Weighted Predictive Control Method Based on Gap Metric Principle

Abstract

The present invention discloses a weighted predictive control method based on the principle of gap measurement. The method comprises the following steps: Step 1, discover the error data points located outside the ellipse domain, perform Jacobian linearization on the original nonlinear system at this point to obtain a new system; Step 2, design the equilibrium point through the elliptical predictive control algorithm; Step 3, improve the recursive Bayesian weighting algorithm according to the relevant properties of gap measurement; Step 4, perform normalization processing on the conditional probability obtained in Step 3 to obtain the control input weight values of each sub-controller. This method adds a protection mechanism on the basis of the elliptical set switching predictive control algorithm, forces the data points that deviate from the ellipse domain to be "pulled" back into the ellipse set, enables the algorithm to continue, saves computer storage, and improves the stability of the algorithm.

Description

Technical Field

[0001] The present invention belongs to the field of industrial automatic control, and relates to a weighted correction algorithm for error data parameters, in particular to a recursive Bayesian weighted algorithm for correcting error data located outside the elliptical set based on gap measurement. Background Art

[0002] In current industrial production, the requirements for automatic control technology are becoming increasingly strict. The core issues of automatic control technology are mainly how to establish a mathematical model that fits the controlled object and find a suitable control method to make the system reach an ideal equilibrium state within a certain period of time. Process control has a large weight in industrial production control. For example, chemical reaction kettles, coordinated control of thermal power units, etc. The controlled objects mostly have the characteristics of strong nonlinearity and large-scale changes in working conditions. Model Predictive Control (MPC) is a computer control technology and is currently the only advanced control algorithm that is truly applied to industrial control processes. It has a very wide application in linear time-invariant (LTI) systems. MPC is a special optimal control. Due to its unique rolling optimization mechanism, whether it is a single-variable or multi-variable LTI system, it has excellent control effects. Compared with infinite-time domain optimal control, it can well handle some linear state, input, and output constraints, and transform them into quadratic programming (QP) problems. In each prediction time domain, solve the open-loop QP problem to obtain the control input for the next moment, and then calculate the latest control input according to the latest state or output information, that is, calculate round by round in a rolling manner to form a special feedback mechanism. Therefore, when facing tracking control problems, it can quickly, accurately, and stably reach the control target, and has good robustness and anti-interference ability in the face of external disturbances. The MPC control algorithm is mainly divided into three parts: constructing a prediction model, rolling optimization within the control time domain, and a feedback correction mechanism. Summary of the Invention

[0003] Based on the solution of non-linear control problems in industrial production by elliptical set switching predictive control, the present invention provides a weighted predictive control method based on the principle of gap measurement. When the algorithm cannot run due to human error or external interference causing a certain point of data to deviate from the elliptical invariant set in practical applications, the weighted algorithm can be used to correct the error data and "pull" it into the elliptical invariant set.

[0004] The object of the present invention is achieved through the following technical solutions:

[0005] A weighted predictive control method based on the principle of gap measurement, comprising the following steps:

[0006] Step 1: Discover the error data points located outside the ellipse domain, denoted as x(0). At this point, perform Jacobian linearization on the original nonlinear system to obtain a new system L0;

[0007] Step 2: Design the equilibrium points through the elliptical predictive control algorithm. Assume there are N equilibrium points, and each equilibrium point corresponds to a linear sub-model L i , i = 0, 1,..., N. The state feedback matrix K i = Y i / G i -1 , where Y i and G i are variables obtained by solving the optimization problem respectively. At time k, the subsystem corresponds to a state feedback controller u i (k) = K i (k). The subsystem states within the given elliptical domain can all converge to the equilibrium points;

[0008] Step 3: According to the relevant properties of the gap metric, improve the recursive Bayesian weighting algorithm. The improved algorithm equation is:

[0009]

[0010] In the formula, p i,k represents the conditional probability of the matching degree between the i-th sub-model of the system at time k and the linear model corresponding to the current point not within the ellipse domain , δ i,k is the gap metric value between the i-th sub-model at time k and this sub-model, M is the convergence coefficient of the recursive formula. Let the conditional probability of the matching degree between the initial state point corresponding subsystem L(0) and the i-th sub-model

[0011] Step 4: Normalize the conditional probability obtained in Step 3 to obtain the control input weights ω i,k of each sub-controller. The calculation formula is:

[0012]

[0013] The sum of the products of the control inputs of each local linear model and their weights, that is, the control input of the system when the error data is not within the ellipse domain at time k: When compared with the prior art, the present invention has the following advantages:

[0014]

[0015] When compared with the prior art, the present invention has the following advantages:

[0016] 1) During the multi-model predictive control process of a non-linear system, when incorrect data appears, it is possible to promptly detect abnormal data outside the elliptical set and force the state points outside the elliptical set to be "pulled" back into the elliptical set, enabling the algorithm to continue running.

