Active covariance assignment stochastic model predictive control approach for chemical processes

Abstract

The application provides a random model predictive control method for active covariance distribution of a chemical process, determines controlled variables and input variables according to process requirements, establishes a linear discrete system model driven by additive random disturbance, sets a polyhedral chance constraint and a target function; solves an assignable terminal covariance optimization problem offline to obtain a terminal penalty weight and a feedback gain; introduces a control law containing an open-loop component and an error feedback gain, constructs a predictive control model, expands the target function into a quadratic form, and converts the chance constraint into a deterministic second-order cone constraint by using Kantorovich inequality and variable replacement; takes the current process state as input to solve a second-order cone programming online to obtain an optimal control sequence. The application realizes explicit regulation and control of state distribution by actively distributing the terminal covariance, convexes the probability constraint into a second-order cone constraint which can be solved efficiently, and optimizes the economic performance of chemical production under the premise of meeting the operation safety margin.

Description

Technical Field

[0001] This invention relates to the field of chemical system control, and in particular to a stochastic model predictive control method for active covariance allocation in chemical processes. Background Technology

[0002] Proportional-integral-derivative (PID) controllers and linear state feedback controllers are widely used in chemical process control due to their simple structure and robustness, especially suitable for typical equipment such as continuous stirred reactors (CSTRs), distillation columns, and heat exchangers. However, these methods rely on tedious empirical parameter tuning, resulting in high development costs, and they struggle to explicitly handle the physical constraints, safety limitations, and nonlinear characteristics of chemical reaction kinetics in complex chemical systems at the control sampling time. In contrast, Model Predictive Control (MPC) can effectively handle multivariable constraints and complex dynamics, and has become the mainstream optimization control framework for constrained dynamic systems in continuous chemical processes.

[0003] Although nominal MPC can suppress a certain degree of disturbances and model errors by relying solely on state feedback, its design phase does not explicitly consider system uncertainties and lacks quantitative robustness guarantees, making it prone to failure when model errors are large or reaction conditions fluctuate significantly. Therefore, traditional Robust Model Predictive Control (RMPC) constructs conservative constraints based on worst-case assumptions to ensure the reliable operation of chemical systems under bounded uncertainties. However, disturbances in chemical processes often have stochastic characteristics, such as fluctuations in feed concentration, temperature disturbances, or batch variations in raw materials. Worst-case-based RMPC often leads to overly conservative control strategies; furthermore, when disturbances exceed preset boundaries, the system faces the risk of default, potentially causing reaction instability or product quality fluctuations.

[0004] For disturbances with random characteristics, probability distribution modeling is more reasonable because it can accurately characterize the occurrence patterns of disturbances of different amplitudes. Traditional RMPC usually requires both the disturbance and constraint sets to be bounded, making it difficult to handle random disturbances with unbounded support (such as Gaussian white noise), which is particularly significant in continuous reactor control. Since unbounded disturbances theoretically always have a probability of default in the future state of the system, RMPC is prone to losing its recursive feasibility under such conditions.

[0005] In existing stochastic MPC algorithms, the steady-state solution of the discrete Lyapunov equation (such as in LQR-based designs) is typically used as the upper limit of variance when handling terminal variance constraints. While this simplifies computation, it makes the terminal variance an implicit function of the performance weights Q and R. Therefore, when the controller adjusts the weights to optimize the dynamic performance of the chemical system, it unpredictably changes the size of the terminal variance constraint set, forcing the system to make a passive trade-off between desired closed-loop performance and system stability guarantees. Summary of the Invention

[0006] The main objective of this invention is to propose a stochastic model predictive control method for active covariance allocation in chemical processes, targeting state variables and operational variables of reactors, distillation columns, or continuous mixing equipment that are affected by random disturbances, in order to solve the aforementioned technical problems.

