CN116661347A - Multi-agent consistency control method containing uncertain time lag - Google Patents

Multi-agent consistency control method containing uncertain time lag Download PDF

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CN116661347A
CN116661347A CN202310602942.6A CN202310602942A CN116661347A CN 116661347 A CN116661347 A CN 116661347A CN 202310602942 A CN202310602942 A CN 202310602942A CN 116661347 A CN116661347 A CN 116661347A
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following
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CN116661347B (en
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郭胜辉
罗世坚
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Suzhou University of Science and Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/04—Programme control other than numerical control, i.e. in sequence controllers or logic controllers
    • G05B19/042—Programme control other than numerical control, i.e. in sequence controllers or logic controllers using digital processors
    • G05B19/0423—Input/output
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/20—Pc systems
    • G05B2219/25—Pc structure of the system
    • G05B2219/25257—Microcontroller
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

Abstract

The invention discloses a multi-agent consistency control method with uncertain time lag, which comprises the following steps: establishing a required state space equation according to a dynamic model and a communication topological structure of a leader-follower multi-agent system containing uncertain time lags and unknown inputs, and converting the state space equation into a global following error system according to the characteristics of the leader-follower dynamic model; then, designing an amplification system according to the global following error system and the measurable noise; then, an observer is designed by using the augmentation system to realize the estimation of the undetectable part and the measurable noise of the system state part, and the augmented system state information is obtained; finally, a consistency control protocol is designed by using the state information of the augmentation system, the consistency control problem of the lead-following multi-agent is converted into the stability problem of the global following error system, and the consistency control is realized; and the stability judgment of the error system is realized through the unobservable part and the unknown input problem.

Description

Multi-agent consistency control method containing uncertain time lag
Technical Field
The invention relates to the field of multi-agent system cooperative control, in particular to a multi-agent consistency control method with uncertain time lags.
Background
Multi-agent systems (MAS) are widely used in the fields of computer science, system control, etc. because they have the advantages of being distributed, strong in coordination, strong in operability, and strong in robustness. The problem of consistency of the multi-agent system is taken as the basis of distributed coordination control, and more attention is paid to the system, such as the research of energy transmission control and group decision of a smart grid, autonomous configuration of a wireless sensor network, satellite formation, an unmanned aerial vehicle control system and the like.
Most researches mainly surround the combination and expansion of system complexity, topology complexity, connection complexity and different combinations of the system complexity and the topology complexity, such as researching a second-order multi-agent system under directed topology or not, and providing an event trigger control strategy based on a self-adaptive dynamic clock, so that continuous communication is effectively avoided, communication resources are saved, and consistency control is realized; and a fully distributed observer and self-adaptive fault-tolerant control protocol are designed for a multi-agent system with actuator faults, unknown nonlinear dynamics and non-matching interference, so that the consistency problem under the complex condition is solved.
However, most of the existing researches focus on one aspect or part of the aspects, less results focus on time lag, unknown input and nonlinearity, and in an actual system, the complex situations are often unavoidable, so that in order to achieve a more practical result, it is necessary to consider both the unobservable part of the system state in intelligent control and the unknown input problem.
Disclosure of Invention
The invention aims to provide a multi-agent consistency control method with uncertain time lag, which solves the following technical problems:
how to solve the problems of unobservable parts, unknown input and uncertain time lags and interference in the system state, and further to realize stability analysis and control of an error system.
The aim of the invention can be achieved by the following technical scheme:
a multi-agent consistency control method with uncertain time lag comprises the following steps:
s1, establishing a required state space equation according to a dynamic model and a communication topological structure of a leader-follower multi-agent system with uncertain time lag and unknown input, and converting the state space equation into a global following error system according to the characteristics of the leader-follower dynamic model; wherein the lead-following multi-intelligent system comprises a sensor and contains measurable noise;
s2, designing an amplification system according to the global following error system and the measurable noise;
s3, designing an observer by using an augmentation system, and estimating an undetectable part and measurable noise of a system state part to obtain augmentation system state information;
s4, designing a consistency control protocol by using the state information of the augmentation system, converting the problem of consistency control of the lead-following multi-agent into the problem of stability of the global following error system, and realizing consistency control.
