CN113885315A - Design method of distributed observer of linear time-invariant moving target system - Google Patents

Design method of distributed observer of linear time-invariant moving target system Download PDF

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CN113885315A
CN113885315A CN202111312430.3A CN202111312430A CN113885315A CN 113885315 A CN113885315 A CN 113885315A CN 202111312430 A CN202111312430 A CN 202111312430A CN 113885315 A CN113885315 A CN 113885315A
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target system
moving target
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CN113885315B (en
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王晓玲
范真
苏厚胜
石金剑
蒋国平
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a distributed observer design method of a linear time-invariant moving target system, namely, a group of moving sensor networks are used for realizing state estimation of a linear time-invariant moving target system. The method mainly comprises the following steps: firstly, carrying out detectable decomposition on a new system consisting of partial output information and a moving target system; secondly, constructing a top-level distributed observer for each sensor to realize state estimation of the moving target system; finally, a bottom-level "leader-follower" based bee-hive move algorithm is built for each sensor to control its motion. The design method of the distributed observer provided by the invention realizes the motion control of effective tracking of the moving target system and the distributed estimation of the state of the moving target system, wherein sensor networks have no collision and are communicated with each other.

Description

Design method of distributed observer of linear time-invariant moving target system
Technical Field
The invention relates to the field of design of a distributed observer, in particular to a design method of a distributed observer of a linear time-invariant moving target system.
Background
State estimation is a traditional and important problem in control theory and applications, which occurs in systems where state variables are difficult or even impossible to measure, in which case it becomes especially important to construct an observer to estimate the state variables. The main research problem of state estimation is the stability problem of the error between the estimated value and the true state.
In the classical lunberg observer theory, one observer usually estimates the state of the system by using the complete output information of the system, however, in practical engineering applications, there are some high-dimensional output systems, and a single observer is difficult to implement or even unable to implement the measurement of the huge output information; a sensor network consisting of a plurality of sensors is easier to implement for measuring the output information of such a system than a single device using a centralized architecture. The core idea of the distributed observer is to use a sensor network composed of a plurality of sensors to realize the state estimation of a large-scale target system. In the distributed observers, each sensor only needs to measure a part of output information, the sensor transmits the measured part of output information to the corresponding observer, and then the state of the target system is effectively estimated through the distributed information interaction between the observers.
Driven by this advantage of the distributed observer, the distributed observer has been greatly developed in recent years and has emerged with many pioneering results. The existing research is directed to a static linear time-invariant system, for example, CN112052585A is a dimension reduction observer designed based on an adaptive strategy for a static target system, however, considering the phenomena of complex task environment, large area, and continuous change of information to be estimated (such as leakage and diffusion of marine crude oil), the estimation accuracy cannot be satisfied by using a static sensor; in addition, for some moving target systems, a static sensor network cannot achieve effective measurement of its output information at all.
Disclosure of Invention
In order to solve the technical problems, the invention provides a novel distributed observer design method, and constructs a distributed observer based on a 'leader-follower' bee-crowded mobile algorithm so as to realize state estimation of a mobile target system, and simultaneously ensure that no collision exists in a sensor network, the sensor network is communicated, and motion control capable of effectively tracking the mobile target system can be realized.
The invention relates to a method for designing a distributed observer of a linear time-invariant moving target system, which comprises the following steps:
step 1, under the precondition that (C, A) can be detected, utilizing a decomposition matrix UiAnd DiFor each group (C)iA) are detectably decomposed to give (C) respectivelyiThe detectable subsystem and the matrix A corresponding to the undetectable subsystem of A)id、CidAnd Aiu；
Step 2 of using (C) obtained in step 1id，Aid) Construction matrix KiTo achieve state estimation of the detectable subsystems and by introducing weighting matrices to the non-detectable subsystems
Figure BDA0003342549610000029
The state estimation of the non-detectable subsystem is realized, so that a top-level distributed observer is constructed to realize the estimation of the state of the moving target system;
and 3, utilizing the position and speed information of the target system obtained by the distributed observer in the step 2, and further designing a bee-crowded moving algorithm based on a leader-follower at the bottom layer to control the movement of the sensor.