[0007] This invention proposes a stochastic model predictive control method for active covariance allocation in chemical processes, the method comprising the following steps: Step 1: Based on the process requirements of the chemical process, determine the controlled variables and control input variables in order to establish a dynamic model of a linear discrete system driven by additive random disturbances; To define polyhedral chance constraints for the state and input of a linear discrete system dynamics model, and to predict the control objective function; Step 2: Based on the linear discrete system dynamics model, solve the optimization problem about the allocatable terminal covariance matrix offline to obtain the minimum terminal covariance matrix; determine the corresponding terminal penalty weight matrix and terminal state feedback gain matrix based on the minimum terminal covariance matrix. Step 3: Based on the terminal state feedback gain matrix, a control law with deterministic open-loop input components and dynamic error feedback gain is introduced into the dynamic model of a linear discrete system driven by additive random disturbances to construct a predictive control model. Based on the terminal penalty weight matrix and the predictive control model, the predictive control objective function is reconstructed into a convex quadratic form, resulting in a quadratic objective function. By using the Cantley inequality and variable substitution techniques, the chance constraints of non-convex polyhedra are transformed into equivalent deterministic second-order cone constraints. Step 4: Using the current actual state of the chemical system as input, based on the predictive control model, the reconstructed convex quadratic objective function, and the second-order cone constraint, solve the second-order cone programming problem online to obtain the optimal control input sequence; Step 5: Apply the first control quantity of the optimal control input sequence to the chemical system, and repeat steps 4 to 5 at the next sampling time with the updated actual state as input to achieve rolling optimization control.

[0008] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention adopts an active terminal covariance allocation strategy. By solving the linear matrix inequality optimization problem about the allocable terminal covariance matrix offline, it realizes explicit control of the terminal state probability distribution of chemical process. It overcomes the limitation of the coupling between terminal cost function design and closed-loop steady-state terminal set size in traditional stochastic model predictive control. This allows the terminal penalty weight configuration and closed-loop dynamic performance to be designed independently, improving the flexibility of controller design.

[0009] 2. This invention utilizes the Cantlay inequality and variable substitution techniques to reconstruct the non-convex polyhedral chance constraint affected by additive random unbounded perturbation into a deterministic second-order cone constraint. This solves the non-convexity problem of optimization problems caused by probabilistic operational constraints, transforming the online optimization problem into a second-order cone programming problem that can be solved efficiently in real time in industrial controllers, thus ensuring the operational safety margin of chemical processes under random perturbations.

[0010] 3. By establishing a control law that includes deterministic open-loop input components and dynamic error feedback gain, this invention decomposes the system dynamics into nominal dynamics and error dynamics, allowing the planning of the desired trajectory and the suppression of random errors to be handled separately. While achieving the goal of optimal economic operation of the chemical process, it effectively suppresses the dispersed influence of random disturbances on the system state, thereby improving control accuracy and robustness.

[0011] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by means of embodiments of the invention. Attached Figure Description

[0012] Figure 1 This is a flowchart of the stochastic model predictive control method for active covariance allocation in chemical processes proposed in this invention. Figure 2 This is a comparison of elliptical PRS results under different control weights according to the present invention; Figure 3 This is a comparison of the PRS results of polyhedra under different control weights according to the present invention; Figure 4 This is a sample image of 1000 system closed-loop trajectories obtained by sampling at k=1,2,…,7 in scenario 1. Figure 5 For scenario 2, this invention provides 1000 system closed-loop trajectory sample images obtained at times k=1,2,…,7. Figure 6 For scenario 1, 1000 system closed-loop trajectory samples were obtained by sampling at k=1,2,…,7 using the RFMPC method; Figure 7For scenario 1, 1000 system closed-loop trajectory samples were obtained by using the Tube MPC method at times k=1,2,…,7; Figure 8 For scenario 1, the system closed-loop mean trajectory is obtained by sampling at k=1,2,…,7. Detailed Implementation

[0013] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0014] These and other aspects of the embodiments of the present invention will become clear from the following description and accompanying drawings. In these descriptions and drawings, some specific embodiments of the present invention are specifically disclosed to illustrate some ways of implementing the principles of the embodiments of the present invention; however, it should be understood that the scope of the embodiments of the present invention is not limited thereto.

