A fully distributed state observation method with time-constrained convergence

Abstract

The application discloses a completely distributed state observation method with a time convergence, and belongs to the technical field of multi-agent system cooperative control and state estimation. The application solves the problems that the existing method needs to depend on global network topology information and cannot realize controllable real convergence time and poor robustness. The application can make each follower independently and efficiently reconstruct the state and derivative of a leader which cannot be directly obtained without any global network information, and ensure that the estimation error accurately converges to zero within the real time set by a user. Moreover, the state observation method of the application is suitable for dynamic systems with external disturbance and model uncertainty, has good robustness and engineering application prospect, and is suitable for application scenes, such as spacecraft formation, unmanned aerial vehicle cluster and robot cooperation, which need high-precision synchronization and robustness. The method of the application can be applied to multi-agent system cooperative control and state estimation.

Description

Technical Field

[0001] This invention belongs to the field of cooperative control and state estimation technology for multi-agent systems, specifically involving a fully distributed state observation method with specified time convergence. Background Technology

[0002] In the cooperative control of multi-agent systems, there is typically a dynamic leader whose motion provides a reference trajectory for the entire group. However, in practical applications, not all followers can directly receive the leader's state information; only some followers have the ability to communicate with the leader. Therefore, each follower must reconstruct the leader's state through local neighbor interactions, a process that relies on the design of distributed observers.

[0003] In recent years, researchers have proposed various distributed observer architectures to estimate the leader's state variables. While traditional asymptotically convergent observers are simple in structure, their convergence time is unpredictable, making it difficult to meet real-time requirements. Subsequent finite-time and fixed-time observers can complete the estimation within a finite time, but their upper bound on convergence time depends on initial conditions or system parameters, still lacking good adjustability.

[0004] To further improve performance, prescribed-time observers have been proposed, allowing users to set an upper bound on the convergence time of the estimation error by adjusting parameters. However, most existing prescribed-time observers rely on global network topology information (such as the smallest non-zero eigenvalue of the Laplacian matrix), which violates the principle of "fully distributed," meaning that each node should only use local information for computation and updates. Furthermore, most existing methods can only guarantee that the "convergence time upper bound" can be preset, but cannot guarantee that the "actual convergence time" strictly does not exceed the preset value, limiting their application in tasks with strong temporal constraints.

[0005] Although existing research has attempted to construct distributed observers that do not rely on global information, such as methods based on dynamic gain mechanisms, these approaches are still limited by the guarantee of a pre-set time upper bound and have failed to achieve truly controllable convergence time. Furthermore, existing methods often lack robustness in handling situations where leader acceleration or higher-order dynamics are unknown.

[0006] In summary, there is an urgent need for a state observer design that can achieve fully distributed operation and ensure that the estimation error converges precisely to zero at the user-specified time, thus balancing practicality, flexibility, and theoretical rigor. Summary of the Invention

[0007] The purpose of this invention is to address the problems of existing methods relying on global network topology information, failing to achieve controllable true convergence time, and exhibiting poor robustness. In response, this invention proposes a fully distributed state observation method with specified convergence time.

[0008] The technical solution adopted by this invention to solve the above-mentioned technical problems is: a fully distributed time-defined convergence state observation method, the method specifically including the following steps:

[0009] Step 1: Set the preset convergence time constant gain Robust gain Initialize the first Node pairs The estimated value , No. Node pairs The estimated value ;

[0010] in, Indicates the leader's status. The derivative representing the leader's state;

[0011] Step 2: Construct a leader state observer and use it to observe the leader's state.

[0012] Furthermore, the operation process of the state observer is as follows:

[0013] Step Two 1: Initialization ;

[0014] Step 22: Calculation Time of the first Auxiliary estimation error variables for each node and ;

[0015]

[0016]

[0017] in, Indicates the leader's status. The derivative representing the leader's state, express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, Represents the adjustment function. Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node;

[0018] Steps two and three: Calculation Time of the first Consistency error of each node and ;

[0019]

[0020] in, Indicates the first The state consistency error corresponding to each node Indicates the first Consistency error of the state derivatives for each node Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node. Indicates the total number of nodes. Indicates from node To the node Connection weights;

[0021] Step 24: Based on consistency error and Conduct the first Dynamic gain update of each node;

[0022] Step 25, according to , , , , and Conduct the first Update the state estimate of each node;

[0023] in, and Indicates intermediate variables. and Indicates dynamic gain. express The first derivative, express The first derivative;

[0024] Step 26: Solve the differential equation for state estimation and update in Step 25 to obtain... Time of the first The state estimation results of each node for the leader and the Estimation results of the derivatives of each node with respect to the leader state ;

[0025] judge Has the preset convergence time been reached? ;

[0026] If the target is reached, then the estimate obtained in the last iteration will be used as the result of the first iteration. The estimation results of the leader state of each node and the estimation results of the derivative of the leader state;

[0027] If it is not achieved, then let Return to step two.