Preferably, the lead-following multi-agent system generally comprises a system of 1 leader and N agents, the lead-following dynamics model comprising: the system comprises a leader dynamics model and a follower dynamics model, wherein the leader dynamics model is as follows:
wherein ,xi (k)＝Φ(k),k∈{-τ max ,-τ max +1,···,0}；
The ith follower dynamics model can be described as:
wherein ,representing the state vector of the i-th agent,representing inter-agent delay state vectors, τ min Represents the lower bound of τ max Representing the upper bound of τ. />Representing the input vector of the system,/>Representing the output vector of the system->Representing unknown input,/->Representing disturbance of the system->Representing a function comprising a continuous nonlinear function, A d B, C, D, G are full rank parameter matrices with known and appropriate dimensions.
Preferably, the global following error system is:
wherein ,δi (k)＝x 0 (k)-x i (k)，δ i (k) Defined as global state tracking error, x 0 (k) Defined as the leading agent state, x i (k) Defined as the following state of the ith following agent.
Preferably, the augmentation system is designed as follows:
wherein ,M＝[I 0]，/>
due to
As a full order matrix, there are always matrices T and N such that:
then, the augmentation system may be further expressed as follows according to the equation above:
preferably, the creating and designing process of the observer in the step S2 is as follows:
since the augmentation system can be controlled and observed, a proportional-integral observer is designed:
wherein ,is the system state vector χ i (k) Is a nonlinear function>Is recorded as an estimated value of (2)The gain matrix of the observer is denoted as L 1 and L2 。
Preferably, the state information includes a state estimate and an unknown input, and the estimation error is designed therefrom:
wherein ,ei (k) Representing state estimation error, eta i (k) Representing unknown input errorsThe method comprises the steps of carrying out a first treatment on the surface of the Ensure thatAndthe observer can achieve accurate estimates of the system state and unknown inputs.
Preferably, to implement the observer to perform state estimation validity analysis on the state estimation and the unknown input, the following error system is designed, including:
wherein ,according to the above equation, the observer error system of the augmentation system may be further written as:
and the observer estimation error system of the augmentation system may perform a linear transformation:
further simplified to the following form:
definition of the definition
Depending on the nature of the Kronecker product, the estimation error system of the augmentation system observer may be expressed in the form:
preferably, the formula of the consistency control protocol is:
wherein, the coupling strength between the current state and the delay state is respectively defined as alpha and beta, and a table tau (k) is more than or equal to 0 and shows the communication time delay, N i Representing a set of neighbor nodes, a, of node i ij Representing the connection weight between node i and node j, K is the coherence control protocol gain matrix.
Preferably, the global following error system stability analysis formula in step S4 is:
preferably, the consistency control protocol analysis includes analyzing a global following system state: system input u i (k) Global following system state delta i (k) Global following system output state Y i (k) And original system state x i (k)。
The invention has the beneficial effects that:
(1) Aiming at a leader-follower multi-agent system with communication time lag, unknown input, measurable noise and nonlinear terms, the leader-follower dynamics model characteristic is converted into a global following error system; the augmentation system is designed according to the global following error system and the measurable noise, and an observer is designed by utilizing the global following error system, so that the problems of unobservable parts and unknown input in the system state are solved; meanwhile, a protocol for solving the consistency control of the multi-agent system with uncertain time lags is provided based on the observer, and the consistency control problem of the system is converted into the stability problem of an error system.
(2) The invention also obtains a feasible solution by solving the inequality of the linear matrix of the nonlinear term meeting the Lipschitz condition and based on graph theory and Lyapunov-Krasckii function, and can realize the analysis and judgment of the stability problem of an error system.
(3) The invention is also based on the prior art, enhances the robustness of the system and reduces the conservation, and verifies the feasibility and the correctness through experimental simulation; the control of the system state is completed by controlling the movement of the leader, so that the control process is greatly simplified and the control cost is reduced.
