Environment-adaptive formation control method based on free parameterization stable network

Abstract

The application provides an environment adaptive formation control method based on a free parameterization stable network, automatically switches between a compact formation and a standard formation according to real-time distances from a formation geometric center to obstacles, and realizes intelligent reconstruction of a formation shape through a preset distance threshold and a smooth transition mechanism; a free parameterization stable network architecture is used to realize parameterization dynamic operator design of each spatial distributed control unit, distributed information exchange is realized through a sparse interconnection topology isomorphic to a spatial system communication graph; inherent stability properties of the free parameterization stable network structure are used to construct a controller parameter unconstrained scheme suitable for a space environment, so that the stability of a spatial closed loop system can be guaranteed for any parameter value, thereby avoiding feasibility constraints of linear matrix inequalities, and realizing unconstrained gradient learning of a three-dimensional distributed controller.

Description

Technical Field

[0001] This invention relates to the field of formation control technology, specifically to an environment-adaptive formation control method based on a free parameterized stable network. Background Technology

[0002] Environmentally adaptive distributed formation control has broad application prospects in fields such as UAV swarms, robot collaboration, and satellite formation. Although significant progress has been made in distributed formation control research, existing methods are generally based on predefined formation configuration assumptions, making it difficult to cope with challenges such as complex three-dimensional environmental constraints, narrow passage crossings, and dynamic formation reconfiguration in real-world systems. How to design distributed control strategies that can adaptively adjust formation shape according to environmental geometry, achieve intelligent formation reconfiguration while ensuring stability, and avoid complex constraint verification is a key scientific problem restricting its practical application.

[0003] Limitations of formation shape adaptation. Traditional methods typically employ rule-based mode-switching strategies, which struggle to handle continuously changing geometric constraints. While recent research has introduced neural network methods to improve control flexibility, it lacks structured stability guarantees. Particularly in complex geometrically constrained environments, how to achieve smooth and continuous formation adjustment based on the dynamic distance from the formation center to obstacles remains unresolved. Existing neural network control methods face two major challenges in multi-agent scenarios: first, how to effectively integrate environmental geometric information into network parameterization design; and second, how to achieve unconstrained parameter learning while ensuring global stability in a distributed architecture.

[0004] Limitations in stability guarantees. Traditional Lyapunov methods prove system stability by constructing energy functions, but are constrained by complex linear matrix inequalities (LMIs) during online parameter learning, resulting in heavy computational burden and difficulty in guaranteeing real-time performance. Existing research mainly focuses on formation maintenance in open spaces, with insufficient theoretical analysis of environments with geometric constraints such as holes and obstacles. Although optimization-based methods improve adaptability to some extent, they rely on convexity assumptions and have high computational complexity, making them difficult to handle complex scenarios where formation compactness needs to be significantly adjusted. More importantly, existing methods lack a quantitative characterization of the relationship between formation scale changes and system stability, failing to provide theoretical guarantees for control design under extreme geometric constraints. While traditional neural network control provides flexibility for nonlinear system design, its parameterization still requires stability verification through solving LMIs and does not consider the impact of time-varying environmental parameters on system dynamics.

[0005] Insufficient environmental perception and utilization. In practical deployments, most existing work assumes a fixed formation configuration, neglecting the impact of dynamic changes in environmental parameters such as obstacle distance and channel width on the formation shape. Although there have been preliminary explorations in environmental adaptive control, a complete theoretical framework from perception to control is still lacking on how to adjust the formation configuration in real time using geometric perception information to improve the system's traversal capability. In particular, how to incorporate the time-varying nature of environmental parameters into the controller design while ensuring the stability of the closed-loop system when parameters change rapidly remains an unsolved theoretical problem. Summary of the Invention

[0006] To address the aforementioned problems, this invention proposes an environment-adaptive formation control method based on a free parameterized stable network, comprising the following steps:

[0007] S1. Environmental parameterization: Based on the real-time distance from the geometric center of the formation to the obstacle, it automatically switches between compact formation and standard formation, and realizes intelligent reconstruction of the formation shape through a preset distance threshold and a smooth transition mechanism;

[0008] S2. Controller Design: A free parameterized stable network architecture is adopted to realize the parameterized dynamic operator design of each space distributed control unit, and distributed information exchange is realized through a sparse interconnection topology isomorphic to the space system communication graph;

[0009] S3. Unconstrained parameter learning: Utilizing the inherent stability properties of free parameterized stable network structures, an unconstrained controller parameter scheme suitable for space environments is constructed, ensuring the stability of the space closed-loop system for any parameter value. This avoids the feasibility constraints of linear matrix inequalities and enables unconstrained gradient learning of the three-dimensional distributed controller.

[0010] Furthermore, the free parameterized stable network in step S2 adopts an internal model control structure, reconstructs the generalized disturbance by calculating the deviation between the actual state of the system and the predicted state of the model, and designs a control strategy based on the generalized disturbance and environmental parameters.

[0011] Furthermore, the sub-operator dynamics of the free parameterized stable network in step S2 follows a structured discrete-time form, including three equations: internal state evolution, intermediate output generation, and neuron activation, to maintain its stability properties.

[0012] Furthermore, the sparse interconnect topology in step S2 satisfies the following condition:

[0013] (a) The perturbation and parameter input matrices are column-orthogonal to ensure effective information fusion;

[0014] (b) The control output matrix satisfies an orthogonal structure to ensure the linear independence of the control output;

[0015] (c) The feedback matrix from the output to the input of the free parameterized stable network is column-independent to ensure the controllability of the interconnection topology and the effectiveness of energy transfer.

