CN113050681A - Singular group system consistency analysis and control method - Google Patents
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Abstract
The application discloses a singular group system consistency analysis and control method, which comprises the following steps: establishing a singular system dynamics model, wherein the singular system dynamics model comprises a dynamic description formula and a control protocol formula of a group system; establishing a leaderless following structure consistency model according to a singular system dynamics model, and determining a consistent dynamic description formula and an inconsistent dynamic description formula; determining a consistency function of the leaderless follow-up structure consistency model; and carrying out consistency adjustment on the agents in the group system according to the leaderless following structure consistency model. The method determines a display expression of a consistency function of a leaderless follower structure, makes up for an LMI (local mean-average) consistency design rule comprising a nonlinear term in the prior art, determines whether each intelligent body can achieve performance guarantee consistency through inference, depends on a dynamic mode and an initial state, and is irrelevant to given energy budget.
Description
Technical Field
The application relates to the technical field of group systems, in particular to a singularity group system consistency analysis and control method.
Background
Compared with the problems of cooperative control of formation, enclosure, aggregation and the like of a multi-agent system, the consistency is the basis for realizing cooperative cooperation and completing a common formulated task of the multi-agents, and the system is widely applied to multiple fields of unmanned aircraft cooperative control, formation control, distributed sensor networks and the like in recent years. The group system can be divided into a normal system and a singular system according to dynamics. The normal system has only a dynamic model, while the singular system has three models, pulsed, static, and dynamic. When algebraic constraints exist between the coordination variables, each agent can only be modeled as a singular system, while the dynamics of the singular group system are typically high-order. The singular group system usually combines Linear Matrix Inequality (LMI) to derive the consistency design and analysis criteria for the singular group system.
In the existing group system consistency scheme, two factors of consistency adjustment performance and energy consumption are modeled into an optimal or suboptimal problem, different upper and lower energy limits are given, and energy budget is not given in advance.
Disclosure of Invention
The application provides a singular group system consistency analysis and control method, which enables the display expression of a consistency function of a leaderless follower structure to be determined, the LMI consistency design criterion including a nonlinear item in the prior art to be made up, whether the performance guarantee consistency of each intelligent body can be realized or not to be determined by inference depends on a dynamic mode and an initial state, and is irrelevant to given energy budget.
In view of the above, the present application provides a singular group system consistency analysis and control method, which includes:
establishing a singular system dynamics model, wherein the singular system dynamics model comprises a dynamic description formula and a control protocol formula of a group system;
establishing a leader-free following structure consistency model according to the singular system dynamics model, and determining a consistent dynamic description formula and an inconsistent dynamic description formula;
determining a consistency function of the leaderless follower structure consistency model;
and carrying out consistency adjustment on the agents in the group system according to the leaderless following structure consistency model.
Optionally, the establishing a singular system dynamics model includes:
the dynamic description formula of the group system is as follows:
in the formula, xi(t) and ui(t) represents the status control input of the ith agent; y isi(t) represents the state control output of the ith agent; i represents the number of agents; coefficient matrix
E is a matrix of order m,
rank(E)≤m；
the formula of the control protocol of the group system is as follows:
in the formula: 1,2, …, N, HT＝H＞0，si(t) is the protocol state with an initial value of zero; kaAnd KbIs a gain matrix; n is a radical ofiIs a neighbor set of time t i; j. the design is a squarec(t) is a consistent motion energy expenditure term;
for a given energy budget; w is aijIs the interaction weight.
Optionally, the establishing a leaderless following structure consistency model according to the singular system dynamics model, and determining a consistent dynamic description formula and an inconsistent dynamic description formula include:
transforming the singular system dynamics model into a leaderless follow-up structural consistency model by structural decomposition, the leaderless follow-up structural consistency model having a consistent dynamic formula of:
the non-uniform dynamic formula of the leaderless follow-up structural consistency model is as follows:
in the formula: let λ1≤λ2≤…≤λNLet ejJ ∈ {1,2, …, N } under an N-dimensional column vector equal to a j-dimensional element of 1 and other elements of 0.
Optionally, the determining a consistency function of the leaderless follow-up structural consistency model includes:
the consistency function c (t) of the leader-less following structure consistency model is represented as:
in which there is a non-singular matrix
And
satisfies the following conditions:
then construct the column non-singular matrix as:
according to the technical scheme, the method has the following advantages:
the application provides a singular group system consistency analysis and control method, which comprises the following steps: establishing a singular system dynamics model, wherein the singular system dynamics model comprises a dynamic description formula and a control protocol formula of a group system; establishing a leaderless following structure consistency model according to a singular system dynamics model, and determining a consistent dynamic description formula and an inconsistent dynamic description formula; determining a consistency function of the leaderless follow-up structure consistency model; and carrying out consistency adjustment on the agents in the group system according to the leaderless following structure consistency model.