[0028] Furthermore, the adjustment function Defined using the exponential decay form:

[0029]

[0030] Where, constant .

[0031] Furthermore, the auxiliary estimation error variables for the leader's corresponding node are as follows:

[0032]

[0033]

[0034] in, and This represents the auxiliary estimation error variable for the node corresponding to the leader.

[0035] Furthermore, the specific process of step two and four is as follows:

[0036]

[0037]

[0038] in, and Indicates intermediate variables. and Indicates dynamic gain. express The first derivative, express The first derivative.

[0039] Furthermore, the specific process of step two five is as follows:

[0040]

[0041]

[0042] in, express The first derivative, Represents a symbolic function. express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, express The first derivative, express The first derivative.

[0043] Furthermore, the differential equation for solving the state estimation update in step two-five is obtained using a numerical integration method.

[0044] Furthermore, the time-varying gain function for:

[0045]

[0046] in, These are design parameters.

[0047] Furthermore, the time-varying gain function for:

[0048]

[0049] in, These are design parameters.

[0050] Furthermore, the design parameters and satisfy: ,and .

[0051] The beneficial effects of this invention are:

[0052] This invention enables each follower to independently and efficiently reconstruct the leader's state and its derivatives, which are not directly obtainable, without requiring any global network information, and ensures that the estimation error converges precisely to zero within a user-preset real-time period. Furthermore, the state observation method of this invention is applicable to dynamic systems with external disturbances and model uncertainties, exhibiting excellent robustness and promising engineering applications. It is suitable for applications requiring high-precision synchronization and robustness, such as spacecraft formations, UAV swarms, and robot collaboration. Attached Figure Description

[0053] Figure 1 This is a flowchart of the observation process of the leader state observer of the present invention;

[0054] Figure 2 This is a state estimation error diagram;

[0055] Figure 3 This is a graph showing the error in the state estimation of the derivative. Detailed Implementation

[0056] System Modeling and Problem Description: For a system consisting of a leader (number 0) and... A multi-spacecraft system consisting of several followers, with the leader spacecraft as node 0, followed by node 1, node 2, ..., node 3. The follower spacecraft were designated as the 1st, 2nd, ..., 3rd... If there are 10 nodes, then the communication topology of a multi-spacecraft system is a directed graph. describe, Represents a set of nodes. , Let the set of edges in a directed graph be such that the first edge is the edge of the graph. The spacecraft corresponding to the node can receive data from the first node. The information of the spacecraft corresponding to each node, then .

[0057] Assuming only some followers can directly obtain the leader's state, the Laplacian matrix among followers is defined as follows: The relationship matrix between followers and leaders is as follows: Let the matrix There exists a matrix Make , Representation matrix transpose;

[0058] Let the leader's angular velocity be... Its derivative is acceleration Existence and satisfaction , It is a bounded positive constant. Let... and The components are respectively and , .make , ; then order , Finally, , For followers who cannot directly obtain information about the leader, each pair of components needs to be processed separately. , It is a bounded normal number. Each follower who cannot directly obtain information about the leader reconstructs it using local communication. and .

[0059] Specific implementation method one: Combining Figure 1 This embodiment describes a fully distributed state observation method with predetermined time convergence, which specifically includes the following steps:

[0060] Step 1: Set the preset convergence time constant gain Robust gain Initialize the first Node pairs The estimated value , No. Node pairs The estimated value ;

[0061] in, Indicates the leader's status. The derivative representing the leader's state, For example, set it to 5 seconds or 10 seconds. , , Indicates the upper bound of the unknown angular acceleration;

[0062] Step 2: Construct a leader state observer and use the constructed state observer to observe the leader's state;

[0063] The operation process of the state observer is as follows:

[0064] Step Two 1: Initialization ;

[0065] Step 22: Calculation Time of the first Auxiliary estimation error variables for each node and ;

[0066]

[0067]

[0068] in, Indicates the leader's status. The derivative representing the leader's state, express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, Represents the adjustment function. Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node;

[0069] The adjustment function Defined using the exponential decay form:

[0070]

[0071] Where, constant ;

[0072] Take initial time Regulation function The following conditions must be met:

[0073] 1. In hour, Monotonically decreasing;

[0074] 2. When hour, ,in, Here are the design parameters, representing the expected convergence time;

[0075] 3. and Continuous and bounded.

