CN112115593B - Distributed centralized member estimation method based on centrosymmetric polyhedron - Google Patents

Distributed centralized member estimation method based on centrosymmetric polyhedron Download PDF

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CN112115593B
CN112115593B CN202010918168.6A CN202010918168A CN112115593B CN 112115593 B CN112115593 B CN 112115593B CN 202010918168 A CN202010918168 A CN 202010918168A CN 112115593 B CN112115593 B CN 112115593B
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韩渭辛
许斌
范泉涌
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Abstract

The invention provides a distributed collective estimation method aiming at voltage estimation of a grid-connected power generation system, adopts a centrosymmetric polyhedron to describe process disturbance and noise range, can effectively estimate a state collection by utilizing information of a plurality of measuring points, breaks through the limitation that the existing method can only estimate a state value, and serves for grid-connected safety control. In addition, the dynamic error recursion of the voltage estimation is obtained from the initial state range, so that the system state set range at any time is obtained at each measuring node, a basis can be provided for subsequent distributed monitoring of voltage abnormality, and the application range is expanded.

Description

Distributed centralized member estimation method based on centrosymmetric polyhedron
Technical Field
The invention belongs to the field of system state estimation, relates to a distributed collective member estimation method in the field of system operation state estimation, and particularly relates to a distributed collective member estimation method based on a centrosymmetric polyhedron.
Background
Distributed estimation is widely applied to the fields of automatic driving, aircraft network control, grid-connected power generation and the like, and attracts more and more attention. The distributed estimation problem is that each estimator can only obtain partial information aiming at an object to be estimated, and a plurality of estimators jointly estimate the state of the system through mutual information. Distributed estimation is an expansion of centralized estimation, and has attracted wide attention of scholars at home and abroad in recent years. At present, distributed estimation based on a Kalman filter and distributed estimation based on an observer are mainly available, but the distributed estimation and the point estimation are point estimation results and can not realize set interval estimation.
A distributed observer for a linear time invariant system (Wang Lili, A. Stephen Morse, IEEE TRANSACTIONS AUTOMATIC CONTROL, vol. 63 No. 7 of 2018) proposes a design method of the distributed observer for the linear time invariant system, and realizes that all states of an original system can be estimated at each point by utilizing interaction of local measurement information and observer information. However, the estimation method can only realize point estimation under the ideal condition of a linear system, and has poor estimation effect in the case of disturbance.
Disclosure of Invention
The technical problem solved by the invention is as follows: in order to overcome the defects of the existing distributed estimation, the invention provides a distributed collective estimation method based on a centrosymmetric polyhedron, which solves the problem that the distributed collective estimation can not be realized at present,
the technical scheme of the invention is as follows: a distributed centralized membership estimation method based on a centrosymmetric polyhedron comprises the following steps:
the method comprises the following steps: constructing a distributed observer based on a system model to be estimated;
establishing a system model to be estimated as a discrete linear system
Figure BDA0002665769880000021
Wherein
Figure BDA0002665769880000022
The status is indicated by a status indicator,
Figure BDA0002665769880000023
which is indicative of a bounded perturbation,
Figure BDA0002665769880000024
representing bounded measurement noise, y 1 ,…,y N Representing the measured outputs of the respective nodes, A, B, C 1 ，…，C N E is a known matrix associated with the system model;
wherein the ranges of the system initial value, disturbance and measurement noise are described by a centrosymmetric polyhedron as follows:
x(0)∈X(0)＝<p 0 ,H 0 >，w(k)∈W＝<p w ,H w >，v i (k)∈V＝<p i v ,H v >
for the convenience of subsequent calculation, let p be w ＝p w ＝0；
Centrosymmetric polyhedron is used to describe the boundary of variable set, and a vector is given
Figure BDA0002665769880000025
And a set of vectors
Figure BDA0002665769880000026
Then the m-dimensional centrosymmetric polyhedron is defined as:
Figure BDA0002665769880000027
the centrosymmetric polyhedron is m-dimensional hypercube with p as center
Figure BDA0002665769880000028
Forming a mapping by radial transformation, wherein a matrix H = { H = } 1 ,h 2 ,…,h m Represents this linear transformation, called the generator matrix of centrosymmetric polyhedrons; the centrosymmetric polyhedron can also be represented as:
Figure BDA0002665769880000029
wherein
Figure BDA00026657698800000210
Representing two sets of minkowski sums;
Figure BDA00026657698800000211
constructing a distributed observer according to the model (1):
Figure BDA00026657698800000212
wherein a is ij Is a contiguous matrix element, L, of a known observer information interaction topology i And M i A gain matrix is to be designed;
step two: constructing a dynamic estimation error system, and designing a gain matrix of a distributed observer;
first, the estimation error is defined:
Figure BDA00026657698800000213
can be obtained from the formulas (1) and (2)
Figure BDA0002665769880000031
Definition of
Figure BDA0002665769880000032
The global error dynamic system obtained is:
Figure BDA0002665769880000033
wherein Λ = diag { A-L 1 C 1 ,…,A-L N C N }，M＝diag{M 1 ,…,M N },
Figure BDA0002665769880000034
Figure BDA0002665769880000035
Here Θ represents the laplacian matrix of the known observer information interaction topological graph;
solving the following optimization problem constrained by the matrix inequality to obtain the observer matrix gain
maxtr(P)s.t.