Further, the implementation process of step 1 is as follows:
for a continuous linear time invariant moving object system:
Figure BDA0003342549610000021
and its output:
y＝Cx
wherein
Figure BDA0003342549610000022
Respectively referring to the moving object system matrix and the state,
Figure BDA0003342549610000025
is the output of a moving target system, wherein
Figure BDA0003342549610000026
A matrix having a dimension of m × N in euclidean space is represented, assuming that output information of the moving target system is measured by a network composed of N moving sensors, and an output that an ith observer can measure is; by superimposing all local measurements yiAll measurements of all sensors are available, that is:
the distributed observer mainly measures the output of the moving target system through a group of sensors, and each sensor obtains partial measurement information yiAnd exchanging information with adjacent sensors through the topological structure so as to complete the state estimation of the whole moving target system.
Suppose (C, A) is detectable, but each group of (C)iA) is not necessarily detectable, so it is first necessary to detect each group (C)iA) performing the following detectability decomposition:
definition fA(s)＝det(sInA) is a characteristic polynomial of the matrix A, and
Figure BDA0003342549610000023
wherein
Figure BDA0003342549610000027
And
Figure BDA0003342549610000028
respectively, a polynomial rooted in the right closed half-plane and the left open half-plane of the complex plane, then each group (C)iThe undetectable subspace in A) is defined as
Figure BDA0003342549610000031
Wherein
Figure BDA0003342549610000037
Is the kernel space of matrix A; suppose that
Figure BDA0003342549610000038
Has a dimension of vi，
Figure BDA0003342549610000039
Defining a matrix for its orthogonal complement
Figure BDA00033425496100000321
Wherein
Figure BDA00033425496100000311
And
Figure BDA00033425496100000312
are respectively
Figure BDA00033425496100000313
And
Figure BDA00033425496100000314
and satisfy the orthogonal base of
Figure BDA00033425496100000322
Wherein im (U)i) Representation matrix UiA core of (a); in the matrix TiUnder the action of (A), the following steps are carried out:
Figure BDA0003342549610000032
wherein
Figure BDA0003342549610000033
And each pair (C)id，Aid) Are detectable.
Further, the implementation process of step 2 is as follows:
definition of
Figure BDA00033425496100000316
For the i-th sensor to estimate the moving target system state x, then the estimated state of the i-th observer will be updated according to the following dynamics:
Figure BDA0003342549610000034
wherein
Figure BDA0003342549610000035
And KidIs selected such that Aid+KidCidIs of Hurwitz, wij(t) is the element at the ith row and jth column position of the system adjacency matrix, and if there is (i, j) e (t), then there is wij(t) 1, otherwise wij(t)＝0；γiA steady coupling gain designed to satisfy the following condition:
Figure BDA0003342549610000036
wherein gamma is00 is any constant, L (0) is the topological relation between sensors at the initial moment, lambdaminAnd (M) is the minimum eigenvalue of the symmetric matrix M.
Further, the implementation process of step 3 is as follows:
for the system
Figure BDA00033425496100000324
Status of state
Figure BDA00033425496100000317
Contains the position and speed information of the target system, which determines the mobility of the target system; position and speed information defining a moving target system are respectively
Figure BDA00033425496100000318
And
Figure BDA00033425496100000323
and 2m is less than n, without loss of generality, the first two vector elements of the state x are respectively x*And xo. Definition map
Figure BDA00033425496100000320
To describe the topological relationship of information interaction among the N sensors, where V { (1, 2.,. N } represents a set of points, and epsilon (t) { (i, j) | i, j ∈ V } is a set of time-varying edges, representing an unordered vertex pair of undirected adjacency relationships among the sensors, the algorithm is designed based on a "leader-follower" bee-crowded moving algorithm as follows:
step 3-1, designing a motion equation of the ith sensor:
Figure BDA0003342549610000041
Figure BDA0003342549610000045
and
Figure BDA0003342549610000046
the position and velocity, u, of the ith sensor, respectivelyiIs a control input to the sensor. Each sensor can only obtain the estimated state value of the corresponding observer
Figure BDA00033425496100000413
Namely can obtain
Figure BDA0003342549610000048
The first two elements
Figure BDA0003342549610000049
And
Figure BDA00033425496100000414
step 3-2, designing the control input u of the ith sensoriComprises the following steps:
Figure BDA0003342549610000042
wherein
Figure BDA00033425496100000411
Is the neighbor set of the ith sensor, | | | · | |, is a 2-norm,
Figure BDA00033425496100000412
for achieving collision avoidance between control sensors, in which the energy function Ψ (| | q)i-qj| is defined as
Figure BDA0003342549610000043
Wherein R > 0 is the influence or sensing radius of all sensors;
and 3-3, constraining the change of the edge set epsilon (t) among the sensors:
i.e. the distance between two sensors i and j q at a certain momenti-qjIf | is smaller than the sensing radius R, then (i, j) belongs to epsilon (t), otherwise,
Figure BDA0003342549610000044
under the control of the algorithm, the topological relation among the sensors is not changed, and finally the state estimation of the moving target system can be realized.