[0016] Furthermore, the unconstrained parameter learning in step S3 is achieved through an automatic calculation rule. This rule automatically calculates the stability gain parameter based on whether the agent receives external input and the strength of its neighborhood coupling, ensuring that the constructed controller can satisfy the global stability condition for any free parameter.

[0017] Further, step S1 specifically includes:

[0018] S1.1. Calculate the geometric center of the formation by averaging the position components of all agents;

[0019] S1.2. Calculate the Euclidean distance from the geometric center of the formation to the center of the known obstacle;

[0020] S1.3. Define a smooth switching function that performs a linear transformation based on the hyperbolic tangent function and outputs a switching coefficient that smoothly varies between 0 and 1 according to the relative relationship between the distance and a preset threshold.

[0021] S1.4. Based on the switching coefficient, perform a convex combination of the preset standard formation configuration and compact formation configuration to generate the current target formation vector in real time;

[0022] S1.5. Integrate the distance, switching coefficient, and target formation vector into a state-dependent environment parameter vector.

[0023] Furthermore, the smooth switching function in step S1.3 satisfies the following: when the formation distance is much smaller than the close distance threshold, the switching coefficient approaches zero, corresponding to compact formation; when the formation distance is much larger than the long distance threshold, the switching coefficient approaches one, corresponding to standard formation; and a smooth transition is achieved between the two thresholds.

[0024] Furthermore, the standard formation configuration and compact formation configuration in step S1.4 are both preset vectors with fixed geometric configurations in three-dimensional space, and their difference determines the maximum magnitude of adaptive formation adjustment.

[0025] Furthermore, the free parameterized stable network in step S2 inherits the structured dynamics form of the recursive equilibrium network to maintain its Lipschitz boundedness and contraction properties.

[0026] Furthermore, the external input of the free parameterized stable network in step S2 consists of three parts: output feedback from the neighborhood, local disturbance reconstruction signal, and environmental parameters; the final control output of the free parameterized stable network consists of two parts: the network's own output and the direct effect of the environmental parameters.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] (1) An environment-adaptive formation control strategy based on geometry perception is proposed. Based on the real-time distance from the formation center to obstacles, the system automatically switches between compact and standard formations: shrinking the formation to traverse narrow passages when approaching obstacles and resuming standard formation to maintain stability when moving away from obstacles. This strategy achieves intelligent reconstruction of the formation shape through distance thresholds and a smooth transition mechanism.

[0029] (2) A three-dimensional distributed state feedback controller design method based on Free Parametric Stable Network (FPSN) is proposed for the first time. This method uses the FPSN architecture to realize the parameterized dynamic operator design of each space distributed control unit, and realizes distributed information exchange through a sparse interconnection topology isomorphic to the space system communication graph. The core contribution is to utilize the inherent stability property of the FPSN structure to construct an unconstrained controller parameter scheme suitable for the space environment, so that the stability of the space closed-loop system can be guaranteed for any parameter value, thereby avoiding the feasibility constraint of linear matrix inequality (LMI) in traditional distributed control methods and realizing unconstrained gradient learning of the three-dimensional distributed controller. Attached Figure Description

[0030] Figure 1 The comparison shows the adaptive passage of quadcopter formations through different hole sizes, where (a) is the case with a hole radius of 1.2m, (b) is the case with a hole radius of 1.6m, and (c) is the case with a hole radius of 2.2m.

[0031] Figure 2 The training loss evolution for quadrotor formation control based on FPSN is shown in (a) for overall training loss evolution, (b) for state tracking and control effort, (c) for safety-critical loss, (d) for formation and velocity loss, and (e) for final convergence phase.

[0032] Figure 3 The formation shapes at the moment of crossing are shown, where (a) is the case with a hole radius of 1.2m, (b) is the case with a hole radius of 1.6m, and (c) is the case with a hole radius of 2.2m.

[0033] Figure 4The distance between the smallest agents is compared, where (a) is the case with a hole radius of 1.2m, (b) is the case with a hole radius of 1.6m, and (c) is the case with a hole radius of 2.2m.

[0034] Figure 5 This is a schematic diagram of the FPSN control framework of the present invention. Detailed Implementation

[0035] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0036] S1 Distributed Nonlinear System Modeling and Environmental Parameterized Formation Control Problem Definition

[0037] S1.1 Distributed Nonlinear Systems

[0038] This invention considers a network consisting of N interconnected nonlinear subsystems. The coupled network between the subsystems is defined as an undirected communication graph G = (V, ε), where V = {1, …, N} represents the subsystems in the network, and the edge set ε contains pairs {i, j} of subsystems that can communicate with each other. Each subsystem has the following form:

[0039] (1);

[0040] Wherein, the state and input of each subsystem i∈V are respectively represented by and express. It is unknown process noise. . It is an environmental parameter vector. The neighborhood of subsystem i is denoted as... The dimensional parameters of each subsystem are defined as follows: Subsystem state dimension Subsystem The control input dimension, : Environmental parameter dimensions Subsystem The total dimension of the neighborhood states for the quadrotor formation system of this invention: (Position 3D + Velocity 3D) (3D acceleration) (formation shape) (2D of distance and switching coefficient).

