Cooperative dynamic predetermined performance control method for output synchronization of network nonlinear system

Abstract

The invention provides a cooperative dynamic predetermined performance control method for output synchronization of a network nonlinear system, and the method comprises the steps: guaranteeing the effective transmission of information through designing a communication network (S1) between multi-agent systems; secondly, designing a dynamic performance function based on a neighborhood error (S2), giving a stricter permissible limit, and achieving a faster convergence speed; in the stage of establishing the dynamic model (S3) of the strict feedback type nonlinear system, establishing a model containing unknown parameters and disturbance for each agent; (S4), reducing the updating frequency through an event triggering mechanism to guarantee the efficient operation of the system; finally, in a PPC and backstepping control strategy (S5) stage, a finite time differentiator is used for solving the differential explosion problem in a backstepping method, and an output synchronous controller is designed; according to the method, a new performance function is introduced while the system performance is ensured, the calculation burden and the updating frequency are reduced, and the practicability and the efficiency of the system are improved.

Description

Technical Field

[0001] The present invention relates to the field of control, and more specifically, to a cooperative dynamic prescribed performance control method for output synchronization of networked nonlinear systems Background Art

[0002] In recent years, the cooperative control problem of nonlinear multi-agent systems has received extensive attention. With the continuous development of adaptive control technology, the dynamic complexity of agents has also been increasing. Researchers have achieved some results in exploring nonlinear agents with uncertainties and unknown control directions. In the prior art, some people have studied the consensus problem in nonlinear multi-agent systems with unknown parameters by using the parameter estimator method. At the same time, the adaptive neural network control technology has also been widely applied to strict feedback systems with uncertain parameters. And applying the radial basis function neural network (RBFNN) controller to uncertain switched stochastic nonlinear systems is also the focus of research

[0003] The actual control performance of the system in the actual scenario depends to a large extent on the design of the controller and the selection of the control gain. However, although the system can finally achieve convergence, the transient performance during this process cannot be guaranteed. To ensure the transient and steady-state performance of the system, researchers have developed various methods. Typical methods include the barrier Lyapunov function method (BLF), the funnel control method (FC), and the prescribed performance control method (PPC), etc. PPC has the following advantages compared with the FC and BLF methods

[0004] (1) PPC is applicable to a wider range of system types without strict restrictions

[0005] (2) PPC implicitly considers the constraint conditions and does not require a special form of Lyapunov function, thus improving the generalizability and flexibility of the design process

[0006] (3) PPC transforms the constrained control problem into a stability control problem, simplifying the analysis

[0007] By adopting a predetermined performance control method, the transient and steady-state performance of multiple uncertain nonlinear systems are simultaneously ensured. It is worth noting that the previous performance function was fixed and did not adapt cooperatively with the error states, which was imperfect. For this reason, this paper proposes a new cooperative dynamic performance function that is adjusted according to the neighborhood error. Most of the above studies require continuous updates of the controller, but continuous controller updates significantly increase the communication cost, bandwidth utilization, and energy consumption. Considering these factors, event-triggered control has received extensive attention due to its efficiency and reduced communication overhead. However, previous studies on event-triggered control, whether or not they include predetermined performance control, rarely achieve precise output synchronization. On the contrary, in previous performance control studies, the performance function was mainly used to limit the overshoot of the system, but the performance function only changed with time and was independent of the system state.

[0008] Therefore, studying new control methods for nonlinear multi-agent systems is of great significance for improving system performance and reducing communication costs. To solve the above problems, this patent aims to provide a new cooperative control method to improve the performance and efficiency of nonlinear multi-agent systems under uncertain and complex dynamic conditions. Summary of the Invention

[0009] Aiming at the problems in the existing research on event-triggered control that precise output synchronization cannot be achieved and the performance function only changes with time and is independent of the system state, the present invention provides a cooperative dynamic predetermined performance control method for output synchronization of networked nonlinear systems.

