Distributed formation control method for heterogeneous cluster unmanned system based on reinforcement learning

Abstract

The application discloses a kind of distributed formation control methods of heterogeneous cluster unmanned system based on reinforcement learning.The application introduces reinforcement learning method on the basis of traditional optimal control, uses online state and input information to solve algebraic riccati equation iteratively, and all iterations can be carried out by repeatedly using the same state and input information in fixed time interval without using explicit, prior internal dynamics knowledge of system, which can get rid of the strong dependence on accurate system model in existing research on multi-agent system formation control;The optimal control strategy is learned by reinforcement learning method, and the formation performance and energy consumption are balanced, while completing the formation task, the minimum cost is met.

Description

Technical Field

[0001] This invention relates to the field of unmanned swarm systems technology, specifically to a distributed formation control method for heterogeneous unmanned swarm systems based on reinforcement learning. Background Technology

[0002] With the continuous development of artificial intelligence, multi-agent system cooperative control has been widely applied in civilian and military fields. Compared with single agents, multi-agent systems can more efficiently complete complex tasks that a single agent cannot accomplish through cooperation, and have advantages such as stronger adaptability, better fault tolerance, and better parallelism. Formation control, as a major research direction in multi-agent system cooperative control, has wide applications in search and rescue, unmanned aerial vehicle cooperation, intelligent vehicle control, and fleet formation.

[0003] In practical applications of formation control for unmanned swarm systems, the heterogeneity of these systems is caused by the diversity of controlled objects, making traditional distributed control methods no longer applicable. Therefore, a hierarchical control method is proposed to perform hierarchical control on heterogeneous unmanned swarm systems.

[0004] Existing research on formation control of multi-agent systems heavily relies on accurate system models. However, in practical applications, the formation of swarm unmanned systems is affected by external uncertainties, making it difficult to accurately model and obtain system models. This poses a significant challenge to the realization of cooperative formation control. Reinforcement learning, with its ability to find optimal control strategies in unknown environments, has shown great promise in the control design of practical systems. Furthermore, ensuring the energy consumption required for continuous data exchange between agents is a key issue in formation control. Reinforcement learning can gain new experience through continuous trial and error. By adopting different strategies and learning from feedback signals during interaction with the environment, it updates its strategies and ultimately learns an optimal strategy, thereby balancing system performance and energy consumption.

[0005] Based on the above two considerations, combining reinforcement learning algorithms to solve the formation problem of heterogeneous unmanned systems can optimize the formation effect while reducing the need for understanding the system dynamics and enhancing the system's ability to cope with external factors. This is a novel and effective approach to solving the formation problem and has significant research value. Summary of the Invention

[0006] In view of this, the present invention provides a distributed formation control method for heterogeneous cluster unmanned systems based on reinforcement learning, which can be independent of the system model and enhance the system's ability to cope with external factors, effectively achieving optimal formation control under the influence of external uncertainties.

[0007] The present invention relates to a distributed formation control method for a heterogeneous swarm unmanned system based on reinforcement learning, wherein the heterogeneous swarm unmanned system includes one leader and N followers;

[0008] Step 1: Establish a heterogeneous cluster unmanned system model; where the system model with N followers is as follows:

[0009]

[0010] Where i = 1, 2, ..., N, x i It is the state of the i-th follower; u i It is the control input of the i-th follower; y i It is the output of the i-th follower; A i B i C i These represent the state matrix, input matrix, and output matrix of the follower system, respectively.

[0011] The leader's system model is:

[0012]

[0013] Where x0 is the leader's state; y0 is the leader's output; A0 represents the leader system's state matrix; and C0 represents the leader system's output matrix.

[0014] Define the expected formation position of follower i. Let x0 be the leader's position and σ be the positional deviation between the follower and the leader. i The sum; the positional deviation σ between follower i and follower j ij =σ i -σ j ;σ j The positional deviation between follower j and the leader;

[0015] Step 2: Design a distributed observer to estimate the desired formation position of the followers, and obtain the estimated value of the desired position. The distributed observer is:

[0016]

[0017] in, For follower i, the desired position The estimated value; ηi is an estimate of the positional deviation between follower i and the leader; η0 is a constant; δi Δi For σ i The local observation error satisfies

[0018]

