CN108614431B - Hammerstein-Wiener system multi-model decomposition and control method based on included angle - Google Patents

Hammerstein-Wiener system multi-model decomposition and control method based on included angle Download PDF

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CN108614431B
CN108614431B CN201810588733.XA CN201810588733A CN108614431B CN 108614431 B CN108614431 B CN 108614431B CN 201810588733 A CN201810588733 A CN 201810588733A CN 108614431 B CN108614431 B CN 108614431B
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included angle
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杜静静
陈俊风
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Changzhou Campus of Hohai University
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Abstract

The invention discloses a Hammerstein-Wiener system multi-model decomposition and control method based on an included angle, which is characterized in that the included angle is utilized to carry out multi-model decomposition on the Hammerstein-Wiener system, a linear model set similar to the Hammerstein-Wiener system is obtained, a linear controller is designed based on the model set, and a weighting method based on the included angle is utilized to carry out weighting fusion on the obtained linear controller so as to carry out optimization control on the Hammerstein-Wiener system. The invention can effectively overcome the defects that the traditional nonlinear inverse control method can only be used for a static reversible Hammerstein-Wiener system, and the input or output nonlinearity of the system cannot be considered in the design of the controller, so that the closed-loop performance is reduced, and the like.

Description

Hammerstein-Wiener system multi-model decomposition and control method based on included angle
Technical Field
The invention relates to a Hammerstein-Wiener system multi-model decomposition and control method based on an included angle, and belongs to the field of nonlinear system multi-model control.
Background
The Hammerstein-Wiener model is a typical block structure model and is formed by connecting two static nonlinear modules in series with a dynamic linear module. The whole model can be regarded as a Hammerstein model which is connected with a Wiener model in series. This series configuration has a series of advantages: the priori knowledge is easily fused in the model; the modeling cost is low; the approximation precision is high; for ease of control, etc. The Hammerstein-Wiener model has therefore been widely used in the past decade for modeling of nonlinear systems, such as continuous stirred tank reactors, neutralisation reactors, DC motors, photovoltaic power generation systems, etc.
However, other block structure models are also difficult to solve when designing controllers based on the Hammerstein-Wiener model: a common nonlinear inverse control method, namely a method for compensating system nonlinearity by using the inverse of a static link, is only suitable for the condition that a nonlinear link is reversible, but an actual Hammerstein-Wiener system usually has input or output nonlinearity, even has input and output nonlinearity simultaneously. In addition, the non-linear inverse method does not take the non-linear characteristics of the system into account when designing the controller, thereby reducing the closed-loop performance of the system. Therefore, other control methods have been proposed to overcome the disadvantages of the non-linear inversion. For example, Khani et al propose approximating the Hammerstein-Wiener system with a linear model with structural or non-structural uncertainty, and then using RMPC for controller design. Lawrynczuk proposes a nonlinear MPC (MPC) aiming at a Hammerstein-Wiener model, linearizes a system along an expected track, and converts a nonlinear optimization problem into a quadratic programming problem, thereby avoiding nonlinear inverse control. However, continuously linearizing the system can result in a heavy computational load.
Disclosure of Invention
In order to simplify the control problem of the Hammerstein-Wiener system, the invention provides a Hammerstein-Wiener system multi-model decomposition and control method based on an included angle, which is used for designing a multi-mode controller, reducing the calculated amount, overcoming the defects in the prior control technical method, optimizing, improving and popularizing the prior included angle decomposition algorithm, avoiding complex calculation and improving the decomposition efficiency and quality.
