CN108614431A - A kind of Hammerstein-Wiener systems multi model decomposition and control method based on angle - Google Patents

Abstract

The invention discloses a multi-model decomposition and control method of the Hammerstein-Wiener system based on the included angle. The multi-model decomposition of the Hammerstein-Wiener system is carried out by using the included angle to obtain a linear model set of the approximate Hammerstein-Wiener system, and based on the model set Design a linear controller, and use the angle-based weighting method to carry out weighted fusion on the obtained linear controller to optimize the control of the Hammerstein-Wiener system. The invention can effectively overcome the disadvantages that the traditional nonlinear inverse control method can only be used in a statically reversible Hammerstein-Wiener system, and that the input or output nonlinearity of the system cannot be considered in the design of the controller, resulting in a decrease in closed-loop performance.

Description

A Multi-model Decomposition and Control of Hammerstein-Wiener System Based on Angle method

technical field

The invention relates to a Hammerstein-Wiener system multi-model decomposition and control method based on included angle, and belongs to the field of nonlinear system multi-model control.

Background technique

The Hammerstein-Wiener model is a typical block structure model, which consists of two static nonlinear modules connected in series with a dynamic linear module. The whole model can be regarded as a Hammerstein model in series with a Wiener model. This series structure has a series of advantages: easy to integrate prior knowledge into the model; low modeling cost; high approximation accuracy; easy to control and so on. Therefore, the Hammerstein-Wiener model has been widely used in the modeling of nonlinear systems in the past ten years, such as continuous stirring reactors, neutralization reactors, DC motors, photovoltaic power generation systems, etc.

However, when the controller is designed based on the Hammerstein-Wiener model, it also encounters the difficulties of other block structure models: the commonly used nonlinear inverse control method - the method of using the inverse of the static link to compensate the nonlinearity of the system, is only suitable for nonlinear However, the actual Hammerstein-Wiener system often has input or output nonlinearity, or even both input and output nonlinearity. In addition, the nonlinear inverse method fails to take the nonlinear characteristics of the system into consideration when designing the controller, thereby reducing the closed-loop performance of the system. Therefore, scholars have proposed other control methods to overcome the shortcomings of nonlinear inverse. For example, Khani proposed to approximate the Hammerstein-Wiener system with a linear model with structural or non-structural uncertainty, and then use RMPC to design the controller. Lawrynczuk proposed a nonlinear MPC for the Hammerstein-Wiener model, which linearizes the system along the desired trajectory and converts the nonlinear optimization problem into a quadratic programming problem, thus avoiding the nonlinear inverse control. However, sequentially linearizing the system results in heavy computation.

Contents of the invention

In order to simplify the control problem of the Hammerstein-Wiener system, a Hammerstein-Wiener system multi-model decomposition and control method based on the included angle of the present invention is used to design a multi-model controller, reduce the amount of calculation, and overcome the problems existing in the existing control technology methods Defects, optimize, improve and promote the existing angle decomposition algorithm, avoid cumbersome calculations, and improve the efficiency and quality of decomposition.

In order to achieve the above object, technical solution of the present invention is achieved in that way:

A Hammerstein-Wiener system multi-model decomposition and control method based on included angle, the specific steps are as follows:

S1. Use the angle-based gridding algorithm to grid the entire operating space of the Hammerstein-Wiener system, and linearize the Hammerstein-Wiener system at each grid point to obtain n _p linearized models G _i ( i=1,2,…,n _p ), calculate the slope of the static input-output curve of the system at each grid point, and further use the slope to calculate the angle between every two grid points, and obtain the angle matrix as Formula (1) shows:

Among them, θ _ij = |θ _i -θ _j |, i,j=1,2,…,n _p . θ _i and θ _j are the slope angles at the i-th and j-th grid points respectively; θ _ij represents the system The angle between the slopes at the i-th and j-th grid points;

Normalize the included angle matrix to obtain the normalized included angle matrix as shown in formula (2):

in, i=1,2,...,n _p , θ _max is the maximum value of the angle matrix in formula (1), and θ _min is the minimum value of the angle matrix in formula (1);