[0010] To achieve the above object, the present invention adopts the following technical solutions:

[0011] S1: Design the communication network between multi-agent systems to ensure that information can be effectively transmitted between agents;

[0012] S2: Design a dynamic performance function based on neighborhood error to ensure that the system has good transient and steady-state performance; dynamically adjust the convergence range to optimize the system response speed;

[0013] S3: Establish the dynamic model of a strictly feedback nonlinear system; use the method based on preset performance control (PPC) to perform error transformation to ensure that the error converges within the specified range;

[0014] S4: Introduce the event-triggered control (ETC) strategy and design the trigger condition to reduce the update frequency of the controller; thereby reducing communication and computational overhead and ensuring the efficient operation of the system;

[0015] S5: Use a finite-time differentiator to solve the differential explosion problem in the backstepping method; combine the PPC and backstepping control strategies to design an output synchronization controller to ensure that all signals of the closed-loop system are bounded;

[0016] Furthermore, the multi-agent system is a non-linear system, and in the communication network of the system, R + represents a family of signals composed of positive real numbers, and z + represents a family of signals composed of positive integers; L2 represents the family of square-integrable signals, while L ∞ represents the family of bounded signals; sup(s) represents the least upper bound of s, and conversely, inf(s) represents the greatest lower bound of s;

[0017] Furthermore, the non-linear multi-agent system contains N agents, and communicates through a weighted directed graph as follows:

[0018] Node set contains N nodes, and the edge set Π contains a directed edge Π i from node v j to node v ij ;

[0019] The degree matrix D is an N×N diagonal matrix, where d i represents the in-degree of node v i ;

[0020] The matrix B is an N×N diagonal matrix used to indicate whether node i can receive the message of the leader. When b i > 0, it means it can receive, otherwise b i = 0; if agent i can directly communicate with the leader, then b i is set to 1;

[0021] Graph The Laplacian matrix of is defined as where the adjacency matrix is used to represent the connection relationship between agents.

[0022] Furthermore, step S2 designs a dynamic performance function based on neighborhood error to propose a new performance function for improving the output synchronization performance of the multi-agent system. The formula is as follows:

[0023]

[0024] where, is a constant designed by the user, and f(z)≥0 is a function of z; the form of the f(z) is chosen as f(z)=kz 2 +a, where k and a are positive constants to be determined; the improved performance function f(z) is a function of the variable z and time t, and it is not a predetermined function; the performance function changes with the neighborhood error z(t). When the error is large, CDPPC gives a more stringent allowable bound, which leads to faster convergence.

[0025] Furthermore, step S3 establishes the dynamic model of the strict-feedback nonlinear system:

[0026] The dynamic model of the strict-feedback nonlinear system is

[0027] where \(q = 1,\cdots,n - 1\), and \(i = 1,\cdots,N\); and represent the state, control input, and output of agent \(i\), respectively; the nonlinear functions and are known and smooth, where \(\Delta\) iq \((t)\) and \(\Delta\) in \((t)\) are disturbance terms, is an unknown parameter vector; the control coefficients \(g\) iq and \(g\) in are unknown;

[0028] The following assumptions are proposed for the dynamic model of the strict-feedback nonlinear system:

[0029] Assumption 1: The non-zero control coefficients \(g\) ik \((k = 1\cdots n)\) and the disturbances \(\Delta\) ik \((t)(k = 1\cdots n)\) are bounded, i.e., \(|g\) ik |\(\leq G\) ik and \(|\Delta\) ik \((t)| \lt D\) ik , where \(G\) ik and \(D\) ik are positive constants;

[0030] Assumption 2: The signal \(y\) r \((t)\) of the leader and its derivative are both bounded;

[0031] The preset performance control (PPC) method includes:

[0032] Error transformation, which is defined as where \(F\) t represents the transformation function, and \(\xi(t)\) represents the transformation error; the function \(F\) t is smooth, strictly increasing, and satisfies \(-l(t) \lt F_{t} \lt l(t)\); the \(l(t)\) is generated in the following way:

[0033] \(\lambda,h\in R\) + ;

[0034] The above formula has the following characteristics: \(l(t) \gt 0\) and is strictly decreasing; C ∈ R + ; In the actual design process, set l(0) as a sufficiently large positive constant;

[0035] Define the error transformation function F t (ξ(t), l(t)) as The transformation error can be expressed as

[0036] Furthermore, step S4 includes:

[0037] The event-triggered control (ETC) strategy includes:

[0038]

[0039] For the convenience of the subsequent proof process, the following expressions are defined to unify the forms of the fixed threshold and the relative threshold:

[0040]