In order to achieve the purpose, the technical scheme of the invention is realized as follows:
a Hammerstein-Wiener system multi-model decomposition and control method based on included angles comprises the following specific steps:
s1, gridding the whole operation space of the Hammerstein-Wiener system by using an included angle-based gridding algorithm, and linearizing the Hammerstein-Wiener system at each grid point to obtain npA linearized model Gi(i＝1,2,…,np) Calculating the slope of the static input-output curve of the system at each grid point, and further calculating the included angle between every two grid points by using the slope to obtain an included angle matrix as shown in formula (1):
Figure BDA0001690092500000021
wherein, thetaij＝|θi-θj|，i,j＝1,2,…,np.θiAnd thetajSlope angles at the ith and jth grid points, respectively; thetaijRepresenting the included angle of the slopes of the ith and jth grid points of the system;
normalizing the included angle matrix to obtain a normalized included angle matrix as shown in formula (2):
Figure BDA0001690092500000022
wherein the content of the first and second substances,
Figure BDA0001690092500000023
i＝1,2,…,np，θmaxis the maximum value of the angle matrix in equation (1), θminIs the minimum value of the included angle matrix in the formula (1);
s2, selecting a threshold gamma of operation space decomposition according to prior knowledge of a Hammerstein-Wiener system;
s3, defining an initial value for the normalized included angle matrix formula (2), setting i to be 1, setting m to be 0, wherein i represents the ith linearization model, and m represents the number of the current local submodels;
s4, if i is less than or equal to npIf j is equal to i, m is equal to m +1, the process goes to S5, otherwise, the process jumps to S12;
s5, from the ith to the jth linearized model, selecting a nominal model G according to the formula (3) shown below*：
Figure BDA0001690092500000031
Wherein max () represents the maximum value, min () represents the minimum value, and h and l are any values between i and j;
s6, calculating the maximum normalized included angle between the nominal model and other linearized models according to the following formula (4):
Figure BDA0001690092500000032
s7, if
Figure BDA0001690092500000033
Let j equal j +1 and go to S5; otherwise jump to S8;
s8, enabling j to be j-1 to enter S9;
s9, selecting a nominal model G according to the formula (3) from the ith linearized model to the jth linearized model again*；
S10, for the nominal model G selected in S9*Designing a conventional linear controller K to make the nominal model G*Is recorded as the m-th local submodel PmAnd the controller K is recorded as the mth local sub-controller KmObtaining a linear sub-model and a sub-controller of the mth sub-interval;
s11, making i equal to j +1, and jumping to the step S4;
s12, at the end of the whole decomposition process, decomposing the Hammerstein-Wiener system into m subintervals, wherein each subinterval corresponds to a linear submodel to form a submodel set P1,P2,…,PmCorresponding subcontroller set is K1,K2,…,Km；
S13, calculating according to a formula (5) shown below to obtain the output of the multimode controller, and performing optimization control on the Hammerstein-Wiener system;
Figure BDA0001690092500000041
wherein u isi(k) Is the output of the i-th sub-controller,
Figure BDA0001690092500000042
is the weighting function based on the included angle of the ith sub-controller, and k is the current time.
Has the advantages that: the invention provides a Hammerstein-Wiener system multi-model decomposition and control method based on an included angle, which has the following advantages and positive effects compared with the prior art:
(1) the method is suitable for the Hammerstein-Wiener system with input or/and output diversity.
(2) The control performance reduction caused by using an inverse function in the traditional nonlinear inverse method is avoided.
(3) The robustness and the closed-loop control performance of the system are improved.
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FIG. 1 is a schematic diagram of a Hammerstein-Wiener system multi-model decomposition and weighting method of the present invention;
FIG. 2 is a diagram of a multiple model control architecture for the Hammerstein-Wiener system of the present invention;
FIG. 3 is a graph of output values for a closed loop system under an empirical multimode type controller;
FIG. 4 is a graph of input values for a closed loop system under an empirical multimode type controller;
FIG. 5 is a graph of the output of a closed loop system in the multimode controller of the present invention;
FIG. 6 is a graph of input values for a closed loop system in a multi-mode controller of the present invention;
Detailed Description
In order to make those skilled in the art better understand the technical solutions in the present application, the technical solutions in the embodiments of the present application are clearly and completely described below, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
A Hammerstein-Wiener system multi-model decomposition and control method based on included angles takes a single-input-single-output system as an example, and comprises the following specific steps:
s1, gridding the whole operation space of the Hammerstein-Wiener system by using an included angle-based gridding algorithm, and linearizing the Hammerstein-Wiener system at each grid point to obtain npA linearized model Gi(i＝1,2,…,np) Calculating the slope of the static input-output curve of the system at each grid point, and further calculating the included angle between each two grid points (i.e. between the linearized models) by using the slope to obtain an included angle matrix as shown in formula (1):
