CN112052585A - Design method of distributed dimensionality reduction observer of linear time invariant system - Google Patents

Design method of distributed dimensionality reduction observer of linear time invariant system Download PDF

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CN112052585A
CN112052585A CN202010908366.4A CN202010908366A CN112052585A CN 112052585 A CN112052585 A CN 112052585A CN 202010908366 A CN202010908366 A CN 202010908366A CN 112052585 A CN112052585 A CN 112052585A
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王晓玲
苏厚胜
方荣
蒋国平
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a design method of a distributed dimensionality reduction observer of a linear time invariant system, which realizes state estimation of a target LTI system by using a plurality of distributed communication sensors, wherein each sensor can only acquire partial output information of the LTI target system; determining an output matrix of each sensor, and performing detectable decomposition on a matrix pair formed by the output matrix and a system matrix of a target LTI system; designing a distributed dimensionality reduction observer for each sensor based on distributed information interaction among the sensors; giving sufficient conditions containing topological information to ensure the existence of the distributed dimensionality reduction observer; further, a fully distributed dimension reduction observer is designed by using an adaptive strategy. The invention simplifies the design of the distributed observer of the LTI system, reduces the dimension of the distributed observer, avoids the dependence of the design of the distributed dimension-reducing observer on the global topology information, and has better flexibility.

Description

Design method of distributed dimensionality reduction observer of linear time invariant system
Technical Field
The invention belongs to the field of distributed observers, and particularly relates to a design method of a distributed dimension reduction observer of a Linear Time Invariant (LTI) system.
Background
State estimation is an important research content in control theory, and mainly realizes estimation of system state information according to output information of a system. The traditional state estimation problem is mainly researched by designing an observer according to complete output information of a system under the assumption that the system can be detected, so that the real state of the system is estimated. However, in practical engineering, there are some targets with a large geographical range, such as measurement of temperature and humidity in a large-scale plant, detection of irrigation conditions at a remote geographical location, collection of solar field related data, etc., and the measurement and estimation of the related state of such targets are much more difficult than the state estimation of small-scale targets. When the micro-miniature sensors are mature, the state estimation of a large target by utilizing a plurality of sensors is a necessary trend, and the method has good application prospect in the fields of military and civilian.
In the state estimation of a large target system by a plurality of sensors, each sensor can only acquire partial output information of the target system. Such acquisition of partial output information destroys the detectability of the system under the original complete output information, thereby greatly increasing the design difficulty of such observers. To enable state estimation of the non-detectable subsystems in the distributed observer, a coherence protocol in the multi-agent system is introduced, i.e. distributed communication between sensors can enable state estimation of the non-detectable subsystems to the responding subsystems of the LTI target system.
The invention provides a design method of a fully distributed dimensionality reduction observer, which avoids the dependence on communication topology, reduces the requirement on equipment precision and has better flexibility.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a design method of a distributed dimensionality reduction observer of a linear time invariant system, which simplifies the design of the distributed observer of an LTI system, eliminates the dependence of a communication network on global information, and has stronger robustness and better flexibility.
The invention content is as follows: the invention provides a method for designing a distributed dimensionality reduction observer of a linear time invariant system, which specifically comprises the following steps of:
(1) determining an output matrix of each sensor, and performing detectable decomposition on a matrix pair formed by the output matrix and a system matrix of a target LTI system;
(2) designing a distributed dimensionality reduction observer for each sensor;
(3) giving sufficient conditions containing topological information to ensure the existence of the distributed dimensionality reduction observer;
(4) and designing a fully distributed dimensionality reduction observer by using a self-adaptive strategy.