[0041] Among them, in the formula, 0 < γ i1 < 1, m i1 > 0, m i2 > 0 and are all positive design parameters; e i (t) = w i (t) - u i (t) represents the measurement error between w i (t) and u i (t), H i > 0; k ∈ z + , which is the update time of the controller; during the entire time interval , the control signal remains unchanged, that is

[0042] Furthermore, step S5 includes:

[0043] Even further, step S51: Neighborhood error definition and Lyapunov function construction analysis;

[0044] The neighborhood error of the multi-agent system i is defined as follows: Among them, a ij represents the element in the adjacency matrix A, and b i represents the element in the matrix B, indicating the potential connection between the multi-agent system i and the leader; analyze the time derivative of the neighborhood error, and obtain the expression through mathematical analysis and reconstruction; select the Lyapunov function candidate, take the derivative with respect to time, and analyze its properties; integrate the expression of the time derivative to obtain a function expression about time.

[0045] Furthermore, step S52: Subsystem iterative analysis and proof;

[0046] First, define the state and time derivative of the subsystem; conduct mathematical analysis and reconstruction on the time derivative of the subsystem; then select a Lyapunov function for the subsystem, take the derivative with respect to time, and analyze its properties.

[0047] Furthermore, step S53: Introduce a finite-time differentiator to avoid iterative derivation of the virtual controller;

[0048] First, define the finite-time differentiator, including its state and parameters; the finite-time differentiator is defined as follows:

[0049]

[0050] where, π iq,1 and π iq,2 represent the state of the differentiator, k1, k2 > 0 are the parameters of the differentiator, and the sig function is expressed as As long as the initial deviation π iq,1 (0) - α i,q-1 (0) and are bounded, the finite-time differentiator can provide an approximation of arbitrary precision for ; thus, there is the estimation error δ iq is bounded, that is, a constant M iq > 0 can be determined such that |δ iq | ≤ M iq .

[0051] Then analyze the approximation accuracy provided by the finite-time differentiator to ensure that the estimation error is bounded; finally, combine the finite-time differentiator, design the virtual controller and refresh rate, and simplify to obtain the expression of the controller.

[0052] Furthermore, step S54: Conduct the controller design for the nth step of the final layer;

[0053] First, define the state and time derivative of the final layer; rewrite the time derivative according to the assumption 1; construct a Lyapunov function, take the derivative with respect to time, and analyze its properties; apply the event-triggered control strategy to rewrite the time derivative of the Lyapunov function; then introduce the following finite-time differentiator

[0054]

[0055] Finally, through simplification and integral processing, obtain

[0056] For the strict-feedback nonlinear system with unknown control directions, unknown parameters, and unknown disturbances, if Assumption 1 and Assumption 2 hold, the proposed controller can ensure the following properties: the outputs of all agents in the system will eventually synchronize precisely; the neighborhood error of each multi-agent converges to a bound within zero; for the communication network among the multi-agent systems, all signals are bounded in the closed-loop system.

[0057] Furthermore, Step S55: Stability proof;

[0058] First, integrate the time derivative inequality of the Lyapunov function obtained by substituting the designed virtual controller in Step S54 to analyze its boundedness; apply Barbalat's lemma to prove the convergence of the system state; finally, by defining the lower bound of the execution interval, avoid Zeno behavior and ensure that the system will not have infinitely many fast triggering events during actual operation.

[0059] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0060] 1. A new performance function is introduced, which is dynamically adjusted based on the neighborhood error. It has many advantages. For example, when the magnitude of the neighborhood error is large, the instantaneous performance bound is tightened to accelerate the system convergence.

[0061] 2. The finite-time differentiator is used to solve the differential explosion problem in the backstepping method, thereby reducing the computational amount.

[0062] 3. Compared with previous event-triggered control, the present invention achieves cumulative output synchronization, that is, the neighborhood error finally converges to zero instead of within the residual. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 It is a diagram of the overall steps of a cooperative dynamic prescribed performance control method for output synchronization of a network nonlinear system according to the present invention.

[0064] Figure 2 It is a curve diagram showing the difference between the traditional PPC and CDPPC of a cooperative dynamic prescribed performance control method for output synchronization of a network nonlinear system according to the present invention.