Of course, it is not necessary for any one product to practice the invention to achieve all of the advantages set forth above at the same time.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are needed for the description of the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a multi-agent consistency control method for uncertain time-lapse according to the present invention;
FIG. 2 is a topology of a queuing system of the present invention;
FIG. 3 is a diagram of system control inputs under the control protocol formation of the present invention;
FIG. 4 is a global tracking state value for a queuing system according to the present invention;
FIG. 5 is a global following output state value for a queuing system according to the present invention;
FIG. 6 is a schematic diagram of the original system state 1 according to the present invention;
FIG. 7 is a schematic diagram of the original system state 2 according to the present invention;
FIG. 8 is a schematic diagram of the original system state 3 according to the present invention;
FIG. 9 is a diagram of the global tracking state 1 and observer state of the queue system according to the present invention;
FIG. 10 is a diagram of the global tracking state 2 and observer state of the queuing system according to the present invention;
FIG. 11 is a diagram of the global tracking state 3 and observer state of the queuing system according to the present invention;
FIG. 12 is a diagram of the global tracking state 4 and observer state of the queuing system according to the present invention;
FIG. 13 is a diagram of the global tracking state 5 and observer state of the queuing system according to the present invention;
FIG. 14 is an observer estimation error of the present invention;
FIG. 15 is a graph of the present invention comparing an estimated value of an unknown input to an actual value;
FIG. 16 is a graph showing the comparison of the estimated and actual values of the system of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 1, the present invention is a multi-agent consistency control method with uncertain time lag, comprising the following steps:
s1, establishing a required state space equation according to a dynamic model and a communication topological structure of a leader-follower multi-agent system with uncertain time lag and unknown input, and converting the state space equation into a global following error system according to the characteristics of the leader-follower dynamic model; wherein the lead-following multi-intelligent system comprises a sensor and contains measurable noise;
s2, designing an amplification system according to the global following error system and the measurable noise;
s3, designing an observer by using an augmentation system, and estimating an undetectable part and measurable noise of a system state part to obtain augmentation system state information;
s4, designing a consistency control protocol by using the state information of the augmentation system, converting the problem of consistency control of the lead-following multi-agent into the problem of stability of the global following error system, and realizing consistency control.
Through the technical scheme: since in a practical multi-agent system, the time lag phenomenon is often unavoidable and widely exists in a physical system as a typical nonlinearity; most of the current researches aim at realizing consistency control for a part of cases; however, unknown inputs and measurable noise are commonly existing in various actual systems, and in the face of the technical problems, the invention carries out the following research by aiming at a leading-following multi-agent system containing uncertain time lags and unknown inputs, in order to solve the problem of partial undetectable states and measurable noise of the system, and provides a control protocol on the basis to realize the consistency control of the leading-following multi-agent system containing uncertain time lags and unknown inputs by designing observers for estimating the measurable noise of the states of the agents.
In a specific embodiment, the steps are as follows: firstly, establishing a required state space equation according to a dynamic model and a communication topological structure of a leader-follower multi-agent system with uncertain time lags and unknown inputs, and converting the state space equation into a global following error system according to the characteristics of the leader-follower dynamic model; wherein the lead-following multi-intelligent system comprises a sensor and contains measurable noise; the global tracking error system is obtained by the prior graph theory related knowledge, necessary assumptions and theories and utilizing the characteristics of the leader-follower multi-intelligent system;
then, designing an amplification system according to the global following error system and the measurable noise; again, using the augmented system to design an observer to realize the estimation of the undetectable part and the measurable noise of the system state part, and obtaining the augmented system state information; finally, the state information of the augmentation system is utilized, wherein the state of the augmentation system and the estimated state thereof are mainly used for designing a consistency control protocol, so that the problem of the consistency control of the leading-following multi-agent is converted into the problem of the stability of the global following error system, and the consistency control is realized; the method is characterized in that the problem of partial undetectable and unknown input of the original system is processed by utilizing the observer principle, a consistency control method is designed based on the designed observer, the problem of consistency of a generalized system is converted into the problem of stability of an error system, and the problem of stability of the error system is analyzed. In addition, the feasibility of the method is verified by carrying out simulation analysis on the technical scheme.
Firstly, defining according to the prior graph theory knowledge, and ensuring the definition of the following intelligence so as to establish a following error system:
by representing the topology asWherein node set V g ＝{V 1 ,V 2 ,…,V n }，Representing a connected set of nodes, node (i, j) represents a directed arc from a starting node i to a terminating node j. Node j is called the neighbor of node i if (i, j) ∈ε g Otherwise it is not a neighbor. The set of neighbor nodes for a given node is denoted as N i ＝j|(i,j)∈ε g ，/>Representing a discrete matrix weight matrix. If (i, j) epsilon g Then a ij > 0, indicating that node i can receive information from node j, otherwise a ij =0. If the topological graph is an undirected graph, a ij ＝a ji Otherwise a ij ≠a ji . The Laplace matrix may be defined as L a ＝D a -A g, wherein Da ＝diag{d 1 ,d 2 ,…,d n },/>
At the node set N i In (2), a node 0 is introduced, and N is i Is referred to as a subsequent node and the newly introduced node 0 is referred to as a leading node. In this case, the preceding node may affect the following node, but is not affected by it. Arc sum through a leading node and connecting a following nodeCan obtain a G g The topology is extended. In the expanded graph topology G g If there is at least one subsequent node having access to the information of the preceding node, and G g Is connected with the leading node and is called G g Is connected. At G g In the following, the Laplacian matrix may be defined as l=l a +B g, wherein Bg ＝diag{b 1 ,b 2 ,…,b n The connectivity matrix of the preceding node and the following node, e.g. b if the following node i can receive information from the preceding node i > 0; otherwise, it is generally assumed that node i has no self-loop, i.e., a ii ＝0。
By associating an extended topology with the leading following multiple intelligent systems, it can be observed that in the extended topology G g Each node may be considered an intelligent entity, where a leading node represents leading intelligence and a following node represents following intelligence.
