Uncertain nonlinear multi-agent consistency control method based on tracking

Abstract

The invention relates to the technical field of intelligent control, in particular to an uncertain nonlinear multi-agent consistency control method based on tracking. The method comprises the following steps: establishing a multi-agent kinematics model; designing a tracking controller based on an observer, and compensating nonlinear / unknown disturbance; a distributed consistency controller is designed to realize bounded consistency of reference signals; and integrating and optimizing system parameters. The invention provides a tracking-based uncertain nonlinear multi-agent consistency control method, which solves or at least alleviates the limitations of robustness, computational efficiency and information dependency of the existing uncertain nonlinear multi-agent system under strong nonlinearity, uncertainty and external disturbance, and controls a high-order system through simple low-order design. The requirement for an intelligent agent system and the algorithm design complexity are reduced, the application range of the control method is expanded, and meanwhile, the performance optimization balance is achieved through integrated optimization selection of parameters.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of intelligent control, and particularly relates to a tracking-based uncertain nonlinear multi-agent consensus control method. BACKGROUND

[0002] Multi-agent systems have made significant progress in many cutting-edge technology fields due to their inherent distributed collaborative advantages, high flexibility and robustness. From complex robot formation, large-scale task scheduling, to intelligent transportation systems, distributed sensor networks, and even unmanned aerial vehicle swarm control, the depth and breadth of multi-agent system applications continue to expand. Under this background, consensus control, as one of the most basic and crucial group behaviors in multi-agent systems, aims to design distributed protocols to gradually converge or converge in a certain sense to a unified target for all relevant states (such as position, velocity, or attitude) of the agents in the system, thereby providing a solid technical foundation for achieving higher-level collaborative tasks. Although the research on multi-agent system consensus control methods has made certain progress, it is generally assumed that the agent's own dynamics is a simple linear system, and the existence of uncertain factors such as model parameter perturbation, system unmodeled dynamics, and external disturbances will lead to highly nonlinear dynamics of the agent. The existing multi-agent system consensus control methods have certain limitations and deficiencies when facing these problems, which further hinders the performance improvement of multi-agent systems.

[0003] In order to solve the above problems, the skilled in the art has carried out a series of research and put forward a series of technical schemes, the invention patent with publication number CN115268275B discloses a kind of multi-agent system consistency tracking method and system based on state observer, its technical scheme discloses a kind of distributed event-triggered iterative learning control framework based on observer, for solving the problem existing in multi-agent system consistency control method.The scheme aims at solving the robust consistent tracking problem of heterogeneous local Lipschitz nonlinear multi-agent system with unmeasurable state.Specifically, the method relaxes the strict requirement of system on initial state consistency by designing initial state learning strategy;At the same time, event-triggered mechanism is introduced, so as to significantly reduce unnecessary communication frequency and computing resource occupation between agents under the premise of ensuring system performance.The application of multi-agent system in the scene of state unmeasurable and resource limited is expanded to a certain extent by the proposal of this technology, which embodies the active attempt in meeting the performance requirements of part of nonlinear system.However, the processing capacity of nonlinear system of this technical scheme still has its inherent limitations.It mainly depends on local Lipschitz continuity assumption, which makes it difficult to guarantee the robustness and stability of its performance when facing the strong nonlinear, non-smooth or highly uncertain dynamic characteristics commonly existing in actual engineering.In addition, the event-triggered mechanism designed by the patent shows high sensitivity to the dynamic switching of communication topology, and under the frequently changing communication topology structure, the system may face the risk of significant decrease in robustness, and its suppression ability to unknown external disturbance is relatively weak.

[0004] Further, the patent with publication number CN110109351B provides a multi-agent consistency control method based on specified performance. The core of this scheme is to design a performance function to transform the tracking error between agents, and on this basis, combined with the classic arctangent method and dynamic surface technology, an event-triggered adaptive controller is designed for the follower agent. Through this series of technology integration, this method successfully realizes the fast and high-precision consistency control of the multi-agent system, and can impose strict constraints on the transient performance and steady-state error of the system, thereby meeting the stringent requirements of control accuracy and response speed in specific application scenarios. This technology has great advantages in ensuring transient performance and steady-state error, and has a great driving effect on the performance optimization of complex high-order multi-agent systems. However, everything has two sides, and this technology also brings a big technical contradiction. In order to ensure such strict constraints on transient performance and steady-state error, the control algorithm designed by the system will be more complex, especially for high-order multi-agent systems, at which time the computational burden of the designed controller will greatly increase, which not only affects the real-time implementation of the control law, but also may become a bottleneck on resource-limited embedded systems. Since the backstepping method and the dynamic surface method are currently effective methods for solving high-order nonlinear systems, but these two methods themselves have a recursive design structure, any step needs to perform differentiation on the virtual controller (or approximate through a filter), when the system has more uncertainties or needs complex adaptive laws to compensate, the burden will grow in the form of a series. In addition, this method has low requirements for the state information of the leader, which means that in actual applications, when the state information of the leader agent is affected by unpredictable noise or disturbance, the performance degradation will be transmitted to all followers in the form of a series through the virtual controller, which will lead to the growth of tracking error, and even lead to the instability of the entire multi-agent system, which is relatively fragile when facing complex external dynamic environments.