[0041] In operator form, the present invention can represent subsystem (1) as:

[0042] (2);

[0043] in It is a strictly causal operator that defines the dynamics of a subsystem, such that

[0044] .

[0045] By combining the local system dynamics in (1), the dynamic result of the global system is:

[0046] (3);

[0047] Similar to the subsystem, the present invention can rewrite system (3) in operator form as:

[0048] (4);

[0049] The present invention makes the following assumptions about the system to be controlled:

[0050] Assumption 1.1. This invention assumes a causal operator. Such that the mapping (w, u, Θ) → x lies in L2, and the process noise w(t) ~ D follows an unknown distribution D, and . : The L2 sequence space of a dimensional vector is the space of all L2 stable sequence operators that satisfy the energy bounded condition.

[0051] The above assumptions imply that the interconnected system is stable or can achieve L2 stability through local control. This is often true in practical applications, where the system is either pre-stabilized or can be stabilized using a simple controller. While these controllers ensure stability, they typically lead to suboptimal closed-loop performance. The goal of this invention is to improve performance while maintaining stability.

[0052] To control the system, this invention considers the following parameterized nonlinear state feedback causal strategy.

[0053] (5);

[0054] Due to causal relationships, each perturbation sequence is denoted as... This leads to the unique trajectory of the closed-loop system (4)-(5). This means that the mapping from disturbance to state and control, denoted as Φx[F, K] and Φu[F, K] respectively, is well defined for any given system F and controller K.

[0055] Therefore, this invention represents x = Φ x [F,K](w) and u =Φ u [F,K](w) for all .

[0056] S1.2. Environmental parameterization

[0057] A key innovation of this invention is the systematic parameterization of geometric constraints (such as obstacles with narrow passages) into time-varying environmental parameters Θ. t This section details how Θ is constructed through a series of mappings from the global state xt. t The process.

[0058] S1.2.1. Formation Centroid Calculation

[0059] The geometric center of the formation is calculated as follows:

[0060] (6);

[0061] in It is the formation center of mass at time t. From The positional component extracted from it.

[0062] S1.2.2. Distance to the obstacle

[0063] The Euclidean distance from the formation's center of mass to the center of the obstacle is given by the following formula:

[0064] (7);

[0065] in It is the known center location of the obstacle.

[0066] S1.2.3. Smooth Switching Mechanism

[0067] To achieve a smooth transition between different formation configurations, this invention defines a switching function:

[0068] (8);

[0069] in , It is a distance threshold and satisfies d near <d far , It is the midpoint. (Function) Satisfy: When d ≪d near When α≈0 (compact formation), when d≫d far When α≈1 (standard formation), in [d near, d far Smooth transition within.

[0070] S1.2.4. Formation Configuration Interpolation

[0071] The target formation vector is obtained through a convex combination:

[0072] (9);

[0073] in It is the target formation vector. It is a standard formation configuration. It is a compact formation configuration. .

[0074] S1.2.5. Environmental Parameter Vector

[0075] Environment parameters are defined as a state-dependent vector:

[0076] (10);

[0077] in It is an environment parameter mapping. Parameter Θ t Real-time geometry sensing can be encapsulated in the following ways:

[0078] (i) Distance scalar d t (Equation (7)), (ii) Smoothing switching coefficient α t (Equation (8)), (iii) Interpolation target formation f t (Equation (9)). This parameterization enables the system dynamics to adaptively respond to environmental geometric constraints.

[0079] S1.3. Problem Statement

[0080] The goal of this invention in designing the control strategy K(X,Θ) is to address the following challenges:

[0081] C-1) Ensure the stability of the closed-loop mappings Φx[F, K] and Φu[F, K] in the L2 sense.

[0082] C-2) Enhance the performance of the closed-loop system by minimizing the loss function, which is defined as follows:

[0083] (11);

[0084] Among them, for all and all r(xT,uT) is differentiable and ≥ 0.

[0085] C-3) Ensure that the obtained strategy K(x,Θ) can be implemented in a distributed manner using neighbor-to-neighbor communication. Specifically, this invention requires:

[0086] (12);

[0087] Unlike most optimal control and dynamic programming problems that rely on the sum of stage costs over time, the method of this invention is more flexible. This invention only requires that the loss function be differentiable in its parameters. This allows the invention to combine the complex objectives typical of many robotics and reinforcement learning tasks, such as those specified using signal-timing-sequential logic (STL).

[0088] S2. Environmental Adaptive Formation Control

[0089] This invention establishes a theoretical framework for adaptive formation control in environments with complex geometric constraints, including holes and obstacles. Before presenting the main results, this invention establishes relevant preliminary mathematical lemmas.

[0090] S2.1. Preliminary Lemma

[0091] S2.1.1. Lipschitz Property of Geometric Mappings

[0092] Lemma 2.1 (Lipschitz constant for the centroid mapping)

[0093] For the formation center of mass mapping (Equation (6)) gives:

[0094] (13);

[0095] The rate of change of the center of mass is less than or equal to the rate of change of the state. The averaging effect suppresses individual perturbations. In distributed formations, the center of mass, as a measure of overall position, is more stable and predictable than the motion of a single agent.

[0096] Lemma 2.2 (Lipschitz constant of the distance function)

[0097] For distance function (Equation (7)) gives:

[0098] (14);

[0099] The distance change does not exceed the position change, and the distance function is 1-Lipschitz continuous. When At that time, gradient It is a unit vector. When the center of mass moves at a velocity v, the rate of change of the distance to the obstacle does not exceed v, ensuring the predictability of the distance measurement.