[0017] 2) Compared with the traditional method of finding the transfer path to establish the elliptical set, this method greatly saves computer memory and is more suitable for embedded industrial control computers in actual engineering.

[0018] 3) The weighted predictive control algorithm based on the gap metric can improve the automation level in the industrial control process and enhance the accuracy in the multi-model predictive control process. Description of the Drawings

[0019] Figure 1 is a schematic diagram of the convergence of the initial point;

[0020] Figure 2 is a schematic diagram of the initial point not being inside the elliptical set;

[0021] Figure 3 is a flowchart of the algorithm;

[0022] Figure 4 is the simulation result of the initial point not being inside the elliptical set;

[0023] Figure 5 is the simulation result of being out of the elliptical set due to an instantaneous disturbance. Detailed Implementation Manner

[0024] The technical solution of the present invention will be further described below in conjunction with the drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0025] At present, automatic control technology is constantly developing. Elliptical set switching predictive control is widely used in non-linear systems. In the actual control process, it is inevitable that some incorrect data will appear. When the state point is not inside the elliptical set, the algorithm will not be able to proceed. In view of the above problems, the present invention provides a weighted predictive control method based on the gap metric principle. This method adds a protection mechanism to the elliptical set switching predictive control algorithm, forcing the data points outside the elliptical domain to be "pulled" back into the elliptical set, enabling the algorithm to continue, saving computer storage, and improving the stability of the algorithm. The following will be described in detail in combination with Figure 3 the steps of the present invention:

[0026] Step 1: Screen the data. Let the initial point not inside the elliptical domain be x(0). At the point x(0), perform Jacobian linearization on the original non-linear system to obtain a new system L0.

[0027] This step is based on the completion of the nonlinear control model of the studied nonlinear system. By using the Jacobian matrix of the nonlinear system at the equilibrium point, its eigenvalues are analyzed to obtain the characteristics at the equilibrium point, thereby realizing the linearization of the nonlinear system.

[0028] Step 2: Design the equilibrium point through the elliptical predictive control algorithm. Assume there are N equilibrium points, and each equilibrium point corresponds to a linear sub-model L i , i = 0, 1,..., N, and the state feedback matrix K i = Y i / G i -1 , where Y i and G i are variables obtained by solving the optimization problem. At time k, the subsystem corresponds to a state feedback controller u i (k) = K i (k). Within the given elliptical domain, the subsystem states can all converge to the equilibrium point.

[0029] The implementation of this step is based on two important lemmas of the optimization problem: 1. For a system with an initial state designed as x(k), design a constant feedback control law u(k+m|k) = Y i+1 Q t -1 x(k+m|k), m ≥ 0, where Y i+1 and Q t are feasible solutions to the infinite-horizon predictive control optimization problem for the initial state to solve the theorem. Then the elliptical region X i = {x ∈ R n |x T Q t -1 x ≤ 1} is an asymptotically stable elliptical set of the closed-loop system; 2. An elliptical feasible stable region under the infinite-horizon predictive control of Theorem 1 is {x|x T G -1 x ≤ 1}, where G is the optimal solution (γ, dimensionless variable) of the following optimization variable Q t under the condition of satisfying the constraints:

[0030]

[0031] Step 3: According to the relevant properties of the gap metric, improve the recursive Bayesian weighting algorithm. The improved algorithm equation is:

[0032]

[0033] In the formula, p i,k represents the i-th sub-model of the system at time k and the current state not in the elliptical domain The conditional probability of the matching degree of the point corresponding to the linear model, δ i,k is the gap metric value between the i-th sub-model at time k and this sub-model, M is the convergence coefficient of the recurrence formula, and let the conditional probability of the matching degree between the initial state point corresponding subsystem L(0) and the i-th sub-model

[0034] This step improves the algorithm by combining the relevant properties of the gap metric on the basis of the Bayesian weighted algorithm.

[0035] Step 4: Normalize the conditional probability obtained in Step 3 to obtain the control input weights ω of each sub-controller i,k , and the calculation formula is:[[]]END]]

[0036]

[0037] The sum of the products of the control inputs of each local linear model and their weights, that is, the control input of the system when the error data at time k is not in the elliptical domain is:[[]]END]]

[0038]

[0039] Example:[[]]END]]

[0040] Use the continuous stirred tank reactor (CSTR) model to test the weighted elliptical switching prediction model of the present invention, and use the recursive Bayesian weighted algorithm based on the gap metric to perform state transition testing on it.