[0005] The above indicates that the current technology has a potential dilemma in improving the control performance of multi-agent systems under nonlinear, uncertainty, and external disturbance conditions: by increasing the algorithm complexity and stronger assumptions to improve local performance indicators, it often comes at the cost of increasing the algorithm's computational burden, reducing the universal robustness of strong nonlinearities or unknown disturbances, and greater dependence on specific information (such as the state of the leader). This design concept leads to a bloated structure of the controller, excessive computational load, and limited adaptability and universality in dealing with strong nonlinearities, high uncertainty, and complex external disturbances that cannot be fully modeled in the real world when pursuing high precision and strict performance guarantees. In other words, existing methods inadvertently introduce new complexity or vulnerability in another dimension while trying to solve complexity in one dimension. Therefore, how to design a new multi-agent consensus control method that can effectively optimize the processing capability of strong nonlinear dynamic characteristics, significantly enhance the robustness to model uncertainty and unknown external disturbances, and at the same time, ingeniously simplify the control algorithm structure to reduce the overall computational complexity, so as to achieve a better balance between multi-dimensional performance indicators, to meet the urgent needs of efficient and reliable multi-agent collaborative control in complex real-world scenarios, has become a key challenge and technical problem to be solved for those skilled in the art. SUMMARY

[0006] The purpose of the present application is to provide a tracking-based uncertain nonlinear multi-agent consensus control method that can solve or at least alleviate the performance limitations of robustness, computational efficiency, and information dependence of existing uncertain nonlinear multi-agent systems under strong nonlinearities, uncertainty, and external disturbances.

[0007] To achieve the above-mentioned purpose, the present application provides the following technical solution: a tracking-based uncertain nonlinear multi-agent consensus control method, comprising the following steps: Step one, model establishment and transformation of the multi-agent system, the step includes establishing an uncertain nonlinear kinematic model of an agent in a two-dimensional Euclidean plane, and transforming the kinematic model into a second-order nonlinear differential equation form, wherein the uncertain nonlinear kinematic model includes the inertial effect and velocity-dependent damping effect experienced by the agent during motion; Step two, designing a dynamic tracking controller for each of the agents, the dynamic tracking controller aims to make the agent track a time-varying reference signal while effectively suppressing the influence of uncertain nonlinear dynamics and external disturbances within the system, the dynamic tracking controller includes an observer module for real-time estimation of unknown nonlinear terms in the system; Step 3: Design a multi-agent consensus controller, which aims to facilitate the consensus among the agents. The time-varying reference signals of the follower agents converge to the reference signal of the leader agent, and the consensus controller implements coordinated control based on the communication topology between the agents; and Step four: Under the action of the dynamic tracking controller and the consistency controller, appropriate parameters are selected according to the motion state of the multi-agent system to complete the consistency control of the multi-agent system. The parameter selection must ensure that the entire control system meets the stability requirements.

[0008] To further realize the present invention, the following technical solutions may be preferred: Preferably, in step one, an uncertain nonlinear kinematic model is established, including an inertia matrix and an uncertain damping matrix. The inertia matrix is ​​in diagonal matrix form, and its diagonal elements are positive definite constants. The uncertain damping matrix is ​​also in diagonal matrix form, and its diagonal elements are non-negative continuous functions related to the velocity components, used to describe the damping characteristics of the intelligent agent at different velocities.

[0009] Preferably, in step one, the kinematic model is transformed into a second-order nonlinear differential equation form, including defining the agent's output as the position state, the first state variable as the position state, the second state variable as the velocity state, and the new control input as the product of the inverse of the inertia matrix and the original control input, thus transforming the kinematic model into a standard second-order nonlinear system form, which facilitates subsequent controller design.

[0010] Preferably, the design of the dynamic tracking controller in step two includes: defining the tracking error as the difference between the actual output of the agent and the reference signal; defining the virtual tracking error as the difference between the actual speed state and the desired virtual control input; designing a virtual controller to generate the desired virtual control input, wherein the gain function of the virtual controller satisfies a preset input-to-state stability condition to ensure the stability of the tracking error subsystem; and obtaining the tracking error subsystem, wherein the tracking error subsystem describes the dynamic change characteristics of the tracking error.

[0011] Preferably, the design of the dynamic tracking controller in step two further includes: defining the unknown nonlinear term as the value of the system nonlinear function at zero; obtaining a speed tracking error expression, wherein the speed tracking error expression includes a composite nonlinear term, an unknown nonlinear term, and a control input; and the composite nonlinear term in the speed tracking error expression satisfies a specific growth condition, wherein the growth condition includes the growth characteristics of the composite nonlinear term relative to the tracking error, the virtual tracking error, and the rate of change of the reference signal.

[0012] Preferably, the design of the dynamic tracking controller in Step 2 further comprises: designing an unknown term observer, which adopts a first-order filter structure to estimate the unknown nonlinear term in the system in real time, the observer including an internal state vector and observer gain parameters; and obtaining an observation error subsystem, which includes the dynamic variation characteristics of the observation error, the stability of which is affected by the observer gain parameters and the composite nonlinear term.

[0013] Preferably, the design of the dynamic tracking controller in Step 2 further comprises: designing a real controller to generate the final control input, the real controller including a nonlinear feedback term of the virtual tracking error and an observation compensation term of the unknown nonlinear term, the gain function of the real controller satisfying a preset input-to-state stability condition and a multi-loop nonlinear small gain condition, the multi-loop nonlinear small gain condition ensuring the stability of the entire error coupling system and avoiding mutual interference between error subsystems leading to system instability; and obtaining a speed tracking error subsystem, which describes the dynamic variation characteristics of the speed tracking error.

[0014] Preferably, the design of the consistency controller in Step 3 comprises: defining the position measurement topology of the multi-agent system as a directed graph, which describes the communication connection relationship between agents; generating the reference signal of each agent through a first-order integrator system, the state of the first-order integrator system being the reference signal and the input being the output of the consistency controller; and the consistency control requires that the reference signal convergence condition be met, that is, the deviation between the reference signal of all follower agents and the reference signal of the leader agent at steady state is less than a preset threshold.

[0015] Preferably, the specific way of the consistency controller in Step 3 is: by defining the relative position error as the position difference between the follower agent and the leader agent, designing a consistency controller to generate the desired speed input, the consistency controller calculating the control input based on the difference between the relative position error and the weighted average of the relative position errors of the neighbor agents, the control input being used to drive the first-order integrator system of the reference signal, so as to realize the consistency of the reference signal.