Figure BDA0002665769880000036
Wherein
λ 11 ＝-γP+Φ 1 +Φ 1 T
Figure BDA0002665769880000037
λ 13 ＝αWH v
λ 14 ＝-αG+Φ 1 T
Figure BDA0002665769880000038
Figure BDA0002665769880000039
Figure BDA00026657698800000310
Figure BDA00026657698800000311
λ 44 ＝P-G-G T ，P＝diag{P 1 ,…,P N }，W＝diag{W 1 ,…,W N }，G＝diag{G 1 ,…,G N }
Figure BDA00026657698800000312
Wherein P > 0 indicates that the matrix P is positive, the inequality is such that α is a predetermined constant greater than zero, the variables P, W, G are solved by an optimization problem,
Figure BDA00026657698800000313
representation matrix H w The transpose matrix of (a) is,
Figure BDA00026657698800000314
representation matrix H v The transposed matrix of (a), the remaining parameters being known;
the gain is:
Figure BDA0002665769880000041
step three: distributed collective member estimation by using centrosymmetric multi-surface based on estimation error system
The distributed centralized member estimation estimates the range of the system state at each measurement node, namely a centrosymmetric polyhedron is calculated at each moment, and the actual state is ensured to be contained in the centrosymmetric polyhedron; the centrosymmetric polyhedron is obtained by analyzing a distributed observer and a state and error system, and is defined as follows:
Figure BDA0002665769880000042
the addition rule of the two centrosymmetric polyhedron sets is
Figure BDA0002665769880000043
The number multiplication rule of two centrosymmetric polyhedron sets is that L < p, H > = < Lp, LH >, and the central point and the generating matrix are obtained from the properties of the centrosymmetric polyhedron
Figure BDA0002665769880000044
Figure BDA0002665769880000045
The distributed panelist estimation results at each measurement node are thus given by the centrosymmetric polyhedron given in (5).
The further technical scheme of the invention is as follows: in the third step
Figure BDA0002665769880000046
Is obtained by the recursive calculation of the distributed observer,
Figure BDA0002665769880000047
the error dynamic system is obtained by analyzing a centrosymmetric polyhedron, and the specific recursion process is as follows:
Figure BDA0002665769880000048
effects of the invention
The invention has the technical effects that: the invention provides a distributed member estimation method aiming at voltage estimation of a grid-connected power generation system, adopts a centrosymmetric polyhedron to describe process disturbance and noise range, can effectively estimate a set of states by utilizing information of a plurality of measuring points, breaks through the limitation that the existing method can only estimate a state value, and serves for grid-connected safety control.
In addition, the dynamic error recursion of the voltage estimation is obtained from the initial state range, so that the system state set range at any time is obtained at each measuring node, a foundation can be provided for subsequent distributed monitoring of voltage abnormity, and the application range is expanded.
Drawings
FIG. 1 is a flow chart of the method
Detailed Description
Referring to fig. 1, the method comprises the following specific steps: constructing a distributed observer based on a system model to be estimated; step two: constructing a dynamic estimation error system, and designing a gain matrix of a distributed observer; step three: and performing distributed collective member estimation by using a central symmetry polygon based on an estimation error system. Aiming at the problem that the voltage estimation in grid-connected power generation is influenced by uncertainty, the method can estimate the set of system states at each point by constructing the distributed observer and analyzing the error system set, breaks through the limitation that the existing method only can give an estimated value, can improve the voltage distributed estimation effect in grid-connected power generation, and provides a basis for subsequent voltage monitoring and control.
Each step is described in detail below.