[0019] If follower i can directly access the leader, then g i =1, otherwise g i =0; j represents the neighbor of follower i, N i a represents the set of neighbors of follower i; ij Let A0 be the interaction strength between the i-th follower and the j-th follower; A0 is the leader system matrix; η1 is a constant. This is an estimate of the positional deviation between follower i and follower j;

[0020] Step 3: Construct the augmented system of the followers using the vector composed of the system state of the followers and the desired formation position as the augmented system state;

[0021] Based on the aforementioned follower augmentation system, the controller is designed as follows:

[0022]

[0023] in, It is the augmented system state of follower i; It is the control gain matrix, K 1i K represents the feedback gain of follower i. 2i This represents the feedforward gain of follower i;

[0024] According to optimal control theory, the optimal formation control strategy takes the following form:

[0025]

[0026] The optimal formation control gain matrix is: u i * (t) represents the learned optimal formation control strategy;

[0027] Step 4: Using the value iteration method of reinforcement learning, the gain matrix of formation control is solved to learn the optimal formation control strategy; specifically:

[0028] Step 41A, initialize the iteration count k = 0; let the initial Lyapunov matrix of follower i be... Bounded set B q Given an initial iteration count q = 0, and any initial bounded formation control input.

[0029] Step 42A, will Acting on the augmented system, measuring [t0, t l During this period, the system status is recorded, and the formation status block and input data block are obtained. in,

[0030]

[0031]

[0032]

[0033] in, These represent the formation system status respectively. Elements in; For follower i in subinterval [t] m-1 , t m System formation status data; For follower i in subinterval [t] m-1 , t m System data associated with formation control inputs; m = 1, 2, ..., l;

[0034] Step 43A, solve the following equation:

[0035]

[0036] Obtain the data matrix of follower i in the k-th iteration. and formation control gain matrix

[0037] in, I n It is an n×n identity matrix; To augment the state matrix of the system; Let Lyapunov be the matrix of follower i in the k-th iteration;

[0038] Step 44A: Calculate the pre-updated Lyapunov matrix in the (k+1)th iteration. ∈ k Let be the step size for the k-th iteration;

[0039] judge Whether it is valid, It is a set that has a non-empty internal bounded set, satisfying q∈Z + B q The number of iterations, lim q→∞ B q =L n L n Let represent the normed space of all n×n real symmetric matrices formed by the induced norms of matrices; if , let . q←q+1, jump to step 46A; if not true, proceed to step 45A;

[0040] Step 45A, Determine Does it hold true? If it does, then we obtain the Lyapunov matrix. and formation control gain matrix Jump to step 47A; if not true, then let Return to step 6; where, The set threshold;

[0041] Step 46A: Let the iteration count k ← k+1, then return to step 43A;

[0042] Step 47A, let Achieve optimal formation control

[0043] In step 4 of this invention, a reinforcement learning-based policy iteration method can also be used to solve for the gain matrix of formation control and learn the optimal formation control policy; specifically:

[0044] Step 41B, initialize the iteration count k = 0; set the formation control input to... in, e is the initial stable formation control gain matrix of the system; i For noise;

[0045] Step 42B, Acting on the augmented system, measuring [t0, t l During this period, the system status is recorded, and the formation status block and input data block are obtained. in,

[0046]

[0047]

[0048]

[0049] in, These represent the formation system status respectively. Element; For follower i in subinterval [t] m-1 , t m System formation status data; For follower i in subinterval [t] m-1 , t m System data associated with formation control inputs; m = 1, 2, ..., l;

[0050] Step 43B, solve the following equation:

[0051]

[0052] Obtain the Lyapunov matrix generated by follower i in the k-th iteration. and formation control gain matrix

[0053] in, I n It is an n×n identity matrix. For Kronecker product, Where a i ∈R n It is the i-th column of matrix A;

[0054] Step 44B, for k≥1, determine If the condition is met, stop the iteration and obtain the formation control gain matrix. Execute step 45B; otherwise, let the iteration count k ← k+1 and return to step 43B; ε is the set threshold of 0;

[0055] Step 45B, let Achieve optimal formation control

[0056] Preferably, the follower augmentation system is:

[0057]

[0058] in, It is the augmented system state of follower i, x i (t) represents the state of follower i. Represents the desired position of follower i in a distributed observer. The estimated state; It is the derivative of the augmented system state of follower i; A is the state matrix of the augmented system. i A represents the state matrix of follower i, and A0 represents the state matrix of the leader. B is the input matrix of the augmented system. i Let represent the input matrix of follower i.