[0016] Preferably, the consistency control of the plurality of agents in the step four is completed, comprising: establishing a system integration and information flow mechanism, the mechanism ensures that the reference signal generated by the consistency controller can be correctly used by the dynamic tracking controller, and the control input generated by the dynamic tracking controller can effectively drive the motion of the agent; and optimizing the selection of key parameters, the key parameters include observer gain parameters, dynamic tracking controller gain function parameters and consistency controller gain parameters, the parameter selection needs to meet the stability condition of the control system, and the performance requirements in the actual application scene are considered, the observer gain parameters affect the estimation accuracy and speed of the unknown nonlinear term, the dynamic tracking controller gain function parameters affect the tracking performance and anti-interference ability, and the consistency controller gain parameters affect the reference signal convergence speed and system stability.

[0017] The beneficial effects of the present application are: The present application proposes a consistency control method for uncertain nonlinear multi-agent system based on tracking control, establishes an uncertain nonlinear model of multi-agent, designs a dynamic tracking controller based on observer for each agent, designs a consistency controller based on multi-first integrator system, and the two controllers together constitute the total controller of the control method, and thus the consistency control of the multi-agent system is realized. The specific advantages are as follows: (1) The present consistency control method is proposed based on the idea of tracking control, which greatly reduces the requirements for the agent itself, expands the application range, and realizes the design of low-order system through the idea of tracking; (2) The consistency controller in the present consistency control method is designed based on multi-first integrator system, compared with the design of consistency controller based on direct tracking of multi-agent model, the controller design process is greatly simplified. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1 The flow chart of the control method of the present application; Figure 2 The gain directed graph of the error association system of the present application and the schematic diagram of the small gain condition met; Figure 3 The closed-loop system block diagram of the control method of the present application; Figure 4 The schematic diagram of the position measurement topology of the present application; Figure 5 The schematic diagram of the motion trajectory of the multi-agent system of the present application on the plane; Figure 6 The schematic diagram of the evolution of position, velocity and error of the multi-agent system of the present application in the direction; Figure 7The multi-agent system of the present application is in - Evolution of position, velocity, error in direction. DETAILED DESCRIPTION

[0019] In the description of the present application, it should also be noted that, unless otherwise explicitly specified and limited, the terms "arrangement", "installation", "connected", "connected" should be understood broadly, for example, it can be fixedly connected, or it can be detachably connected, or integrally connected; it can be mechanically connected, or it can be electrically connected; it can be directly connected, or it can be indirectly connected through an intermediate medium, or it can be connected inside two elements. For those skilled in the art, the specific meaning of the above terms in the present application can be understood according to the specific circumstances.

[0020] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application. Embodiment one

[0021] The present embodiment discloses a tracking-based uncertain nonlinear multi-agent consensus control method, as shown in Figure 1 The system model establishment, robust dynamic tracking controller design, multi-agent consensus controller design, and system integration and parameter optimization strategy of the present application will be described in detail below according to the logical order of the invention content.

[0022] I. Model establishment and transformation of uncertain nonlinear multi-agent system This part aims to describe in detail the mathematical model construction of the multi-agent system and its transformation process to the standard differential equation form, laying a foundation for the subsequent controller design.

[0023] In a specific embodiment, the present application considers a multi-agent system composed of n+1 agents, including one leader agent (numbered 0) and n follower agents (numbered from 1 to n). All agents move in a two-dimensional Euclidean plane, and their state evolution is represented in an inertial coordinate system.

[0024] 1.1 Establishment of uncertain nonlinear kinematic model of multi-agent in plane For any agent in the set its instantaneous position in the inertial coordinate system is described by a two-dimensional vector , where represents the coordinate component of agent i in the east axis of the inertial coordinate system, and where represents its coordinate component in the north direction of the inertial coordinate frame. Meanwhile, the instantaneous velocity of agent i in its body coordinate frame is represented by a two-dimensional vector where represents the velocity component of agent i along its forward direction, where represents the velocity component of agent i along its lateral direction.

[0025] Based on the above physical quantity definitions, the kinematic model of agent i in the plane is established as a differential equation, which is expressed as follows:

[0026]

[0027] This model captures the dynamic behavior of the agent in the physical space. In this model, where represents the resultant control force and control moment that agent i receives. Specifically, is the eastward control component in the inertial coordinate frame, while is the northward control component. These control inputs are generated by the actuators inside the agent (e.g., propeller thrusters on a UAV, rudders or thrust vector systems on an underwater vehicle, wheeled drive motors on a ground robot), whose goal is to control the position and velocity of the agent.

[0028] where represents the inertia matrix of agent i. This matrix is a diagonal matrix, whose diagonal elements represent the equivalent inertia mass or inertia moment of agent i in the east and north directions, respectively. These values are usually positive constants and represent the agent's ability to resist changes in motion state when subjected to external forces.

[0029] where represents the uncertain damping matrix of agent i. This matrix is also a diagonal matrix, whose diagonal elements are defined as non-negative continuous functions related to the velocity components. These functions describe various resistances in the agent's motion, such as fluid viscous resistance in underwater robots, aerodynamic resistance in UAVs, or ground friction in ground robots. Since resistance usually depends on velocity in a nonlinear manner (e.g., a combination of linear damping, quadratic damping (or even higher order)), and its strength may be disturbed by external environments (e.g., changes in wind speed, changes in water flow, changes in medium density, changes in road roughness), this damping matrix is modeled as an uncertainty function. This mathematical description of uncertain nonlinear damping is the core starting point for the invention to handle strong nonlinear and uncertainty challenges.

[0030] 1.2 Converting the planar kinematic model of multiple agents into standard differential equations​ To simplify the subsequent controller design and theoretical analysis process, this invention transforms the aforementioned physical kinematic model into a standard, control theory-friendly state-space form of differential equations. Before this transformation, a mathematical term used to describe the characteristics of functions is introduced. Class functions and Definition of a function class. For a continuous function... ,if It is strictly increasing and satisfies Then it is called for Function class. If yes Class function and simultaneously satisfy Then it is called for Class functions. These functions are the cornerstone for constructing gain functions, Lyapunov functions, and analyzing system input-to-state stability (ISS).