[0100] Lemma 2.3 (Lipschitz constant of the switching function)

[0101] The Lipschitz constant of the switching function s(d) (Equation (8)) is:

[0102] (15);

[0103] The rate of change of the switching coefficient is determined by the width of the transition interval; the wider the interval, the smoother the switching. S This determines the formation's sensitivity to changes in obstacle distance. A smaller L... S This implies a more gradual formation adjustment.

[0104] Lemma 2.4 (Lipschitz constant for formation interpolation)

[0105] Formation interpolation mapping (Equation (9)) satisfies

[0106] (16);

[0107] The rate of change of formation shape is proportional to the rate of change of switching coefficient, with the proportionality constant being the difference between the two formations. The magnitude of formation adjustment is determined by the design of F and C. If the difference between the two is small, the adjustment process is smoother.

[0108] S2.1.2. Combinatorial Properties

[0109] Corollary 2.1 (Lipschitz constant of the combinatorial mapping)

[0110] Formation shape mapping The Lipschitz constant is:

[0111] (17);

[0112] After four layers of mapping, the total amplification factor of the formation shape change is L. h L h This is the sensitivity coefficient of the formation adaptive strategy, which determines the formation's response speed to the flight trajectory. Typical value: for N=4, d far -d near = 0.5m, with L h ≈ 1.415.

[0113] S2.2. Main Results

[0114] Theorem 2.1 (Feasibility and Stability Compatibility of Environment-Adaptive Formation Control)

[0115] Consider the geometry-aware formation adaptive strategy defined by equations (6)-(10). Under assumption 1.1, this strategy satisfies the following properties:

[0116] (i) Smooth Reconfiguration:

[0117] (18);

[0118] in is the Lipschitz constant for formation mapping.

[0119] (ii) Geometric Feasibility:

[0120] (19);

[0121] in For the formation radius, This is the bound for formation tracking error.

[0122] (iii) Stability Compatibility:

[0123] (20);

[0124] The combined Lipschitz constant is:

[0125] (twenty one);

[0126] (iv) Convergence Time Bound:

[0127] (twenty two);

[0128] S2.3. Corollary

[0129] Corollary 2.2 (Agent Spacing Constraint)

[0130] In the compact formation area (d <d near The maximum agent spacing satisfies:

[0131] (twenty three);

[0132] The maximum distance between any two agents does not exceed the diameter of the hole, ensuring that all agents can safely pass through simultaneously. Verify the sufficiency of the formation compression: all agents can simultaneously reside within a circle of diameter 2R.

[0133] Corollary 2.3 (Small Gain Stability Condition)

[0134] If the system dynamics satisfy Assumption 1.1, then the sufficient condition for the stability of the closed-loop system is:

[0135] (twenty four);

[0136] The formation adaptive mechanism does not violate the L2 stability of the closed-loop system, the condition of which is given by the small gain theorem. It provides a verifiable stability criterion, eliminates the need for online LMI solving, and is suitable for real-time control.

[0137] S3. Design of Free-Parameterized Stable Networks

[0138] This invention establishes a general controller design framework applicable to systems containing external parameters. The theoretical results of this invention are applicable to arbitrary environmental parameters. The fact that all of them are true demonstrates the universality of the framework.

[0139] S3.1. Internal Model Control Framework

[0140] To parameterize all stable controllers, this invention employs an internal model control (IMC) architecture, which designs the controller by reconstructing generalized disturbances.

[0141] S3.1.1. Disturbance Reconstruction

[0142] The reconstruction perturbation is defined as:

[0143] (39);

[0144] In operator form, the present invention can express the control strategy as:

[0145] (40);

[0146] in It is a generalized disturbance of reconstruction. It is a stable L2 operator.

[0147] By reconstructing the generalized disturbance w r All stability control parameters are parameterized as L2 operators R acting on the reconstructed disturbance. This framework guarantees u=R(w) r The stability of the closed-loop system under (Θ).

[0148] S3.2. Dynamics of Distributed Subsystems

[0149] S3.2.1. General Dynamics of Suboperators

[0150] Define the i-th sub-operator R [i]The internal dynamics are:

[0151] (41);

[0152] in It is the internal state of the sub-operator. It is the external input of the sub-operator (from the interconnect). It is the internal output of the sub-operator. It is a state transition function. It is the output function.

[0153] Define the i-th sub-operator The dimension parameters are as follows:

[0154] : Internal state dimension (hidden layer dimension); External input dimension; : Internal output dimension.

[0155] S3.2.2. Dissipation and Storage Functions

[0156] Suboperators satisfy the energy dissipation condition:

[0157] (42);

[0158] in It is the storage function (Lyapunov function) of the i-th sub-operator. It is the supply rate function, which measures the rate at which energy flows into the system.

[0159] S3.2.3. Definition of Supply Ratio

[0160] Define the supply rate of the suboperator relative to the quadratic form with respect to three inputs and one output as:

[0161] (43);

[0162] in It is the L2 gain parameter of the sub-operator relative to the perturbation input. It is the L2 gain parameter of the sub-operator relative to the input parameters, the negative definiteness of the quadratic form (with respect to the output). This ensures the dissipation of energy.