The method comprises the steps of determining the influence of given energy budget on a consistency criterion and a consistency function, designing a gain matrix of a control protocol by using a two-step method (adopting a consistency dynamic description formula and an inconsistent dynamic state to solve step by step), determining a display expression of the consistency function by using a first equivalent form, and checking (a step of solving the consistency function and the energy budget) by using a generalized Riccati equation and a linear matrix inequality to make up the problem that the LMI consistency design criterion in the prior art comprises a nonlinear item.
Drawings
FIG. 1 is a flow chart of a method of one embodiment of a singular group system consistency analysis and control method of the present application;
FIG. 2 is a schematic diagram of a leaderless follow-communication topology according to an embodiment of the present application;
FIG. 3 is a diagram illustrating a state trajectory of a singular group system under a leaderless following condition in an embodiment of the present application;
fig. 4 is a schematic diagram of an energy trajectory of a singular group system under a leaderless following condition in an embodiment of the present application.
Detailed Description
According to the method, firstly, high-order linear agents are selected as research objects, and when algebraic constraints exist among coordination variables, each agent can only be modeled as a singular system. Compared with a normal system only having a dynamic mode, the singular system also comprises a pulse mode and a static mode. In this case, the pulse mode needs to be eliminated, and the state related to the static mode is also consistent.
The communication model v (g) { v ═ v) between N agents can be described in this application with reference to graph theory1,v2…,vN}. G ═ (v (G), e (G)) represents a node set, and e (G) ═ eij＝(vi,vj) Represents a set of communication relationships, eijRepresenting the communication channel between agents i and j, with a neighbor set of agent i denoted Ni＝{j:(vj,vi) E (G) }. Communication topolaplacian matrix
lji＝-wji(i≠j)，wijIs the interaction weight between agents i and j.
In order to make the technical solutions of the present application better understood, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
Fig. 1 is a flowchart of a method of an embodiment of a singular group system consistency analysis and control method according to the present application, as shown in fig. 1, where fig. 1 includes:
101. establishing a singular system dynamics model, wherein the singular system dynamics model comprises a dynamic description formula and a control protocol formula of a group system;
it should be noted that, considering a network formed by N agents in m-th order, the dynamic description formula of the group system of the ith agent is:
in the formula, xi(t)、ui(t) and yi(t) respectively representing state control inputs and outputs of the ith agent; coefficient matrix
rank (E) is less than or equal to m. The dynamic description formulation of the swarm system represents the state of the agent, i.e., the relationship between the control input signal and the output signal.
To solve the consistency problem of the group system, a control protocol formula with energy constraint can be selected as follows:
in the formula: 1,2, …, N, HT＝H＞0，si(t) is the protocol state with an initial value of zero; kaAnd KbIs a gain matrix; n is a radical ofiI is a neighbor set of (time t); j. the design is a squarec(t) is a consistent motion energy expenditure term;
for a given energy budget; w is aijIs the interaction weight.
102. Establishing a leaderless following structure consistency model according to the singular system dynamics model;
it should be noted that, since the present application mainly solves the problem of consistency of the group system: because the dynamic description formula of the singular system dynamic model constructed by the singular matrix E and the control protocol formula of dynamic output feedback both have singularity, and the control protocol formula is formed by the protocol state difference between adjacent intelligent agents and meets the separation theorem, the gain matrix K is enabled to beaAnd KbCan be designed independently.
In one specific embodiment, the singular system dynamics model is converted into a leader-free following structure consistency model through structure decomposition, and a consistent dynamic formula of the leader-free following structure consistency model is as follows:
the non-uniform dynamic formula of the leaderless following structural consistency model is:
in the formula: let λ be less than or equal to λ3≤…≤λNLet ej(j e {1,2, …, N }) is equal to an N-dimensional column vector with j-dimensional element 1 and other elements 0.
The dots in (1) indicate the differential meaning, and the dots with horizontal lines indicate the fixed topology WTX of1(t) the total content of, in particular,
and
respectively expressed as:
in the formula, WTRepresenting a fixed topology, ImRepresenting an identity matrix.
103. Determining a consistency function of the leaderless follow-up structure consistency model;
it should be noted that the expression of the consistency function is:
let r ═ rank (E), there is a nonsingular matrix
And
satisfies the formula:
if the group system dynamic description formula of the singular system dynamic model and the control protocol formula realize the leaderless consistency of the limited energy, no pulse is generated for (E, A),
is not unusual. Thus, the following non-singular matrices can be constructed, namely:
if the group system dynamic description formula of the singular system dynamics model and the control protocol formula realize the leaderless consistency of the limited energy, the consistency function c (t) satisfies the formula:
the above equation shows that the display expression of the consistency function is independent of the protocol state and the given energy budget, and is assumed in the group system dynamic control protocol equation of the singular system dynamics modelProtocol state si(t) is 0.
The process on the energy budget is then:
the relationship matrix followed without leader is IN-N-111TSo there is a scalar α such that:
matrix IN-N-111TThere is one eigenvalue of zero and N-1 eigenvalues greater than zero. Then can obtain
Can ensure
It can be stated that whether the respective intelligence of the present application can achieve performance consistency depends on the dynamic mode and the initial state, irrespective of the given energy budget.