[0031] Next, the present invention defines The output of agent i is physically represented by its position in the inertial coordinate system. At the same time, new state variables are introduced. and .in, This represents the position and state vector of agent i. This represents the velocity state vector of agent i. To further simplify the form of the control input, this invention defines a new control input. Through the inertia matrix Physical control torque Decoupling is an acceleration command that directly acts on the system velocity. This allows the controller design to be directly targeted at acceleration rather than the original torque.

[0032] Based on the redefinition of the state variables and control inputs described above, the planar kinematic model of the intelligent agent established in the first sub-step of step one is rewritten in the form of the following second-order nonlinear differential equation:

[0033]

[0034]

[0035] Among them, core items This term represents the inherent nonlinear dynamic characteristics of agent i, which includes uncertain damping effects and is the main source of nonlinearity and uncertainty in this system. Its specific component functions... and The nonlinear effects under the influence of the eastward and northward velocity components are described respectively.

[0036] In order to include The nonlinear dynamics in the process are rigorously handled, and this invention requires that this function satisfy the following properties:

[0037] in, Let be any real number, .function This is a non-negative continuous function whose value depends on the agent's velocity state, used to represent the gain characteristics of the nonlinear term. for A class function represents the growth characteristic of a nonlinear term with respect to the state difference. For example, it can be set to... It is a constant. in ,or Etc. This property shows that, although It is nonlinear, but its changing characteristics can be represented by a... The model is defined by a class function and a non-negative continuous function. This definition provides a mathematical foundation for handling the complexity of controller design, especially in the presence of non-global Lipschitz continuity, ensuring the applicability of theoretical analysis and the robustness of controller design. Through this model transformation and property description, the present invention can effectively capture the real dynamics of the agent and lay the foundation for subsequent tracking and consistency controller design, thereby effectively addressing the strong nonlinear challenges mentioned in the background art.

[0038] II. Design of a Robust Dynamic Tracking Controller for Agent i This section details a method for designing a robust dynamic tracking controller for each agent i, with the core objective of enabling the agent to track a time-varying reference signal. This controller effectively suppresses the uncertain nonlinear dynamics within the system and the effects of external disturbances. It combines the backstepping method with observer technology and employs rigorous design and analysis based on input-to-state stability (ISS) theory and multi-loop nonlinear small-gain conditions.

[0039] 2.1 Design a virtual controller for agent i to obtain Error subsystem First, this invention defines the time-varying reference signal that the intelligent agent i wishes to track as... This signal represents the ideal position trajectory that agent i is expected to reach in the inertial coordinate system. To quantify tracking performance, this invention defines the position tracking error of agent i as... ,in and These are the position tracking error components for the east and north directions, respectively.

[0040] Considering that the agent model includes a first-order integrator dynamic This invention will use state variables This is considered as a virtual control input in the backstepping design. Based on this, the virtual tracking error is defined. ,in The desired virtual control input is represented by the speed that agent i expects to achieve.

[0041] Tracking error By taking the derivative over time and combining it with the agent model and the definition of virtual error, we can obtain its dynamic equation:

[0042] To ensure The present invention designs a virtual controller that can asymptotically converge to zero or a preset small range boundary. :

[0043] in, This represents the Hadamard product, which is the product of corresponding elements.

[0044] It is a nonlinear gain vector. Its components... (for These functions are designed to be non-decreasing, continuously differentiable positive functions. These functions directly affect... The convergence rate of the error and its impact on subsequent errors and the rate of change of the reference signal The ability to suppress disturbances.

[0045] To ensure the input-to-state stability of the system under nonlinear coupling, The function must satisfy the following inequality conditions:

[0046] in, , , All are arbitrarily specified local Lipschitz continuous. Function-like.

[0047] Represents virtual error arrive The input gain of the subsystem serves to characterize right The impact on the stability of the subsystem.

[0048] Represents the rate of change from the reference signal arrive - The input gain of the subsystem is used to characterize the effect of changes in the reference signal on the tracking error.

[0049] This is a way to ensure - The gain function of a subsystem that converges asymptotically when there are no other inputs.

[0050] By designing these Class functions and The function, this invention ensures The subsystem receives and When used as input, it can maintain the input to a stable state (ISS), that is, the output (error) is bounded under bounded input.

[0051] Virtual controller Substitution From the expression, we can obtain the following: -Error subsystem:

[0052] This subsystem represents the position tracking error. In nonlinear gain Under the influence of virtual tracking error and the rate of change of the reference signal The dynamic behavior of the influence.

[0053] 2.2 Designing a real controller for agent i and obtaining... -and -Error Subsystem This invention addresses virtual tracking errors. Perform control design. Regarding the time derivative, and in conjunction with the agent model and virtual controller From the definition and its derivative, we can obtain:

[0054] To better handle nonlinear terms, this invention will... The term is decomposed into two parts. Definition This term represents the uncertain dynamic component experienced by the agent when the velocity is zero; it is typically an unknown constant or a slowly varying perturbation. The remaining nonlinear terms are combined with the derivative terms of the virtual controller into a composite function. .then, The expression can be rewritten as:

[0055] in, .for Quantity, .

[0056] This composite function Includes agent nonlinear dynamics Part of the virtual controller The differential term, and its relation to the error. and the rate of change of the reference signal The complex coupling relationship between them. In the traditional backstepping method, the calculation Need to Performing differentiation can introduce the computational explosion problem. This invention effectively manages this complexity through subsequent observer design and gain function selection.