Further, the step (1) includes the steps of:
(11) performing detectable decomposition on each sensor to obtain a detectable subsystem and a non-detectable subsystem of each sensor:
Figure BDA0002662346680000021
yi(t)＝Cix(t)，
wherein,
Figure BDA0002662346680000022
it is the state of the system that is,
Figure BDA0002662346680000023
is the measurement output, piIs an output matrix CiIs of and satisfies
Figure BDA0002662346680000024
By using
Figure BDA0002662346680000025
A matrix with dimension m × n in euclidean space; according to the detectability decomposition of the system, obtaining (C)iA) non-detectable subspace
Figure BDA0002662346680000026
Eyes are merged
Figure BDA0002662346680000027
Has a dimension of vi(ii) a Definition matrix
Figure BDA0002662346680000028
Wherein U isiIs that
Figure BDA0002662346680000029
An orthogonal base of (a); note the book
Figure BDA00026623466800000210
Is composed of
Figure BDA00026623466800000211
Of orthogonal complement of, defining a matrix
Figure BDA00026623466800000212
Wherein
Figure BDA00026623466800000213
Is that
Figure BDA00026623466800000214
An orthogonal base of (a); the matrix defined above
Figure BDA00026623466800000215
And UiOn the basis of (2), an orthogonal matrix is defined
Figure BDA00026623466800000216
In matrix EiObtaining a detectable moiety x of the ith sensoridAnd a non-detectable moiety xiuNamely:
Figure BDA00026623466800000217
wherein
Figure BDA00026623466800000218
Is a matrix obtained by decomposition, and (C)id，Aid) Is detectableOf (1); i isnAn identity matrix representing n dimensions;
(12) for detectable moiety x of the ith sensoridPerforming nonsingular decomposition to obtain xidMeasurement output information y iniI.e. presence of non-singular matrices
Figure BDA00026623466800000219
Such that:
Figure BDA00026623466800000220
wherein,
Figure BDA00026623466800000221
and has:
Figure BDA00026623466800000222
wherein,
Figure BDA00026623466800000223
further, the step (2) comprises the steps of:
(21) for a strongly connected directed graph containing N nodes
Figure BDA0002662346680000031
Defining its Laplace matrix as
Figure BDA0002662346680000032
(22) Find a vector θ ═ θ1，θ2，...，θN]TSo that it satisfies thetai＞0(i＝1，...，N)，
Figure BDA0002662346680000033
(23) At thetaiDefining an N-dimensional diagonal matrix theta for diagonal elements, and defining a new matrixOf the symmetric matrix
Figure BDA0002662346680000034
(24) Selecting matrix KiSo that
Figure BDA0002662346680000035
Is of Hurwitz;
(25) subsystem xidDesigning a dimensionality reduction observer:
Figure BDA0002662346680000036
wherein,
Figure BDA0002662346680000037
is subsystem xid2Is observed, then xidThe observation state of (a) is:
Figure BDA0002662346680000038
wherein,
Figure BDA0002662346680000039
and is
Figure BDA00026623466800000310
(26) Subsystem x with the help of a coherence protocol of a multi-agent systemiuThe following observer was designed:
Figure BDA00026623466800000311
wherein, γi> 0 is coupling gain, and the use of the coherency protocol in the algorithm is primarily to compensate for subsystem xiuIs not detectable;
(27) designing a dimension reduction observer of the ith sensor:
Figure BDA00026623466800000312
wherein
Figure BDA00026623466800000313
(28) The state of the ith sensor is estimated as:
Figure BDA00026623466800000314
wherein,
Figure BDA00026623466800000315
further, the step (3) is realized as follows:
according to the Lyapunov stability theory, obtaining the value range of the coupling gain:
Figure BDA0002662346680000041
wherein the matrix U is UiIs a diagonal matrix of diagonal elements, | | | | - | represents a two-norm,
Figure BDA0002662346680000042
denotes the kronecker product, lambdamin(. cndot.) represents a minimum eigenvalue; here gamma isiIs selected so that the estimate of each sensor can converge to the state value of the LTI target system.
Further, the step (4) comprises the steps of:
(41) designing a fully distributed dimensionality reduction observer of the ith sensor based on an adaptive strategy:
Figure BDA0002662346680000043
wherein, γi(0) Is greater than 0, and
Figure BDA0002662346680000044
(42) the status of the ith sensor is estimated as
Figure BDA0002662346680000045
According to the Lyapunov stability theory, the estimated value of each sensor can converge to the state value of the LTI target system.
Has the advantages that: compared with the prior art, the invention has the beneficial effects that: 1. the invention provides a design method for realizing the state estimation of an LTI system with continuous time by utilizing a plurality of sensors which can be in distributed communication with each other, and the method reduces the precision requirement on sensor equipment, thereby reducing the economic cost of the equipment; 2. the method avoids the dependence of the design of the distributed observer on a communication topological structure by introducing a self-adaptive strategy on the coupling gain.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic diagram of an LTI target system consisting of three inertias;
FIG. 3 is a schematic diagram of a communication topology of three sensors;
FIG. 4 is a diagram of state estimation of a target system by three sensors at a constant coupling gain;
FIG. 5 is a diagram of state estimation of a target system by three sensors under adaptive coupling gain;
fig. 6 is a graph of the magnitude of the adaptive coupling gain.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the invention provides a method for designing a distributed dimension reduction observer of a linear time invariant system, which specifically comprises the following steps as shown in figure 1:
s1: an output matrix for each sensor is determined and a detectable decomposition is performed on a matrix pair formed by the output matrix and a system matrix of the target LTI system.