[0163] The dimensions of the identity matrix are explained below:

[0164] identity matrix : , , Output dimension for sub-operator (corresponding to three-dimensional acceleration control);

[0165] identity matrix : , , The external input dimension of the sub-operator;

[0166] identity matrix : , , Define the suboperator for the quadratic supply rate of the three inputs and one output for the environmental parameter dimension (defined in Equation (10)).

[0167] S3.3. Sparse Interconnect Topology

[0168] S3.3.1. Interconnection Matrix Definition

[0169] Define the interconnection topology between sub-operators and with external inputs as follows:

[0170] (44);

[0171] in It is the feedback matrix from FPSN output to FPSN input (neighborhood communication). It is the coupling matrix perturbed to the FPSN input. It is the coupling matrix from environmental parameters to FPSN input. It is the mapping from the FPSN output to the final control. It is a direct mapping from environmental parameters to final control.

[0172] S3.3.2. Sparse Interconnection Topology Assumption

[0173] To ensure the desirable properties of sparse interconnect topologies, this invention makes the following assumptions:

[0174] Assumption 3.12. (Interconnection Matrix Structure)

[0175] The interconnection matrix satisfies the following conditions:

[0176] (a) The perturbation and parameter input matrices are column-orthogonal:

[0177] (45);

[0178] (b) The control output matrix satisfies an orthogonal structure:

[0179] (46);

[0180] in, .

[0181] (c) The feedback matrix from the FPSN output to the input is column-independent:

[0182] (47);

[0183] Assumption 3.12(a) ensures that disturbance and parameter information can be effectively integrated into the FPSN input. Assumption 3.12(b) ensures the linear independence of the control output. Assumption 3.12(c) ensures the controllability of the interconnect topology and the efficiency of energy transfer. In distributed networks, this assumption is equivalent to requiring that the output of each FPSN contributes independently to the global system, avoiding the degradation of the control channel.

[0184] S3.3.3. Definition of Outgoing Join Sets

[0185] Define which agents receive external input (perturbations or parameters), and divide them into two categories for separate processing:

[0186] (48);

[0187] in, It is a subset of intelligent agents that receive perturbation inputs. It is a subset of intelligent agents that receive environmental parameter input. It is a subset of all agents that receive any external input. Only agents in J can perceive external changes.

[0188] S3.4. From REN to FPSN

[0189] This invention uses the Recursive Equilibrium Network (REN) architecture as the basic framework and proposes a Free Parameterized Stable Network (FPSN). The FPSN maintains the structured dynamics of the REN to inherit its inherent stability properties, but by introducing a free parameter α to realize the automatic calculation of the L2 gain, it eliminates the linear matrix inequality (LMI) feasibility constraint in traditional distributed control methods.

[0190] S3.4.1. Dynamics of REN

[0191] The standard form of REN is:

[0192] (49);

[0193] in It is the internal state (hidden layer) of REN. It is the neuronal activation input of REN. It is the intermediate output of REN. It is the external input of REN (from the interconnect). It is an activation function (such as ReLU) that operates element-wise.

[0194] FPSN inherits the structured discrete-time dynamics form of REN, including three equations: state evolution, intermediate output, and activation generation, to maintain its Lipschitz boundedness and contraction properties. The innovation of FPSN lies in the introduction of the free parameter α in the interconnect layer (Equation (50)) and gain calculation (Equation (55)).

[0195] 3.4.2. Interconnection Equations of FPSN

[0196] The external input of the FPSN consists of three parts:

[0197] (50);

[0198] The first equation defines the external input of the FPSN as consisting of three parts: output feedback from the neighborhood, local disturbance, and environmental parameters. The second equation defines the final control output as consisting of two parts: the output of the FPSN and the direct effect of the environmental parameters. Sparse This ensures that information flows only between neighboring areas.

[0199] S3.5. Lyapunov Stability Analysis

[0200] S3.5.1. Global Supply Rate

[0201] The sum of the supply rates of all subsystems does not exceed the global performance metric, which guarantees network-level L2 stability.

[0202] (51);

[0203] The left side represents the total energy dissipation of all sub-operators, and the right side represents the global L2 performance matrix. It is the upper bound of the disturbance. It is the upper bound of the parameter, and the negative sign on the control item indicates a performance requirement (the control output is penalized).

[0204] S3.5.2. LMI Stability Conditions

[0205] Lemma 3.13. (Supply Rate Aggregation Lemma)

[0206] Under assumption 3.12, if each subsystem satisfies the dissipation inequality (42), then the global supply rate satisfies:

[0207] (52);

[0208] in, It is a weighted suboperator supply rate matrix.

[0209] Substitute the interconnection equation (44) into the supply rate quadratic form. Stack all subsystems to obtain the global form. Substitute the interconnection equation back in and simplify using the orthogonality of assumption 3.12 to obtain the explicit form of Ψ.

[0210] This lemma aggregates the dissipative properties of distributed subsystems into a global quadratic form, serving as a bridge between subsystem stability (Equation (42)) and network-level LMI conditions (Equation (52)).

[0211] S3.5.3. Lyapunov Function Construction

[0212] Define the global Lyapunov function as the weighted sum of all sub-operator Lyapunov functions:

[0213] (53);

[0214] in It is the global Lyapunov function (total energy). It is the weighting coefficient (adjustable parameter) of the i-th sub-operator. It is the Lyapunov matrix (symmetric positive definite) of the i-th suboperator.

