A computing method of variable-ellipse set model switching predictive control with constraints

Abstract

The application discloses a calculation method of a variable-ellipse set model switching predictive control with constraints, and belongs to the technical field of industrial automatic control. The method is as follows: a first equilibrium point is selected; a linear submodel at the first equilibrium point is obtained; a neighborhood which can be approximated by the equilibrium point corresponding linear submodel is estimated, and output saturation constraints are obtained; control input saturation constraints are set, and the maximum feasible ellipse set of the subsystem is estimated; the initial state of the maximum feasible ellipse set is judged; an optimization problem is solved in the maximum feasible ellipse set, the state feedback matrix of the subsystem at the k moment is obtained, and a state feedback controller is designed in the maximum feasible ellipse domain. The application solves the problem that the neighborhood estimation algorithm cannot guarantee that the neighborhood is a convex set, and can effectively reduce the system conservativeness. By using the variable-ellipse set model switching predictive control algorithm, the state feedback control law is used to replace the free variable in the system switching process, and the calculation amount of the system is reduced.

Description

TECHNICAL FIELD

[0001] The application relates to a calculation method of a variable-ellipse set model switching predictive control with constraints, and belongs to the technical field of industrial automatic control. BACKGROUND

[0002] In the industrial automatic control process, the rolling optimization and feedback correction link of the multi-model predictive control (MPC) of a system with linear states and control inputs have good processing capacity and robustness, are low in model requirement, and are strong in flexibility, but in the actual control technical field, few control systems are linear, and often face nonlinear systems, and at present, no computer tool can process any nonlinear system. In addition, in the actual process control, the system state transfer range is large, and a single model is used to approximate the entire system, which can cause huge errors and even cause the collapse of the control process. Therefore, the switching mode of the multi-model predictive control is particularly important.

[0003] At present, the control switching mode of the multi-model predictive control of a large working condition nonlinear system mainly includes an ellipse invariant set and a polyhedral invariant set, wherein the ellipse invariant set is more widely applied in the actual engineering field. However, the ellipse invariant set model predictive control still cannot be flexibly applied to different nonlinear control systems due to its low generalization, and in the traditional ellipse invariant set multi-model switching predictive control algorithm, the ellipse set X i of the system state needs to be determined at the beginning, and the invariant ellipse set is too conservative in the actual application process, so the research on the model switching mode of the multi-model predictive control is still a hot issue. SUMMARY

[0004] In order to solve the problems in the background art, the application provides a calculation method of a variable-ellipse set model switching predictive control with constraints.

[0005] In order to achieve the above object, the application adopts the following technical scheme: a calculation method of a variable-ellipse set model switching predictive control with constraints, the method comprising the following steps:

[0006] S1: let the ellipse set X i i=1, select an ideal working condition equilibrium point as the first equilibrium point E p 1 {x p 1 , u p 1}, wherein: x p i is a static equilibrium point, u p i is the value of the control variable at the equilibrium point, and i is a positive integer; and the first equilibrium point E p1 linear sub-model L i ;

[0007] S2: estimating equilibrium point E p i corresponding linear sub-model L i approximated neighborhood through equilibrium point E p i system control output range at the point, obtaining output saturation constraint;

[0008] S3: setting control input saturation constraint, estimating maximum feasible ellipsoid set ψ of the sub-system i ={x|x T G i -1 x≤1}, wherein: x is state vector, G i is symmetric positive definite matrix describing system variable range, ability of system under given condition to meet stability and performance requirement, x T is transposition vector of state vector;

[0009] S4: if initial state x(0) of maximum feasible ellipsoid set is in ψ i , then let N=i, N is number of linear sub-model set, and then execute S5; otherwise, within the boundary of maximum feasible ellipsoid set ψ i , according to rational state transition curve, select next equilibrium point E p i+1 {x p i+1 ,u p i+1}, return to S2;

[0010] S5: solve optimization problem within maximum feasible ellipsoid set ψ i , obtain state feedback matrix K of the sub-system at k moment i =Y i Q t -1 , wherein: Y i is controllability matrix Q t is observability matrix, and state feedback controller is designed within maximum feasible ellipsoid domain ψ i .