[0015] Please see Figure 1 This embodiment provides a stochastic model predictive control method for active covariance allocation in chemical processes, the method comprising the following steps: Step 1: Based on the process requirements of the chemical process, determine the controlled variables and control input variables in order to establish a dynamic model of a linear discrete system driven by additive random disturbances; Furthermore, polyhedral chance constraints on the state and input, as well as a predictive control objective function, are set for the dynamic model of the linear discrete system. In a preferred embodiment of the present invention, the dynamic model of a linear discrete system driven by additive random perturbation has the following relationship: ; in, This represents the state vector at time k+1. This represents the state vector at time k. , The dimension of the system state vector; This represents the control input variable at time k. , This indicates the dimension of the control input vector; This represents an additive perturbation with zero mean and known variance, or a random unbounded perturbation with bounded variance. , The dimension of the additive random perturbation vector; These represent the system's state matrix, disturbance matrix, and control input matrix, respectively. .

[0016] In a preferred embodiment of the present invention, in step 1, the polyhedral chance constraint has the following relationship: ; in, Represents a probability operator. Represents the set of non-negative integers. Let these represent the set of state constraints and the set of control input constraints of the system, respectively. These represent the maximum probability limits for the system state and control input to violate constraints, respectively. Defined as ; in, This represents the total number of state inequality constraints. Let represent the transpose of the coefficient vector of the state constraint of the i-th polyhedron. Denotes the boundary constants of the state constraint of the i-th polyhedron. , Indicates the transpose operator; Defined as ; in, This indicates the number of polyhedral hyperplanes controlling the input constraints. Let represent the transpose of the coefficient vector of the control input constraint of the i-th polyhedron. This represents the boundary constant of the control input constraint for the i-th polyhedron. , The number of polyhedral hyperplanes representing the joint state and control input constraints.

[0017] In a preferred embodiment of the present invention, the overall objective is to find a control sequence that can guide the system's trajectory from a random initial state to a desired terminal distribution, while satisfying chance constraints and minimizing the cost function. Therefore, the predictive control objective function is defined as follows: ; in, This represents the predictive control objective function. Indicates the length of the prediction time domain. This represents the state vector at the initial moment. This represents the stage cost at time k. Represents the terminal cost function. This represents the terminal state vector at the end of the prediction time domain. The mathematical expectation operator for a random perturbation w; The control sequence is defined as: ; in, This represents the complete control input sequence in the prediction time domain. This represents the transpose of the control input vector at time n-1 (i.e., the last step) in the prediction time domain; The random initial state is defined as: ; in, This represents the random state vector at the initial moment. The mean vector representing the initial state. The covariance matrix representing the initial state; The desired terminal distribution is defined as:

[0018] in, This represents the terminal state vector at the end of the prediction time domain. This represents the expected mean vector of terminal states. Represents the desired terminal state covariance matrix; The stage cost at time k is defined as: ; in, This represents the state vector at time t. This represents the control input vector at time t. Denotes the state-weighted square norm. This represents the weighted square norm of the control input. and These represent the state weight matrix and the control input weight matrix, respectively. ; The terminal cost function is defined as: ; in, This represents the weighted square norm of the terminal state.

[0019] Step 2: Based on the linear discrete system dynamics model, solve the optimization problem about the allocatable terminal covariance matrix offline to obtain the minimum terminal covariance matrix; determine the corresponding terminal penalty weight matrix and terminal state feedback gain matrix based on the minimum terminal covariance matrix. In a preferred embodiment of the present invention, based on a linear discrete system dynamics model, an optimization problem concerning the allocatable terminal covariance matrix is ​​solved offline to obtain the minimum terminal covariance matrix; based on the minimum terminal covariance matrix, the corresponding terminal penalty weight matrix and terminal state feedback gain matrix are determined, specifically including the following steps: Based on the system dynamics matrix and the upper bound matrix of the perturbation covariance, an offline optimization problem is constructed with the terminal covariance matrix as the decision variable. The corresponding process has the following relationship: ;

[0020] in, Represents the terminal covariance matrix. Describing the Frobenius norm, express Moore-Penrose pseudo-inverse; It should be noted that in the above offline optimization problem, the feasibility conditions of the equality and inequality constraints strictly depend on the controllability of the chemical system dynamics. Specifically, the linear matrix inequality (LMI) has a bounded feasible solution if and only if the nominal system has a stabilizable property. If the current chemical system has uncontrollable and unstable divergent modes, it is impossible to find a covariance upper bound that satisfies the conditions, and the offline optimization problem will have no solution.

