CN108828949B - Distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming - Google Patents

Distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming Download PDF

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CN108828949B
CN108828949B CN201810799985.7A CN201810799985A CN108828949B CN 108828949 B CN108828949 B CN 108828949B CN 201810799985 A CN201810799985 A CN 201810799985A CN 108828949 B CN108828949 B CN 108828949B
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CN108828949A (en
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戴姣
刘春生
孙景亮
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Nanjing University of Aeronautics and Astronautics
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

The invention discloses a distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming, which comprises the following steps of firstly, constructing a communication topology of a multi-agent system through communication links among agents and expressing the communication topology by a directed graph G (V, E and A); secondly, establishing a local field consistency error equation, and defining an intelligent agent v based on an optimal control theory and a minimum value principleiHas a fault-free cooperative control input quantity of uiObtaining a distributed optimal cooperative control law; then, executing a distributed optimal cooperative control law; and finally, designing a distributed optimal cooperative fault-tolerant control law of the intelligent agent based on fault compensation. The method can overcome the defects of the fault-tolerant control method of the conventional nonlinear multi-agent system, and has a good application prospect in the fault-tolerant control of unmanned aerial vehicle formation.

Description

Distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming
Technical Field
The invention relates to the field of fault-tolerant control of a multi-agent system, in particular to a distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming.
Background
With the rapid development of science and technology, in recent years, multi-agent systems have been greatly popular in various fields such as biological field, physical field, control field, and computer field due to their unique advantages. When the multi-agent system is actually operated, faults are easy to occur, and the multi-agent system is mainly divided into two types: communication failures and actuator failures. At present, many expert scholars have studied on fault-tolerant control of multiple intelligent agents, but most research results aim at communication faults occurring in a multi-intelligent-agent system, and rarely relate to actuator faults of a single intelligent agent. Therefore, how to realize reconfiguration control and fault management under the condition of considering coordination with other agents and self fault and damage is an important problem in the design of a multi-agent control system.
In addition, most of the existing fault-tolerant control achievements aiming at actuator faults in the multi-agent system are based on a linear system, and the problem of optimality is rarely considered while cooperative fault-tolerant control is realized. Therefore, it is important to develop a distributed optimal cooperative fault-tolerant control method for a nonlinear multi-agent system. The premise of designing the nonlinear optimal cooperative fault-tolerant control law is to solve a nonlinear Hamilton-Jacobi-Bellman (HJB) equation, however, since the HJB equation is a nonlinear partial differential equation in nature, it is difficult or even impossible to obtain an analytic solution thereof. Therefore, how to efficiently solve the HJB equation becomes a critical problem for designing a distributed optimal cooperative fault-tolerant control law. The self-adaptive dynamic programming technology utilizes a neural network approximation method to approximate a performance index function, is widely applied to solving the problem of nonlinear optimization in recent years, and has wide application prospect.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming, overcomes the defects of the fault-tolerant control method of the existing nonlinear multi-agent system, and has a good application prospect in the fault-tolerant control of unmanned aerial vehicle formation.
The technical scheme is as follows: the invention relates to a distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming, which comprises the following steps of:
(1) constructing a communication topology of the multi-agent system through communication links among agents based on graph theory;
(2) establishing a local field consistency error equation based on a consistency theory;
(3) deducing a distributed optimal cooperative control law under the condition of no fault;
(4) executing a distributed optimal cooperative control law;
(5) and deducing a distributed optimal cooperative fault-tolerant control law.
The step (1) comprises the following steps:
(11) the communication topology of the multi-agent system is represented by a directed graph:
G＝(V,E,A)
wherein V ═ { V ═ V0,v1,v2,...vNDenotes all agents, v0Representing leader node, viRepresents the ith following node, i 1.. N, E { (v)i,vj):vi,vjE.g. V represents the set of communication links between following nodes, element (V) in Ei,vj) Representative node vjCan directly obtain the node viN, a weighted adjacency matrix, with the transmitted information i, j ═ 1
Figure BDA0001736867050000021
If (v)i,vj) E is E, then aij1, otherwise, aij＝0；
(12) Defining a laplacian matrix:
Figure BDA0001736867050000022
wherein lijA matrix element representing the ith row and the jth column, wherein L is expressed as D-A,
Figure BDA0001736867050000023
as an in-degree matrix, the matrix elements
Figure BDA0001736867050000024
Is a node viThe degree of entry of (c).