[0059] Beneficial effects:

[0060] The distributed observer proposed in this invention can better estimate the true desired formation position and the positional deviation between followers and the leader, thus eliminating the need for each follower to directly obtain information from the leader. Furthermore, a formation control method based on value iteration / policy iteration is proposed to realize the preset formation, which can overcome the strong dependence on precise system models in existing research on formation control of multi-agent systems.

[0061] Formation control and formation maintenance are the core issues of formation control. However, general formation methods suffer from low formation efficiency and poor control performance, failing to consider the optimal performance of the system. Traditional adaptive controllers typically cannot achieve optimality in the sense of minimizing the cost function. This invention learns the optimal control strategy through reinforcement learning, balancing formation performance and energy consumption, and minimizing cost while completing the formation task.

[0062] Traditional optimal control design methods are offline, requiring system model identification before solving the relevant algebraic Riccati equations. They are slow to respond to changes in system parameters and cannot promptly address uncertainties and changes in system dynamics. This invention introduces reinforcement learning into traditional optimal control, using online state and input information to iteratively solve the algebraic Riccati equations. Furthermore, all iterations can be performed by repeatedly using the same state and input information within fixed time intervals, without relying on explicit, prior knowledge of the system's internal dynamics.

[0063] Most existing research on formation control of heterogeneous swarm unmanned systems is based on the premise that the system model is known, relying on system matrix information and failing to consider external factors such as disturbances. This makes it incapable of handling situations with external disturbances, unknown system models, or time-varying system models. This invention designs a distributed formation control method for heterogeneous swarm unmanned systems based on reinforcement learning. Attached Figure Description

[0064] Figure 1 This is a flowchart of the distributed formation control method for heterogeneous cluster unmanned systems based on reinforcement learning proposed in this invention.

[0065] Figure 2 The present invention presents a flowchart of a model-free distributed optimal formation control algorithm based on policy iterative learning.

[0066] Figure 3 This is a flowchart of the model-free distributed optimal formation control algorithm based on value iterative learning proposed in this invention. Detailed Implementation

[0067] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0068] This invention provides a distributed formation control method for heterogeneous cluster unmanned systems based on reinforcement learning, the flowchart of which is shown below. Figure 1 As shown, the specific steps include the following:

[0069] Step 1: Establish a heterogeneous cluster unmanned system model. Assume it consists of N followers and 1 leader, and they communicate through a topology graph.

[0070] A system model with N followers is established as follows:

[0071]

[0072] Where i = 1, 2, ..., N, x i ∈R n Let x represent the state of the i-th follower, and n represent the state x of the i-th follower. i dimensionality; The derivative of the i-th follower's state; u i ∈R m It represents the control input of the i-th follower, where m represents the control input u of the i-th follower. i dimensionality; y i ∈R n It represents the output of the i-th follower, where n represents the output y of the i-th follower. i The dimension of A; i ∈R n ×n B represents a system matrix of dimension n×n. i ∈R n×m Let C represent an input matrix of dimension n×m. i ∈R n×n This represents the system output matrix with dimension n×n.

[0073] The leader's system model is established as

[0074]

[0075] Where, x0∈R n It represents the leader's state and the expected trajectory that followers need to track. n represents the dimension of the leader's state x0. The derivative of the leader's state; y0∈Rn is the leader's output, where n represents the dimension of the leader's output y0; A0∈R n×n Let C0 ∈ R be the leader system matrix of dimension n×n. n×n This represents the leader output matrix with dimension n×n.

[0076] In the desired formation, the positional deviation σ between the i-th follower and the leader is defined. i Satisfying A0σ i =0, therefore the expected formation position of the i-th follower is defined as Furthermore, the positional deviation between the i-th follower and the j-th follower is defined as σ. ij =σ i -σ j .

[0077] Step 2: Considering that not all followers can directly obtain the leader's state information, design a distributed observer to estimate the expected positions of the followers. The distributed observer is designed as follows:

[0078]

[0079] in For follower i, the desired position The estimated value; This is an estimate of the positional deviation between the i-th follower and the leader; Let ηi be the estimated positional deviation between the i-th follower and the j-th follower; η0 and η1 represent known constants that can be arbitrarily chosen; j represents the neighbors of follower i, and N is the distance between them. i a represents the set of neighbors of follower i; ij δ represents the interaction strength between the i-th follower and the j-th follower. Δi For σ i The local observation error satisfies

[0080]

[0081] If follower i can directly access the leader, then g i =1, otherwise g i =0.