[0065] Figure 3 It is a diagram of the design tables of the controller, estimator, and various variables of a cooperative dynamic prescribed performance control method for output synchronization of a network nonlinear system according to the present invention.

[0066] Figure 4 It is a communication topology diagram of a cooperative dynamic prescribed performance control method for output synchronization of a network nonlinear system according to the present invention.

[0067] Figure 5This is the simulation diagram of the state output trajectory of the intelligent agent for the collaborative dynamic prescribed performance control of the output synchronization of a network nonlinear system in the present invention.

[0068] Figure 6 This is the simulation diagram of the neighborhood error and its specified performance bounds for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0069] Figure 7 This is the simulation diagram of the Nussbaum function variable for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0070] Figure 8 This is the unknown parameter i = 1, 2, 3, 4 estimation simulation diagram for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0071] Figure 9 This is the unknown parameter i = 1, 2, 3, 4 estimation simulation diagram for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0072] Figure 10 This is the simulation diagram of the controller input estimation for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0073] Figure 11 This is the simulation diagram of the controller event trigger interval for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention. Detailed implementation manner

[0074] The present invention will be further described in detail below in conjunction with the embodiments and the drawings, but the implementation manners of the present invention are not limited thereto.

[0075] Such as Figure 1 This is the overall step diagram for the collaborative dynamic prescribed performance control method of the output synchronization of a network nonlinear system in the present invention.

[0076] S1: Design the communication network between multi-agent systems to ensure that information can be effectively transmitted between agents;

[0077] S2: Design a dynamic performance function based on neighborhood error to ensure that the system has good transient and steady-state performances; dynamically adjust the convergence range to optimize the system response speed;

[0078] S3: Establish the dynamic model of a strict-feedback nonlinear system; use the method based on preset performance control (PPC) to perform error transformation to ensure that the error converges within the specified range;

[0079] S4: Introduce the event-triggered control (ETC) strategy, design trigger conditions to reduce the update frequency of the controller; thereby reducing communication and computational overhead and ensuring the efficient operation of the system;

[0080] S5: Use a finite-time differentiator to solve the problem of differential explosion in the backstepping method; combine the PPC and backstepping control strategies to design an output synchronization controller to ensure that all signals of the closed-loop system are bounded;

[0081] The multi-agent system is a nonlinear system. In the communication network of the system, R + represents the family of signals composed of positive real numbers, and z + represents the family of signals composed of positive integers; L2 represents the family of quadratically integrable signals, while L ∞ represents the family of bounded signals; sup(s) represents the least upper bound of s, and conversely, inf(s) represents the greatest lower bound of s;

[0082] The nonlinear multi-agent system contains N agents, and they communicate through a weighted directed graph as follows:

[0083] Node set contains N nodes. The edge set Π contains the directed edge Π i from node v j to node v ij ;

[0084] The degree matrix D is an N×N diagonal matrix, where d i represents the in-degree of node v i ;

[0085] The matrix B is an N×N diagonal matrix used to indicate whether node i can receive the leader's message. When b i > 0, it means it can receive, otherwise b i = 0; if agent i can directly communicate with the leader, then b i is set to 1;

[0086] Graph The Laplacian matrix of is defined as where the adjacency matrix is used to represent the connection relationship between agents.

[0087] Step S2 designs a new performance function based on the neighborhood error to improve the output synchronization performance of the multi-agent system. The formula is as follows:

[0088]

[0089] where, A constant designed for the user, f(z)≥0 is a function of z; the form of the said f(z) is selected as f(z)=kz 2 +a, where k and a are positive constants to be determined; the improved performance function f(z) is a function of the variable z and time t, and it is not a predefined function; the performance function varies with the neighborhood error z(t). When the error is large, CDPPC gives a more stringent allowable bound, which leads to faster convergence.

[0090] Compared with the prior art, the improved performance function is a function of the variable z(t) and time t. Therefore, it is not a predefined function. The performance function varies with the neighborhood error z(t). As Figure 2 is the difference curve graph between the traditional PPC and CDPPC of the cooperative dynamic predefined performance control method for output synchronization of a network nonlinear system according to the present invention. When the error is large, CDPPC gives a more stringent allowable bound, which leads to faster convergence.