The method comprises the following steps: constructing a distributed observer based on a system model to be estimated;
establishing a system model to be estimated as a discrete linear system
Figure BDA0002665769880000051
Wherein
Figure BDA0002665769880000052
The status is represented by a number of time slots,
Figure BDA0002665769880000053
which is indicative of a bounded perturbation,
Figure BDA0002665769880000054
representing bounded measurement noise, y 1 ,…,y N Representing the measured outputs of the respective nodes, A, B, C 1 ，…，C N And E is a known matrix associated with the system model.
Wherein the ranges of the system initial value, disturbance and measurement noise are described by a central symmetry polyhedron as follows:
x(0)∈X(0)＝<p 0 ,H 0 >，w(k)∈W＝<p w ,H w >，v i (k)∈V＝<p i v ,H v >
for the convenience of subsequent calculation, let p be w ＝p w ＝0。
Centrosymmetric polyhedron is used to describe the boundary of variable set, and a vector is given
Figure BDA0002665769880000061
And a set of vectors
Figure BDA0002665769880000062
Then m-dimensional central symmetryThe polyhedron is defined as:
Figure BDA0002665769880000063
the centrosymmetric polyhedron is m-dimensional hypercube with p as center
Figure BDA0002665769880000064
Forming a mapping by radial transformation, wherein a matrix H = { H = } 1 ,h 2 ,…,h m Denotes this linear transformation, called the generator matrix of the centrosymmetric polyhedron. The centrosymmetric polyhedron can also be represented as:
Figure BDA0002665769880000065
wherein
Figure BDA0002665769880000066
Representing two sets of minkowski sums.
Figure BDA0002665769880000067
Constructing a distributed observer according to the model (1):
Figure BDA0002665769880000068
wherein a is ij Is a contiguous matrix element, L, of a known observer information interaction topology i And M i A gain matrix is to be designed.
Step two: constructing a dynamic estimation error system, and designing a gain matrix of a distributed observer;
first, the estimation error is defined:
Figure BDA0002665769880000069
can be obtained from the formulas (1) and (2)
Figure BDA00026657698800000610
Definition e (k) = [ e = [) 1 T (k) … e N T (k)] T The global error dynamic system can be obtained as follows:
Figure BDA00026657698800000611
wherein Λ = diag { A-L 1 C 1 ,…,A-L N C N }，M＝diag{M 1 ,…,M N },
Figure BDA00026657698800000612
Figure BDA00026657698800000613
Here Θ represents the laplacian matrix of the known observer information interaction topology.
Solving the following optimization problem constrained by the matrix inequality to obtain the observer matrix gain
maxtr(P)s.t.
Figure BDA0002665769880000071
Wherein
λ 11 ＝-γP+Φ 1 +Φ 1 T
Figure BDA0002665769880000072
λ 13 ＝αWH v
λ 14 ＝-αG+Φ 1 T
Figure BDA0002665769880000073
Figure BDA0002665769880000074
Figure BDA0002665769880000075
Figure BDA0002665769880000076
λ 44 ＝P-G-G T ，P＝diag{P 1 ,…,P N }，W＝diag{W 1 ,…,W N }，G＝diag{G 1 ,…,G N }
Figure BDA0002665769880000077
Wherein P > 0 indicates that the matrix P is positive, the inequality is such that α is a predetermined constant greater than zero, the variables P, W, G are solved by an optimization problem,
Figure BDA0002665769880000078
representation matrix H w The transpose matrix of (a) is,
Figure BDA0002665769880000079
representation matrix H v The transpose matrix of (a), the remaining parameters being known.
The gain is:
Figure BDA00026657698800000710
step three: distributed collective member estimation by using centrosymmetric multi-surface based on estimation error system
The distributed centralized member estimation estimates the range of the system state at each measurement node, namely, a central symmetry polyhedron is calculated at each moment, and the fact that the actual state is contained in the central symmetry polyhedron is guaranteed. The centrosymmetric polyhedron is obtained by analyzing a distributed observer and a state and error system, and is defined as follows:
Figure BDA0002665769880000081
the addition rule of the two centrosymmetric polyhedron sets is
Figure BDA0002665769880000082
Wherein
Figure BDA0002665769880000083
Is obtained by the recursive calculation of the distributed observer,
Figure BDA0002665769880000084
the error dynamic system is obtained by analyzing a centrosymmetric polyhedron, and the specific recursion process is as follows:
Figure BDA0002665769880000085
the number multiplication rule of two centrosymmetric polyhedron sets is that L < p, H > = < Lp, LH >, and the central point and the generating matrix are obtained from the properties of the centrosymmetric polyhedron
Figure BDA0002665769880000086
Figure BDA0002665769880000087
The distributed panelist estimate at each measurement node is therefore given by the centrosymmetric polyhedron given in (5).