[0011] Compared with prior art, the present application has the advantages of:

[0012] The present application determines the ellipse set in advance in the traditional ellipse invariant set multi-model predictive control process, and the variable ellipse set multi-model predictive control system does not need to obtain the accurate ellipse set at the initial moment. Compared with the great conservativeness of the ellipse invariant set in the application process, the present application solves the problem that the neighborhood estimation algorithm cannot guarantee that the neighborhood is a convex set, and effectively reduces the system conservativeness. By using the variable ellipse set model switching predictive control algorithm, the state feedback control law is used to replace the free variable in the system switching process, and the calculation amount of the system is reduced. BRIEF DESCRIPTION OF DRAWINGS

[0013] Figure 1 is a flow chart of the present application;

[0014] Figure 2 is a static equilibrium point distribution diagram, wherein (a) represents a control input u and state vector x2 relationship curve, and (b) represents a state vector x1 and state vector x2 relationship curve;

[0015] Figure 3 is a top view of the gap degree measurement relationship between each point in the neighborhood and the equilibrium point;

[0016] Figure 4 is a system state transition diagram;

[0017] Figure 5 is a switching sequence switching process diagram;

[0018] Figure 6 is a system state diagram;

[0019] Figure 7 is a control input response curve diagram. DETAILED DESCRIPTION

[0020] The technical solutions in the present application will be described clearly and completely in the embodiments of the present application combined with the drawings. Obviously, the described embodiments are only a part of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present application.

[0021] The present application improves the ellipse invariant set of the sub-model switching mode on the basis of the nonlinear system ellipse invariant set multi-model predictive switching control, and proposes a calculation method of the variable ellipse set model switching predictive control with constraints, which comprises the following steps:

[0022] S1: According to the expected equilibrium point, estimate the maximum ellipse set of the first sub-model, and let the ellipse set X i i=1, select the equilibrium point of the ideal working condition as the first equilibrium point E p 1 {xp 1 u p 1}, where: x p i For the static equilibrium point, u p i To determine the value of the control variable at this equilibrium point, where i is a positive integer; and using the Jacobian linearization method, the first equilibrium point E is obtained. p 1 Linear submodel L at the location i ;

[0023] This step is based on establishing a nonlinear mathematical model of the target nonlinear system, and using the Jacobi linearization method to transform the linearized system into a linear sub-model, which will facilitate the calculation of elliptic sets and stability analysis.

[0024] S2: Estimate the equilibrium point E using a neighborhood estimation algorithm based on gap metric. p i Corresponding linear submodel L i The neighborhood can be approximated by the equilibrium point E. p i By determining the system's control output range, output saturation constraints are obtained.

[0025] S201: A neighborhood estimation algorithm based on gap metric to determine the equilibrium point E. p i The scope of the domain, and use this to analyze the system at the equilibrium point E. p i Nearby stability and dynamic characteristics;

[0026] S202: Linear submodel L obtained through S1 i Determine the system's output constraints based on the model and input constraints;

[0027] S203: Analyze the system's behavior at the equilibrium point by estimating the output constraints, and design corresponding control strategies to improve system stability and prevent the system output from exceeding the controllable range.

[0028] S3: Set saturation constraints for the control input and estimate the maximum feasible elliptic set ψ of the subsystem. i ={x|x T G i -1 x≤1}, where: x is the state vector, G i The ability of x to describe the stability and performance requirements of a system under given conditions using a symmetric positive definite matrix with a variable range of system properties. T It is the transpose of the state vector;

[0029] The step is based on the switching system asymptotic stability control theorem of linear matrix inequality, solves the problem that the neighborhood estimation algorithm cannot guarantee that the neighborhood is a convex set, and reduces the conservativeness of the system.

[0030] S4: If the initial state x(0) of the maximum feasible ellipse set is in the neighborhood ψ i , then let N=i, N is the number of linear sub-model sets, and then perform S5; otherwise, in the maximum feasible ellipse set ψ i , according to the rational state transition curve, select the next equilibrium point E p i+1 {x p i+1 , u p i+1}, return to S2;

[0031] The step makes a simple judgment on whether the target processed is in the predetermined boundary range, and the next step of calculation is performed on the data points meeting the predetermined boundary, and the points located outside the predetermined boundary range are returned to the above step two to reconstruct a new predetermined boundary for judgment.