[0215] S3.5.4. Lyapunov Difference

[0216] Calculate the change of the global Lyapunov function over a time step:

[0217] (54);

[0218] in It is a one-step difference of the Lyapunov function. If (a certain) If the difference is greater than 0, the system is asymptotically stable. The negativity of the difference ensures a monotonically decreasing energy.

[0219] S3.6. Main Results

[0220] Theorem 3.14. (Automatic L2 Gain Calculation Rule) The L2 gain parameter of the agent is automatically calculated based on whether it receives external input, as follows:

[0221] (55);

[0222] in It is the L2 gain of the i-th agent. It's about free parameters. nonlinear functions, It is the strongest influence of neighborhood disturbances. This has the strongest influence from the neighborhood parameters. Furthermore, the synthesis parameter is defined as:

[0223] (56);

[0224] Where h It is the total perturbation coupling strength from the neighborhood, h It is the total parameter coupling strength from the neighborhood. It is the square term of the free parameter (any positive number can be chosen). ).

[0225] In equation (55) The automatic calculation rules ensure that any free parameter is calculated. >0, thus constructed And Ψ necessarily satisfy the LMI condition (52). This is precisely the meaning of free parameterization—the parameter space is unconstrained, but stability is automatically guaranteed. Parameter It plays a normalization role in the L2 gain calculation of Theorem 3.14. A larger neighborhood coupling strength ( A more conservative gain bound is needed; while the free parameters It provides the flexibility of online adjustment, allowing designers to balance stability margins and control performance.

[0226] S5. Simulation Verification

[0227] To test the performance of the designed FPSN-based environment-adaptive formation controller, simulation verification was performed in a formation system containing four quadcopter UAVs. The system physical parameters and initial states are listed in Table 1 (System Parameter Configuration) and Table 2 (Formation Configuration and Obstacle Parameters), respectively. The communication topology adopts a fully connected undirected graph containing four nodes and six edges.

[0228] 1) Quadrotor system configuration

[0229] The simulation environment includes four quadcopter UAVs performing a formation obstacle crossing mission. The system adopts a discrete-time dynamics model (Equation 28), and the main parameters are shown in Table 1.

[0230] Table 1 System Parameter Configuration

[0231]

[0232] 2) Formation configuration and obstacle parameters

[0233] The system supports three obstacle size configurations. The geometric parameters and formation settings for each scenario are shown in Table 2. All experiments performed a traversal task from the initial position (x=-5m) to the target position (x=5m), passing through circular obstacles on the x=0 plane.

[0234] Table 2 Formation Configuration and Obstacle Parameters

[0235]

[0236] The system is based on the formation's center of mass X t To the center of the obstacle, P = [0,0,0] T distance Dynamically adjust the formation shape. When d t <d near At that time, the formation switches to a compact configuration c to traverse narrow passages; when d t >d far At that time, the standard formation w is restored to maintain stability and safety margin. During the transition interval [d] near ,d far Within ], the target formation vector f t Convex combination interpolation is performed using Formula 8 to ensure smooth changes in formation shape.

[0237] The three sets of experiments covered compact (1.2 m), medium (1.6 m), and loose (2.2 m) scenarios, respectively, verifying the adaptive capability of the FPSN controller under different geometric constraints. The hole width in experimental group 1 (measured formation width of 1.81 m during crossing) was close to the theoretical value for compact formation. The result is approximately 2.83 m, demonstrating the effectiveness of the method under the limit constraints. Table 3 shows a comparison of the performance of the three sets of experiments.

[0238] Table 3

[0239]

[0240] Table 3 summarizes the key performance indicators of the three sets of experiments under the optimal number of training rounds.

[0241] Safety metrics demonstrate that zero collisions were guaranteed in all scenarios. The minimum spacing increased with the size of the hole (0.78m → 1.58m → 1.77m). The minimum spacing of 0.78m in the 1.2m hole scenario was significantly higher than the safety threshold of 0.3m, verifying the effectiveness of the collision avoidance loss function in extreme compression scenarios. The minimum spacing in the 1.6m and 2.2m hole scenarios (1.58m and 1.77m) was significantly higher than the safety threshold of 0.8m, reflecting sufficient safety margin. The minimum spacing occurred at 7.45 seconds (step 149), corresponding to the final formation after the formation reached the target position.

[0242] Formation width during passage: A 1.2m hole group with a width of 1.67m (compression ratio 0.418) demonstrates extreme passability; a 1.6m hole group with a width of 2.37m (compression ratio 0.593) reflects a moderate compression strategy; and a 2.2m hole group with a width of 3.61m (compression ratio 0.903) is close to the standard formation configuration, prioritizing formation stability under relaxed constraints. The formation width is reasonably adapted to the obstacle size, verifying the FPSN controller's ability to adaptively adjust the formation geometry.

[0243] Example 2

[0244] Figure 1 The diagram illustrates formation trajectories for three different hole sizes: (a) small hole (radius 1.2m), (b) medium hole (radius 1.6m), and (c) large hole (radius 2.2m). Four UAVs start from a standard formation (4×4m square), traverse a circular obstacle (marked by a red circle) located in the z=0 plane, and ultimately reach the target position. The gray dashed line represents the initial position, and the colored solid line represents the flight trajectory.