[0076] Steps two and three: Calculation Time of the first Consistency error of each node and ;

[0077]

[0078] in, Indicates the first The state consistency error corresponding to each node Indicates the first Consistency error of the state derivatives for each node Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node. Indicates the total number of nodes. Indicates from node To the node The connection weights, Indicates the first The set of neighbors of a node. It means that it can give the first A set of nodes that send information, when hour, The value is 1, otherwise, The value is 0;

[0079] It should be noted that the auxiliary estimation error variables for the leader's corresponding node are as follows:

[0080]

[0081]

[0082] in, and The auxiliary estimation error variable representing the node corresponding to the leader;

[0083] Step 24: Based on consistency error and Conduct the first Dynamic gain update for each node:

[0084]

[0085]

[0086] in, and Indicates intermediate variables. and Indicates dynamic gain. express The first derivative, express The first derivative;

[0087] Step 25, according to , , , , and Conduct the first Update the state estimate of each node:

[0088]

[0089]

[0090] in, express The first derivative, Represents a symbolic function. express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, express The first derivative, express The first derivative;

[0091] Step 26: Solve the differential equation for the state estimation update using numerical integration methods (such as the Euler method or the Runge-Kutta method) to obtain... Time of the first The state estimation results of each node for the leader and the Estimation results of the derivatives of each node with respect to the leader state ;

[0092] in, and It is a time-varying gain function;

[0093]

[0094]

[0095] in, and These are design parameters, and the design parameters satisfy... ,and ;

[0096] judge Has the preset convergence time been reached? ;

[0097] If the target is reached, then the estimate obtained in the last iteration will be used as the result of the first iteration. The estimation results of the leader state of each node and the estimation results of the derivative of the leader state can be used as inputs for subsequent controllers.

[0098] If it is not achieved, then let Return to step two.

[0099] The convergence theory of the observer proves:

[0100] To verify the effectiveness of the observer proposed in this invention, the present invention provides the following convergence proof, demonstrating that the estimation error can be minimized within the user-preset real time. It converges precisely to zero.

[0101] Consider the dynamics of observation error system:

[0102]

[0103] in, , , forming a vector , .

[0104] Constructing Lyapunov functions:

[0105]

[0106] in, , , Let be the positive definite matrix given above, and be the constant. ;

[0107] right By taking the derivative and combining it with graph theory properties and inequality techniques, we can obtain:

[0108]

[0109] Because when hour, And in The integral diverges over the integral, therefore, in time, It is exactly true and remains zero.

[0110] Similarly, construct It can be proven that it satisfies:

[0111]

[0112] Solving this differential inequality reveals that there exists a constant. , making ,Right now Consistent and stable;

[0113] And because , so when hour, Therefore, when hour, convergence.

[0114] Note ,and Non-singular (guaranteed by connectivity), therefore when hour, Established.

[0115] Similar analysis Subsystem, utilizing , can be appropriate hour, Convergence, and then .

[0116] Ultimately, due to For all Established, and ,therefore:

[0117]

[0118] That is, the estimation error at the exact time It converges precisely to zero and remains zero thereafter.

[0119] In summary, the method of this invention is particularly suitable for collaborative control tasks of complex systems with high requirements for response speed, synchronization accuracy, and autonomy. This invention has the following advantages:

[0120] 1. Presettable True Convergence Time: Unlike traditional methods that only guarantee an adjustable "upper bound on the convergence time," this invention introduces an adjustment function and a time-varying gain mechanism to ensure that the estimation error is within a certain time frame. It converges precisely to zero and remains at zero thereafter, achieving strict convergence within the pre-set time.

[0121] 2. Fully Distributed Implementation: All parameters of the observer designed in this invention (including gain adjustment) depend only on local information (the state difference between itself and its neighbors), without requiring any knowledge of the global eigenvalues ​​of the communication network (such as the smallest non-zero eigenvalue of the Laplace matrix), making it suitable for deployment in large-scale, dynamically changing network environments.

[0122] 3. Strong robustness: By introducing a symbolic function term This effectively suppresses the impact of external disturbances caused by leader acceleration and improves the stability of the observer under uncertainty.

[0123] 4. No need for initial error boundary knowledge: The observer can achieve preset time convergence under any bounded initial conditions, without requiring the initial range of the estimation error as a priori, which enhances its practicality.

[0124] 5. Modular design: The state observer of this invention can be deployed independently according to the state dimension, which is convenient to extend to high-dimensional state spaces or multi-degree-of-freedom systems (such as three-dimensional attitude estimation).