[0091] Step S3: Establish the dynamic model of the strict-feedback nonlinear system:

[0092] The dynamic model of the strict-feedback nonlinear system is

[0093] where, q = 1,..., n - 1, and i = 1,…, N; and represent the state, control input, and output of agent i respectively; the nonlinear functions and are known and smooth, where Δ iq (t) and Δ in (t) are disturbance terms, is an unknown parameter vector; the control coefficients g iq and g in are unknown;

[0094] The following assumptions are proposed for the dynamic model of the strict-feedback nonlinear system:

[0095] Assumption 1: The non-zero control coefficients g ik (k = 1...n) and the disturbances Δ ik (t)(k = 1...n) are bounded, that is, |g ik |≤G ik and |Δ ik (t)|<D ik , where G ik and D ik are positive constants;

[0096] Assumption 2: The signal y of the leaderr (t) and its derivative are all bounded;

[0097] The described preset performance control (PPC) method includes:

[0098] Error transformation, which is defined as where F t represents the conversion function, and ξ(t) represents the conversion error; the function F t is smooth, strictly increasing, and satisfies -l(t) < F_t < l(t); the l(t) is generated in the following manner:

[0099] λ, h ∈ R + ;

[0100] The above formula has the following characteristics: l(t) > 0 and is strictly decreasing; C ∈ R + ; in the actual design process, l(0) is set to a sufficiently large positive constant;

[0101] Define the error transformation function F t (ξ(t), l(t)) as The conversion error can then be expressed as

[0102] Step S4 includes:

[0103] The event-triggered control (ETC) strategy includes:

[0104]

[0105] For the convenience of the subsequent proof process, the following expressions are defined to unify the forms of the fixed threshold and the relative threshold:

[0106]

[0107] where, in the formula, 0 < γ i1 < 1, m i1 > 0, m i2 > 0 and are all positive design parameters; e i (t) = w i (t) - u i (t) represents the measurement error between w i (t) and u i (t), H i > 0; k ∈ z + , is the update time of the controller; in the entire time interval the control signal remains unchanged, that is,

[0108] To more clearly show the design of the controller in step S5, Figure 3 This is a design chart of the controller, estimator, and various variables for the cooperative dynamic prescribed performance control method for output synchronization of a networked nonlinear system according to the present invention.

[0109] Step S5 includes:

[0110] Step S51, neighborhood error definition and Lyapunov function construction analysis; the neighborhood error of multi-agent system i is defined as follows: where a ij represents the element in the adjacency matrix A, and b i represents the element in matrix B, indicating the potential connection between multi-agent system i and the leader;

[0111] Define V i1 as where ξ i1 is the error transformed from z i1 i.e., The time derivative of V i1 is

[0112] The time derivative of V i1 obtained through mathematical analysis and reconstruction is:

[0113]

[0114] Select the Lyapunov function candidate as:

[0115]

[0116] The above formula is differentiated with respect to time to obtain:

[0117]

[0118] By substituting the virtual controller α Figure 3 designed in i1 and the refresh rate into the time derivative of we get

[0119]

[0120] Integrating both sides of the above formula, we get

[0121]

[0122] When there is

[0123] Step S52, the subsystem iteratively analyzes and proves that z i2 ∈L2; define z iq = x iq -α i,q-1 (2 ≤ q ≤ n - 1) and The time derivative of the said V iq is:

[0124]

[0125] V iq The time derivative of V is obtained through mathematical analysis and reconstruction as:

[0126]

[0127] Select the following Lyapunov function:

[0128]

[0129] The above formula Deriving with respect to time gives:

[0130]

[0131] Step S53, introduce a finite-time differentiator to avoid the iterative derivation of the virtual controller; the finite-time differentiator is defined as follows:

[0132]

[0133] where, π iq,1 and π iq,2 represent the states of the differentiator, k1, k2 > 0 are the parameters of the differentiator, and the sig function is expressed as As long as the initial deviation π iq,1 (0) - α i,q-1 (0) and are bounded, the finite-time differentiator can provide an approximation of with arbitrary precision; thus there is The estimation error δ iq is bounded, that is, a constant M iq > 0 can be determined such that |δ iq | ≤ M iq ;

[0134] Combined with the said finite-time differentiator, by substituting the virtual controller α Figure 3 designed in iq and the refresh rate into the derivative of with respect to time and simplifying, we get:

[0135]

[0136] Integrating both sides gives

[0137]

[0138] where, when there is

[0139] Step S54, perform the controller design for the nth step on the final layer; prove that z iq ∈ L2; define z in = x in - α i,n-1 and the time derivative of the said V in is:

[0140] Rewrite the time derivative of the said V in according to Assumption 1:

[0141]

[0142] Construct a Lyapunov function

[0143] The above formula is differentiated with respect to time to obtain:

[0144]

[0145] Rewrite the above formula according to the said event-triggered control (ETC) strategy:

[0146]

[0147] By substituting Figure 3 the virtual controller α iq designed in and the refresh rate into the above formula, it is simplified to obtain:

[0148]

[0149] Then introduce the following finite-time differentiator

[0150]

[0151] Substitute to obtain:

[0152] Simplify to obtain:

[0153] Integrating both sides of the above formula gives:

[0154]

[0155] For a strict-feedback nonlinear system with unknown control directions, unknown parameters, and unknown disturbances, if Assumption 1 and Assumption 2 hold, the proposed controller can ensure the following properties:

[0156] The outputs of all agents in the system will eventually synchronize precisely; the neighborhood error of each multi-agent converges to a bound within zero; for the communication network between the multi-agent systems, all signals are bounded in the closed-loop system;

[0157] Step S55, stability proof; Integrate both sides to obtain:

[0158]

[0159] Since ζ in (t), is bounded, then is bounded; applying Barbalat's lemma, we have Similarly, z iq ∈L2∩L ∞ , where 2 ≤ q ≤ n - 1; Next, we prove that ξ i1 ∈L2;

[0160] From the

[0161]

[0162] formula, we get

[0163]

[0164] Integrating both sides of the first inequality in the above formula according to the lemma, we obtain

[0165]

[0166] Using lemma analysis, it can be concluded that is bounded; according to Barbalat's lemma, we have lim t→∞ ξ i1 (t) = 0;

[0167] The ,, if and lim t→∞ ξ i1 (t) = 0. Then, define Z1(t) = [z (t),..., z 11 (t)] N1 (t)] T = L[x 11 (t),..., x N1 (t)]T ., according to the lemma

[0168] Because such that for all k ∈ Z + , Therefore, the said e i (t) = w i (t) - u i (t), Obtain:

[0169] where the "*" symbol represents multiplication, and further obtain:

[0170] In the closed-loop system, since is a polynomial function of the signals in the system, and all the signals in the closed-loop system are bounded, so there exists a constant κ i > 0, such that By defining e i (t) = w i (t) - u i (t), the lower bound of the execution interval is obtained must satisfy That is, Zeno behavior is successfully avoided.

[0171] In the specific simulation part, we use a ship model to verify the designed controller. This model has certain practical application value. As Figure 4 shows the communication topology between agents.

[0172] The dynamic formula of the ship model is as follows:

[0173]

[0174] where θ i = [-1 / T i , -W i / T i T are unknown control coefficients and parameters.

[0175] Set the initial values of the states of the ship multi-agent to

[0176] [x 11 (0), x 21 (0), x 31 (0), x 41 (0)] = [0.2, 1, -1.5, -1.7], [l1(0), l2(0), l3(0), l4(0)] = [1, 1, 2, 2] ​

[0177]

[0178] For simplicity, the initial values of all other estimators are uniformly set to 0. The smoothing functions l1(t) = l2(t) = l3(t) = l4(t) = l(t) are chosen as

[0179] The performance function is set as:

[0180] wherein,...

[0181] The unknown parameter selects M i = 20 + 0.5i, T i = 1 + 0.02i, W i = 0.4 - 0.05i. Note that these coefficients are unknown to the controller and their values are randomly assigned for simulation. Let H i = 10, γ i1 = 0.5, m i1 = 0.1, m i2 = 2, ε(t) = e -t . The trajectory of the leader is defined as y r (t) = 0.8sin(8t).