In order to better realize the distributed estimation of the voltage of the grid-connected power generation system under the influence of process disturbance and uncertainty and improve the effect of voltage safety monitoring, the specific implementation mode of the invention is explained by combining the distributed collective estimation of the voltage of a single-phase grid-connected power generation system:
executing the step one: constructing a distributed observer based on a system model to be estimated;
establishing a model of a single-phase grid-connected power generation system to be estimated, and rewriting the model into a discrete linear system
Figure BDA0002665769880000088
Where ω =100 π rad/s denotes the fundamental angular frequency, T =0.0025s denotes the sampling time,
Figure BDA0002665769880000089
Δu＝10 -3 U 0 sin (ω kT) denotes the voltage difference over the sampling interval, with a measurement error v i (k)＝0.002(1+0.1i)U 0 sin (ω kT). The matrix parameter in the formula (6) is
Figure BDA0002665769880000091
C 1 ＝[1.1 0]，C 2 ＝[1.2 0]，C 3 ＝[1.3 0]。
Wherein the ranges of system disturbance and measurement noise are described by a centrosymmetric polyhedron as:
w(k)∈W＝<p w ,H w >，v i (k)∈V＝<p i v ,H v >
for the convenience of subsequent calculation, let p be w ＝p w ＝0，
Figure BDA0002665769880000092
The voltage range of the system is estimated according to the output values of the plurality of measuring points, and reference is provided for safety monitoring and reliable control of grid-connected power generation.
Constructing a distributed observer according to the model:
Figure BDA0002665769880000093
wherein a is ij Is a contiguous matrix element, L, of a known observer information interaction topology i And M i A gain matrix is to be designed.
And (5) executing the step two: constructing a dynamic estimation error system, and designing a gain matrix of a distributed observer;
first, the estimation error is defined:
Figure BDA0002665769880000094
from the formulae (6) and (7) can be obtained
Figure BDA0002665769880000095
Definition e (k) = [ e = [) 1 T (k) … e N T (k)】 T The global error dynamic system can be obtained as follows:
Figure BDA0002665769880000096
wherein Λ = diag { A-L 1 C 1 ,…,A-L N C N }，M＝diag{M 1 ,…,M N },
Figure BDA0002665769880000097
Figure BDA0002665769880000098
Here Θ represents the laplacian matrix of the known observer information interaction topology.
Solving the following optimization problem constrained by the matrix inequality to obtain the observer matrix gain
max tr(P)s.t.
Figure BDA0002665769880000101
P＞0
Wherein
λ 11 ＝-γP+Φ 1 +Φ 1 T
Figure BDA0002665769880000102
λ 13 ＝αWH v
λ 14 ＝-αG+Φ 1 T
Figure BDA0002665769880000103
Figure BDA0002665769880000104
Figure BDA0002665769880000105
Figure BDA0002665769880000106
λ 44 ＝P-G-G T ，P＝diag{P 1 ,…,P N }，W＝diag{W 1 ,…,W N }，G＝diag{G 1 ,…,G N }
Figure BDA0002665769880000107
Where P > 0 indicates that the matrix P is positive, the inequality where α is a predetermined constant greater than zero, the variables P, W, G are solved by an optimization problem, and the remaining parameters are known.
The gain is:
Figure BDA0002665769880000108
here, the gain matrix is obtained as:
Figure BDA0002665769880000109
Figure BDA00026657698800001010
and step three is executed: distributed collective member estimation by using centrosymmetric polyhedral shape based on estimation error system
The distributed centralized member estimation estimates the range of the system state at each measurement node, namely, a central symmetry polyhedron is calculated at each moment, and the fact that the actual state is contained in the central symmetry polyhedron is guaranteed. The centrosymmetric polyhedron is obtained by analyzing a distributed observer and a state and error system, and is defined as follows:
Figure BDA0002665769880000111
wherein
Figure BDA0002665769880000112
Is obtained by the recursive calculation of the distributed observer,
Figure BDA0002665769880000113
the error dynamic system is obtained by analyzing a centrosymmetric polyhedron, and the specific recursion process is as follows:
Figure BDA0002665769880000114
from the properties of the centrosymmetric polyhedron, the central point and the generation matrix are obtained as
Figure BDA0002665769880000115
Figure BDA0002665769880000116
The distributed panelist estimation results at each measurement node are thus given by the centrosymmetric polyhedron given by (11).