As one embodiment of the present invention, a lead-following multi-agent system generally includes a system consisting of 1 leader and N agents, the lead-following dynamics model including: the system comprises a leader dynamics model and a follower dynamics model, wherein the leader dynamics model is as follows:
wherein ,xi (k)＝Φ(k),k∈{-τ max ,-τ max +1,···,0}
The ith follower dynamics model can be described as:
wherein ,representing the state vector of the i-th agent,representing inter-agent delay state vectors, τ min Represents the lower bound of τ max Representing the upper bound of τ. />Representing the input vector of the system,/>Representing the output vector of the system->Representing unknown input,/->Representing disturbance of the system->Representing a function comprising a continuous nonlinear function, A d B, C, D, G are full rank parameter matrices with known and appropriate dimensions.
Through the technical scheme: by way of problem description, consider that a lead-following multi-agent system is typically composed of 1 leader and N agents; the dynamics model of the design leader is:
wherein xi (k)＝Φ(k),k∈{-τ max ,-τ max +1, …, 0), from a dynamic model perspective, the ith slave vehicle's kinematic model can be described as:
wherein ,representing the state vector of the i-th agent,representing inter-agent delay state vectors, τ min Represents the lower bound of τ max An upper bound representing τ; />Representing the input vector of the system,/>The output vector of the system is represented,representing unknown input,/->Representing a disturbance of the system; />Representing a function comprising a continuous nonlinear function; a, A d B, C, D, G are full rank parameter matrices with known and appropriate dimensions.
Before presenting the results, the following is a necessary assumption:
suppose 1: the system rank [ A B ] is controllable, and rank [ A C ] is considerable.
Suppose 2: nonlinear term functionMeeting Lipschitz conditions, i.e
wherein xi (k),x j (k)∈R n ,γ＞0。
Suppose 3: system disturbance omega i (k) The following conditions are satisfied:
wherein Is a non-negative scalar.
Suppose 4: communication time lags in real systems are often unavoidable, let τ (k) represent an uncertain time lag in the communication process, and satisfy:
τ min ＜τ(k)＜τ max .
wherein 0＜τmin ＜τ max The method comprises the steps of carrying out a first treatment on the surface of the Define the initial state as x (k) =Φ (k), k e [ - τ [ max 0]Where Φ (k) represents the initial function of the state.
Note 1: the necessary assumptions given here are widely used in the literature of distributed coordinated control directions and have popularity.
1. In hypothesis 1, it is assumed that the original system can be observed and controlled, which is a necessary condition for system design control, obtained from prior art analysis.
2. In hypothesis 2, assume that the nonlinear term satisfies the Lipschitz condition, and then use the Lipschitz inequality to convert the error equation to an inequality, providing conditions for building the LMI, are obtained from prior art analysis.
3. In hypothesis 3, the unknown input is assumed, which is a condition that ensures that the control system will not fail after external disturbances and that control protocol stabilization can be used, is obtained according to prior art analysis.
4. In assumption 4, the communication delay is assumed to be bounded, which is a condition that ensures that the control system does not collapse when subjected to the communication delay and that stability can be achieved by using the control protocol, which is obtained according to prior art analysis.