The invention has the beneficial effects that: the invention provides a design method of a linear time-invariant distributed observer of a moving target system under the conditions of joint detectability of local measurement values and connectivity of an initial topology associated with a sensor; in the distributed observer design, each sensor is controlled by two sets of dynamics: one is a distributed state estimation algorithm based on consistency, and the other is a bee-holding mobile cluster algorithm for guiding the movement of the sensor; the invention successfully applies the bee-hive moving algorithm to the construction of the distributed moving observer, realizes the state estimation of the target system, and simultaneously realizes the purposes of tracking motion of the sensor to the moving target system, no collision in the sensor, connectivity maintenance of a topological structure and the like.
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In order that the present invention may be more readily and clearly understood, a more particular description of the invention briefly described above will be rendered by reference to specific embodiments that are illustrated in the appended drawings.
FIG. 1 is a schematic view of
Figure BDA0003342549610000053
A schematic diagram of the trajectory of (a);
FIG. 2 is a schematic diagram of a cluster of mobile sensors;
FIG. 3 is a velocity trace diagram for all sensors;
max in FIG. 4(i，j)∈ε(t)||qi-qj||，min(i，j)∈ε(t)||qi-qjA schematic diagram of the trace, |;
FIG. 5 is x*，qiSchematic diagram of the trajectory of (1).
Detailed Description
The invention relates to a method for designing a distributed moving observer of a linear time-invariant moving target system, which comprises the following steps:
step 1, under the precondition that (C, A) can be detected, utilizing a decomposition matrix UiAnd DiFor each group (C)iA) performing a detectability decomposition to obtain a matrix A of corresponding detectable and non-detectable moieties, respectivelyid、Cid、Aiu；
Step 2, based on the steady coupling gain, (C) obtained in step 1 is used respectivelyid，Aid) Construction matrix KiStabilization of detectable moieties and weighting matrices
Figure BDA0003342549610000054
Controlling the convergence of the error of the undetectable part, thereby constructing a top-level fully distributed observer to realize the complete estimation of the state of the moving target system;
and 3, controlling the movement of the sensor by utilizing the position and speed information of the moving target system obtained by the observer in the step 2 and designing a bottom layer based on a bee-crowded moving algorithm of a leader-follower.
The implementation process of the step 1 is as follows:
for a continuous linear time invariant moving object system:
Figure BDA0003342549610000051
and its output:
y＝Cx
wherein
Figure BDA0003342549610000052
Respectively to the system matrix and the system state,
Figure BDA0003342549610000055
is the output of a moving target system, wherein
Figure BDA0003342549610000064
Representing a matrix with dimension m × N in euclidean space, assuming that the output information of the moving target system is measured by a network of N moving sensors, and the output that the i-th observer can measure is
Figure BDA00033425496100000619
By superimposing all local measurements yiAll measurements of all sensors are available, that is:
the distributed observer mainly measures the output of the moving target system through a group of sensors, and each sensor obtains partial measurement information yiAnd exchanging information with adjacent sensors through the topological structure so as to complete the state estimation of the whole moving target system.