[0057] Furthermore, according to the first sub-step The properties and the second sub-step of step one The Lipschitz-like properties that are satisfied determine the existence of three Class function , , , making The absolute value of satisfies the following inequality:

[0058] This inequality provides a solution for composite nonlinear terms. A strict upper bound estimate, the bound of which is determined by the error. , and the rate of change of the reference signal of Defined by class functions. This definition avoids [the need for] [specific definitions]. The accurate modeling of the term relies solely on its upper bound properties, which greatly simplifies controller design.

[0059] To estimate unmeasurable unknown constants or slowly varying perturbations This invention designs a state observer:

[0060]

[0061] in, Let be the internal state vector of the observer, whose dynamics are described by the above differential equation.

[0062] For unknown terms The real-time estimate. This observer utilizes measurable control inputs. and virtual tracking error To dynamically estimate This avoids the complex parameter identification process in traditional adaptive methods.

[0063] It is a component containing positive definite constants. The gain vector. These gain constants determine the convergence speed of the observer estimate and its robustness to measurement noise. Typically, larger ones are chosen. The value can achieve faster estimation convergence, but it may amplify measurement noise and cause oscillations in the control output, so a careful trade-off is required.

[0064] The observation error is defined as ,express The deviation between the actual value and the estimated value. This is due to observation error. Differentiate and combine The expression can be obtained as follows: -Observation error subsystem:

[0065] This error dynamic indicates that the observation error Convergence is affected by the observer gain The provided exponential decay term and composite nonlinear term The impact. If , and If something is inherently bounded, then the observation error can be confined to a bounded region.

[0066] The present invention designs the following real controller:

[0067] This controller consists of two parts: Part One It is a nonlinear feedback control term whose function is to directly stabilize the virtual tracking error. Through nonlinear gain The controller can provide strong correction when there is a large error, while maintaining smoothness when there is a small error to avoid chattering.

[0068] Part Two It is a compensation term based on observer estimates, used to offset unknown nonlinear terms in real time. The resulting impact. This compensation mechanism is the key means of this invention to handle model uncertainty, unmodeled dynamics, and external disturbances, significantly improving the robustness of the system.

[0069] in, It is a nonlinear gain vector, whose components (for It is designed to be a non-decreasing, continuously differentiable positive function.

[0070] To ensure the closed-loop stability of the system, The function must satisfy the following inequality conditions:

[0071] in, , , , , All are arbitrarily specified local Lipschitz continuouss that satisfy a specific small gain condition. Function-like.

[0072] Here , , The aforementioned is used to define of Function-like terms. These complex combinations reflect... The subsystem is affected not only by its own state and control actions, but also by external factors. , and the rate of change of the reference signal The effects of cross-coupling.

[0073] The key small gain conditions are as follows:

[0074]

[0075]

[0076] in, Let represent the identity function. These conditions ensure the entire closed-loop system (including ) , and The three interconnected subsystems ultimately achieve the key mathematical constraints for input-to-state stability (ISS). These constraints require that the gains between the subsystems, under combined action, cannot exceed the identity function, thus avoiding gain explosion or instability. Through careful selection of these... Class functions and controller gains can ensure that these strict mathematical conditions are met.

[0077] controller From the expression, we can obtain the following: -Error subsystem:

[0078] Thus, this invention yields two interrelated error subsystems: -Error subsystem and - Observation error subsystem. Together, they describe the dynamics of the velocity tracking error and unknown term estimation error of agent i under the action of the controller.

[0079] 2.3 Dynamic tracking controller integration and closed-loop system stability analysis of agent i Combining the controller designs of the first and second steps, this invention designs the following dynamic tracking controller for each agent i: First, based on the agent's actual location and expected reference signal Calculate the position tracking error and design the desired virtual speed accordingly:

[0080]

[0081] Next, using the agent's actual speed And the above-mentioned expected virtual speed Calculate the speed tracking error and combine it with the estimated value of the unknown term. Generate actual physical control inputs:

[0082]

[0083] Meanwhile, in order to update the estimates of unknowns in real time, the observer state and the estimates of unknowns need to be continuously updated:

[0084]

[0085] Under the action of this dynamic tracking controller, the closed-loop system of agent i is transformed into an error-correlated system, whose dynamic behavior is described by the following set of differential equations:

[0086]

[0087]

[0088] To demonstrate the stability of this correlation system, the present invention addresses each component. Three locally Lipschitz continuous inputs to state-stable (ISS) Lyapunov functions are introduced respectively: , and These functions provide a measure of distance from the origin in the error space, and their selection is based on the validity of the ISS property proven in control theory.

[0089] Based on the gain function in the second sub-step and Design conditions and By defining the Lyapunov function, we can obtain the following Lyapunov function conditions and their time derivative inequalities, thereby enabling a quantitative analysis of system stability: for : exist Class function and , making when When, its time derivative satisfies:

[0090] This inequality shows that when the position error is large enough, Monotonically decreasing, ensuring The subsystem receives from and Given a bounded input, it can achieve bounded convergence on its own. This is achieved by choosing a sufficiently large... Gain is used to drive the dynamic realization of error.

[0091] for : exist Class function , and , making when When, its time derivative satisfies:

[0092] This inequality shows that when the speed error is large enough, Monotonically decreasing, ensuring The subsystem receives from Given a bounded input, it can achieve bounded convergence. Here... The gain must be large enough to overcome The impact.

[0093] for : exist Class function , and , making when When, its time derivative satisfies:

[0094] This inequality shows that when the observation error is large enough, Monotonically decreasing, ensuring The subsystem receives from Given a bounded input, it can achieve bounded convergence, such as... Figure 2 As shown.