Lemma 1: l (L) a Undirected graph G of (2) g Has the following characteristics:
1、L a all eigenvalues λ i Not less than 0, and only has one zero characteristic value, and the corresponding characteristic vector is 1 n ；
2、rank(L a )＝n-1。
Lemma 2 (Jensen inequality): for any non-negative matrixVector->Positive integer alpha 1 、α 2 Satisfy alpha 2 ≥α 1 1, the following inequality holds:
and (3) lemma 3: according to the prior art, given a matrix Y, D, E of appropriate dimensions, where Y is a symmetric matrix, the following inequality can be satisfied:
Y+DFE+E T F T D T ＜0.
for all meeting F T F.ltoreq.I, one sufficient condition is that ε > 0 is present to satisfy the following inequality:
Y+εED T +ε -1 E T E＜0.
depending on the relaxation characteristics of the multi-agent system, the relaxation error can be defined as:
δ i (k)＝x 0 (k)-x i (k) (3)
wherein ,δi (k)Defined as global state tracking error, x 0 (k) Defined as the front vehicle state, x i (k) Defined as the following state of the ith vehicle. Based on formulas (1), (2), and (3), the expression of the global state following error can be obtained as follows:
wherein ,
the global output following error is defined as:
Y i (k)＝y 0 (k)-y i (k) (5)
wherein ,Yi (k) Defined as global output error, y 0 (k) Defined as the output of the lead vehicle, y i (k) Defined as the output of the ith slave follower. According to equations (1), (2) and (5), the global output error can be expressed as:
as an embodiment of the present invention, the global following error system is:
wherein ,δi (k)＝x 0 (k)-x i (k)，δ i (k) Defined as global state tracking error, x 0 (k) Defined as the front vehicle state, x i (k) Defined as the following state of the ith vehicle.
Through the technical scheme: the global tracking error system can be expressed as:
and (2) injection: through the above transformation, the problem of the consistency of individual followers and leaders in the system is re-expressed as a problem of the stability of the global state following error system (7).
As an embodiment of the present invention, the augmentation system is designed as follows:
wherein ,M＝[I 0]，/>
due to
As a full order matrix, there are always matrices T and N such that:
then, the augmentation system may be further expressed as follows according to the equation above:
through the technical scheme: by setting the state estimation, the following augmentation system can be designed according to the following error system and the measurable noise:
wherein ,M＝[I 0]，/>
as a result of the fact that,
as a full order matrix, there are always matrices T and N such that:
then, according to the above equation, the augmentation system may be further expressed as
If a given system is controllable and observable, any state information of the system must be reflected from the system output, and the state information of the system can be estimated by designing an observer.
And 4, lemma: since the original system is observable and controllable, the augmentation system is also controllable and observable according to the augmentation system design in equation (9).
The pattern: due to A epsilon R n×n ,B∈R n×m ,E∈R n×l ,C∈R s×n ,F∈R s×l ThenTB＝B 5 And A is 5 ∈R (n+l)×(n+l) ，χ i (k)∈R n+l ，/>B 5 ∈R (n+l)×m And the system parameter matrixes are all full rank; the following formula can be derived from the criteria of the rank.
Thus, the augmentation system (9) may be controlled and observed.
As an embodiment of the present invention, the creation design process of the observer in step S2 is as follows:
since the augmentation system can be controlled and observed, a proportional-integral observer is designed, where the proportional-integral observer is obtained based on the prior art:
wherein ,is the system state vector χ i (k) Is used for the estimation of the estimated value of (a). Nonlinear function->Is recorded as an estimated value of (2)The gain matrix of the observer is denoted as L 1 and L2 。
As one embodiment of the invention, the state information includes a state estimate and an unknown input, and the estimation error is designed from this: : represented as L according to the gain matrix of the observer 1 and L2 The method comprises the steps of carrying out a first treatment on the surface of the The state estimate and unknown input error may be defined as follows:
if there is an appropriate observer gain L 1 and L2 So that and />This means that the designed observer (10) meets the requirements and can be used as an observer for the system (9). Wherein e i (k) Representing state estimation error, eta i (k) Representing an unknown input error. By ensuring that both converge to zero at infinity, the observer is able to achieve accurate estimates of the system state and unknown inputs.
As one embodiment of the present invention, to achieve the observer to perform state estimation on state estimation and unknown input, the following error system is designed according to the state estimation error and the unknown input validity analysis, according to formulas (9) and (10):
wherein ,the observer error system describing the augmentation system for estimating the error may be further written as:
the observer estimation error system of the augmentation system may perform the following linear transformation:
further simplified to the following form:
definition of the definition
Depending on the nature of the Kronecker product, the estimation error of the augmented system observer can be expressed in the form: />
As mentioned above, if there is a suitable gain matrix L, such thatThis indicates that the proportional-integral observer (10) can implement a state estimation of the system (9).