[0021] In practical engineering applications, those skilled in the art can use the standard convex optimization solution toolbox based on the interior point method to solve this LMI problem. If the solver returns an infeasible state, it can guide process engineers to check whether there are actuator saturation or uncontrollable physical structural defects in the system.

[0022] The offline optimization problem is solved to obtain the minimum terminal covariance matrix; Based on the minimum terminal covariance matrix, the terminal penalty weight matrix required for the predictive control objective function and the corresponding terminal state feedback gain matrix are determined by solving the discrete-time Lyapunov equation.

[0023] Specifically, based on the minimum terminal covariance matrix, the terminal penalty weight matrix required for the predictive control objective function and the corresponding terminal state feedback gain matrix are determined by solving the discrete-time Lyapunov equation; the specific calculation process is as follows: First, the discrete-time algebraic Riccati equation is solved for the nominal system to obtain the steady-state weight matrix. The corresponding process has the following relationship: ; in, The solution to the discrete-time algebraic Riccati equation is represented by the steady-state weight matrix. Based on the obtained steady-state weight matrix, the corresponding terminal state feedback gain matrix is ​​calculated, and the corresponding process has the following relationship: ; in, This represents the terminal state feedback gain matrix; Finally, the terminal state feedback gain matrix Substituting into the closed-loop system, the terminal penalty weight matrix is ​​calculated by solving the discrete-time Lyapunov equations. The corresponding process has the following relationship: .

[0024] Step 3: Based on the terminal state feedback gain matrix, a control law with deterministic open-loop input components and dynamic error feedback gain is introduced into the dynamic model of a linear discrete system driven by additive random disturbances to construct a predictive control model. Based on the terminal penalty weight matrix and the predictive control model, the predictive control objective function is reconstructed into a convex quadratic form, resulting in a quadratic objective function. By using the Cantley inequality and variable substitution techniques, the chance constraints of non-convex polyhedra are transformed into equivalent deterministic second-order cone constraints. In a preferred embodiment of the present invention, based on the terminal state feedback gain matrix, a control law with deterministic open-loop input components and dynamic error feedback gain is introduced into the dynamic model of a linear discrete system driven by additive random disturbances to construct a predictive control model, specifically including the following steps: Define the expected state as Define the state error as ; in, Represents the expected state. Indicates state error; Using the terminal state feedback gain matrix as the constant error feedback gain, and based on the deterministic open-loop input vector, state error, and constant error feedback gain, a control law is constructed. The corresponding process has the following relationship: ; in, This represents the deterministic open-loop control component to be optimized online. This represents the terminal state feedback gain matrix; Substituting the control law into the dynamics model of the linear discrete system and taking the expectation of the state, we obtain the nominal dynamics and error dynamics. The corresponding process has the following relationship: ; ; in, Indicates nominal dynamics. Indicates error dynamics; Nominal dynamics and error dynamics constitute the predictive control model.

[0025] In a preferred embodiment of the present invention, based on the terminal penalty weight matrix and the predictive control model, the predictive control objective function is reconstructed into a convex quadratic form to obtain the quadratic objective function. The corresponding process has the following relationship: ; in, This represents the predictive control objective function that includes both stage costs and terminal costs. Indicates the length of the prediction time domain. This represents the state vector at the predicted time-domain terminal. This represents the terminal penalty weight matrix. .