Performing detectable decomposition on each sensor to obtain a detectable subsystem and a non-detectable subsystem of each sensor:
Figure BDA0002662346680000051
yi(t)＝Cix(t)，
wherein,
Figure BDA0002662346680000052
it is the state of the system that is,
Figure BDA0002662346680000053
is the measurement output, piIs an output matrix CiIs of and satisfies
Figure BDA0002662346680000054
Here we use
Figure BDA0002662346680000055
Representing a matrix of dimension m x n in euclidean space. According to the detectability decomposition of the system, obtaining (C)iA) non-detectable subspace
Figure BDA0002662346680000056
And is
Figure BDA0002662346680000057
Has a dimension of vi. Definition matrix
Figure BDA0002662346680000058
Wherein U isiIs that
Figure BDA0002662346680000059
The orthogonal basis of (2). Note the book
Figure BDA00026623466800000510
Is composed of
Figure BDA00026623466800000511
Of orthogonal complement of, defining a matrix
Figure BDA00026623466800000512
Wherein
Figure BDA00026623466800000513
Is that
Figure BDA00026623466800000514
The orthogonal basis of (2). The matrix defined above
Figure BDA00026623466800000515
And UiOn the basis of (2), an orthogonal matrix is defined
Figure BDA00026623466800000516
In matrix EiBy decomposition, we obtain the detectable moiety x of the ith sensoridAnd a non-detectable moiety xiuNamely:
Figure BDA00026623466800000517
wherein,
Figure BDA00026623466800000518
is a matrix obtained by decomposition, and (C)id，Aid) Is detectable. It should be noted that in the introduction of the technical method of the present invention, we use 0 to denote a constant zero and a zero matrix of corresponding dimension, and the specific meaning is determined by the specific case. In addition, InAn n-dimensional identity matrix is represented.
(12) For detectable moiety x of the ith sensoridPerforming nonsingular decomposition to obtain xidMeasurement output information y iniI.e. presence of non-singular matrices
Figure BDA00026623466800000519
Such that:
Figure BDA00026623466800000520
wherein,
Figure BDA00026623466800000521
and has:
Figure BDA0002662346680000061
wherein,
Figure BDA0002662346680000062
and S2, designing a distributed dimension reduction observer for each sensor.
A sensor network containing N sensors is abstracted into a graph, wherein each node in the graph represents one sensor, and therefore a strongly connected directed graph containing N nodes is obtained
Figure BDA0002662346680000063
We define its Laplace matrix as
Figure BDA0002662346680000064
Find a vector, make it full theta ═ theta1，θ2，...，θN]TFoot thetai＞0(i＝1，...，N)，
Figure BDA0002662346680000065
At thetaiDefining an N-dimensional diagonal matrix theta for diagonal elements, and defining a new symmetric matrix
Figure BDA0002662346680000066
Selecting matrix KiSo that
Figure BDA0002662346680000067
Is of Hurwitz.
Subsystem xidDesigning a dimensionality reduction observer:
Figure BDA0002662346680000068
x is thenidIs observed in
Figure BDA0002662346680000069
Wherein,
Figure BDA00026623466800000610
and,
Figure BDA00026623466800000611
subsystem x with the help of a coherence protocol of a multi-agent systemiuDesigning an observer:
Figure BDA00026623466800000612
wherein, γi> 0 is coupling gain, and the use of the coherency protocol in the algorithm is primarily to compensate for subsystem xiuIs introduced without being detectable.
Designing a dimension reduction observer of the ith sensor:
Figure BDA00026623466800000613
wherein
Figure BDA00026623466800000614
The state of the ith sensor is estimated as:
Figure BDA00026623466800000615
wherein,
Figure BDA0002662346680000071
and S3, giving sufficient conditions containing topology information to ensure the existence of the distributed dimension reduction observer.
According to the Lyapunov stability theory, obtaining the value range of the coupling gain:
Figure BDA0002662346680000072
wherein the matrix U is UiIs a diagonal matrix of diagonal elements, InIs an n-dimensional unit matrix, | | | | | - | represents a two-norm,
Figure BDA0002662346680000073
denotes the kronecker product, lambdamin(. cndot.) represents the minimum eigenvalue. Here gamma isiIs chosen so that the estimates for each sensor can converge to the state values of the LTI target system.
