascade continuous stirred tank reactor based on the controller.

2. The tracking control method for the state-dependent constrained lower cascade continuous stirred tank reactor time-delay system according to claim 1, characterized in that, The step of obtaining the time-varying asymmetric Lyapunov function of the cascade continuous stirred tank reactor includes: Determine the dependence constraint boundary function based on the non-linear time-delay system model, and obtain the error constraint boundary function based on the dependence constraint boundary function; Based on the non-linear time-delay system model and the desired trajectory signal, obtain the tracking error; Based on the error constraint boundary function, the tracking error, and the desired trajectory signal, determine the time-varying asymmetric Lyapunov function of the continuous stirred tank reactor.

3. The tracking control method for the state-dependent constrained lower cascaded continuous stirred tank reactor time-delay system according to claim 1, characterized in that The cascade continuous stirred tank reactor includes a first reactor and a second reactor connected in cascade. In the first reactor or the second reactor, the representation formula of the time-varying asymmetric Lyapunov function includes: where, V B1 is the time-varying asymmetric Lyapunov function of the first reactor or the second reactor; s1(z1) is the first sign 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 first error constraint upper bound function; y d is the desired trajectory signal; e1 is the first tracking error; t is the time; is the first estimation error; Γ1 is the first constant gain matrix; T is the transpose operation.

4. The tracking control method for the state-dependent constrained lower cascade continuous stirred tank reactor time-delay system according to claim 1, characterized in that, The step of obtaining the controller of the continuous stirred tank reactor includes: Based on the first derivative function of the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain the first parameter; Based on the approximation of the neural network, perform an approximation process on the first parameter to obtain the second parameter; Based on the second parameter and the first derivative function of the time-varying asymmetric Lyapunov function, obtain the virtual controller and the first adaptive law.

5. The tracking control method for the state-dependent constrained lower cascade continuous stirred tank reactor time-delay system according to claim 1, characterized in that, The cascade continuous stirred tank reactor includes a first reactor and a second reactor connected in cascade. The representation formula of the non-linear time-delay system model includes: Among them, and represent the expected values of R A and R B respectively. 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; represents an unknown smooth non - linear function, represents an unknown control gain function, is an unknown non - linear time - delay function, τ i (t) is the time - delay term, τ i (t) has an upper bound of a positive constant τ max , there exists a positive constant satisfying V is the reactor volume; K is the reaction constant; D is the reaction residence time; F is the recycle flow rate; P is the feed rate; Q is a non - linear function with uncertainty and external disturbance; t is the time; d is the time - delay term, used to characterize the time - delay in the reaction process between the first reactor and the second reactor.

6. The tracking control method for the state-dependent constrained lower cascade continuous stirred tank reactor time-delay system according to claim 5, characterized in that, The cascade continuous stirred tank reactor includes a first reactor and a second reactor connected in cascade. In the cascaded first reactor and second reactor, the step of obtaining the controller of the continuous stirred tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function further includes: Based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function, obtain the Lyapunov function: V = V B2 +V L ; where, V is the Lyapunov function; V B2 is the time-varying asymmetric Lyapunov function of the cascaded first reactor and the second reactor; V L is the Lyapunov-Krasovskii function; V B1 is the time-varying asymmetric Lyapunov function of the cascaded first reactor or the second reactor; s2(z2) is the second sign discriminant function, which is used to meet the characteristic requirements of the time-varying asymmetric Lyapunov function, where z2 represents the input second independent variable. When z2>0, s2(z2)=1; when z2<0, s2(z2)=0; and is the second error constraint boundary function, is the second error constraint lower bound function, is the second error constraint upper bound function; e2 is the second tracking error; t is time; is the second estimation error; p is the feed rate, which is a positive constant and satisfies 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-delay term; d is the integral operation; is the system state variable; Based on the first derivative function of the Lyapunov function, obtain the actual controller and the second adaptive law: where, u is the actual controller; is the second adaptive law; g2 is a positive constant used to characterize the lower bound of, i.e., 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.

7. The tracking control method for the state-dependent constraint lower cascade continuous stirred tank reactor time-delay system according to claim 1, characterized in that The step of obtaining the non-linear time-delay system model of the cascade continuous stirred tank reactor subject to state-dependent constraints based on the mechanism model includes: When the cascade continuous stirred tank reactor is in an equilibrium state, obtain the expected value of the reactant concentration of the reactor based on the mechanism model; Based on the expected value, obtain the reaction concentration of the reactor, and based on the reaction concentration, obtain the non-linear time-delay system model.

8. The tracking control method for the state-dependent constraint lower cascade continuous stirred tank reactor time-delay system according to claim 1, characterized in that, 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: Wherein, A is the first reactor; B is the second reactor; R is the reactant concentration in 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 non-linear function with complex behavior; Q is a non-linear function with uncertainty and external disturbance; t is time; d is the time-delay term, which is used to characterize the time-delay in the reaction process between the first reactor and the second reactor.

9. The tracking control method for the state-dependent constrained lower cascade continuous stirred tank reactor time-delay system according to claim 7, characterized in that, When the continuous stirred-tank reactor is in an equilibrium state: q A = R A + R A (t - d A ); Q A ＝0； Q B ＝0。 10. A tracking control system for a time-delay system of a cascaded continuous stirred tank reactor with state-dependent constraints at the lower level, characterized in that, Applied to the cascade continuous stirred-tank reactor, the tracking control system includes: A time-delay system acquisition module (10), which is used to acquire the mechanism model of the time-delay system of the cascade continuous stirred-tank reactor, and based on the mechanism model, acquire the non-linear time-delay system model of the time-delay system of the cascade continuous stirred-tank reactor subject to state-dependent constraints; A function acquisition module (20), which is used to acquire the time-varying asymmetric Lyapunov function of the cascade continuous stirred-tank reactor based on the non-linear time-delay system model; A controller acquisition module (30), which is used to acquire the controller of the continuous stirred-tank reactor based on the time-varying asymmetric Lyapunov function and the Lyapunov-Krasovskii function; A tracking control module (40), which is used to perform tracking control on the continuous stirred-tank reactor based on the controller.

Images