[0125] Experimental Section

[0126] Taking spacecraft formation flying as an example: the primary satellite (leader) sends an attitude angular velocity signal, which only a few secondary satellites can receive. The remaining secondary satellites, based on the observer and neighboring satellite information of this invention, respond within a specified time (…). At the specified time (seconds), the precise reconstruction of the primary star's angular velocity and its rate of change is achieved, which is used for attitude synchronization control, significantly improving the formation response speed and stability. For ease of performance evaluation and simulation verification, the following estimation and error variables are defined: For the ... One follower ( ) and state components ,make Indicates the first The first follower to the leader Dimensional state The estimated value; This represents the corresponding state estimation error. Similarly, let... Indicates the first The first follower to the leader dimensional derivative state The estimated value; This represents the corresponding derivative state estimation error.

[0127] The observer using the method of this invention can reduce the estimation error. and At the preset time The time interval converges precisely to zero and remains zero thereafter. The state estimation errors and derivative state estimation errors of each follower can be obtained separately through... Figure 2 and Figure 3 The simulation curves are clearly displayed, with the horizontal axis representing time. The vertical axis represents the estimation error for each dimension.

[0128] The above examples of the present invention are merely illustrative of the computational model and process of the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is impossible to exhaustively list all possible implementations here. Any obvious variations or modifications derived from the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A fully distributed, time-limited convergent state observation method, characterized in that, The method specifically includes the following steps: Step 1: Set the preset convergence time constant gain Robust gain Initialize the first Node pairs The estimated value , No. Node pairs The estimated value ; in, Indicates the leader's status. The derivative representing the leader's state; Step 2: Construct a leader state observer and use it to observe the leader's state.

2. The fully distributed, time-limited convergent state observation method according to claim 1, characterized in that, The operation process of the state observer is as follows: Step Two 1: Initialization ; Step 22: Calculation Time of the first Auxiliary estimation error variables for each node and ; in, Indicates the leader's status. The derivative representing the leader's state, express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, Represents the adjustment function. Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node; Steps two and three: Calculation Time of the first Consistency error of each node and ; in, Indicates the first The state consistency error corresponding to each node Indicates the first Consistency error of the state derivatives for each node Indicates the first Error variables for auxiliary state estimation of each node, Indicates the first Error variables for the estimation of auxiliary state derivatives of each node. Indicates the total number of nodes. Indicates from node To the node Connection weights; Step 24: Based on consistency error and Conduct the first Dynamic gain update of each node; Step 25, according to , , , , and Conduct the first Update the state estimate of each node; in, and Indicates intermediate variables. and Indicates dynamic gain. express The first derivative, express The first derivative; Step 26: Solve the differential equation for state estimation and update in Step 25 to obtain... Time of the first The state estimation results of each node for the leader and the Estimation results of the derivatives of each node with respect to the leader state ; judge Has the preset convergence time been reached? ; If the target is reached, then the estimate obtained in the last iteration will be used as the result of the first iteration. The estimation results of the leader state of each node and the estimation results of the derivative of the leader state; If it is not achieved, then let Return to step two.

3. The fully distributed, time-limited convergent state observation method according to claim 2, characterized in that, The adjustment function Defined using the exponential decay form: Where, constant .

4. The fully distributed, time-limited convergent state observation method according to claim 3, characterized in that, The auxiliary estimation error variables for the leader's corresponding node are as follows: in, and This represents the auxiliary estimation error variable for the node corresponding to the leader.

5. The fully distributed, time-limited convergent state observation method according to claim 4, characterized in that, The specific process of step two or four is as follows: in, and Indicates intermediate variables. and Indicates dynamic gain. express The first derivative, express The first derivative.

6. The fully distributed, time-limited convergent state observation method according to claim 5, characterized in that, The specific process of step two five is as follows: in, express The first derivative, Represents a symbolic function. express Time of the first Node pairs The estimated value, express Time of the first Node pairs The estimated value, express The first derivative, express The first derivative.

7. The fully distributed, time-limited convergent state observation method according to claim 6, characterized in that, The numerical integration method is used to solve the differential equation for state estimation update in step two-five.

8. The fully distributed, time-limited convergent state observation method according to claim 7, characterized in that, The time-varying gain function for: in, These are design parameters.

9. The fully distributed, time-limited convergent state observation method according to claim 8, characterized in that, The time-varying gain function for: in, These are design parameters.

10. A fully distributed state observation method with predetermined time convergence according to claim 9, characterized in that, The design parameters and satisfy: ,and .

Images