[0082] Step 3: Construct the augmented system based on the follower system state and the desired formation position vector.

[0083]

[0084] in It is the augmented system state of follower i, x i (t) represents the state of follower i. Represents the desired position of follower i in a distributed observer. The estimated state; It is the derivative of the augmented system state of follower i; A is the state matrix of the augmented system. i A represents the state matrix of follower i, and A0 represents the state matrix of the leader. B is the input matrix of the augmented system. i Let represent the input matrix of follower i.

[0085] Controller Design

[0086]

[0087] in, It is the augmented system state of follower i; It is the control gain matrix, K 1i K represents the feedback gain of follower i. 2i This represents the feedforward gain of follower i.

[0088] Define the expected output of follower i The output formation tracking error is then...

[0089]

[0090] Among them, y i It is the output of follower i; It is the expected output of follower i; C is the state matrix of the error system. i C1 represents the output matrix of follower i, and C2 represents the output matrix of leader i.

[0091] Cost function J i design

[0092]

[0093] Among them, e j It is the output formation tracking error of follower i; u i Q is the control input for follower i; i With R i Let the weight matrix corresponding to follower i be a symmetric positive definite matrix that satisfies Q. i =Q i T ≥0, R i =R i T >0, and Observable.

[0094] Value function V i design

[0095]

[0096] Among them, P i Let Lyapunov be the Lyapunov matrix of follower i, which is a symmetric positive definite matrix and satisfies the ARE equation.

[0097]

[0098] The weight matrix

[0099] According to optimal control theory, the optimal formation control strategy can be derived as follows:

[0100]

[0101] The optimal formation control gain matrix is: u i * (t) represents the learned optimal formation control strategy.

[0102] Step 4: To approximate the solution of the Lyapunov matrix P i We designed a model-free distributed optimal formation control algorithm based on reinforcement learning.

[0103] Furthermore, the model-free distributed optimal formation control algorithm based on reinforcement learning designed in step 4 mainly includes two methods: policy iteration and value iteration.

[0104] Step 4.1a: Design a model-free distributed optimal formation control algorithm using the strategy iteration method.

[0105] ① For follower i, assume the system initially has a stable formation control gain. Given, define a new matrix.

[0106]

[0107] in It is the state matrix of the augmented system. It is the input matrix of the augmented system.

[0108] ② Consider the following new closed-loop system

[0109]

[0110] ③ Design value function

[0111]

[0112] Where k represents the number of algorithm iterations, and Lyapunov matrix

[0113] ④ Bellman equations for valued functions

[0114]

[0115] in, It is the augmented system state of follower i at time t. It is the augmented system state at time t+T, where T is the time interval; u i It is the formation control strategy for follower i; R i It is a given weight matrix; It is the Lyapunov matrix of follower i in the k-th iteration, and the data matrix in the k-th iteration. Gain matrix

[0116] The above equation will involve the control input matrix. item Replace with in As another unknown matrix and Simultaneous solution. Depends on the unknown matrix. and item Replaced with This integral term can be obtained through online measurement of the system state. In summary, the above equation can replace the requirement for the system matrix with online measured state and input information.

[0117] ⑤ The system generates data

[0118]

[0119] in, Represents a column vector. These represent the formation system status respectively. Element. Representation matrix The m-th element in the n-th row of the array.

[0120] ⑥ System data collection. Define formation state and input data storage units.

[0121]

[0122]

[0123]

[0124] Where 0 ≤ t0 < t1 < ... < t l In the time interval [t0, t] l The system collects the formation state and input data generated by the system, and divides the interval into several sub-intervals with a certain sampling step size; t0 is the initial time of data collection, t1 is the time after the system has passed the sampling step size at the initial data collection time t0, and so on, t l The termination time for system data collection; These represent the positions of the corresponding follower i in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 , t l The system formation status data collected within the system; These represent the positions of the corresponding follower i in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 , t l The system collects system data associated with formation control inputs.

[0125] ⑦ Equivalence relation. By combining ⑤ and ⑥, we can obtain the following equivalence relation.

[0126]

[0127] in Represents a row vector. Represents the formation controller gain matrix of follower i The value learned in the (k+1)th iteration.