[0026] As a preferred embodiment of the present invention, for the convex reconstruction of nonlinear optimization problems caused by probabilistic operational constraints in chemical process control, the nonconvex safety constraints are transformed into deterministic second-order cone constraints that are easy for industrial controllers to solve in real time and efficiently using the Cantley inequality and specific variable substitution techniques. Specifically, the following steps are included: The total budget for state violation probabilities is allocated to the risk budgets of each state constraint surface, satisfying... ,in, This represents the risk budget allocated to the i-th state constraint surface; The total budget for input violation probabilities is allocated to the risk budget of each input constraint surface, satisfying... ,in, This represents the risk budget allocated to the i-th input constraint surface; Introducing an auxiliary gain matrix ,in, Represents the auxiliary gain matrix. Indicates the constant error feedback gain. Represents the identity matrix; The mathematical background and motivation for introducing the auxiliary gain matrix lies in the fact that this transformation essentially converts the state feedback form of the control law into a feedforward equivalent form. In stochastic MPC, to overcome the non-convexity caused by probabilistic constraints, it is necessary to transform the control input vector at time t... The whole is parameterized as an affine variable By replacing the closed-loop dynamic parameters in this way, the mutual influence between the deterministic open-loop control components and the random error covariance on the system state evolution can be effectively decoupled. This transforms the originally nonlinear covariance propagation dynamics into a matrix inequality form that is linear with respect to the decision variables, laying a theoretical foundation for subsequent efficient convex optimization solutions.

[0027] Using the Cantley inequality, the constraint limits are independently derived for each constraint surface. Based on the auxiliary gain matrix, the control law and the mean and covariance of the state under error dynamics are substituted to obtain a deterministic second-order cone form constraint expression.

[0028] Specifically, the transformation from polyhedral chance constraints to deterministic second-order cone constraints using the Cantelli inequality is derived as follows: For the i-th state chance constraint surface Introducing state error This is transformed into a probabilistic constraint on the error. .

[0029] Define random variables ,because ,in, To express the expectation of the variable, according to Cantelli's inequality... ,in, This represents a positive threshold. This represents the variance of the variable Z. Let... Simplifying, we obtain the deterministic inequality: ; in, , This indicates the extraction of the variance of the k-th variable. Represents the error variable in a compact form. The covariance matrix represents the state error sequence.

[0030] The derivation process for the transformation of the deterministic second-order cone constraints in the state part and the input part is the same, so I will not go into details here.

[0031] Through the above derivation, the probabilistic constraints were successfully reconstructed into a deterministic second-order cone constraint space.

[0032] The transformed deterministic state second-order cone constraint form is as follows: ; in, This represents the risk budget allocated to the i-th state constraint surface. This represents the block selection matrix at time k. The expected vector representing the initial state. Indicates the open-loop input sequence direction. The number of polyhedral hyperplanes representing state constraints; The transformed deterministic input second-order cone constraint form is: ; in, This represents the risk budget allocated to the i-th control input constraint surface. This represents the covariance matrix associated with the sequence of auxiliary variables.

[0033] Step 4: Using the current actual state of the chemical system as input, based on the predictive control model, the reconstructed convex quadratic objective function, and the second-order cone constraint, solve the second-order cone programming problem online to obtain the optimal control input sequence; Step 5: Apply the first control quantity of the optimal control input sequence to the chemical system, and repeat steps 4 to 5 at the next sampling time with the updated actual state as input to achieve rolling optimization control.

[0034] To verify the effectiveness of this invention, a system mathematical model was established, and then simulations were performed using a computing platform and the Yalmip optimization modeling framework to achieve optimal stochastic model predictive control based on active covariance allocation for a linear discrete system driven by additive stochastic disturbances. In this example, the simulation uses the following linear discrete time-invariant system: ; In the formula, and Represent the initial value of the system error and the matrix with zero mean and covariance, respectively. Gaussian white noise.

[0035] This case considers the following finite-time stochastic model predictive control problem: ; in, This represents the predicted state vector for the i-th future step at the current moment. This represents the control input vector predicted for the i-th future step at the current time k. This represents the terminal state vector predicted at the end of the prediction time domain at the current time k; where the input and state constraints are... and The initial conditions are .