And S4, designing a fully-distributed dimensionality reduction observer by using an adaptive strategy on the basis of the S3.
Designing a fully distributed dimensionality reduction observer of the ith sensor based on an adaptive strategy:
Figure BDA0002662346680000074
wherein, γi(0) Is greater than 0, and
Figure BDA0002662346680000075
the status of the ith sensor is estimated as
Figure BDA0002662346680000076
According to the Lyapunov stability theory, the estimated value of each sensor can converge to the state value of the LTI target system.
In the following, we will further illustrate the method of use of the present invention with reference to specific examples. For the target system given in fig. 2, the system matrix is:
Figure BDA0002662346680000077
where k and J are the torsional stiffness and moment of inertia. In this example, we will use 3 sensors to estimate the state of the inertia system, wherein the output information obtained by the three sensors is determined by the following output matrix:
C1＝[1 0 -1 0 0 0]，
C2＝[0 0 1 0 0 0]，
C3＝[0 0 1 0 -1 0].
while the distributed communication topology among the three sensors is shown in fig. 3. The details of the corresponding steps are as follows:
to (C)iAnd a) (i ═ 1, 2, 3) is detectably decomposed to yield:
Figure BDA0002662346680000081
Figure BDA0002662346680000082
thus, there are:
Figure BDA0002662346680000083
Figure BDA0002662346680000084
and A is3d＝A1d，A3r＝A1r，A3u＝A1u，C3d＝C1d。
Selecting
Figure BDA0002662346680000085
Figure BDA0002662346680000086
The information interaction topology among the three sensors is shown in fig. 3.
Vector taking
Figure BDA0002662346680000091
Definition matrix
Figure BDA0002662346680000092
Selecting
Figure BDA0002662346680000093
K3＝K1。
The constant gain is selected as gamma1＝γ2＝γ3Fig. 4 shows the estimation of the target state by the distributed dimensionality reduction observer at a constant coupling gain for this example, where the state trajectories of the target system are marked with an "x" in the figure, and the state trajectories of the 3 distributed dimensionality reduction observers are marked with a solid line. As can be seen from fig. 4, all 3 distributed dimension reduction observations can achieve an estimation of the state of the target system.
In the design of the distributed dimensionality reduction observer under the adaptive coupling gain strategy, the initial state values of the adaptive coupling gains of 3 observers are respectively selected as gamma1(0)＝0.1，γ2(0)＝0.2，γ3(0) 0.3. Fig. 5 shows the estimation of the target state by the distributed dimensionality reduction observer under the adaptive coupling gain of the example, wherein the state tracks of the target system are marked by an x in the figure, and the state tracks of the 3 distributed dimensionality reduction observers are marked by a solid line. From FIG. 5It can be seen that 3 distributed dimension reduction observations can achieve the estimation of the state of the target system. The locus of the adaptive coupling gain parameters is illustrated in fig. 6, and it can be seen from fig. 6 that the adaptive coupling gain parameters of each eventually converge to a constant, and that the constants may be different.

Claims (5)

1. A design method of a distributed dimensionality reduction observer of a linear time invariant system is characterized by comprising the following steps of:
(1) determining an output matrix of each sensor, and performing detectable decomposition on a matrix pair formed by the output matrix and a system matrix of a target LTI system;
(2) designing a distributed dimensionality reduction observer for each sensor;
(3) giving sufficient conditions containing topological information to ensure the existence of the distributed dimensionality reduction observer;
(4) and designing a fully distributed dimensionality reduction observer by using a self-adaptive strategy.