[0095] In summary, for each agent i, under the action of the designed dynamic tracking controller, its closed-loop system is transformed into an error-correlated system. By selecting... and Function and observer gain This ensures that these correlated gains satisfy the aforementioned multi-loop small gain condition. These conditions guarantee that despite the complex nonlinear couplings between subsystems, the overall system remains input-to-state stable. Specifically, for each component... The designed dynamic tracking controller for agent i can ensure that the closed-loop system changes at the rate of change of the reference signal. As input, the error state vector As a state, it is the input that stabilizes the state. This means that as long as the rate of change of the reference signal... If the tracking error remains bounded, then... , and observation error All of these will remain bounded, thus achieving robust tracking of uncertain nonlinear dynamics and external disturbances, effectively overcoming the limitations of existing technologies in terms of insufficient ability to handle strong nonlinearities and weak ability to suppress unknown disturbances.

[0096] III. Design of Multi-Agent Consensus Controller This section details a method for designing a distributed consensus controller for multi-agent systems, the core objective of which is to ensure that the reference signal of all follower agents is consistent. The reference signal that can asymptotically approach or converge to the leader agent 0 in a certain sense This provides a unified action benchmark for collaborative tasks across the entire multi-agent system.

[0097] First, this invention defines the position measurement topology of a multi-agent system as a directed graph. .

[0098] in, This represents a set of intelligent agents, consisting of a leader agent 0 and n follower agents 1 to n.

[0099] Let represent the set of all directed edges. If there exists an edge from agent j to agent i... This indicates that agent i can obtain the position information of agent j. This topology can be fixed or time-varying, and its dynamic characteristics will directly affect the performance and robustness of the consensus controller.

[0100] Considering the time-varying reference signal of agent i in step two... This invention defines it as being generated by the following first-order integrator system:

[0101]

[0102] in, Let be the state vector of the integrator, and its physical meaning is the ideal position trajectory that agent i expects to track.

[0103] This is an inherently bounded input vector, whose physical meaning is the ideal speed that agent i hopes to achieve. This is the quantity that the consistency controller needs to be designed for in this step. Through control... It can be indirectly controlled This dynamics allows for consistency.

[0104] This invention defines agent 0 as the leader agent, whose reference signal is... (Depend on The goal of achieving convergence for the other n follower agents is to generate (the system of n+1 first-order integrators). Consistent control of the agents is achieved through tracking, with the core objective of ensuring that the aforementioned system of n+1 first-order integrators achieves consistent control within a finite time or asymptotically. Specifically, it requires the following conditions to be met: in, For a sufficiently small constant. This condition indicates that, over time, the reference signal of each follower agent is a sufficiently small constant. Reference signal with leader agent 0 The deviations will eventually converge to a pre-defined, sufficiently small constant boundary. Within this bounded consistency, it is more practical in engineering because it allows the system to achieve coordination with a small range of errors, rather than demanding theoretically absolute zero-error convergence.

[0105] To design a consistency controller, this invention addresses each follower agent. The error signal relative to the leader agent 0 is defined as follows: This indicates the deviation between the follower's reference position and the leader's reference position.

[0106] This indicates the deviation between the speed expected by followers and the speed expected by the leader.

[0107] Based on the above definition of error, the consistency problem of a first-order integrator system can be transformed into the following error system:

[0108] This invention addresses the position measurement topology of multi-agent systems. By defining the relative position error of the agent (here) It is the actual physical location, not the reference location. This means that the consensus controller directly uses the agent's actual position information to generate a reference velocity, and then the tracking controller ensures that the actual physical position can track this reference velocity, thereby achieving overall consistency. The following distributed consensus controller was designed to generate the desired velocity input. (or its deviation) ): in, The relative position error of the leader agent 0 is zero because it is the reference benchmark.

[0109] Let be a positive definite constant, representing the gain of the consensus controller. The magnitude of these gains determines how quickly the follower agent converges to the average position of its neighbors. Larger gains generally mean faster convergence, but may also introduce system oscillations or make the system more sensitive to communication noise.

[0110] Let be a positive definite constant, representing the influence weight of agent j on agent i. In common communication topologies, if agent i can receive the location information of agent j (i.e., ... ),but The denominator is usually positive (e.g., a constant 1), otherwise it is 0. This represents the sum of the weights of all of agent i's neighbors, used to normalize neighbor information and ensure the appropriate amplitude of the control signal.

[0111] This represents the set of neighbors of the follower agent i, meaning that agent i can directly obtain information from these neighboring agents.

[0112] This consensus controller is a distributed protocol, where each follower agent only needs to utilize its own relative position error. and the relative position error of its neighboring agent j The desired speed deviation can then be calculated. This characteristic of local information interaction significantly reduces the system's communication burden and computational concentration, improving its scalability and robustness. The controller drives the reference signal of each follower agent to converge to a weighted average of its neighbors, ultimately achieving bounded consistency between the reference signals of all follower agents and the leader agent. Then, these... As the essentially bounded input to the tracking controller of each agent in step two, the driver Then, the tracking controller in step two ensures that the actual intelligent agent... track This achieves physical consistency across the entire system. This layered design effectively decouples the complex agent dynamics from the consistency protocol, a key strategy in this invention that maintains high performance while reducing algorithm complexity.

[0113] IV. Under the action of the designed tracking controller and consensus controller, appropriate parameters are selected based on the motion states of the multiple agents to achieve consensus control of the multiple agents. The overall block diagram of the closed-loop system is shown below. Figure 3 .

[0114] This section details the application of the theoretical framework designed in the previous three steps to a practical multi-agent system, and the refinement of parameter configuration and system integration to achieve robust and consistent control of uncertain nonlinear multi-agent systems. This step is the engineering implementation phase of the entire invention method, requiring comprehensive consideration of system state, environmental conditions, and desired performance indicators.

[0115] 4.1 Establishment of System Integration and Information Flow Mechanism The core of this invention is to achieve seamless connection and efficient information flow between the dynamic tracking controller designed in step two and the multi-agent consensus controller designed in step three. The specific implementation mechanism is as follows: 1. The output of the consensus controller serves as the input of the tracking controller: The consensus controller designed in step three is designed for each follower agent. Based on its own relative positional error and that of its neighbors Calculate and output the expected speed deviation. .because ,and The essential bounded inputs of the leader agent (whose values ​​are known or can be obtained through communication, for example, the leader can broadcast its desired speed) (Given to all followers), thus each follower agent can obtain its own desired velocity input. .this That is, the input of the first-order integrator system defined in step three, used to generate the reference signal for agent i. .