Figure BDA0001690092500000051
wherein, thetaij＝|θi-θj|，i,j＝1,2,…,np.θiAnd thetajSlope angles at the ith and jth grid points, respectively; thetaijRepresenting the included angle of the slopes of the ith and jth grid points of the system, namely the included angle between the ith and jth linearized models;
normalizing the included angle matrix to obtain a normalized included angle matrix as shown in formula (2):
Figure BDA0001690092500000052
wherein the content of the first and second substances,
Figure BDA0001690092500000053
i＝1,2,…,np，θmaxis the maximum value of the angle matrix in equation (1), θminIs the minimum value of the included angle matrix in the formula (1);
s2, selecting a threshold gamma of operation space decomposition according to prior knowledge of a Hammerstein-Wiener system;
s3, defining an initial value for the normalized included angle matrix formula (2), setting i to be 1, setting m to be 0, wherein i represents the ith linearization model, and m represents the number of the current local submodels;
s4, if i is less than or equal to npIf j is equal to i, m is equal to m +1, the process goes to S5, otherwise, the process jumps to S12;
s5, from the ith to the jth linearized model, selecting a nominal model G according to the formula (3) shown below*：
Figure BDA0001690092500000061
Wherein max () represents the maximum value, min () represents the minimum value, and h and l are any values between i and j;
s6, calculating the maximum normalized included angle between the nominal model and other linearized models according to the following formula (4):
Figure BDA0001690092500000062
s7, if
Figure BDA0001690092500000063
Let j equal j +1 and go to S5; otherwise jump to S8;
s8, enabling j to be j-1 to enter S9;
s9, selecting a nominal model G according to the formula (3) from the ith linearized model to the jth linearized model again*；
S10, for the nominal model G selected in S9*Designing a conventional linear controller K (the linear controller K is designed as a conventional technical means in the invention and can be a PID controller, an MPC controller or an LQ controller), and designing a nominal model G*Is recorded as the m-th local submodel PmAnd the controller K is recorded as the mth local sub-controller KmObtaining a linear sub-model and a sub-controller of the mth sub-interval;
s11, making i equal to j +1, and jumping to the step S4;
s12, at the end of the whole decomposition process, decomposing the Hammerstein-Wiener system into m subintervals, wherein each subinterval corresponds to a linear submodel to form a submodel set P1,P2,…,PmCorresponding subcontroller set is K1,K2,…,Km；
S13, calculating according to a formula (5) shown below to obtain the output of the multimode controller, and performing optimization control on the Hammerstein-Wiener system;
Figure BDA0001690092500000071
wherein u isi(k) Is the output of the i-th sub-controller,
Figure BDA0001690092500000072
is the weighting function based on the included angle of the ith sub-controller, and k is the current time.
Example 1:
the decomposition and control method is described below by way of example, and a numerical model is simulated and analyzed.
v＝u+0.5u2-0.7u3
Figure BDA0001690092500000073
y＝x-0.5x2+0.2x3
The output non-linearity of the system is not reversible, so that the traditional non-linear inverse method cannot be used. The method is adopted to decompose and control the system based on the included angle, and comprises the following specific steps:
s1, using a gridding algorithm based on an included angle to gridde the whole operation space of a SISO Hammerstein-Wiener system to obtain 82 grid points, linearizing the Hammerstein-Wiener system at each grid point to obtain an 82 linearization model, calculating the slope of a static input and output curve of the system at each grid point, further calculating the included angle between every two grid points, and obtaining an included angle matrix theta ═ theta [ [ theta ] ]ij]82×82Wherein the maximum value of the angle matrix is thetamaxMinimum value of thetaminFor theta ═ thetaij]82×82Normalized to obtain an included angle matrix of
Figure BDA0001690092500000081
Wherein
Figure BDA0001690092500000082
And S2, selecting the threshold gamma to be 0.26 according to the prior knowledge of the Hammerstein-Wiener system.
And S3, setting i to be 1, m to be 0, wherein i represents the ith linearization model, and m represents the number of local submodels.
And S4, if i is less than or equal to 82, setting j to i, and m to m +1, otherwise, jumping to S12.
S5, from the ith to the jth linearized model according to a formula
Figure BDA0001690092500000083
Selecting a nominal model G*；
S6, according to a formula
Figure BDA0001690092500000084
Calculating the maximum normalized included angle between the nominal model and other linearized models;
s7, if
Figure BDA0001690092500000085
J is made j +1 and S5 is skipped, otherwise S8 is skipped;
s8, enabling j to be j-1;
s9, the ith to jth linearization models are again expressed according to the formula
Figure BDA0001690092500000086
Selecting a nominal model G*。
S10, nominal model G is compared*Designing a linear PID controller K to normalize the model G*Is recorded as the m-th local submodel PmAnd the controller K is recorded as the mth local sub-controller Km(ii) a I.e. the linear sub-model P of the mth sub-intervalmAnd a sub-controller KmAll produced.