[0128] when When the column is full rank, there is a least-squares solution.

[0129]

[0130] In summary, the model-free distributed optimal formation control algorithm based on policy iterative learning is designed as follows:

[0131]

[0132] Most existing research on formation control of multi-agent systems is based on the premise that the system model is known, relying on system matrix information and neglecting external factors such as disturbances. This results in slow response to changes in system parameters and an inability to handle situations with external disturbances, unknown system models, or time-varying system models. The proposed algorithm introduces a policy iteration method based on traditional optimal control, using online state and input information to iteratively solve the algebraic Riccati equation. Furthermore, all iterations can be performed by repeatedly using the same state and input information within fixed time intervals, without relying on explicit, prior knowledge of the system's internal dynamics. Simultaneously, the algorithm achieves faster formation error convergence than traditional formation control, requires less energy, and has higher formation efficiency.

[0133] However, the strategy iteration algorithm depends on the initial stabilization strategy of the system. To solve this problem, this patent further proposes a model-free distributed optimal formation control algorithm based on value iteration learning.

[0134] Step 4.1b: Design a model-free distributed optimal formation control algorithm using the value iteration method.

[0135] ① Consider a closed-loop system for follower i.

[0136]

[0137] in It is the state matrix of the augmented system. It is the input matrix of the augmented system.

[0138] ②Design value function

[0139]

[0140] Where k represents the number of algorithm iterations, and Lyapunov matrix

[0141] ③ Bellman equations for valued functions

[0142]

[0143]

[0144] in, It is the augmented system state of follower i at time t. It is the augmented system state at time t+T, where T is the time interval; u i It is the formation control strategy for follower i; R i It is a given weight matrix; It is the Lyapunov matrix of follower i in the k-th iteration, and the data matrix in the k-th iteration. Formation control gain matrix

[0145] In the above formula, it depends on the unknown matrix. item Replaced with This will involve the control input matrix item Replace with in As another unknown matrix and Solve simultaneously.

[0146] ④ The system generates data

[0147]

[0148] in, Represents a column vector. These represent the formation system status respectively. Element.

[0149] ⑤ System data collection. Define formation state and input data storage units.

[0150]

[0151]

[0152]

[0153] Where 0 ≤ t0 < t1 < ... < t l In the time interval [t0, t] lThe system collects the formation state and input data generated by the system, and divides the interval into several sub-intervals with a certain sampling step size; t0 is the initial time of data collection, t1 is the time after the system has passed the sampling step size at the initial data collection time t0, and so on, t l The termination time for system data collection; These represent the positions of the corresponding follower i in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 , t l The system formation status data collected within the system; These represent the positions of the corresponding follower i in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 , t l The system collects system data associated with formation control inputs.

[0154] ⑥ Equivalence relation. By combining ④ and ⑤, we can obtain the following equivalence relation.

[0155]

[0156] in This represents a row vector.

[0157] When θ i When the column is full rank, there is a least-squares solution.

[0158]

[0159] In summary, the model-free distributed optimal formation control algorithm based on value iterative learning is designed as follows:

[0160]

[0161]

[0162] Compared to the policy iteration algorithm proposed in 4.1a, the model-free distributed optimal formation control algorithm based on value iterative learning designed above does not depend on the initial stabilization policy and releases the constraints on the initial conditions. In addition, in the case of unknown system dynamics model, this algorithm eliminates the assumptions about the knowledge of the system dynamics by iteratively solving the LQR problem of the linear continuous-time system online, thus making it a truly model-free algorithm.