[0041] The first case: As Figure 1 shown, the initial equilibrium point x(0) is not in the elliptical domain. Take x(0)1 = [0.22354, 5.5], that is Figure 2 the x(0)1 point in

[0042]

[0043] Among them:[[]]END]]

[0044]

[0045] The gap metrics between L(0) and the established sub-models are respectively:[[]]END]]

[0046] [δ 1,0 δ 2,0 δ 3,0 δ 4,0 δ 5,0 = [0.3141 0.9941 1.0000 0.9522 0.5395]

[0047] The initial probabilities of each sub-model are obtained by using the weighted recursive Bayesian formula as follows:

[0048] [p 1,0 p 2,0 p 3,0 p 4,0 p 5,0 = [0.9999 4.81×10 -20 2.68×10 -20 2.85×10 -18 6.65×10 -5

[0049] Then, the weights of the state feedback controllers of each sub-model are further obtained by the above normalization formula, and finally the control input at the initial moment is obtained:

[0050]

[0051] When the system control input at the current moment is obtained, the system state observation value at the next moment can be obtained. Return to the above steps, and the final simulation results are as Figure 4 shown.

[0052] From the simulation results, it can be verified that the recursive Bayesian weighted algorithm based on gap metric will cause the state points outside the ellipse set to migrate towards the ellipse domain and finally enter the ellipse neighborhood.

[0053] The second case: During the control process, the system state point is instantaneously disturbed and detached from the ellipse domain.

[0054] Select the system initial point x(0) = x p3 = [0.2345, 4.705] T , and the state point is within the ellipse domain. At t = 38s, a negative pulse with an amplitude of 1 is added to the system state x2, so that the system state changes from [0.2994, 4.058] T to [0.2994, 3.058] T instantaneously, and the system state is detached from the ellipse set of the fourth subsystem, obtaining the linear model L(k):

[0055]

[0056] where:

[0057]

[0058] The gap metrics between L(k) and the established sub-models are respectively:

[0059] [δ 1,k δ 2,k δ 3,k δ​4,k δ 5,k = [0.2966 1.0000 1.0000 0.9766 0.1512]

[0060] The initial probability values of each sub-model are obtained by using the weighted recursive Bayesian formula as follows:

[0061] [p 1,k p 2,k p 3,k p 4,k p 5,k = [0.0372 5.83×10 -22 5.83×10 -22 5.91×10 -22 0.9628]

[0062] Similarly, the simulation results are as Figure 5 shown.

[0063] During the operation of the algorithm, when the system state suddenly changes and gets out of the entire elliptical domain, the weighted algorithm will, according to the comparison of the gap metric, force the state to migrate back into the elliptical domain. Although through a rather tortuous path, the system state can find and enter the handover domain and finally converge to the desired equilibrium point.

Claims

1. A weighted predictive control method based on the principle of gap measurement, characterized in that The method includes the following steps: Step 1. Discover the error data points located outside the ellipse domain, and denote it as , and perform Jacobian linearization on the original nonlinear system at this point to obtain the new system L0; Step 2: Design the equilibrium point through the elliptical predictive control algorithm. Assume there are equilibrium points, and the linear sub-model corresponding to each equilibrium point is , . The state feedback matrix of the subsystem is , where and are variables obtained by solving the optimization problem respectively. At moment, the subsystem corresponds to a state feedback controller , and the states of the subsystem can converge to the equilibrium point within the given elliptical domain. Step 3. According to the relevant properties of gap measurement, improve the recursive Bayesian weighting algorithm, and the improved algorithm equation is: In the formula, represents the system The conditional probability of the matching degree between the i-th sub-model and the point not in the elliptical domain at time , is The gap metric value between the i-th sub-model and the sub-model at time is the convergence coefficient of the recurrence formula. Let the conditional probability of the matching degree between the initial state point and the i-th sub-system be ; Step 4. Normalize the conditional probabilities obtained in Step 3 to obtain the control input weights of each sub-controller , and the calculation formula is as follows: The sum of the products of the control inputs of each local linear model and their weights is the control input of the system at time k when the error data is not within the elliptical domain : 。 2. The weighted predictive control method based on the gap measurement principle according to claim 1, characterized in that The implementation of the second step is based on two important lemmas of the optimization problem:

1. For an initial state of a system design , design the constant feedback control law , where , is a feasible solution to the infinite-horizon predictive control optimization problem of the theorem for the initial state. Then the elliptical region is an asymptotically stable elliptical set of the closed-loop system; 2. An elliptical feasible stable region under the infinite-horizon predictive control of Theorem 1 is , where is the optimal solution of the following optimization variables under the constraint conditions: , is a dimensionless variable.

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