The step (2) comprises the following steps:
(21) the following node model in the nonlinear multi-agent system is described with affine nonlinear dynamics as follows:
Figure BDA0001736867050000025
wherein the content of the first and second substances,
Figure BDA0001736867050000026
representing an agent viIs determined by the state vector of (a),
Figure BDA0001736867050000027
denotes xi(t) a first derivative with respect to time,
Figure BDA0001736867050000028
representing an agent viThe control input vector of (a) is,
Figure BDA0001736867050000029
respectively being agents viThe system state function and the input function of (c),
Figure BDA00017368670500000210
representing an agent viIs detected to be in a fault with the unknown actuator,
Figure BDA00017368670500000211
representing a column vector, the superscript n representing the dimension,
Figure BDA00017368670500000212
representing an n × m dimensional matrix;
(22) defining Agents viHas a local domain coherence error of ei：
Figure BDA0001736867050000031
Deriving formula (2) over time to yield:
Figure BDA0001736867050000032
wherein the content of the first and second substances,
Figure BDA0001736867050000033
representative of a consistency error eiThe first derivative with respect to time is,
Figure BDA0001736867050000034
the distributed optimal cooperative control law in the step (3) is as follows:
Figure BDA0001736867050000035
wherein, superscript denotes the optimal value of the variable, superscript-1 denotes the inversion operation, superscript T denotes the transposition of the matrix, R denotes the number of the inverse operationsii>0 is a preset positive definite symmetric matrix,
Figure BDA0001736867050000036
function representing performance indicator Ji(ei) For consistency error eiPartial derivatives of (a).
The step (4) comprises the following steps:
(41) designing and evaluating network approximate intelligent agent v according to neural network approximation methodiOf the optimum performance indicator function
Figure BDA0001736867050000037
Figure BDA0001736867050000038
Wherein the content of the first and second substances,
Figure BDA0001736867050000039
to represent
Figure BDA00017368670500000310
In the form of an approximation of (a),
Figure BDA00017368670500000311
to evaluate the network approximation weight vector, σi(ei) Activating a function vector for evaluating the network;
(42) obtaining an agent v based on the above formulaiApproximately distributed optimal cooperative control law of
Figure BDA00017368670500000312
Comprises the following steps:
Figure BDA00017368670500000313
wherein the content of the first and second substances,
Figure BDA00017368670500000314
for activating a function sigmai(ei) For error state eiThe partial derivatives of (a) are, i.e.,
Figure BDA00017368670500000315
design of
Figure BDA00017368670500000316
The update law is as follows:
Figure BDA00017368670500000317
wherein the content of the first and second substances,
Figure BDA00017368670500000318
to represent
Figure BDA00017368670500000319
Derivative with respect to time, λiRepresents the learning rate of weight, F1iAnd F2iWhich is representative of the design parameters of the device,
Figure BDA00017368670500000320
Figure BDA0001736867050000041
neural network output error
Figure BDA0001736867050000042
The optimal cooperative fault-tolerant control law in the step (5) is as follows:
Figure BDA0001736867050000043
wherein the content of the first and second substances,
Figure BDA0001736867050000044
for fault compensation, i ═ 1.. N, the designed fault compensation update rate is:
Figure BDA0001736867050000045
wherein the content of the first and second substances,
Figure BDA0001736867050000046
for compensating for faults
Figure BDA0001736867050000047
Derivative with respect to time, beta represents the fault-compensated learning rate
Has the advantages that: compared with the prior art, the invention has the beneficial effects that: 1. the invention considers the fault-tolerant control problem of the nonlinear multi-agent system under the condition that the actuator fails, and based on the self-adaptive dynamic programming, the designed distributed optimal cooperative fault-tolerant control scheme not only can enable each agent to follow the leader node under the condition that the agent fails, but also ensures the minimization of respective performance indexes, effectively solves the problem of solving the nonlinear HJB equation, and realizes the on-line learning of the control law; 2. the method has good practical significance and application prospect in fault-tolerant control of unmanned aerial vehicle formation.
Drawings
FIG. 1 is a flow chart of the present invention.
Detailed Description
The present invention is described in further detail below with reference to the attached drawing figures.