Suppose (C, A) is detectable, but each group of (C)iA) is not necessarily detectableMeasured, therefore, it is first necessary to perform for each group (C)iA) performing the following detectability decomposition:
definition fA(s)＝det(sInA) is a characteristic polynomial of the matrix A, and
Figure BDA00033425496100000620
wherein
Figure BDA0003342549610000067
And
Figure BDA0003342549610000068
respectively, a polynomial rooted in the right closed half-plane and the left open half-plane of the complex plane, then each group (C)iThe undetectable subspace in A) is defined as
Figure BDA0003342549610000061
Wherein
Figure BDA0003342549610000069
Is the kernel space of matrix A; suppose that
Figure BDA00033425496100000610
Has a dimension of vi，
Figure BDA00033425496100000611
Defining a matrix for its orthogonal complement
Figure BDA00033425496100000621
Wherein
Figure BDA00033425496100000613
And
Figure BDA00033425496100000614
are respectively
Figure BDA00033425496100000615
And
Figure BDA00033425496100000616
and satisfy the orthogonal base of
Figure BDA00033425496100000622
Wherein im (U)i) Representation matrix UiThe image space of (a); in the matrix TiUnder the action of (A), the following steps are carried out:
Figure BDA0003342549610000062
wherein
Figure BDA0003342549610000063
And each pair (C)id，Aid) Are detectable.
The implementation process of the step 2 is as follows:
gain gamma based on steady couplingiDefinition of
Figure BDA00033425496100000618
For the estimated state of the i-th sensor on the moving target system state x, then the estimated state of the i-th observer will be updated according to the following dynamics:
Figure BDA0003342549610000071
wherein
Figure BDA0003342549610000072
And KidIs selected such that Aid+KidCidIs of Hurwitz, wij(t) is the element at the ith row and jth column position of the system adjacency matrix, and if there is (i, j) ∈ ε (t), then there is wij(t) 1, otherwise wij(t)＝0；γiA steady coupling gain designed to satisfy the following condition:
Figure BDA0003342549610000073
wherein gamma is00 is any constant, L (0) is the topological relation between sensors at the initial moment, lambdamin(M) refers to the minimum eigenvalue of matrix M.
The implementation process of the step 3 is as follows:
for the system
Figure BDA00033425496100000717
Status of state
Figure BDA0003342549610000077
Contains the position and speed information of the moving target system, which determines the mobility of the target system; defining position and speed information of the power system as
Figure BDA0003342549610000078
And
Figure BDA00033425496100000718
and 2m is less than n, without loss of generality, the first two vector elements of the state x are respectively x*And xoDefinition of the figure
Figure BDA00033425496100000710
To describe the topological relationship of information interaction between the N sensors, V { (1, 2.,. N } represents a set of points, and epsilon (t) { (i, j) | i, j ∈ V } is a time-varying edge set, and represents an unordered vertex pair of undirected adjacency relationship between the sensors, the algorithm is designed based on the "leader-follower" bee-crowded moving algorithm as follows:
step 3-1, designing a motion equation of the ith sensor:
Figure BDA0003342549610000074
Figure BDA00033425496100000711
and
Figure BDA00033425496100000712
the position and velocity, u, of the ith sensor, respectivelyiIs a control input to the sensor. Each sensor can only obtain the estimated state value of the corresponding observer
Figure BDA00033425496100000719
Namely can obtain
Figure BDA00033425496100000714
The first two elements
Figure BDA00033425496100000715
And
Figure BDA00033425496100000720
step 3-2, designing the control input u of the ith sensoriComprises the following steps:
Figure BDA0003342549610000075
wherein
Figure BDA0003342549610000084
Is the neighbor set of the ith sensor, | | | · | | is a 2 norm, ψ (| q)i-qjI) control collision avoidance between sensors, defined as
Figure BDA0003342549610000081
Where R > 0 is the influence or sensing radius of all sensors.
Step 3-3, the change design of the edge set epsilon (t) among the sensors follows the following principle:
(1) the initial edge satisfies: epsilon (0) { (i, j) | | | | qi(0)-qj(0) L < r, i, j belongs to V }. R < R is a constant.
(2) If it is not
Figure BDA0003342549610000085
And | qi-qj | is less than R, (i, j) will be added as a new edge to ε (t).
(3) If | q |i-qjIf | is not less than R, then
Figure BDA0003342549610000086
Specifically, we can use a symmetric index function σ (i, j) (t) ∈ {0, 1} to determine whether there is an edge between sensors i and j at time t.