S2. According to the prior knowledge of the Hammerstein-Wiener system, select the threshold γ of the operation space decomposition;

S3. Define the initial value to the normalized included angle matrix formula (2), set i=1, m=0, i represents the i-th linearization model, and m represents the number of the current local sub-model;

S4. If _i≤np , then j=i, m=m+1, enter S5, otherwise jump to S12;

S5. From the i-th linearization model to the j-th linearization model, select a nominal model G ^* according to the formula (3) shown below:

Among them, max() means seeking the maximum value, min() means seeking the minimum value, and h and l are any values between i and j;

S6. Calculate the maximum normalized angle between the nominal model and other linearized models according to the formula (4) shown below:

S7. If Then make j=j+1 and jump to S5; otherwise jump to S8;

S8. Let j=j-1 enter S9;

S9. Select a nominal model G ^* according to formula (3) from the i-th to the j-th linearization model again;

S10. For the nominal model G ^* selected in S9, design a conventional linear controller K, denote the nominal model G ^* as the mth local sub-model P _m , and denote the controller K as the mth local sub-controller K _m , that is, to obtain the linear sub-model and sub-controller of the mth sub-interval;

S11. Make i=j+1, and jump to the S4 step;

S12. So far the whole decomposition process is over, the Hammerstein-Wiener system is decomposed into m sub-intervals, each sub-interval corresponds to a linear sub-model to form a sub-model set P ₁ , P ₂ ,...,P _m , and the corresponding sub-controller set is K ₁ ,K ₂ ,…,K _m ;

S13. Calculate the output of the multi-model controller according to the formula (5) shown below, and optimize the control of the Hammerstein-Wiener system;

where u _i (k) is the output of the ith sub-controller, is the angle-based weighting function of the i-th sub-controller, and k is the current moment.

Beneficial effects: the present invention provides a Hammerstein-Wiener system multi-model decomposition and control method based on included angle, compared with the prior art, it has the following advantages and positive effects:

(1) Suitable for Hammerstein-Wiener systems with input or/and output diversity.

(2) It avoids the reduction of control performance caused by the use of inverse functions in traditional nonlinear inverse methods.

(3) The robustness and closed-loop control performance of the system are improved.

Description of drawings

Fig. 1 is Hammerstein-Wiener system multi-model decomposition and weighting method schematic diagram of the present invention;

Fig. 2 is a Hammerstein-Wiener system multi-model control structure diagram of the present invention;

Fig. 3 is the output value diagram of the closed-loop system under the empirical multi-model controller;

Fig. 4 is the input value diagram of the closed-loop system under the empirical multi-model controller;

Fig. 5 is the output figure of closed-loop system in multi-model controller of the present invention;

Fig. 6 is the input value figure of closed-loop system in multi-model controller of the present invention;

Detailed ways

In order to enable those skilled in the art to better understand the technical solutions in the application, the technical solutions in the embodiments of the application are clearly and completely described below. Obviously, the described embodiments are only part of the embodiments of the application, and Not all examples. Based on the embodiments in this application, all other embodiments obtained by persons of ordinary skill in the art without creative efforts shall fall within the scope of protection of this application.

A multi-model decomposition and control method of a Hammerstein-Wiener system based on an included angle, taking a single-input-single-output system as an example, the specific steps are as follows:

S1. Use the angle-based gridding algorithm to grid the entire operating space of the Hammerstein-Wiener system, and linearize the Hammerstein-Wiener system at each grid point to obtain n _p linearized models G _i ( i=1,2,...,n _p ), calculate the slope of the static input-output curve of the system at each grid point, and further use the slope to calculate the The included angle, the obtained included angle matrix is shown in formula (1):

Among them, θ _ij = |θ _i -θ _j |, i,j=1,2,…,n _p . θ _i and θ _j are the slope angles at the i-th and j-th grid points respectively; θ _ij represents the system The angle between the slopes at the i-th and j-th grid points, that is, the angle between the i-th and j-th linearized models;