[0032] S5: Solve the optimization problem in the maximum feasible ellipse set ψ i , and obtain the state feedback matrix K i of the subsystem at time k i = Y t Q -1 , wherein: Y i is the controllability matrix Q t is the observability matrix, and a state feedback controller is designed in the maximum feasible ellipse domain ψ i .

[0033] The step is based on the state feedback matrix, and a feedback controller is designed, which generates a control input signal according to the current system state and K i , so that the system remains stable in the ellipse domain, so as to achieve the design purpose.

[0034] The application is improved on the basis of the traditional ellipse invariant set switching predictive control algorithm, so that the state can be transferred from the initial point to the desired equilibrium point in the stable domain, the conservativeness of the traditional algorithm is reduced, the stability of the control system is improved, and the convergence speed of the algorithm is accelerated, and the control target can be achieved under the condition of meeting the constraint condition.

[0035] Embodiment 1

[0036] The continuous stirred tank reactor (CSTR) model is used to test the variable ellipse set switching predictive control of the application, and the CSTR model is as follows:

[0037]

[0038] In formula (1):

[0039] x1 represents the concentration of the reactant;

[0040] x2 represents the temperature of the reactant;

[0041] dx1 / dt represents the derivative of x1;

[0042] dx2 / dt represents the derivative of x2;

[0043] x 1f x1 represents the concentration of the reactant in the feed;

[0044] x 2f x2 represents the temperature of the reactant in the feed;

[0045] q represents the flow rate of the feed;

[0046] φ represents the molar volume of the reactant;

[0047] γ represents the thermal expansion coefficient of the reactant;

[0048] δ represents the flow rate of the coolant;

[0049] t represents the coolant temperature;

[0050] β represents the transfer coefficient of the reaction heat;

[0051] The parameters used are shown in Table 1 below:

[0052] Table 1 Non-dimensional model parameters

[0053]

[0054] Consider that the system operating condition is shifted from the static equilibrium point x p 3 to the static equilibrium point x p 1 , a large range shift is achieved, and the static equilibrium point distribution is shown in the accompanying Figure 2 , let the initial state of the system x(0) = x p 3 = [0.234, 4.705] T , the desired equilibrium point is x e = x p 1 = [0.8558, 0.8863] T .

[0055] First, let x p 1 = x eThe CSTR system is discretized linearly with a sampling time of 0.02 s to obtain a linear submodel L1:

[0056]

[0057] In formula (2):

[0058] x represents a state vector of the system at a certain time;

[0059] y represents an output vector of the system at a certain time;

[0060] k represents that the system is at time k;

[0061] u represents an input vector of the system at a certain time;

[0062] A represents a state transition matrix, which describes how the system state evolves without input, and

[0063]

[0064] B represents an input matrix, which describes how the input affects the evolution of the system state, and

[0065]

[0066] C represents an output matrix, which describes how the state vector is mapped to the output vector, and C = [0 1];

[0067] D represents a feedforward matrix, which describes how the input vector affects the output vector, and D = 0;

[0068] In the second step, the neighborhood estimation algorithm is used to estimate the neighborhood Φ1 of the linear submodel L1 to obtain an approximate neighborhood. The system output x2 can be obtained from the output equation of formula (1), and the saturation constraint of the output can be determined according to the appendix Figure 3 .

[0069] In the third step, an infinite time domain performance index function is introduced:

[0070]

[0071] Let Q = I (unit matrix) and R = 0.1 in the performance index, and simultaneously satisfy the saturation constraint of the control input |u(k+m|k)|≤2, m = 1, ···, n, solve the optimization problem, and obtain the state feedback control law of the system at each time.

[0072] In the fourth step, the maximum feasible ellipsoid set ψ i of the subsystem is estimated according to Lemma 3.2 T G i -1 x≤1}, wherein

[0073] Step 5, due to the initial point Therefore, according to the model set selection algorithm, the next equilibrium point x is selected on the ideal state transition curve. p 2 =[0.72,1.968] T Return to step two to perform the next equilibrium point and ellipse set selection calculation. When x(0)∈Ψ i When the time is reached, the algorithm ends, and the VESMPC algorithm is complete.