Figure BDA0003342549610000082
Under the control of the algorithm, the topological relation among the sensors is not changed, and finally the state estimation of the moving target system can be realized.
Selecting a moving target system matrix as follows:
Figure BDA0003342549610000083
we use a sensor network of N-8 sensors. The initial topology of the sensor network is connected, and the adjacency matrix is:
Figure BDA0003342549610000091
wherein C isi(i ═ 1.., 8.) satisfies row full rank, but (C)iA) is not detectable, but (col)i∈v(Ci) A) is detectable. In pair (C)iA) can be obtained after detectable decomposition:
C1d＝[0 0 0 1]，
C2d＝[0 0 0 0 0 0 1.7321]，
C3d＝C1d，
C4d＝C1d，
C5d＝[0 1]，
C6d＝[0 0 0 0 0 1]，
C7d＝[0 0 0 0 1.4142]，
C8d＝[0 0 1.4142].
Figure BDA0003342549610000101
Figure BDA0003342549610000102
Figure BDA0003342549610000103
Figure BDA0003342549610000104
Figure BDA0003342549610000105
Figure BDA0003342549610000106
Figure BDA0003342549610000107
Figure BDA0003342549610000108
to ensure Aid+KidCidI 1, 8 satisfies hurwitzeness, we choose to
Figure BDA0003342549610000109
Figure BDA00033425496100001010
Initial state xiAnd
Figure BDA0003342549610000111
is randomly selected from [0, 1 ]]In this example, we choose R4 and γ 110.
FIG. 1 shows
Figure BDA0003342549610000112
For the state estimation of x, verifying that each mobile sensor can reconstruct the complete state of the mobile target system;
FIG. 2 depicts the position of the mobile sensor at four different times, and it can be seen that the topological relationship of the 8 sensors remains unchanged at all times;
fig. 3 shows the velocity trace of the sensors, demonstrating that each sensor ultimately maintains the same velocity as the moving target system.
Further, as can be seen from fig. 4, the distance between any two sensors satisfies | | qi-qj||(i，j)∈ε(t)E (0, R) so that collisions between sensors are avoided.
As can be seen from fig. 5, the positions of all sensors are eventually close to the target system
Figure BDA0003342549610000113
Both of these figures verify that the moving observer works well in the design of the distributed observer.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and all equivalent variations made by using the contents of the present specification and the drawings are within the protection scope of the present invention.

Claims (4)

1. A method for designing a distributed observer of a linear time-invariant moving target system is characterized by comprising the following steps:
step 1, under the precondition that (C', A) is detectable, utilizing a decomposition matrix UiAnd DiFor each group (C)iA) are detectably decomposed to give (C) respectivelyiThe detectable subsystem and the matrix A corresponding to the undetectable subsystem of A)id、CidAnd Aiu；
Step 2 of using (C) obtained in step 1id，Aid) Construction matrix KiTo achieve state estimation of the detectable subsystems and by introducing weighting matrices to the non-detectable subsystems
Figure FDA0003342549600000011
The state estimation of the non-detectable subsystem is realized, so that a top-level distributed observer is constructed to realize the estimation of the state of the moving target system;
and 3, utilizing the position and speed information of the target system obtained by the distributed observer in the step 2, and further designing a bottom layer based on a 'leader-follower' bee-possessed movement algorithm to control the movement of the sensor.