[0036] For linear discrete systems driven by additive random disturbances, a stochastic model predictive controller based on active covariance allocation and second-order cone programming is established to achieve explicit regulation and closed-loop optimal control of the state distribution of the chemical system, as detailed below: To quantitatively analyze the effectiveness of the active terminal variance allocation method of this invention in suppressing system uncertainty, Probabilistic Reachable Sets (PRS) are introduced as an evaluation index. PRS is used to characterize the distribution range of state errors in a stochastic system under closed-loop control and is often used to achieve constraint tightening of the nominal state. , Indicates nominal constraint limits. Represents the original set of state constraints. This represents the Pontryagin difference operator. This represents a probabilistically reachable set. In step 2, this invention aims to optimize and reduce the size of the problem by solving the covariance optimization problem of allocable terminals offline. This allows for a larger nominal constraint limit to be secured for the system. Since the magnitude of terminal variance directly affects the coverage of PRS, PRS can serve as an intuitive and fair comparative tool for evaluating the effectiveness of active covariance allocation strategies.

[0037] Since the chance constraint only provides a lower bound guarantee for the probability level, a Monte Carlo sampling method is used for statistical verification to accurately assess the true probability of system state default. Each sampling starts with the same initial conditions, and multiple samples are taken for the system error. The probability of satisfying the chance constraint is calculated by dividing the number of samples within the set at each time step by the total number of samples. This invention achieves explicit control over the state probability distribution by introducing an error covariance term into the objective function and combining it with the online second-order cone programming solution in step 4. In practical applications of control systems, this ability to effectively suppress the diffusion of system state variance has significant theoretical and practical implications for ensuring the long-term safe operation of chemical plants under random disturbances.

[0038] like Figure 2 , Figure 3 As shown, the comparison results of elliptical and polyhedral PRS under different control weights are presented. In the figure, Q is the state penalty weight matrix, and R is the control input penalty weight matrix. It can be seen that under all weight combinations, the PRS obtained by the terminal variance allocation method of this invention is significantly smaller. This indicates that the method exhibits stronger uncertainty contraction capability against common disturbances in chemical production, such as feed component fluctuations or random environmental temperature disturbances. A smaller PRS means that the algorithm can achieve a more compact probabilistic constraint execution effect; in other words, for the nominal dynamic model of the chemical process, the method of this invention can obtain a larger safe operating feasible region, thereby significantly reducing the conservatism of the control design and enabling the system to operate closer to the optimal yield point.

[0039] like Figure 4 , Figure 5 As shown, 1000 closed-loop trajectory samples of a chemical system obtained by downsampling under scenarios 1 (deterministic initial error) and 2 (random initial error, simulating sensor measurement uncertainty) are presented. In the figure, the light blue solid line represents the actual state trajectory of the reactor, the dark blue line represents the nominal predicted trajectory, the green dots represent the trajectory endpoints, and the dashed ellipse represents the preset 3δ error range of the state of each sampling point with respect to variance. Figure 4It can be seen that, under the condition of no initial start-up error, the method of the present invention can constrain the actual system state within the safe operating envelope (such as the equipment withstand pressure limit or the maximum allowable heat of reaction) with a very high probability, and effectively guide the trajectory to converge smoothly to the process steady-state target point. Figure 5 Furthermore, it is shown that even with the introduction of additional initial perturbations, the state distribution of the chemical system remains well constrained within the confidence region, with no large-scale constraint violations or equipment alarms, verifying the robust adaptability of the method of this invention to different initial production conditions. Monte Carlo estimation results show that the minimum probability of the state satisfying the constraints is... This verifies that the state-chance constraints for key process parameters are strictly satisfied throughout the entire time domain.

[0040] Depend on Figure 6 , Figure 7 The results verify the effectiveness and superiority of the method of the present invention under random perturbation conditions. Figure 6 The study presents 1,000 trajectory samples of the RFMPC method. It can be seen that the trajectory distribution is relatively divergent, especially near the constraint boundary, where there is a significant risk of exceeding the limit. In actual chemical production, this can easily lead to substandard product quality or safety interlock shutdown. Figure 7 The Tube MPC method is shown, and although it ensures an extremely low violation rate through pipeline constraints, the trajectory is forcibly constrained within a very narrow range, exhibiting strong control conservatism. This means that significant production efficiency and economic benefits are sacrificed for safety. In contrast, the method of this invention achieves excellent variance suppression and state envelope within a 95% confidence interval. The overall sampling trajectory can more closely approximate the process limit boundaries without exceeding them, effectively balancing the control performance and operational safety of the chemical process.