And (3) injection: since the above process cannot calculate L 1 ,L 2 As a result of (a), the process is proved to be incomplete. The matrix equation will be solved directly using the LMI toolbox in Matlab to obtain L 1 ,L 2 And the specific equations are listed here below and show the observation effect of the designed observer (10) on the system (9) by means of the simulation image, when L is obtained 1 ,L 2 After the value of (2), the proof will be completed in theorem 1.
As one embodiment of the present invention, the formula of the coherence control protocol is:
wherein, the coupling strength between the current state and the delay state is respectively defined as alpha and beta, and a table tau (k) is more than or equal to 0 and shows the communication time delay, N i Representing a set of neighbor nodes, a, of node i ij Representing the connection weights between node i and node j, K is the cooperation forming the control gain matrix.
Through the technical scheme: by setting a consistency design, the following distributed formation control protocol can be designed according to the estimation result of the augmentation system observer:
wherein the coupling strength between the current state and the delay state is respectively defined as alpha and beta, tau (k) is equal to or larger than 0 and represents communication time delay, N i Representing a set of neighbor nodes, a, of node i ij Representing the connection weights between node i and node j, K is the cooperation forming the control gain matrix.
The state synchronization error and the unknown input synchronization error may be defined as follows:
from (9) and (16), the following results can be obtained
wherein ,/>
if a proper feedback gain matrix K exists, makeIt is stated that under the control protocol (16), the leading heel system (1), (2) is able to implement the formation control; this will ultimately translate the problem of consistency of the ICV system into that of an error system and is solved by a linear transformation.
Symmetry a according to undirected topology ij ＝a ji Then there is
Further can obtain
Then, by sorting equations (18), (19), the following global following error system stability analysis formula can be obtained:
definition of the definition
and /> and Wherein q= [ I0 ]]And based on the requirements of the collaborative formation control, assume Δε i (k)＝ε i (k+1)-ε i (k)＝0。
Based on the topological properties of the undirected graph and the Laplacian matrix, the Kronecker product can further write the error system into the form:
as above, there is a suitable K such thatI.e. the error (18) of each preamble-heel system tends to stabilize. This shows that the observer-based design of the control protocol (16) enables the systems (1) and (2) to achieve a cooperative formation of control.
And (4) injection: the above procedure does not calculate the result of K, proving that the procedure is still incomplete. The LMI toolbox in Matlab can be used directly to solve (23) to obtain K; simulation results show that the formation control protocol (16) based on the observer design effectively meets the formation control requirements of the system; once K is obtained in theorem 1, the attestation process may be completed.
Theorem: for a multi-agent consisting of 1 leader (1) and n followers (2), assuming that the original system satisfies hypothesis 1, the nonlinear term satisfies hypothesis 2, the external interference satisfies hypothesis 3, the time lag satisfies hypothesis 4, and given H ∞ Performance metric iota, characteristic value lambda describing Laplacian matrix i (i=0, 1, …, N), and non-negative scalar constants γ and κ, if the following conditions are satisfied: symmetric positive definite matrix P with proper dimension 1 ,P 2 And an adaptive matrix R 1 ,R 2 ,R 3 ,W 1 ,W 2 ,W 3 So that LMI (22) is established, wherein
/>
At this time, the observer estimation error system (15) satisfies H ∞ Performance index |xi (k) | is less than or equal to iota|deltad (k) |. The observer gain matrix L is defined by l=p 1 -1 U 1 Given. The coherence control gain K can be obtained using the analytical formula (15) and formula (22) of the LMI, i.eAnd (3) proving: in order to construct the Lyapunov-Krasovskii function, firstly, an error system (16) is obtained according to a generalized system constructed by a leader-follower system and a generalized system observer; subsequently, the consistency problem (22) of the leader-follower system is converted into a stability problem of the error system; to this end, the Lyapunov-Krasovskii function is constructed as:
wherein
Thus, the time difference for V (k) can be derived as follows:
when assumption 3 holds, i.e. nonlinear functionWhen the Lipschitz condition is satisfied, the following formula can be obtained:
regarding the Lipschitz condition, the estimation error system can be further written as:
thenAnd |e (k) | is less than or equal to |ζ (k) |, therefore +.>Finally obtainRegarding the Lipschitz condition, the synchronization error system can be further written as:
thenIn the same way->As can be seen from the quotation 1, there is an orthogonal matrix pi, so that +.>
L a ＝ΠΛΠ T .