2. The method for designing the distributed observer of the linear time-invariant moving target system according to claim 1, wherein the step 1 is implemented by:
for a continuous linear time invariant moving object system:
Figure FDA0003342549600000012
and its output:
y＝Cx
wherein
Figure FDA0003342549600000013
Respectively referring to the moving object system matrix and the state,
Figure FDA0003342549600000014
is the output of a moving target system, wherein
Figure FDA0003342549600000015
Representing a matrix with dimension m × N in euclidean space, measuring output information of the moving target system through a network of N moving sensors, and an output that an ith observer can measure is
Figure FDA0003342549600000016
By superimposing all local measurements yiAll the measured values of all the sensors can be obtained;
suppose (C, A) is detectable, but each group of (C)iA) is not necessarily detectable, so it is first necessary to detect each group (C)iA) performing the following detectability decomposition:
definition fA(s)＝det(sInA) is a characteristic polynomial of the matrix A, and
Figure FDA0003342549600000017
wherein
Figure FDA0003342549600000018
And
Figure FDA0003342549600000019
respectively, the right closed half plane and the left open half plane of the complex plane, then each group (C)iThe undetectable subspace in A) is defined as
Figure FDA0003342549600000021
Wherein
Figure FDA0003342549600000022
Is the kernel space of matrix A; suppose that
Figure FDA0003342549600000023
Has a dimension of vi，
Figure FDA0003342549600000024
Defining a matrix for its orthogonal complement
Figure FDA0003342549600000025
Wherein
Figure FDA0003342549600000026
And
Figure FDA0003342549600000027
are respectively
Figure FDA0003342549600000028
And
Figure FDA0003342549600000029
and satisfy the orthogonal base of
Figure FDA00033425496000000218
Period im (U)i) Representation matrix UiThe image space of (a); in the matrix TiUnder the action of (A), the following steps are carried out:
Figure FDA00033425496000000210
CiTi＝[Cid 0].
wherein
Figure FDA00033425496000000211
And each pair (C)id，Aid) Are detectable.
3. The method for designing the distributed observer of the linear time-invariant moving target system according to claim 1, wherein the step 2 is implemented by:
definition of
Figure FDA00033425496000000212
For the i-th sensor to estimate the moving target system state x, then the estimated state of the i-th observer will be updated according to the following dynamics:
Figure FDA00033425496000000213
wherein
Figure FDA00033425496000000214
And KidIs selected such that Aid+KidCidIs of Hurwitz, wij(t) is the element at the ith row and jth column position of the system adjacency matrix, and if there is (i, j) ∈ ε (t), then there is wij(t) 1, otherwise wij(t)＝0；γiTo satisfy the steady coupling gain of the following condition:
Figure FDA00033425496000000215
wherein gamma is00 is any constant, L (0) is the topological relation between sensors at the initial moment, lambdaminAnd (M) is the minimum eigenvalue of the symmetric matrix M.
4. The method for designing the distributed observer of the linear time-invariant moving target system according to claim 1, wherein the step 3 is implemented by:
for moving target system
Figure FDA00033425496000000216
Status of state
Figure FDA00033425496000000217
Contains the location and velocity information of the system, which determines the mobility of the target system; position and speed information defining a moving target system are respectively
Figure FDA0003342549600000031
And
Figure FDA0003342549600000032
and 2m is less than n, without loss of generality, the first two vector elements of the state x are respectively x*And x◇(ii) a Definition map
Figure FDA0003342549600000033
To describe the topological relationship of information interaction among the N sensors, where V { (1, 2.,. N } represents a set of points, and epsilon (t) { (i, j) | i, j ∈ V } is a set of time-varying edges, representing an unordered vertex pair of undirected adjacency relationships among the sensors, the algorithm is designed based on a "leader-follower" bee-crowded moving algorithm as follows:
step 3-1, designing a motion equation of the ith sensor:
Figure FDA0003342549600000034
Figure FDA0003342549600000035
and
Figure FDA0003342549600000036
the position and velocity, u, of the ith sensor, respectivelyiIs a control input to the sensor. Each sensor can only obtain the estimated state value of the corresponding observer
Figure FDA0003342549600000037
Namely can obtain
Figure FDA0003342549600000038
The first two elements
Figure FDA0003342549600000039
And
Figure FDA00033425496000000310
step 3-2, designing the control input u of the ith sensoriComprises the following steps:
Figure FDA00033425496000000311
wherein
Figure FDA00033425496000000312
Is the neighbor set of the ith sensor, | | | · | |, is a 2-norm,
Figure FDA00033425496000000313
for achieving collision avoidance between control sensors, in which the energy function Ψ (| | q)i-qj| is defined as
Figure FDA00033425496000000314
Wherein R > 0 is the influence or sensing radius of all sensors;
and 3-3, constraining the change of the edge set epsilon (t) among the sensors:
i.e. the distance between two sensors i and j q at a certain momenti-qjIf | is smaller than the sensing radius R, then (i, j) belongs to epsilon (t), otherwise,
Figure FDA00033425496000000315
under the control of the algorithm, the topological relation among the sensors is not changed, and finally the state estimation of the moving target system can be realized.
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