[0182] The simulation results are as Figures 5 to 11 shown. Figure 5 It shows the outputs of the four ship agents, indicating that they are precisely synchronized with the reference y r (t). Additionally, Figure 6 it shows that the neighborhood error of each agent remains within the specified bounds and finally converges to zero, that is, transient performance and steady-state accuracy are ensured simultaneously. Furthermore, Figure 6 it also shows that the CDPPC has a dynamic boundary, which varies with the neighborhood error and is more stringent than the existing PPC. It will push for a better boundary. This is the main difference between this article and other articles. It can be seen from Figure 7 -8 that the parameter estimators i2 in the Nussbaum function N(ζ ) and the variable ζ i2 are bounded. The trajectories of the parameter estimators are as Figure 9 shown and they are also bounded. Figure 10 It shows that the input of the agent is bounded. Figure 11 It describes the triggering interval of each agent, revealing that the input signal is not updated continuously. The controller is only updated when the triggering condition is satisfied, which reduces the computational load and update frequency.

[0183] Although embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the claims and their equivalents.

Claims

1. A cooperative dynamic prescribed performance control method for output synchronization of a networked nonlinear system, characterized in that Including: S1: Design the communication network among multi-agent systems to ensure the effective transmission of information among agents; S2: Design a dynamic performance function based on neighborhood error to ensure good transient and steady-state performance of the system; Dynamically adjust the convergence range to optimize the system response speed; S3: Establish the dynamic model of a strict-feedback nonlinear system; Use the preset performance control (PPC) method to perform error transformation to ensure the convergence of the error within a specified range; S4: Introduce the event-triggered control (ETC) strategy and design the trigger condition to reduce the update frequency of the controller; thereby reducing communication and computational overhead and ensuring the efficient operation of the system; S5: Use a finite-time differentiator to solve the differential explosion problem in the backstepping method; Combine the PPC and backstepping control strategies to design an output synchronization controller to ensure the boundedness of all signals in the closed-loop system; 2. The collaborative dynamic prescribed performance control method for output synchronization of a networked nonlinear system according to claim 1, wherein Step S1 of designing the communication network among multi-agent systems includes: The multi-agent system is a non-linear system. In the communication network of the system, R + represents a family of signals composed of positive real numbers, and z + represents a family of signals composed of positive integers; L2 represents the family of square-integrable signals, while L ∞ represents the family of bounded signals; sup(s) represents the least upper bound of s, and conversely, inf(s) represents the greatest lower bound of s; The non-linear multi-agent system contains N agents and communicates through a weighted directed graph as follows: Node set contains N nodes, and the edge set Π contains a directed edge Π i from node v j to node v ij ; The degree matrix D is an N×N diagonal matrix, where d i represents the in-degree of node v i ; Matrix B is an N×N diagonal matrix used to indicate whether node i can receive messages from the leader. When b i > 0, it means that it can receive, otherwise b i = 0; if agent i can communicate directly with the leader, then b i is set to 1; Figure The Laplacian matrix of is defined as where the adjacency matrix is used to represent the connection relationships among agents.

3. A cooperative dynamic prescribed performance control method for output synchronization of a network nonlinear system according to claim 1, wherein, Step S2 of designing a dynamic performance function based on neighborhood error includes: The step S2 of designing a dynamic performance function based on neighborhood error proposes a new performance function for improving the output synchronization performance of multi-agent systems, and the formula is as follows: Among them, is a constant designed for the user, and f(z)≥0 is a function of z; the form of the said f(z) is chosen as f(z)=kz 2 +a, where k and a are positive constants to be determined; the improved performance function f(z) is a function of the variable z and time t, and it is not a predefined function; the performance function varies with the neighborhood error z(t), and when the error is large, CDPPC gives a more stringent allowable limit, which leads to faster convergence.