FIG. 1 is a flow chart of the present invention, which mainly comprises the following steps:
step 1: establishing communication topology of multi-agent system by using relevant theoretical knowledge of graph theory
Consider a multi-agent system with a leading agent and N following agents, where a directed graph G ═ V, E, a is used to represent the communication topology in the system. Wherein V ═ { V ═ V0,v1,v2,...vNDenotes the set of all agents, v0Representing leader node, viRepresents the ith following node, i ═ 1.. N; e { (v)i,vj):vi,vjE.g. V represents the set of communication links between following nodes, element (V) in Ei,vj) Representative node vjCan directly obtain the node viInformation transferred, i, j ═ 1.. N; weighted adjacency matrix
Figure BDA0001736867050000048
Figure BDA0001736867050000049
Dimension N × N, if (v)i,vj) E is E, then the element a in Aij1, i.e. if node vjCan directly obtain the node viThe information to be transmitted is then aij1, otherwise, aij＝0。
In addition, define Ni＝{j∈V:(vi,vj) E is as node viRepresents a node viCan obtain all belongings to NiInformation of the node of (a); defining leader node adjacency matrix B ═ diag { B }1,b2,...bnWhere diag denotes the diagonal matrix, matrix element bi1 stands for node viCan directly obtain the information transmitted by the leader node, otherwise bi＝0。
Defining a Laplace matrix
Figure BDA0001736867050000051
Wherein lijRepresenting the elements of matrix L at row i and column j. The expression of L is L ═ D-a, where,
Figure BDA0001736867050000052
is an in-degree matrixElements of a matrix
Figure BDA0001736867050000053
Is a node viIn degree of (v), representing node viThe number of adjacent nodes.
Step 2: based on consistency theory, establishing a local field consistency error equation
The model of the following node in a multi-agent system is described by the following affine nonlinear dynamics:
Figure BDA0001736867050000054
wherein the content of the first and second substances,
Figure BDA0001736867050000055
representing an agent viIs determined by the state vector of (a),
Figure BDA0001736867050000056
denotes xi(t) first derivative with respect to time
Figure BDA0001736867050000057
Representing an n-dimensional column vector, superscript n representing node viThe number of the state quantities of (2),
Figure BDA0001736867050000058
representing an agent viThe control input vector of (a) is,
Figure BDA0001736867050000059
respectively being agents viBoth being continuous functions and having fi(0)＝0，
Figure BDA00017368670500000510
Representing a matrix of dimensions n x m,
Figure BDA00017368670500000511
representing an agent viIs detected by the sensor.
The leader node signals
Figure BDA00017368670500000512
And is
Figure BDA00017368670500000513
Is continuous, where r (t) represents the state vector of the leader node,
Figure BDA00017368670500000514
the first derivative of r (t) with respect to time is indicated.
Defining a node viHas a local domain coherence error of eiThe specific expression is as follows:
Figure BDA00017368670500000515
deriving formula (2) over time to yield:
Figure BDA00017368670500000516
wherein the content of the first and second substances,
Figure BDA00017368670500000517
representative of a consistency error eiThe first derivative with respect to time is,
Figure BDA00017368670500000518
and step 3: derivation of distributed optimal cooperative control law under fault-free condition
In the event that no actuator failure has occurred,
Figure BDA0001736867050000061
the expression of (a) is as follows:
Figure BDA0001736867050000062
defining a node viPerformance index function J ofi(ei) Comprises the following steps:
Figure BDA0001736867050000063
wherein Q isi(ei) A representation of ≧ 0 and an error state eiA semi-positive definite function of correlation. Rii>0,Rij>0 is a predetermined positive definite symmetric matrix uiFor an agent viThe superscript T represents transposing the matrix.
Define the Hamilton function as:
Figure BDA0001736867050000064
wherein the content of the first and second substances,
Figure BDA0001736867050000065
function J representing performance indicatorsi(ei) For consistency error eiPartial derivatives of, i.e.
Figure BDA0001736867050000066
Obtaining an agent v according to the principle of minimumsiThe distributed optimal cooperative control law is as follows:
Figure BDA0001736867050000067
the upper scale indicates the optimum value of the variable (same below), and the upper scale-1 indicates the inversion operation (same below).
Will be in the above formula
Figure BDA0001736867050000068
And (5) is substituted, simple operation is carried out, and then an HJB equation is obtained:
Figure BDA0001736867050000069
wherein the content of the first and second substances,
Figure BDA00017368670500000610
therefore, as long as the nonlinear HJB equation can be solved, the distributed optimal cooperative control law can be obtained. In fact, however, it is difficult or even impossible to obtain an analytical solution for the non-linear HJB equation. Therefore, the nonlinear HJB equation is approximately solved by adopting a self-adaptive dynamic programming method.