[0116] 2. Reference signal generation: Each agent i receives its corresponding desired velocity input. Then, by integrating it, its time-varying reference signal is obtained. ,Right now ,in Meanwhile, in order to provide the tracking controller in step two... (Right now ), The value is directly used as The input is given to the tracking controller. This direct mapping avoids numerical differentiation of the reference signal, thus reducing the risk of noise introduction.

[0117] 3. Tracking controller execution: The dynamic tracking controller of each agent i (step two) receives its current actual position. Actual speed and the aforementioned generated consistency reference signal and its rate of change Based on these inputs, the controller first calculates the position tracking error. Expected virtual speed Then calculate the speed tracking error. Meanwhile, its internal unknown observer is based on and current control input Real-time estimation of unknown and uncertain dynamic components Ultimately, the tracking controller outputs the actual physical control input. This input is converted into actual torque through the agent's actuator module. , the dynamic system acting on agent i.

[0118] 4. State Feedback and Iteration: When agent i receives control input... After that, its position and speed The state is updated based on the kinematic model established in step one. The updated state is fed back through the sensor as the input for the next control cycle, forming a complete closed-loop control loop for continuous real-time control and state updates.

[0119] 4.2 Optimization and Configuration Strategies for Key Parameters The numerous parameters involved in this invention directly determine the system's performance, stability margin, and robustness. This step details the selection strategy.

[0120] 1. Observer Gain Choice: Observer gain vector Positive definite constant components in Mainly affects unknown items The estimated speed. Choose a larger one. Values ​​can speed up the estimation of errors. This improves the convergence speed, thus more effectively compensating for uncertainties. However, excessive gain may amplify measurement noise (especially when...). When affected by noise, this can cause oscillations in the control output, and may even lead to system instability. Therefore, in practical applications, The selection of the value needs to be balanced between estimating the speed and suppressing noise.

[0121] 2. Nonlinear gain function and Design: Design: its weight ( () is a non-decreasing, continuously differentiable positive function that must satisfy the inequality condition in the first sub-step of step two. These functions are usually designed to be of the form of or (in (The constant) to provide linear or superlinear feedback, or a nonlinear function containing an exponential term to provide smooth control with small errors and strong recovery with large errors. The specific form and coefficient selection need to be consistent with the pre-selected... Class function , , They must match to satisfy the inequality conditions. In practical applications, The coefficient should be large enough to ensure rapid convergence of position errors, but excessive gain should be avoided to prevent oscillations.

[0122] Design: its weight ( It is also a non-decreasing, continuously differentiable positive function, but its design conditions must satisfy the inequality conditions in the second sub-step of step two and the multi-loop nonlinear small-gain condition. It must be ensured that it conforms to the defined... of Class function , , and the small gain condition , , , Mutual coordination. For example, when it is known... When setting the parameter range, you can do so by setting a sufficiently large value. To ensure the inequality holds, while also ensuring the small gain condition, for example... To be satisfied The gain should not be infinite, but should be coordinated with the gains of other subsystems.

[0123] 3. Consistency Gain and topological weights Choice: Topological weights This reflects the communication topology. If agent j is a neighbor of agent i, and agent i can obtain the location information of agent j, then... Typically, a positive value (e.g., 1) is assigned, otherwise 0. In more complex application scenarios, different weights can be assigned based on the reliability of the communication link, data transmission bandwidth, information timeliness, or spatial distance between agents.

[0124] Consistent gain It is a positive definite constant and determines the uniform convergence speed of the reference signal. A larger value... A higher value can speed up the consensus process, but may lead to more aggressive control inputs and increase the risk of instability when topology changes or communication delays occur. Its selection typically requires considering both the system's actual requirements for consensus convergence speed and its tolerance for communication disturbances.

[0125] 4.3 Performance Evaluation and Verification After selecting the parameters, the performance of the entire multi-agent consensus control system was verified through a series of simulations and experiments, and the parameters were fine-tuned based on the evaluation results. This included testing under different operating conditions, disturbance levels, and topologies.

[0126] 4.4 Achieving Multi-Agent Consistency Control Finally, with optimized and validated controller parameter configurations, the control algorithm is deployed in a practical multi-agent system. Each agent acquires its own position, velocity, and other state information through its sensors, obtains the position information of neighboring agents (or the leader agent) through the communication module, and then independently runs the controller algorithms designed in steps two and three to generate its own control inputs. Ultimately, they collaboratively achieve the overall multi-agent consensus control task. This process is continuous and real-time, ensuring that the agents can dynamically adapt to environmental changes and maintain cooperative behavior.

[0127] Through the above parameter selection and system integration, this invention can effectively cope with the strong nonlinearity, uncertainty and external disturbances in multi-agent systems, and achieve high-precision, high-robustness and low computational complexity multi-agent consistency control, thereby overcoming the trade-off dilemma between existing technologies in multi-dimensional performance indicators.

[0128] Example 2 The following is an example: Consider a differential equation model that satisfies... , ,in Using the control design method of this invention, the corresponding parameters and functions in the tracking controller can be designed as follows: ， ， Based on such Figure 4 The parameters in the consistency controller of the position measurement topology shown can be designed as follows: ， Consider four grain leveling robots, whose inertia matrix and damping matrix are respectively... and This verifies that, through transformation, the grain leveling robot model meets the requirements of a differential equation model. Therefore, by selecting the initial state of the observer, the initial velocity of the four grain leveling robots as 0, and the initial positions of the four grain leveling robots as follows: ， ， and The simulation results are as follows: Figures 5-7 As shown. By Figures 5-7 It is understood that the control method of the present invention enables the consistent control of uncertain nonlinear multi-agent systems.