[0163] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A distributed formation control method for a heterogeneous swarm unmanned system based on reinforcement learning, wherein the heterogeneous swarm unmanned system comprises one leader and N followers; characterized in that, include: Step 1: Establish a heterogeneous cluster unmanned system model; where the system model with N followers is as follows: in, , It is the first The state of a follower; It is the first The control input of a follower; It is the first The output of each follower; , , These represent the state matrix, input matrix, and output matrix of the follower system, respectively. The leader's system model is: in, It is the state of a leader; It is the leader's output; Represents the state matrix of the leader system. This represents the output matrix of the leader system; Define followers Expected formation position For the position of leader Positional deviation between the follower and the leader and; followers With followers Positional deviation between ; For followers Positional discrepancy with the leader; Step 2: Design a distributed observer to estimate the desired formation position of the followers, and obtain the estimated value of the desired position. The distributed observer is: in, For followers For desired position The estimated value; For followers An estimate of the positional deviation from the leader; It is a constant; For about The local observation error satisfies If followers If the leader can be accessed directly, then ,otherwise ; Indicates follower Neighbors Indicates follower The set of neighbors; For the first The first follower and the first The intensity of interaction between followers; A 0 represents the leader system matrix; It is a constant; For followers With followers The estimated value of the positional deviation between them; Step 3: Construct the augmented system of the followers using the vector composed of the system state of the followers and the desired formation position as the augmented system state; The follower augmentation system is as follows: in, Followers The augmented system state, Indicates follower state, Represents the follower in a distributed observer For desired position The estimated state; Followers The derivative of the augmented system state; It is the state matrix of the augmented system. Indicates follower The state matrix, The state matrix representing the leader; It is the input matrix of the augmented system. Indicates follower The input matrix; Based on the aforementioned follower augmentation system, the controller is designed as follows: in, Followers The augmented system state; It is the control gain matrix. Indicates follower Feedback gain, Indicates follower Feedforward gain; Design cost function for: in, Followers The output formation tracking error, , Followers The output; Followers The expected output; It is the state matrix of the error system. Indicates follower The output matrix, Represents the leader's output matrix; Followers Control input; and Indicates follower The corresponding weight matrix is ​​a symmetric positive definite matrix that satisfies ,and Observable; According to optimal control theory, the optimal formation control strategy takes the following form: The optimal formation control gain matrix is: ; This represents the learned optimal formation control strategy; Step 4: Using the value iteration method of reinforcement learning, the gain matrix of formation control is solved to learn the optimal formation control strategy; specifically: Step 41A, Initialize the number of iterations ; to command followers The initial Lyapunov matrix , bounded set Initial iteration count Given any initial bounded formation control input ; Step 42A, will Acting on the augmentation system, measuring During this period, the system status is obtained, and the formation state block and input data block are acquired. ;in, in, ; These represent the formation system status respectively. Elements in; For followers In subinterval System formation status data; For followers In subinterval System data associated with formation control inputs; m=1,2,…, l ; Step 43A, solve the following equation: Gain followers In the Data matrix in the next iteration and formation control gain matrix ; in, , for identity matrix; ; For the augmented system's state matrix; For the first Followers in the next iteration Lyapunov matrix; Step 44A, calculate the first... The pre-updated Lyapunov matrix in the next iteration ; For the first The step size of the next iteration; judge Whether it is valid, It is a set that has a non-empty internal bounded set, satisfying , express The number of iterations, , Represents all matrices composed of their induced norms. A real symmetric normed space; if true, then let , If the condition is not met, proceed to step 46A; if not, proceed to step 45A. Step 45A, Determine Does it hold true? If it does, then we obtain the Lyapunov matrix. and formation control gain matrix Proceed to step 47A; if not, then... Return to step 6; where, The set threshold; Step 46A, let the number of iterations be... Return to step 43A; Step 47A, let To achieve optimal formation control .

2. The method as described in claim 1, characterized in that, In step 4, a reinforcement learning-based policy iteration method is used to solve for the gain matrix of formation control and learn the optimal formation control policy; specifically: Step 41B, Initialize the number of iterations ; Set the formation control input to ;in, The initial stable formation control gain matrix of the system; For noise; Step 42B, Acting on the augmentation system, measuring During this period, the system status is obtained, and the formation state block and input data block are acquired. ;in, in, ; These represent the formation system status respectively. Element; For followers In subinterval System formation status data; For followers In subinterval System data associated with formation control inputs; m=1,2,…, l ; Step 43B, solve the following equation: Gain followers In the The Lyapunov matrix generated in the next iteration and formation control gain matrix ; in, , for identity matrix For Kronecker product, , ,in It is the first of matrix A List; Step 44B, for ,judge If the condition is met, stop the iteration and obtain the formation control gain matrix. Execute step 45B; otherwise, reduce the iteration count. Return to step 43B; The threshold value is set to 0. Step 45B, let To achieve optimal formation control .

3. The method as described in claim 1, characterized in that, Value function Designed as follows: in, For followers The Lyapunov matrix is ​​a symmetric positive definite matrix that satisfies the ARE equation: in, It is the state matrix of the augmented system; The input matrix of the augmented system; the weight matrix. .

Images