The invention is not described in detail and is part of the common general knowledge of a person skilled in the art.

Claims (1)

1. A distributed membership estimation method based on a centrosymmetric polyhedron is characterized by comprising the following steps:
the method comprises the following steps: constructing a distributed observer based on a system model to be estimated;
establishing a model of a single-phase grid-connected power generation system to be estimated, and rewriting the model into a discrete linear system
Figure FDA0003997743210000011
Where ω =100 π rad/s denotes the fundamental angular frequency, T =0.0025s denotes the sampling time,
Figure FDA0003997743210000012
Δu＝10 -3 U 0 sin (ω kT) represents the voltage difference between sampling intervals with a measurement error of
Figure FDA0003997743210000017
The matrix parameter in formula (1) is
Figure FDA0003997743210000013
C 1 ＝[1.1 0]，C 2 ＝[1.2 0]，C 3 ＝[1.3 0]；
Wherein the ranges of system disturbance and measurement noise are described by a centrosymmetric polyhedron as:
w(k)∈W＝<p w ,H w >，v i (k)∈V＝<p i v ,H i v >；
for the convenience of subsequent calculation, let p be w ＝p i v ＝0，
Figure FDA0003997743210000014
Estimating the voltage range of the system according to the output values of the plurality of measuring points, and providing reference for safety monitoring and reliable control of grid-connected power generation;
constructing a distributed observer according to the model:
Figure FDA0003997743210000015
wherein a is ij Is a contiguous matrix element, L, of a known observer information interaction topology i And M i A gain matrix is to be designed;
step two: constructing a dynamic estimation error system, and designing a gain matrix of a distributed observer;
first, the estimation error is defined:
Figure FDA0003997743210000016
can be obtained from the formulas (1) and (2)
Figure FDA0003997743210000021
Definition e (k) = [ e = [) 1 T (k) … e N T (k)] T The global error dynamic system can be obtained as follows:
Figure FDA0003997743210000022
wherein Λ = diag { A-L 1 C 1 ,…,A-L N C N }，M＝diag{M 1 ,…,M N },
Figure FDA0003997743210000023
Figure FDA0003997743210000024
Here Θ represents the laplacian matrix of the known observer information interaction topological graph;
solving the following optimization problem constrained by the matrix inequality to obtain the observer matrix gain
max tr(P)s.t.
Figure FDA0003997743210000025
P＞0
Wherein
λ 11 ＝-γP+Φ 1 +Φ 1 T
Figure FDA0003997743210000026
λ 13 ＝αWH v
λ 14 ＝-αG+Φ 1 T
Figure FDA0003997743210000027
Figure FDA0003997743210000028
λ 33 ＝-(H i v ) T H v
Figure FDA0003997743210000029
λ 44 ＝P-G-G T ，P＝diag{P 1 ,…,P N }，W＝diag{W 1 ,…,W N }，G＝diag{G 1 ,…,G N }
Figure FDA00039977432100000210
Where P > 0 denotes that the matrix P is positive, in which inequality α is a constant greater than zero, and the matrix variable P to be solved 1 ,…,P N ，W 1 ,…,W N ，G 1 ,…,G N ，Y 1 ,…,Y N The method is obtained by solving a matrix inequality constraint optimization problem, and other parameters are known;
the gain is:
Figure FDA0003997743210000031
solving a gain matrix as follows:
Figure FDA0003997743210000032
Figure FDA0003997743210000033
step three: based on an estimation error system, performing distributed collective member estimation by using a centrosymmetric multi-surface shape;
the distributed centralized member estimation estimates the range of the system state at each measurement node, namely a centrosymmetric polyhedron is calculated at each moment, and the actual state is ensured to be contained in the centrosymmetric polyhedron; the centrosymmetric polyhedron is obtained by analyzing a distributed observer and a state and error system, and is defined as follows:
Figure FDA0003997743210000034
wherein
Figure FDA0003997743210000035
Is obtained by the recursive calculation of the distributed observer,
Figure FDA0003997743210000036
the error dynamic system is obtained by the analysis of a centrosymmetric polyhedron;
from the properties of the centrosymmetric polyhedron, the central point and the generation matrix are obtained as
Figure FDA0003997743210000037
Figure FDA0003997743210000038
The distributed set membership estimation result at each measurement node is given by a centrosymmetric polyhedron given by (5).
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