[0129] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A tracking-based uncertain nonlinear multi-agent consensus control method, characterized in that, Includes the following steps: Step one involves establishing and transforming a model of the multi-agent system. This step includes establishing... An uncertain nonlinear kinematic model of an intelligent agent on a two-dimensional Euclidean plane is constructed, and the kinematic model is transformed into a second-order nonlinear differential equation. The uncertain nonlinear kinematic model includes the inertial effect and velocity-related damping effect experienced by the intelligent agent during motion. Step two, for the above Each of the agents is designed with a dynamic tracking controller, which is designed to enable the agent to track a time-varying reference signal while effectively suppressing the effects of uncertain nonlinear dynamics within the system and external disturbances. The dynamic tracking controller includes an observer module for real-time estimation of unknown nonlinear terms in the system. Step 3: Design a multi-agent consensus controller, which aims to facilitate the consensus among the agents. The time-varying reference signals of the follower agents converge to the reference signal of the leader agent, and the consensus controller achieves coordinated control based on the communication topology between the agents. as well as Step four: Under the action of the dynamic tracking controller and the consistency controller, appropriate parameters are selected according to the motion state of the multi-agent system to complete the consistency control of the multi-agent system. The parameter selection must ensure that the entire control system meets the stability requirements.

2. The method according to claim 1, characterized in that, In step one, an uncertain nonlinear kinematic model is established, including an inertia matrix and an uncertain damping matrix. The inertia matrix is ​​in diagonal form, and its diagonal elements are positive definite constants. The uncertain damping matrix is ​​also in diagonal form, and its diagonal elements are non-negative continuous functions related to the velocity components, which are used to describe the damping characteristics of the agent at different velocities.

3. The method according to claim 1, characterized in that, In step one, the kinematic model is transformed into a second-order nonlinear differential equation form. This includes defining the agent's output as the position state, the first state variable as the position state, the second state variable as the velocity state, and the new control input as the product of the inverse of the inertia matrix and the original control input. This transforms the kinematic model into a standard second-order nonlinear system form, which facilitates subsequent controller design.

4. The method according to claim 1, characterized in that, The design of the dynamic tracking controller in step two includes: defining the tracking error as the difference between the actual output of the agent and the reference signal; defining the virtual tracking error as the difference between the actual speed state and the desired virtual control input; designing a virtual controller to generate the desired virtual control input, wherein the gain function of the virtual controller satisfies a preset input-to-state stability condition to ensure the stability of the tracking error subsystem; and obtaining the tracking error subsystem, which describes the dynamic change characteristics of the tracking error.

5. The method according to claim 4, characterized in that, The design of the dynamic tracking controller in step two further includes: defining the unknown nonlinear term as the value of the system's nonlinear function at zero; obtaining a speed tracking error expression, which includes a composite nonlinear term, an unknown nonlinear term, and a control input; and ensuring that the composite nonlinear term in the speed tracking error expression satisfies a specific growth condition, which includes the growth characteristics of the composite nonlinear term relative to the tracking error, the virtual tracking error, and the rate of change of the reference signal.

6. The method according to claim 5, characterized in that, The design of the dynamic tracking controller in step two also includes: designing an unknown term observer, which adopts a first-order filter structure for real-time estimation of unknown nonlinear terms in the system, and the observer includes an internal state vector and an observer gain parameter; and obtaining an observation error subsystem, which includes the dynamic change characteristics of the observation error, and whose stability is affected by the observer gain parameter and the composite nonlinear term.

7. The method according to claim 6, characterized in that, The design of the dynamic tracking controller in step two further includes: designing a real controller to generate the final control input, wherein the real controller includes a nonlinear feedback term for the virtual tracking error and an observation compensation term for the unknown nonlinear term, and the gain function of the real controller satisfies a preset input-to-state stability condition and a multi-loop nonlinear small gain condition, wherein the multi-loop nonlinear small gain condition ensures the stability of the entire error-related system and avoids mutual interference between error subsystems leading to system instability; and obtaining a speed tracking error subsystem, wherein the speed tracking error subsystem describes the dynamic change characteristics of the speed tracking error.

8. The method according to claim 1, characterized in that, The design of the consensus controller in step three includes: defining the position measurement topology of the multi-agent system as a directed graph, which describes the communication connections between agents; generating reference signals for each agent through a first-order integrator system, where the state of the first-order integrator system is the reference signal and the input is the output of the consensus controller; and the consensus control requires that the reference signal convergence condition be met, that is, the deviation between the reference signals of all follower agents and the reference signal of the leader agent in steady state is less than a preset threshold.

9. The method according to claim 1, characterized in that, The specific method of the consensus controller in step three is as follows: by defining the relative position error as the position difference between the follower agent and the leader agent, a consensus controller is designed to generate the desired velocity input. The consensus controller calculates the control input based on the difference between the relative position error and the weighted average of the relative position errors of its neighboring agents. The control input is used to drive the first-order integrator system of the reference signal, thereby achieving the consensus of the reference signal.

10. The method according to claim 1, characterized in that, Step four involves achieving consensus control of the multi-agent system, including: establishing a system integration and information flow mechanism to ensure that the reference signal generated by the consensus controller can be correctly used by the dynamic tracking controller, and that the control input generated by the dynamic tracking controller can effectively drive the agent's movement; and optimizing the selection of key parameters, including the observer gain parameter, the dynamic tracking controller gain function parameter, and the consensus controller gain parameter. The parameter selection must meet the stability conditions of the control system and consider the performance requirements in the actual application scenario. The observer gain parameter affects the estimation accuracy and speed of the unknown nonlinear term, the dynamic tracking controller gain function parameter affects the tracking performance and anti-interference capability, and the consensus controller gain parameter affects the convergence speed of the reference signal and the system stability.

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