And 4, step 4: performing a distributed optimal cooperative control law
According to the neural network approximation method, the invention designs a single-layer evaluation network to approximate a single intelligent agent viOptimal performance indicator function
Figure BDA00017368670500000611
Its ideal approximation can be expressed as:
Figure BDA0001736867050000071
wherein, WciTo evaluate the ideal weight vector, σ, of the networki(ei) For evaluating the network activation function, ∈ci(ei) Representing the approximation error.
Due to the ideal weight vector WciIs often unknown and therefore is represented in a practical approximate way, of the form:
Figure BDA0001736867050000072
wherein the content of the first and second substances,
Figure BDA0001736867050000073
represents
Figure BDA0001736867050000074
In the form of an approximation of (a),
Figure BDA0001736867050000075
approximate weight vectors for the evaluation network. Thus, the network weight error is evaluated as
Figure BDA0001736867050000076
Based on the above formula, an approximate distributed optimal cooperative control law can be obtained
Figure BDA0001736867050000077
Comprises the following steps:
Figure BDA0001736867050000078
wherein the content of the first and second substances,
Figure BDA0001736867050000079
for activating a function sigmai(ei) For error state eiThe partial derivatives of (a) are, i.e.,
Figure BDA00017368670500000710
by combining equation (7) and equation (10), the output error of the evaluation network can be obtained as:
Figure BDA00017368670500000711
wherein the content of the first and second substances,
Figure BDA00017368670500000712
therefore, it is necessary to design and evaluate the neural network weight update rate so that the error of the evaluation network weight is
Figure BDA00017368670500000713
Approaching 0, i.e. obtaining an error function
Figure BDA00017368670500000714
And (4) minimizing.
Comprehensively considering the stability of a closed loop system, designing based on a gradient descent method
Figure BDA00017368670500000715
The update law is as follows:
Figure BDA00017368670500000716
wherein the content of the first and second substances,
Figure BDA00017368670500000717
to represent
Figure BDA00017368670500000718
Derivative with respect to time. Lambda [ alpha ]iRepresenting the weight learning rate. F1iAnd F2iRepresenting design parameters.
Figure BDA00017368670500000719
And 5: derivation of distributed optimal cooperative fault-tolerant control law
The distributed optimal cooperative control method is only suitable for a fault-free system, the stability of a closed-loop system is comprehensively considered, and a distributed optimal cooperative fault-tolerant control law U is designed based on a fault compensation methodiThe following were used:
Figure BDA0001736867050000081
wherein the content of the first and second substances,
Figure BDA0001736867050000082
for fault compensation, i is 1.
To compensate for errors
Figure BDA0001736867050000083
Approaching 0, i.e. obtaining an error function
Figure BDA0001736867050000084
Minimizing, and meanwhile, designing a continuously differentiable Lyapunov function expressed as L in order to ensure the boundedness of each intelligent closed-loop system in the learning processiTo enable it to satisfy
Figure BDA0001736867050000085
The design fault compensation update rate is as follows:
Figure BDA0001736867050000086
wherein the content of the first and second substances,
Figure BDA0001736867050000087
for compensating for faults
Figure BDA0001736867050000088
The derivative over time, β, represents the fault-compensated learning rate.
At present, the invention on the aspect of multi-agent fault-tolerant control is few, and the invention which considers the system optimization while realizing the fault-tolerant control is rare. The method has important application reference value for fault-tolerant control of the unmanned aerial vehicle formation control system under the condition of actuator failure.