[0245] Three sets of experiments demonstrate geometric adaptive characteristics: small hole scenario ( Figure 1 (a) The formation compresses to a width of 1.67m at the obstacle (compression ratio 0.418), and the four drones closely converge to pass through the narrow passage before resuming their standard formation; (middle hole scene) Figure 1 (b) The crossing width is 2.37m (compression ratio 0.593), and the contraction is moderate; large hole scene ( Figure 2 (c) The formation maintains 3.61m (compression ratio 0.903), close to the standard formation size, with only slight compression when approaching obstacles. Progressive variation validates that the FPSN adaptively adjusts its formation geometry according to obstacle size.

[0246] Observations from 3D views and planar projections show that the trajectories of all agents remain smooth and continuous, without abrupt changes or oscillations. Even in the scenario with small holes, the formation's rapid contraction-expansion process before and after obstacles (comparison of trajectories inside and outside the red circle) remains smooth, demonstrating the high-frequency response capability and stability of the FPSN. All three sets of experiments achieved zero collisions and zero position errors (Table IV shows that all UAVs accurately reached the target position), verifying the algorithm's convergence.

[0247] Figure 2 shows the evolution curves of each loss term during 7000 training rounds. (a) represents the overall training loss evolution, (b) represents state tracking and control effort, (c) represents safety-critical loss, (d) represents formation and velocity loss, and (e) represents the final convergence phase. Figure 2As shown, the total loss decreases rapidly in the first 500 rounds, then plateaus after 1000 rounds, eventually converging to 69952. The optimal round occurs at round 6257 (marked by the orange dashed line). The learning rate decay strategy (reducing from 0.01 to 0.005 when epochs > 4000) causes a second decrease in the curve after 4000 rounds; the green shaded area represents the convergence phase.

[0248] The evolution characteristics of each sub-item loss are as follows: (1) The state tracking loss (green line) drops sharply in the first 500 rounds and then stabilizes at a low level, while the control loss (red line) shows an initial peak and then decays rapidly; (2) The collision avoidance loss (cyan) and boundary constraint loss (purple) fluctuate and converge on a logarithmic scale. The fluctuations are caused by different initial disturbances, but none of the trajectories violate safety constraints; (3) The formation loss (yellow line) and velocity loss (purple line) converge rapidly to near zero in the first 1000 rounds and then remain stable; (4) The total loss in the last 2000 rounds has an average of 225874, a standard deviation of 51602, a relative standard deviation of 22.8%, and no divergence trend.

[0249] According to equation (55), the gain parameters of the four UAVs converge to γ ​​under the optimal training rounds. [1] = 0.83, γ [2] =0.85, γ [3] = 0.82, γ [4] = 0.84, which satisfies the stability condition of Theorem 4.14.

[0250] Figure 3 shows the top-view formations of four UAVs passing through the obstacle plane under three different hole sizes, where (a) is the case with a hole radius of 1.2m, (b) is the case with a hole radius of 1.6m, and (c) is the case with a hole radius of 2.2m. Red circles mark the hole boundaries, and dashed lines connect them to show the formation topology.

[0251] The formation widths and compression ratios for the three sets of experiments are as follows: 1.2m hole formation width 1.67m (compression ratio 0.418), 1.6m hole width 2.37m (compression ratio 0.593), and 2.2m hole width 3.61m (compression ratio 0.903, close to the standard formation width of 4m). The formation width is reasonably adapted to the obstacle size, with the most compact scene width being only 41.8% of the standard formation width.

[0252] Despite significant differences in formation scale across the three scenarios (1.67m vs 3.61m, a difference of 116%), all scenarios maintained rectangular topological invariance—the four drones remained at the four vertices of the rectangle, differing only in scaling factor. The angle between the diagonals connected by dashed lines remained approximately 90°, verifying shape consistency. The four drones maintained central symmetry in the yz plane (about the center of the obstacle). Even in the scenario with the smallest hole, where the formation was extremely compressed, the relative distance between the agents remained uniform, with no individual agents falling behind or vying for space, ensuring the integrity of the formation.

[0253] Figure 4 The time evolution curves of the minimum distance between any two UAVs are shown for three hole sizes, where (a) is the case with a hole radius of 1.2m, (b) is the case with a hole radius of 1.6m, and (c) is the case with a hole radius of 2.2m. The red dashed line represents the safety threshold of 0.8m, and the yellow asterisk marks the minimum distance value during the crossing stage.

[0254] The collision avoidance performance of the three sets of experiments is as follows: (1) The minimum spacing during the crossing phase of the 1.2m hole scene is 1.15m (margin 0.35m), the curve has a U-shaped feature, the initial and final phase distances are about 4.0m and 3.0m respectively, corresponding to standard formation, and the crossing phase (6-10 seconds) drops to the bottom of 1.15m, corresponding to compact formation; (2) The minimum spacing during the 1.6m hole scene is 1.42m (margin 0.62m, 77% improvement for smaller holes), the U-shaped bottom is gentler (about 9 seconds), and the curve transition is smooth without abrupt changes; (3) The minimum spacing during the 2.2m hole scene remains at 1.50m (margin 0.70m), the curve is generally high, and only slightly decreases at the crossing moment (about 8 seconds).

[0255] All three curves exhibit a symmetrical U-shaped structure, with the axis of symmetry located between 7 and 9 seconds (obstacle position x=0). The distance decreases monotonically when approaching the obstacle (0-8 seconds) and increases monotonically after passing through it (8-15 seconds). In the small hole scene, the valley floor is narrower and deeper (minimum 1.15m, valley floor width approximately 4 seconds), while in the large hole scene, the valley floor is wider and shallower (minimum 1.50m, valley floor width approximately 6 seconds).