2. The method of designing a distributed dimension reduction observer for a linear time invariant system according to claim 1, wherein said step (1) comprises the steps of:
(11) performing detectable decomposition on each sensor to obtain a detectable subsystem and a non-detectable subsystem of each sensor:
Figure FDA0002662346670000011
yi(t)＝Cix(t),
wherein,
Figure FDA0002662346670000012
it is the state of the system that is,
Figure FDA0002662346670000013
is the measurement output, piIs an output matrixCiIs of and satisfies
Figure FDA0002662346670000014
By using
Figure FDA0002662346670000015
A matrix with dimension m × n in euclidean space; according to the detectability decomposition of the system, obtaining (C)iA non-detectable subspace u of A)iAnd u isiHas a dimension of vi(ii) a Definition matrix
Figure FDA0002662346670000016
Wherein U isiIs u columniAn orthogonal base of (a); note the book
Figure FDA0002662346670000017
Is uiOf orthogonal complement of, defining a matrix
Figure FDA0002662346670000018
Wherein
Figure FDA0002662346670000019
Is that
Figure FDA00026623466700000110
An orthogonal base of (a); the matrix defined above
Figure FDA00026623466700000111
And UiOn the basis of (2), an orthogonal matrix is defined
Figure FDA00026623466700000112
In matrix EiObtaining a detectable moiety x of the ith sensoridAnd a non-detectable moiety xiuNamely:
Figure FDA00026623466700000113
wherein
Figure FDA00026623466700000114
Figure FDA00026623466700000115
Is a matrix obtained by decomposition, and (C)id，Aid) Is detectable; i isnAn identity matrix representing n dimensions;
(12) for detectable moiety x of the ith sensoridPerforming nonsingular decomposition to obtain xidMeasurement output information y iniI.e. presence of non-singular matrices
Figure FDA00026623466700000116
Such that:
Figure FDA00026623466700000117
wherein,
Figure FDA00026623466700000118
and has:
Figure FDA0002662346670000021
wherein,
Figure FDA0002662346670000022
3. the method of designing a distributed dimension reduction observer for a linear time invariant system according to claim 1, wherein said step (2) comprises the steps of:
(21) for a strongly connected directed graph containing N nodes
Figure FDA0002662346670000023
Defining its Laplace matrix as
Figure FDA0002662346670000024
(22) Find a vector θ ═ θ1，θ2，...，θN]TSo that it satisfies thetai＞0(i＝1，...，N)，
Figure FDA0002662346670000025
(23) At thetaiDefining an N-dimensional diagonal matrix theta for diagonal elements, and defining a new symmetric matrix
Figure FDA0002662346670000026
(24) Selecting matrix KiSo that
Figure FDA00026623466700000215
Is of Hurwitz;
(25) subsystem xidDesigning a dimensionality reduction observer:
Figure FDA0002662346670000027
wherein,
Figure FDA0002662346670000028
is subsystem xid2Is observed, then xidThe observation state of (a) is:
Figure FDA0002662346670000029
wherein,
Figure FDA00026623466700000210
and is
Figure FDA00026623466700000211
(26) Subsystem x with the help of a coherence protocol of a multi-agent systemiuThe following observer was designed:
Figure FDA00026623466700000212
wherein, γi> 0 is coupling gain, and the use of the coherency protocol in the algorithm is primarily to compensate for subsystem xiuIs not detectable;
(27) designing a dimension reduction observer of the ith sensor:
Figure FDA00026623466700000213
wherein
Figure FDA00026623466700000214
(28) The state of the ith sensor is estimated as:
Figure FDA0002662346670000031
wherein,
Figure FDA0002662346670000032
4. the method for designing the distributed dimension reduction observer of the linear time-invariant system according to claim 1, wherein the step (3) is implemented as follows:
according to the Lyapunov stability theory, obtaining the value range of the coupling gain:
Figure FDA0002662346670000033
wherein the matrix U is UiIs a diagonal matrix of diagonal elements, | | | | - | represents a two-norm,
Figure FDA0002662346670000034
denotes the kronecker product, lambdamin(. cndot.) represents a minimum eigenvalue; here gamma isiIs selected so that the estimate of each sensor can converge to the state value of the LTI target system.
5. The method of designing a distributed dimension reduction observer for a linear time invariant system according to claim 1, wherein said step (4) comprises the steps of:
(41) designing a fully distributed dimensionality reduction observer of the ith sensor based on an adaptive strategy:
Figure FDA0002662346670000035
wherein, γi(0) Is greater than 0, and
Figure FDA0002662346670000036
(42) the status of the ith sensor is estimated as
Figure FDA0002662346670000037
According to the Lyapunov stability theory, the estimated value of each sensor can converge to the state value of the LTI target system.
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CN112947084A (en) * 2021-02-08 2021-06-11 重庆大学 Model unknown multi-agent consistency control method based on reinforcement learning

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CN112947084A (en) * 2021-02-08 2021-06-11 重庆大学 Model unknown multi-agent consistency control method based on reinforcement learning
CN112947084B (en) * 2021-02-08 2022-09-23 重庆大学 Model unknown multi-agent consistency control method based on reinforcement learning

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