Normalize the included angle matrix to obtain the normalized included angle matrix as shown in formula (2):

in, i=1,2,...,n _p , θ _max is the maximum value of the angle matrix in formula (1), and θ _min is the minimum value of the angle matrix in formula (1);

S2. According to the prior knowledge of the Hammerstein-Wiener system, select the threshold γ of the operation space decomposition;

S3. Define the initial value to the normalized included angle matrix formula (2), set i=1, m=0, i represents the i-th linearization model, and m represents the number of the current local sub-model;

S4. If _i≤np , then j=i, m=m+1, enter S5, otherwise jump to S12;

S5. From the i-th linearization model to the j-th linearization model, select a nominal model G ^* according to the formula (3) shown below:

Among them, max() means seeking the maximum value, min() means seeking the minimum value, and h and l are any values between i and j;

S6. Calculate the maximum normalized angle between the nominal model and other linearized models according to the formula (4) shown below:

S7. If Then make j=j+1 and jump to S5; otherwise jump to S8;

S8. Let j=j-1 enter S9;

S9. Select a nominal model G ^* according to formula (3) from the i-th to the j-th linearization model again;

S10. To the nominal model G ^* selected in S9, design a conventional linear controller K (designing linear controller K in the present invention is conventional technical means, can be PID controller, MPC controller or LQ controller), will The nominal model G ^* is denoted as the mth local sub-model P _m , and the controller K is denoted as the mth local sub-controller K _m , that is, the linear sub-model and sub-controller of the m-th sub-interval are obtained;

S11. Make i=j+1, and jump to the S4 step;

S12. So far the whole decomposition process is over, the Hammerstein-Wiener system is decomposed into m sub-intervals, each sub-interval corresponds to a linear sub-model to form a sub-model set P ₁ , P ₂ ,...,P _m , and the corresponding sub-controller set is K ₁ ,K ₂ ,…,K _m ;

S13. Calculate the output of the multi-model controller according to the formula (5) shown below, and optimize the control of the Hammerstein-Wiener system;

where u _i (k) is the output of the ith sub-controller, is the angle-based weighting function of the i-th sub-controller, and k is the current moment.

Example 1:

The following examples are given to illustrate the above decomposition and control methods, and the numerical model is simulated and analyzed.

v=u+0.5u ² -0.7u ³

y=x-0.5x ² +0.2x ³

The output nonlinearity of the above system is irreversible, so the traditional nonlinear inversion method cannot be used. Using the present invention to decompose and control the above-mentioned system based on the included angle, the specific steps are as follows:

S1. Use the angle-based gridding algorithm to grid the entire operating space of the SISO Hammerstein-Wiener system to obtain 82 grid points, and linearize the Hammerstein-Wiener system at each grid point to obtain 82 Linearize the model, calculate the slope of the static input-output curve of the system at each grid point, and further calculate the angle between every two grid points, and obtain the angle matrix Θ=[θ _ij ] _82×82 , where The maximum value of the included angle matrix is θ _max , the minimum value is θ _min , and Θ=[θ _ij ] _82×82 is normalized, and the obtained included angle matrix is in

S2. According to the prior knowledge of the Hammerstein-Wiener system, select the threshold γ=0.26.

S3. Set i=1, m=0, i represents the i-th linearization model, and m represents the number of local sub-models.

S4. If i≤82, set j=i, m=m+1, otherwise jump to S12.

S5. From the i-th to the j-th linearization model according to the formula Select a nominal model G ^* ;

S6. According to the formula Calculate the maximum normalized angle between the nominal model and other linearized models;

S7. If Then make j=j+1 and jump to S5. Otherwise, jump to S8;

S8. Let j=j-1;

S9. From the i-th to the j-th linearization model again according to the formula

Choose a nominal model G ^* .

S10. For the nominal model G ^* , design a linear PID controller K, record the nominal model G ^* as the mth local sub-model P _m , and record the controller K as the mth local sub-controller K _m ; That is, both the linear sub-model P _m and the sub-controller K _m of the mth sub-interval are generated.

S11. Make i=j+1, and jump to step S4.