[0256] The relative safety margin η is defined as (dmin - dsafe) / dsafe. The margins for the three experimental crossing phases are η1 = 44%, η2 = 78%, and η3 = 88%, respectively. The margin increases with the hole radius, verifying that the FPSN dynamically adjusts its strategy according to environmental constraints: minimizing the formation size under strict constraints ensures crossing feasibility, while maximizing the safety margin under relaxed constraints improves robustness. No collision events occurred throughout the entire time domain (distance < 0.8m), verifying the inherent safety guarantee.

[0257] Figure 5 This is a schematic diagram of the FPSN control framework of the present invention. The left side shows the multi-agent system and its output state. The middle section is the environment parameterization module, which generates parameter vectors through distance calculation, smooth switching functions, and formation interpolation. (Equations 6-9); The right side is the FPSN controller module, which includes sparse interconnect topology (Equation 36), REN dynamics (Equation 35), and gain calculation (Equations 41-42). It generates control signals by fusing three types of inputs: neighborhood output, disturbance reconstruction, and environmental parameters. The bottom is a closed-loop system. The core innovation of FPSN lies in establishing free parameters and... The explicit mapping of the gain ensures that the closed-loop stability holds automatically for any free parameters, thereby avoiding the feasibility verification of linear matrix inequalities and realizing unconstrained gradient learning.

[0258] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. An environment-adaptive platoon control method based on free parameterization stable network, characterized in that, The method comprises the following steps: S1. Automatically switching between compact formation and standard formation according to the real-time distance from the geometric center of the formation to the obstacle, and realizing intelligent reconstruction of the formation shape through a preset distance threshold and a smooth transition mechanism; S2. Realizing parameterized dynamic operator design of each spatially distributed control unit by using a free parameterized stable network architecture, and realizing distributed information exchange through a sparse interconnection topology isomorphic to a spatial system communication graph; S3. Using the inherent stability properties of the free parameterized stable network structure, constructing a parameter-unconstrained scheme suitable for the controller in the space environment, so that the stability of the closed-loop system in space is guaranteed for any parameter value, thereby avoiding the feasibility constraints of linear matrix inequalities, and realizing unconstrained gradient learning of the three-dimensional distributed controller.

2. The method of claim 1, wherein, The free parameterized stable network in step S2 adopts an internal model control structure, reconstructs the generalized disturbance by calculating the deviation between the actual state of the system and the predicted state of the model, and designs a control strategy based on the generalized disturbance and the environmental parameters.

3. The method of claim 1, wherein, The sub-operator dynamics of the free parameterized stable network in step S2 follow a structured discrete-time form, including three equations of internal state evolution, intermediate output generation and neuron activation, to maintain its stability properties.

4. The method of claim 1, wherein, The sparse interconnection topology in step S2 satisfies the following conditions: (a), the disturbance and parameter input matrix is column orthogonal, which ensures effective fusion of information; (b), the control output matrix satisfies the orthogonality structure, which ensures the linear independence of the control output; (c), the feedback matrix of the free parameterized stable network output to the input is column independent, which ensures the controllability of the sparse interconnection topology and the effectiveness of energy transmission.

5. The method of claim 1, wherein, The parameter-unconstrained learning in step S3 is realized by automatically calculating rules, which automatically calculate the stability gain parameters according to whether the agent receives external input and the coupling strength of its neighborhood, to ensure that for any free parameter, the constructed controller can satisfy the global stability condition.

6. The method of claim 1, wherein, Step S1 specifically comprises: S1.

1. Calculate the geometric center of the formation by averaging all agent position components; S1.

2. Calculate the Euclidean distance from the geometric center of the formation to the center of the known obstacle; S1.

3. Define a smooth switching function based on a hyperbolic tangent function, which outputs a switching coefficient that varies smoothly between 0 and 1 according to the relative relationship between the distance and the preset threshold; S1.

4. According to the switching coefficient, convexly combine the preset standard formation configuration and compact formation configuration to generate the current target formation vector in real time; S1.

5. Integrate the distance, switching coefficient and target formation vector into a state-dependent environmental parameter vector.

7. The method of claim 6, wherein, The smooth switching function in step S1.3 satisfies: when the formation distance is much smaller than the near distance threshold, the switching coefficient tends to zero, corresponding to the compact formation; when the formation distance is much larger than the far distance threshold, the switching coefficient tends to one, corresponding to the standard formation; and a smooth transition is realized between the two thresholds. The standard formation configuration and the compact formation configuration in step S1.4 are both preset vectors with fixed geometric configurations in three-dimensional space, and the difference determines the maximum amplitude of the adaptive adjustment of the formation.

8. The method of claim 6, wherein, ​ 9. The method of claim 1, wherein, The free parameterization stable network in step S2 inherits the structured dynamic form of the recursive balanced network, and maintains the Lipschitz bound and contraction properties.

10. The method of claim 1, wherein, The external input of the free parameterization stable network in step S2 is composed of three parts: the output feedback from the neighborhood, the local disturbance reconstruction signal and the environmental parameters; the final control output of the free parameterization stable network is composed of two parts: the output of the network itself and the direct action of the environmental parameters.

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