Where Λ=diag { λ } 1 ,λ 2 ,…,λ n }，{λ 1 ,λ 2 ,…,λ n The matrix L is Laplacian a And lambda is the characteristic value of (1) 1 =0. Definition:
the variables ζ (k), ε (k), θ (k),Δd (k) is converted to Γ (k), Θ (k), respectively,/->ζ (k), ψ (k). Definition:
o 1 @[I,0,0,0,0,0,0,0] T ,L,o 8 ＝[0,0,0,0,0,0,0,I] T
substituting (16) into V by Laplace theorem 1 (k) Can be further written as:
from V 2 (k) Can be further written as:
from V 3 (k) Can be further written as:
from V 4 (k) And processed by the lemma 2, can be further written as:
wherein, according to the quotation 2, the upper bound of t can be obtained as
Substituting formula (22) into V 5 (k) In (c) can be further written as:
from V 6 (k) Can be further written as:
from V 7 (k) Can be further written as:
from V 8 (k) And is processed by the primer 2After that, it can be further written as:
/>
wherein, can easily prove
Definition of the definition and />They are as follows:
define the following performance indexes
Wherein, by replacing the variables ζ (k), Δd (k) with ζ (k), ψ (k), and taking into account x under zero initial value conditions i (k)＝Φ(k)＝x(0),i＝1,2,…,N,k∈-τ max ,-τ max +1, …,0}, can be obtained
The following can be concluded:
definition of the definition
ρ＝[Ξ T (k),Ξ T (k-τ(k)),Ξ T (k-τ max ),
Γ T (k),Γ T (k-τ(k)),Γ T (k-τ max ),Θ T (k),
Must be present
Thereby having the following characteristics
ΔV+Ξ T (k)Ξ(k)-ι 2 Ψ T (k)Ψ(k)≤ρ T Φρ
According to the lemma 3, Φ < 0 is equivalent to (23) being true for Φ. State observer gain L 1 ,L 2 And forming the control gain K, and γ and H in the Lipschitz hypothesis ∞ May be obtained by solving an unknown matrix in the LMI toolbox. Thus, when (23) is established, the theorem given in the article can be proved to be established, and J < 0 can be obtained, thereby explaining that the performance index |ζ (k) |+|Δd (k) | is established on the error system (14).
As one embodiment of the invention, the coherence control protocol analysis includes analyzing global following system states: system input u i (k) Global following system state delta i (k) Global following system output state Y i (k) And original system state x i (k) The method comprises the steps of carrying out a first treatment on the surface of the Simulation verification analysis shows that the following vehicle can finally achieve consistency with the track of the leading vehicle.
The embodiment of the invention also comprises the step of carrying out simulation verification on the technical scheme:
in the leader-follower multi-agent system, since the leader is not influenced by the follower and can influence the follower, the system state control can be completed by controlling the movement of the leader in simulation verification, so that the control process is greatly simplified and the control cost is reduced; thus to verify the effectiveness of the design method, a leader-follower multi-agent system consisting of 1 leader with system model (\ref { formula1 }) and 5 followers with system model (\ref { formula2 }) will be considered, the communication topology of the agent being as shown in FIG. 2, wherein agent 0 represents the leader and agents 1,2,3,4,5 represent the followers.
The parameter matrix for a given preamble-heel system is as follows:
ι＝0.7,γ＝0.05,τ max ＝5,τ min =1, α=0.5, β=0.5, the fdi attack input vector is assumed to be:the continuous nonlinear function is assumed to be: />The external disturbance of the system is assumed to be: />
wherein The parameter matrix of the preamble heel system is as follows:
by equation ofOne of the parameter sets T, N can be found as follows:
substituting the above data into a linear matrix inequality (\reffrormula 23) and using Matlab toolbox can obtain L 1 ,L 2 ,K,κ。
κ＝4.1666×10 6 ,K＝[1.8933 0.9818 1.5268 1.7158]
According to hypothesis 4, hypothesis τ max ＝5,τ min ＝1，τ min ＜τ(k)＜τ max For communication delay, the initial state of the system is x (t) =phi (k), k epsilon [ -5,0]Where Φ (k) is the initial function of the state.
FIG. 3 shows the system input u under the formation of the control protocol (16) i (k) A. The invention relates to a method for producing a fibre-reinforced plastic composite Figures 4-8 illustrate global following system state delta i (k) Global following system output state Y i (k) And original system state x i (k) The following vehicle is shown to be consistent with the track of the leading vehicle finally; FIGS. 9-13 illustrate trajectories of a following system state and an observer state thereof; FIG. 14 shows all design observer estimation errors; FIG. 15 illustrates a comparison of a trajectory of unknown input estimates obtained using an observer with actual values; fig. 16 shows a trace of a measurable noise estimate obtained using an observer versus an actual value.