[0074] Following the steps in Example 1, five equilibrium points were designed to establish five corresponding ellipse sets, as shown below:

[0075]

[0076] The corresponding elliptic sets are as follows:

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] Let the current controller action number be flag, and the switching order be flag5 to flag1.

[0083] The present invention is compared with the traditional Invariant Elliptic Switching Predictive Control (IVESMPC), and the comparison results are attached. Figures 4-7 As shown, attached Figure 4 Line 1 represents the ideal state transition curve, line 2 represents the actual state transition curve under the control of the VESMPC algorithm of this invention, and line 3 represents the actual state transition curve under the control of the traditional IVSMPC algorithm. (The last sentence appears to be incomplete and possibly refers to an appendix.) Figure 4 It can be observed that the innovative algorithm of this invention is closer to the ideal transfer curve and the transfer is faster.

[0084] Appendix Figure 7 The two lines in the middle represent VESMPC and IVESMPC respectively, with appended... Figure 6 The two dashed lines represent x1 and x2 respectively. It can also be observed that the VESMPC algorithm of this invention has a faster convergence speed, and the transient impacts encountered in the VESMPC algorithm are smaller than those in the traditional IVESMPC algorithm, thus improving the system stability.

[0085] It will be obvious to a person skilled in the art that the application is not limited to the details of the foregoing exemplary embodiments and can be implemented in other forms without departing from the spirit or essential characteristics of the application. The embodiments are considered in all respects to be illustrative and not restrictive, the scope of the application being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalents of the claims are therefore intended to be embraced therein. No reference signs in the claims should be considered as limiting the scope of the claims to the features to which the reference signs are attached.

[0086] Furthermore, it should be understood that although the description is made on the basis of the embodiments, not every embodiment contains only one independent technical solution, and the description of the specification is only for the sake of clarity, and those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that those skilled in the art can understand.

Claims

1. A computational method for switching predictive control of a constrained variable elliptic set model, characterized in that: The method includes the following steps: S1: Let the elliptic set X i If i=1, choose the equilibrium point where the ideal working condition is located as the first equilibrium point E. p 1 {x p 1 u p 1 }, where: x p i For the static equilibrium point, u p i To determine the value of the control variable at this equilibrium point, where i is a positive integer; the first equilibrium point E is obtained. p 1 Linear submodel L at the location i ; S2: Estimated equilibrium point E p i Corresponding linear submodel L i The neighborhood that can be approximated is the equilibrium point E. p i By determining the system's control output range, output saturation constraints are obtained. S3: Set saturation constraints for the control input and estimate the maximum feasible elliptic set ψ of the subsystem. i ={x|x T G i -1 x≤1}, where: x is the state vector, G i The ability of x to describe the stability and performance requirements of a system under given conditions using a symmetric positive definite matrix with a variable range of system properties. T It is the transpose of the state vector; S4: If the initial state of the maximum feasible elliptic set is x(0)∈ψ i If so, let N = i, where N is the number of linear sub-model sets, and then execute S5; otherwise, in the maximum feasible elliptic set ψ i Within the boundary, based on the rational state transition curve, select the next equilibrium point E. p i+1 {x p i +1 ,u p i+1 }, return to S2; S5: In the maximum feasible elliptic set ψ i Solving the optimization problem internally yields the state feedback matrix K of the subsystem at time k. i =Y i Q t -1 , where: Y i The controllability matrix Q t Let be the observability matrix, and within the maximum feasible elliptic domain ψ i Internally designed state feedback controller.

2. The calculation method for switching predictive control of a constrained variable elliptic set model according to claim 1, characterized in that: S2 includes the following steps: S201: A neighborhood estimation algorithm based on gap metric to determine the equilibrium point E. p i The scope of the domain, and use this to analyze the system at the equilibrium point E. p i Nearby stability and dynamic characteristics; S202: Linear submodel L obtained through S1 i Determine the system's output constraints based on the model and input constraints; S203: Analyze the system's behavior at the equilibrium point by estimating the output constraints, and design corresponding control strategies to improve system stability and prevent the system output from exceeding the controllable range.

Images