S12. So far the whole decomposition process is over, the Hammerstein-Wiener system is decomposed into 2 sub-intervals, each sub-interval has a linear sub-model to form the sub-model set P ₁ , the sub-PID controller set corresponding to P ₂ is K ₁ , K ₂ .

S13. Finally, the output of the multi-model controller according to the formula Calculated to optimize the control of the Hammerstein-Wiener system, where u _i (k) is the output of the i-th sub-controller, is the angle-based weighting function of the i-th sub-controller.

Figure 3 and Figure 4 are the input and output values of the closed-loop system under the empirical multi-model controller, respectively, where ref represents the reference input, y1(k) is the system closed-loop output, and u1(k) is the system control input, as shown in Figure 3 and Fig. 4 shows that the output response speed of the system is slow; Fig. 4 and Fig. 6 are the input and output of the multi-model controller under the decomposition and control method of the closed-loop system of the present invention, ref (k) is a reference input, and y2 (k) is The closed-loop output of the system, u2(k) is the system control input. Obviously, the effect of outputting the tracking reference signal is much faster than the former, and the tracking accuracy is much higher, which is both fast and accurate.

The gridding algorithm based on the included angle and the weighting function based on the included angle mentioned in the present invention belong to conventional technical means mastered by those skilled in the art, so they are not described in detail.

The above description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the invention. Both modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention will not be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (1)

1. a kind of Hammerstein-Wiener system multi-model decomposition and control method based on included angle, it is characterized in that, concrete steps are as follows: S1. Use the angle-based gridding algorithm to grid the entire operating space of the Hammerstein-Wiener system, and linearize the Hammerstein-Wiener system at each grid point to obtain n _p linearized models G _i ( i=1, 2,...,n _p ), calculate the slope of the static input-output curve of the system at each grid point, and further use the slope to calculate the angle between every two grid points, and obtain the angle matrix as Formula (1) shows: where, θ _ij =|θ _i -θ _j |, i, j=1, 2,..., n _p . θ _i and θ _j are the slope angles at the i and j grid points respectively; θ _ij represents the system The angle between the slopes at the i-th and j-th grid points; Normalize the included angle matrix to obtain the normalized included angle matrix as shown in formula (2): in, i=1, 2,..., n _p , θ _max is the maximum value of the angle matrix in the formula (1), and θ _min is the minimum value of the angle matrix in the formula (1); S2. According to the prior knowledge of the Hammerstein-Wiener system, select the threshold γ of the operation space decomposition; S3. define the initial value to the normalized included angle matrix formula (2), set i=1, m=0, i represents the i linearization model, and m represents the number of the current local sub-model; S4. If _i≤np , then j=i, m=m+1, enter S5, otherwise jump to S12; S5. From the i-th linearization model to the j-th linearization model, select a nominal model G ^* according to the formula (3) shown below: Among them, max() means seeking the maximum value, min() means seeking the minimum value, and h and l are any values between i and j; S6. Calculate the maximum normalized angle between the nominal model and other linearized models according to the formula (4) shown below: S7. If Then make j=j+1 and jump to S5; otherwise jump to S8; S8. Let j=j-1 enter S9; S9. Select a nominal model G ^* from the i-th to the j-th linearization model again according to formula (3): S10. For the nominal model G ^* selected in S9, design a conventional linear controller K, denote the nominal model G ^* as the mth local sub-model P _m , and denote the controller K as the mth local sub-controller K _m , that is, to obtain the linear sub-model and sub-controller of the mth sub-interval; S11. Make i=j+1, and jump to step S4; S12. So far the whole decomposition process is over, the Hammerstein-Wiener system is decomposed into m sub-intervals, each sub-interval corresponds to a linear sub-model to form a sub-model set P ₁ , P ₂ ,..., P _m , and the corresponding sub-controller set is K ₁ , K ₂ ,..., K _m ; S13. Calculate the output of the multi-model controller according to the formula (5) shown below, and optimize the control of the Hammerstein-Wiener system; where u _i (k) is the output of the ith sub-controller, is the angle-based weighting function of the i-th sub-controller, and k is the current moment.