[0041] like Figure 8 The comparison results of the mean trajectory further intuitively confirm this point. The mean trajectory of the present invention can more closely approach the constraint boundary and quickly converge to the target region without violating the constraints. This shows that the method of the present invention can effectively suppress random fluctuations while making fuller use of the operating space through active adjustment of covariance. Based on the above simulation verification results, it can be seen that the stochastic MPC method based on active variance allocation proposed in this invention not only ensures the efficiency of online solution through second-order cone programming reconstruction (steps 3 and 4), but also effectively ensures the strict satisfaction of the safety operation constraints of chemical plants. It achieves an excellent balance between improving economic benefits and avoiding production risks, and has significant practical application significance in the automated control of modern complex chemical processes.

[0042] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A stochastic model predictive control method for active covariance allocation for chemical processes, characterized in that, The method includes the following steps: Step 1: Based on the process requirements of the chemical process, determine the controlled variables and control input variables in order to establish a dynamic model of a linear discrete system driven by additive random disturbances; To define polyhedral chance constraints for the state and input of a linear discrete system dynamics model, and to predict the control objective function; Step 2: Based on the linear discrete system dynamics model, solve the optimization problem of the allocatable terminal covariance matrix offline to obtain the minimum terminal covariance matrix; The corresponding terminal penalty weight matrix and terminal state feedback gain matrix are determined based on the minimum terminal covariance matrix. Step 3: Based on the terminal state feedback gain matrix, a control law with deterministic open-loop input components and dynamic error feedback gain is introduced into the dynamic model of a linear discrete system driven by additive random disturbances to construct a predictive control model. Based on the terminal penalty weight matrix and the predictive control model, the predictive control objective function is reconstructed into a convex quadratic form, resulting in a quadratic objective function. By using the Cantley inequality and variable substitution techniques, the chance constraints of non-convex polyhedra are transformed into equivalent deterministic second-order cone constraints. Step 4: Using the current actual state of the chemical system as input, based on the predictive control model, the reconstructed convex quadratic objective function, and the second-order cone constraint, solve the second-order cone programming problem online to obtain the optimal control input sequence; Step 5: Apply the first control quantity of the optimal control input sequence to the chemical system, and repeat steps 4 to 5 at the next sampling time with the updated actual state as input to achieve rolling optimization control.

2. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 1, characterized in that, In step 1, the dynamic model of the linear discrete system driven by additive random perturbation has the following relationship: ； in, This represents the state vector at time k+1. This represents the state vector at time k. , The dimension of the system state vector; This represents the control input variable at time k. , This indicates the dimension of the control input vector; This represents an additive perturbation with zero mean and known variance, or a random unbounded perturbation with bounded variance. , The dimension of the additive random perturbation vector; Let represent the state matrix, disturbance matrix, and control input matrix of the system dynamics model, respectively. .

3. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 2, characterized in that, In step 1, the polyhedral chance constraint has the following relationship: ； in, Represents a probability operator. Represents the set of non-negative integers. Let these represent the set of state constraints and the set of control input constraints of the system, respectively. These represent the maximum probability limits for the system state and control input to violate constraints, respectively. Defined as ; in, This represents the total number of state inequality constraints. Let represent the transpose of the coefficient vector of the state constraint of the i-th polyhedron. Denotes the boundary constants of the state constraint of the i-th polyhedron. , Indicates the transpose operator; Defined as ; in, This indicates the number of polyhedral hyperplanes controlling the input constraints. Let represent the transpose of the coefficient vector of the control input constraint of the i-th polyhedron. This represents the boundary constant of the control input constraint for the i-th polyhedron. , The number of polyhedral hyperplanes representing the joint state and control input constraints.

4. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 3, characterized in that, In step 1, the predictive control objective function has the following relationship: ； in, This represents the predictive control objective function. Indicates the length of the prediction time domain. This represents the state vector at the initial moment. This represents the stage cost at time k. Represents the terminal cost function. This represents the terminal state vector at the end of the prediction time domain. The mathematical expectation operator for a random perturbation w; The stage cost at time k is defined as: ； in, This represents the state vector at time t. This represents the control input vector at time t. Denotes the state-weighted square norm. This represents the weighted square norm of the control input. and These represent the state weight matrix and the control input weight matrix, respectively. ; The terminal cost function is defined as: ； in, This represents the weighted square norm of the terminal state.

5. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 4, characterized in that, In step 2, based on the linear discrete system dynamics model, an offline optimization problem concerning the allocatable terminal covariance matrix is ​​solved to obtain the minimum terminal covariance matrix. Based on the minimum terminal covariance matrix, the corresponding terminal penalty weight matrix and terminal state feedback gain matrix are determined, specifically including the following steps: Based on the system dynamics matrix and the upper bound matrix of the perturbation covariance, an offline optimization problem is constructed with the terminal covariance matrix as the decision variable. The corresponding process has the following relationship: ； ； in, Represents the terminal covariance matrix. Describing the Frobenius norm, express Moore-Penrose pseudo-reverse, Represents the identity matrix; The offline optimization problem is solved to obtain the minimum terminal covariance matrix; Based on the minimum terminal covariance matrix, the terminal penalty weight matrix required for the predictive control objective function and the corresponding terminal state feedback gain matrix are determined by solving the discrete-time Lyapunov equation.

6. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 5, characterized in that, In step 3, based on the terminal state feedback gain matrix, a deterministic open-loop input component and dynamic error feedback gain control law are introduced into the dynamic model of the linear discrete system driven by additive random disturbances to construct a predictive control model. This specifically includes the following steps: Define the expected state as Define the state error as ;in, Represents the expected state. Indicates state error; Using the terminal state feedback gain matrix as the constant error feedback gain, and based on the deterministic open-loop input vector, state error, and constant error feedback gain, a control law is constructed. The corresponding process has the following relationship: ； in, This represents the deterministic open-loop control component to be optimized online. This represents the terminal state feedback gain matrix; Substituting the control law into the dynamics model of the linear discrete system and taking the expectation of the state, we obtain the nominal dynamics and error dynamics. The corresponding process has the following relationship: ； ； in, Indicates nominal dynamics. Indicates error dynamics; Nominal dynamics and error dynamics constitute the predictive control model.

7. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 6, characterized in that, In step 3, based on the terminal penalty weight matrix and the predictive control model, the predictive control objective function is reconstructed into a convex quadratic form, resulting in a quadratic objective function. The corresponding process has the following relationship: ； in, This represents the predictive control objective function that includes both stage costs and terminal costs. Indicates the length of the prediction time domain. This represents the state vector at the predicted time-domain terminal. This represents the terminal penalty weight matrix. .

8. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 7, characterized in that, In step 3, the non-convex polyhedral chance constraint is transformed into an equivalent deterministic second-order cone constraint using the Cantley inequality and variable substitution technique. This specifically includes the following steps: The total budget for state violation probabilities is allocated to the risk budgets of each state constraint surface, satisfying... ,in, This represents the risk budget allocated to the i-th state constraint surface; Input the total budget of the probability of violation. Risk budget allocated to each input constraint surface ,satisfy ,in, This represents the risk budget allocated to the i-th input constraint surface; Introducing an auxiliary gain matrix ,in, Represents the auxiliary gain matrix. Indicates the constant error feedback gain; Using the Cantley inequality, the constraint limits are independently derived for each constraint surface. Based on the auxiliary gain matrix, the control law and the mean and covariance of the state under error dynamics are substituted to obtain a deterministic second-order cone form constraint expression.

9. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 8, characterized in that, The transformed deterministic state second-order cone constraint form is: ； in, This represents the block selection matrix at time k. The expected vector representing the initial state. Indicates the open-loop input sequence direction. The number of polyhedral hyperplanes representing state constraints. The covariance matrix representing the state error sequence; The transformed deterministic input second-order cone constraint form is: ； in, This represents the covariance matrix associated with the sequence of auxiliary variables.

10. The stochastic model predictive control method for active covariance allocation in chemical processes according to claim 9, characterized in that, The state vector includes at least one of the temperature, concentration, and pressure of the chemical reactor, and the input vector includes at least one of the coolant flow rate, feed flow rate, and heating power; the additive random perturbation characterizes process noise and external environmental fluctuations.

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