104. And carrying out consistency adjustment on the agents in the group system according to the leaderless following structure consistency model.
It should be noted that the application adjusts the consistency of the agents in the group system by designing a leaderless follow-up structure consistency model.
On the premise that the change of topology along with time is not considered, a multi-agent system is modeled into a high-order singular group system, and a leaderless following dynamic output feedback consistency control protocol with energy budget is provided. Firstly, because the group system is singular and has certain separation characteristics, the influence of a given energy budget on the consistency criterion and the consistency function is determined, secondly, a gain matrix of the control protocol is designed by a two-step method, a display expression of the consistency function is determined by a first equivalent form, and the LMI consistency design criterion in the prior art is compensated to comprise a nonlinear term by the generalized Riccati equation and the linear matrix inequality test. Finally, it is determined by inference whether each agent can achieve performance consistency depending on the dynamic mode and initial state, regardless of a given energy budget.
In a specific embodiment, as shown in fig. 2, a communication topology between agents that are not followed by leader in the present application may be set to 1 or 0 (i.e. 1 is a communication connection between the agents, otherwise 0) for the communication network between the agents.
Setting initial values, E, A, B and C parameters, Q values, H values and gain matrix K of six agents without leadershipaAnd KbThe singular group system state trajectory and energy trajectory without leadership following are shown in fig. 3-4 below.
Fig. 3 depicts the state trajectories of the singular group system in the leaderless following state, and it can be seen from fig. 3 that initially, the state trajectories of the six agents are all inconsistent, but as time goes on, the trajectories of the six agents slowly approach to be consistent (the circle trajectories are marked as the motion trajectories of the consistent function), which indicates that the consistency of the leaderless following structure is realized. FIG. 4 represents given energy budget for uniform motion of an agent
And energy consumption function JcFrom FIG. 4, it can be seen that the energy function converges to a finite value
It is clear to those skilled in the art that, for convenience and brevity of description, the specific working processes of the above-described systems, apparatuses and units may refer to the corresponding processes in the foregoing method embodiments, and are not described herein again.
The terms "comprises," "comprising," and "having," and any variations thereof, in this application are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
It should be understood that in the present application, "at least one" means one or more, "a plurality" means two or more. "and/or" for describing an association relationship of associated objects, indicating that there may be three relationships, e.g., "a and/or B" may indicate: only A, only B and both A and B are present, wherein A and B may be singular or plural. The character "/" generally indicates that the former and latter associated objects are in an "or" relationship. "at least one of the following" or similar expressions refer to any combination of these items, including any combination of single item(s) or plural items. For example, at least one (one) of a, b, or c, may represent: a, b, c, "a and b", "a and c", "b and c", or "a and b and c", wherein a, b, c may be single or plural.
The above embodiments are only used for illustrating the technical solutions of the present application, and not for limiting the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions in the embodiments of the present application.
Claims (4)
1. A singular group system consistency analysis and control method is characterized by comprising the following steps:
establishing a singular system dynamics model, wherein the singular system dynamics model comprises a dynamic description formula and a control protocol formula of a group system;
establishing a leader-free following structure consistency model according to the singular system dynamics model, and determining a consistent dynamic description formula and an inconsistent dynamic description formula;
determining a consistency function of the leaderless follower structure consistency model;
and carrying out consistency adjustment on the agents in the group system according to the leaderless following structure consistency model.
2. The singular group system consistency analysis and control method as claimed in claim 1, wherein said establishing a singular system dynamics model comprises:
the dynamic description formula of the group system is as follows:
in the formula, xi(t) and ui(t) represents the status control input of the ith agent; y isi(t) represents the state control output of the ith agent; i represents the number of agents; coefficient matrix
E is a matrix of order m,
rank(E)≤m；
the formula of the control protocol of the group system is as follows:
in the formula: 1,2, …, N, HT＝H＞0，si(t) is the protocol state with an initial value of zero; kaAnd KbIs a gain matrix; n is a radical ofiIs a neighbor set of time t i; j. the design is a squarec(t) is a consistent motion energy expenditure term;
for a given energy budget; w is aijIs the interaction weight.
3. The singular group system consistency analysis and control method as claimed in claim 1, wherein said establishing a leaderless following structure consistency model from said singular system dynamics model, determining consistent dynamic description formulas and inconsistent dynamic description formulas, comprises:
transforming the singular system dynamics model into a leaderless follow-up structural consistency model by structural decomposition, the leaderless follow-up structural consistency model having a consistent dynamic formula of:
the non-uniform dynamic formula of the leaderless follow-up structural consistency model is as follows:
in the formula: let λ1≤λ2≤…≤λNLet ejJ ∈ {1,2, …, N } under an N-dimensional column vector equal to a j-dimensional element of 1 and other elements of 0.
4. The singular group system consistency analysis and control method as claimed in claim 1, wherein said determining a consistency function of said leader-free following structural consistency model comprises:
the consistency function c (t) of the leader-less following structure consistency model is represented as:
in which there is a non-singular matrix
And
satisfies the following conditions:
then construct the column non-singular matrix as:
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