4. A cooperative dynamic prescribed performance control method for output synchronization of a networked nonlinear system according to claim 1, characterized in that, Step S3 of establishing the dynamic model of a strict-feedback nonlinear system: The dynamic model of the strict-feedback nonlinear system is where \(q = 1,\ldots,n - 1\), and \(i = 1,\ldots,N\); and represent the state, control input, and output of agent \(i\), respectively; the nonlinear functions and are known and smooth, where \(\Delta\) iq (t) and \(\Delta\) in (t) are disturbance terms, is an unknown parameter vector; the control coefficients \(g\) iq and \(g\) in are unknown; The dynamic model of the strict-feedback nonlinear system proposes the following assumptions: Hypothesis 1: Non-zero control coefficient g ik (k = 1...n) and disturbance Δ ik (t) (k = 1...n) are bounded, i.e., |g ik | ≤ G ik And |Δ ik (t)| < D ik , where G ik and D ik are positive constants; Hypothesis 2: The signal y r (t) and its derivative are both bounded; The preset performance control (PPC) method includes: Error transformation, which is defined as where F t represents a conversion function, and ξ(t) represents the conversion error; the function F t is smooth, strictly increasing, and satisfies -l(t) < F_t < l(t); the l(t) is generated in the following manner: λ, h ∈ R + ; The above formula has the following characteristics: l(t) ≥ 0 and is strictly decreasing; C ∈ R + ; in the actual design process, l(0) is set to a sufficiently large positive constant; The defined error transformation function F t (ξ(t), l(t)) is The transformation error can be expressed as 5. A cooperative dynamic prescribed performance control method for output synchronization of a networked nonlinear system according to claim 1, characterized in that, The step S4 includes: The event-triggered control (ETC) strategy includes: For the convenience of subsequent proof processes, the following expressions are defined to unify the forms of fixed thresholds and relative thresholds: where, in the formula, 0 < γ i1 < 1, m i1 > 0, m i2 > 0 and are all positive design parameters; e i (t) = w i (t) - u i (t) represents the measurement error between w i (t) and u i (t), H i > 0; k ∈ z + , is the update time of the controller; within the entire time interval the control signal remains unchanged, that is 6. A cooperative dynamic prescribed performance control method for output synchronization of a networked nonlinear system according to claim 1, characterized in that The step S5 includes: Step S51: Definition of neighborhood error and construction analysis of Lyapunov function; The neighborhood error of multi-agent system i is defined as follows: where a ij represents the element in the adjacency matrix A, and b i represents the element in matrix B, indicating the potential connection between multi-agent system i and the leader; analyze the time derivative of the neighborhood error, and obtain the expression through mathematical analysis and reconstruction; select the Lyapunov function candidate, take the derivative with respect to time, and analyze its properties; integrate the expression of the time derivative to obtain a function expression with respect to time; Step S52: Iterative analysis and proof of subsystems; First, define the state and time derivative of the subsystem; conduct mathematical analysis and reconstruction on the time derivative of the subsystem; then select a Lyapunov function for the subsystem, take the derivative with respect to time, and analyze its properties; Step S53: Introduce a finite-time differentiator to avoid the iterative derivation of virtual controllers; First, define the finite-time differentiator, including its state and parameters; The finite-time differentiator is defined as follows: where, π iq,1 and π iq,2 represent the states of the differentiator, k1, k2 > 0 are the parameters of the differentiator, and the sig function is expressed as As long as the initial deviation π iq,1 (0) - α i,q-1 (0) and are bounded, the finite-time differentiator can provide an approximation of with arbitrary precision; thus, there is an estimation error δ iq that is bounded, meaning that a constant M iq > 0 can be determined such that |δ iq | ≤ M iq ; Then analyze the approximation accuracy provided by the finite-time differentiator to ensure the boundedness of the estimation error; Finally, combine the finite-time differentiator to design the virtual controller and refresh rate, and simplify to obtain the expression of the controller; Step S54: Design the controller for the nth step of the final layer; First, define the state and time derivative of the final layer; rewrite the time derivative according to the assumption 1; construct a Lyapunov function, take the derivative with respect to time, and analyze its properties; apply the event-triggered control strategy to rewrite the time derivative of the Lyapunov function; then introduce the following finite-time differentiator Finally, through simplification and integration processing, we obtain For the strict-feedback nonlinear system with unknown control directions, unknown parameters, and unknown disturbances, if Assumption 1 and Assumption 2 hold, the proposed controller can ensure the following properties: the outputs of all agents in the system will eventually synchronize precisely; the neighborhood error of each multi-agent converges to a bound within zero; for the communication network among the multi-agent systems, all signals are bounded in the closed-loop system; Step S55: Stability proof; First, integrate the time derivative inequality of the Lyapunov function obtained by substituting the designed virtual controller for the said Step S54, and analyze its boundedness; apply Barbalat's lemma to prove the convergence of the system state; finally, by defining the lower bound of the execution interval, avoid Zeno behavior and ensure that there will be no infinitely many fast triggering events in the actual operation of the system.

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