Claims (2)

1. A distributed optimal cooperative fault-tolerant control method based on self-adaptive dynamic programming is characterized by comprising the following steps:
(1) constructing a communication topology of the multi-agent system through communication links among agents based on graph theory;
(2) establishing a local field consistency error equation based on a consistency theory;
(3) deducing a distributed optimal cooperative control law under the condition of no fault;
(4) executing a distributed optimal cooperative control law;
(5) deducing a distributed optimal cooperative fault-tolerant control law;
the step (2) comprises the following steps:
(21) the following node model in the nonlinear multi-agent system is described with affine nonlinear dynamics as follows:
Figure FDA0003003206320000011
wherein the content of the first and second substances,
Figure FDA0003003206320000012
representing an agent viIs determined by the state vector of (a),
Figure FDA0003003206320000013
denotes xi(t) a first derivative with respect to time,
Figure FDA0003003206320000014
representing an agent viThe control input vector of (a) is,
Figure FDA0003003206320000015
respectively being agents viThe system state function and the input function of (c),
Figure FDA0003003206320000016
representing an agent viIs detected to be in a fault with the unknown actuator,
Figure FDA0003003206320000017
representing a column vector, the superscript n representing the dimension,
Figure FDA0003003206320000018
representing an n × m dimensional matrix;
(22) defining Agents viLocal area ofA sexual error of ei：
Figure FDA00030032063200000115
Deriving formula (2) over time to yield:
Figure FDA0003003206320000019
wherein the content of the first and second substances,
Figure FDA00030032063200000110
representative of a consistency error eiFirst derivative of time, matrix element
Figure FDA00030032063200000111
Is a node viThe degree of penetration of the (c) is,
Figure FDA00030032063200000112
the distributed optimal cooperative control law in the step (3) is as follows:
Figure FDA00030032063200000113
wherein, superscript denotes the optimal value of the variable, superscript-1 denotes the inversion operation, superscript T denotes the transposition of the matrix, R denotes the number of the inverse operationsii>0 is a preset positive definite symmetric matrix,
Figure FDA00030032063200000114
function representing performance indicator Ji(ei) For consistency error eiPartial derivatives of (d);
the step (4) comprises the following steps:
(41) designing and evaluating network approximate intelligent agent v according to neural network approximation methodiOf the optimum performance indicator function
Figure FDA0003003206320000021
Figure FDA0003003206320000022
Wherein the content of the first and second substances,
Figure FDA0003003206320000023
to represent
Figure FDA0003003206320000024
In the form of an approximation of (a),
Figure FDA0003003206320000025
to evaluate the network approximation weight vector, σi(ei) Activating a function vector for evaluating the network;
(42) obtaining an agent v based on the above formulaiApproximately distributed optimal cooperative control law of
Figure FDA0003003206320000026
Comprises the following steps:
Figure FDA0003003206320000027
wherein R isii>0 is a preset positive definite symmetric matrix,
Figure FDA0003003206320000028
for activating a function sigmai(ei) For error state eiThe partial derivatives of (a) are, i.e.,
Figure FDA0003003206320000029
design of
Figure FDA00030032063200000210
The update law is as follows:
Figure FDA00030032063200000211
wherein the content of the first and second substances,
Figure FDA00030032063200000212
to represent
Figure FDA00030032063200000213
Derivative with respect to time, λiRepresents the learning rate of weight, F1iAnd F2iWhich is representative of the design parameters of the device,
Figure FDA00030032063200000214
Figure FDA00030032063200000215
neural network output error
Figure FDA00030032063200000216
Figure FDA00030032063200000217
Figure FDA00030032063200000218
Qi(ei) More than or equal to 0 is related to the error e of cooperative consistencyiA semi-positive definite matrix of (a);
the optimal cooperative fault-tolerant control law in the step (5) is as follows:
Figure FDA00030032063200000219
wherein the content of the first and second substances,
Figure FDA00030032063200000220
for fault compensation, i-1,. N,the design fault compensation update rate is as follows:
Figure FDA00030032063200000221
wherein R isiiFor a predetermined positive definite symmetric matrix,
Figure FDA0003003206320000031
for compensating for faults
Figure FDA0003003206320000032
The derivative over time, β, represents the fault-compensated learning rate.
2. The distributed optimal cooperative fault-tolerant control method based on the adaptive dynamic programming as claimed in claim 1, wherein the step (1) comprises the following steps:
(11) the communication topology of the multi-agent system is represented by a directed graph:
G＝(V,E,A)
wherein V ═ { V ═ V0,v1,v2,...vNDenotes all agents, v0Representing leader node, viRepresents the ith following node, i 1.. N, E { (v)i,vj):vi,vjE.g. V represents the set of communication links between following nodes, element (V) in Ei,vj) Representative node vjCan directly obtain the node viN, a weighted adjacency matrix, with the transmitted information i, j ═ 1
Figure FDA0003003206320000033
If (v)i,vj) E is E, then aij1, otherwise, aij＝0；
(12) Defining a laplacian matrix:
Figure FDA0003003206320000034
wherein lijA matrix element representing the ith row and the jth column, wherein L is expressed as D-A,
Figure FDA0003003206320000035
as an in-degree matrix, the matrix elements
Figure FDA0003003206320000036
Is a node viThe degree of entry of (c).
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