And S11, changing i to j +1, and jumping to the step S4.
S12, at the end of the whole decomposition process, decomposing the Hammerstein-Wiener system into 2 subintervals, forming a linear submodel set P by each subinterval1,P2Corresponding sub-PID controller set is K1,K2。
S13. finally, the output of the multi-mode controller is according to a formula
Figure BDA0001690092500000091
Calculated, and optimized control is carried out on the Hammerstein-Wiener system, wherein ui(k) Is the output of the i-th sub-controller,
Figure BDA0001690092500000092
is the weighting function based on the included angle of the ith sub-controller.
FIGS. 3 and 4 are the input and output values of the closed-loop system under the empirical multimode controller, respectively, wherein ref represents the reference input, y1(k) is the closed-loop output of the system, u1(k) is the control input of the system, and the slow response speed of the output of the system is known from FIGS. 3 and 4; fig. 4 and 6 show the input and output of the multi-mode controller under the decomposition and control method of the present invention of the closed-loop system, ref (k) is the reference input, y2(k) is the system closed-loop output, u2(k) is the system control input, and obviously the effect of outputting the tracking reference signal is much faster and much higher tracking accuracy than the former one, which is both fast and accurate.
The grid algorithm based on the included angle and the weighting function based on the included angle in the present invention belong to the conventional technical means grasped by those skilled in the art, and therefore, detailed description is not provided.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Two modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to 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 Hammerstein-Wiener system multi-model decomposition and control method based on included angles is characterized by comprising the following specific steps:
s1, gridding the whole operation space of the Hammerstein-Wiener system by using an included angle-based gridding algorithm, and linearizing the Hammerstein-Wiener system at each grid point to obtain npA linearized model Gi(i＝1,2,…,np) Calculating the slope of the static input-output curve of the system at each grid point, and further calculating the included angle between every two grid points by using the slope to obtain an included angle matrix as shown in formula (1):
Figure FDA0002916215320000011
wherein, thetaij＝|θi-θj|，i,j＝1,2,…,np.θiAnd thetajSlope angles at the ith and jth grid points, respectively; thetaijRepresenting the included angle of the slopes of the ith and jth grid points of the system;
normalizing the included angle matrix to obtain a normalized included angle matrix as shown in formula (2):
Figure FDA0002916215320000012
wherein the content of the first and second substances,
Figure FDA0002916215320000013
i＝1,2,…,np，θmaxis the maximum value of the angle matrix in equation (1), θminIs the minimum value of the included angle matrix in the formula (1);
s2, selecting a threshold gamma of operation space decomposition according to prior knowledge of a Hammerstein-Wiener system;
s3, setting i to be 1, m to be 0, wherein i represents the ith linearization model, and m represents the number of the current local submodels;
s4, if i is less than or equal to npIf j is equal to i, m is equal to m +1, the process goes to S5, otherwise, the process jumps to S12;
s5, from the ith to the jth linearized model, selecting a nominal model G according to the formula (3) shown below*：
Figure FDA0002916215320000021
Wherein max () represents the maximum value, min () represents the minimum value, and h and l are any values between i and j;
s6, calculating the maximum normalized included angle between the nominal model and other linearized models according to the following formula (4):
Figure FDA0002916215320000022
s7, if
Figure FDA0002916215320000023
Let j equal j +1 and go to S5; otherwise jump to S8;
s8, enabling j to be j-1 to enter S9;
s9, selecting a nominal model G according to the formula (3) from the ith linearized model to the jth linearized model again*；
S10, for the nominal model G selected in S9*Designing a conventional linear controller K to make the nominal model G*Is recorded as the m-th local submodel PmAnd the controller K is recorded as the mth local sub-controller KmObtaining a linear sub-model and a sub-controller of the mth sub-interval;
s11, making i equal to j +1, and jumping to the step S4;
s12, at the end of the whole decomposition process, decomposing the Hammerstein-Wiener system into m subintervals, wherein each subinterval corresponds to a linear submodel to form a submodel set P1,P2,…,PmCorresponding subcontroller set is K1,K2,…,Km；
S13, calculating according to a formula (5) shown below to obtain the output of the multimode controller, and performing optimization control on the Hammerstein-Wiener system;
Figure FDA0002916215320000024
wherein u isi(k) Is the output of the i-th sub-controller,
Figure FDA0002916215320000025
is the weighting function based on the included angle of the ith sub-controller, and k is the current time.
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