The foregoing is merely illustrative and explanatory of the principles of the invention, as various modifications and additions may be made to the specific embodiments described, or similar thereto, by those skilled in the art, without departing from the principles of the invention or beyond the scope of the appended claims.

Claims (10)

1. The multi-agent consistency control method with uncertain time lag is characterized by comprising the following steps of:
s1, establishing a required state space equation according to a dynamic model and a communication topological structure of a leader-follower multi-agent system with uncertain time lag and unknown input, and converting the state space equation into a global following error system according to the characteristics of the leader-follower dynamic model; wherein the lead-following multi-intelligent system comprises a sensor and contains measurable noise;
s2, designing an amplification system according to the global following error system and the measurable noise;
s3, designing an observer by using an augmentation system, and estimating an undetectable part and measurable noise of a system state part to obtain augmentation system state information;
s4, designing a consistency control protocol by using the state information of the augmentation system, converting the problem of consistency control of the lead-following multi-agent into the problem of stability of the global following error system, and realizing consistency control.
2. The method of claim 1, wherein the lead-following multi-agent system generally comprises 1 leader and N agents, and wherein the lead-following dynamics model comprises: the system comprises a leader dynamics model and a follower dynamics model, wherein the leader dynamics model is as follows:
wherein ,xi (k)＝Φ(k),k∈{-τ max ,-τ max +1,···,0}；
The ith follower dynamics model can be described as:
wherein ,representing the state vector of the i-th agent,representing inter-agent delay state vectors, τ min Represents the lower bound of τ max Representing the upper bound of τ. />Representing the input vector of the system,/>Representing the output vector of the system->Representing unknown input,/->Representing disturbance of the system->Representing a function comprising a continuous nonlinear function, A d B, C, D, G are full rank parameter matrices with known and appropriate dimensions.
3. The multi-agent consistent control method of claim 1 wherein said global tracking error system is:
wherein ,δi (k)＝x 0 (k)-x i (k)，δ i (k) Defined as global state tracking error, x 0 (k) Defined as the leading agent state, x i (k) Defined as the following state of the ith following agent.
4. The multi-agent consistent control method of claim 1 wherein said augmentation system is designed as follows:
wherein ,M＝[I 0]，/>
due to
As a full order matrix, there are always matrices T and N such that:
then, the augmentation system may be further expressed as follows according to the equation above:
5. the method for controlling the consistency of multiple agents with uncertain time lags according to claim 4, wherein the creating and designing process of the observer in the step S2 is as follows:
since the augmentation system can be controlled and observed, a proportional-integral observer is designed:
wherein ,is the system state vector χ i (k) Is a nonlinear function>Is recorded as an estimated value of (2)The gain matrix of the observer is denoted as L 1 and L2 。
6. The multi-agent consistent control method of claim 1 wherein said state information includes state estimates and unknown inputs and from which estimation errors are designed:
wherein ,ei (k) Representing state estimation error, eta i (k) Representing an unknown input error; ensure thatAndthe observer can achieve accurate estimates of the system state and unknown inputs.
7. The method of claim 6, wherein for the observer to perform a state estimation validity analysis on the state estimation and the unknown input, the following error system is designed, comprising:
wherein ,according to the above equation, the observer error system of the augmentation system may be further written as:
and the observer estimation error system of the augmentation system may perform a linear transformation:
further simplified to the following form:
definition of the definition
Depending on the nature of the Kronecker product, the estimation error system of the augmentation system observer may be expressed in the form:
8. the multi-agent consistent control method of claim 1 wherein said consistent control protocol is formulated as:
wherein, the coupling strength between the current state and the delay state is respectively defined as alpha and beta, and a table tau (k) is more than or equal to 0 and shows the communication time delay, N i Representing a set of neighbor nodes, a, of node i ij Representing the connection weight between node i and node j, K is the coherence control protocol gain matrix.
9. The method for controlling the uniformity of multiple agents with uncertain time lags according to claim 1, wherein the global following error system stability analysis formula in step S4 is:
10. the multi-agent consistent control method of claim 1 wherein said consistent control protocol analysis includes analyzing global following system conditions: system input u i (k) Global following system state delta i (k) Global following system output state Y i (k) And original system state x i (k)。
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