A fixed-time adaptive control method for high-order nonlinear multi-agent systems

Abstract

The present application relates to the field of high-order nonlinear system control, and discloses a fixed-time adaptive control method for a high-order nonlinear multi-agent system, including the following steps: S1, describing the state of the multi-agent system and modeling the topological structure; S2, approximating the unknown nonlinear function in the system; S3, approximating the input saturation characteristic by a smoothing function; S4, designing a control law that satisfies the full-state constraint; S5, designing a fixed-time adaptive controller; S6, designing an adaptive law to estimate and compensate for disturbances; S7, designing a filtering error compensation mechanism to make the filtering error converge within a fixed time. The present invention can achieve high-precision and high-stability control in a multi-agent system by designing a fixed-time adaptive control method based on a neural network, combined with an obstacle Lyapunov function and a backstepping method, to ensure that the states of all agents converge within a set time and remain within a predetermined constraint range.

Description

Technical Field

[0001] The present invention relates to the technical field of high-order nonlinear system control, and in particular to a fixed-time adaptive control method for a high-order nonlinear multi-agent system. Background Art

[0002] Multi-agent systems (MAS) rely on efficient coordination and information sharing mechanisms between internal agents, show strong adaptability to dynamically changing environments and demands, can efficiently allocate and collaboratively perform complex tasks, and show great application potential and value in key areas such as mobile robots, satellite clusters, and intelligent transportation systems. In MAS research, designing efficient control algorithms to enhance system consistency is one of the main challenges. Compared with low-order systems, high-order systems have more far-reaching research value and wider application potential, especially in the deepening of basic theories and the expansion of practical applications, due to their strict requirements on stability and wider application range.

[0003] In the study of high-order nonlinear MAS consistency problems, the iterative derivation of virtual control laws often leads to a "complexity explosion". Dynamic surface control (DSC) effectively alleviates this problem; however, its practical implementation is challenging due to the need to obtain initial values. The introduction of a first-order sliding mode differentiator as an approximation of the control signal effectively circumvents these difficulties. In addition, the presence of nonlinear functions significantly affects the stability of the system. Neural networks (NN) and fuzzy logic systems (FLS) are often used to solve nonlinear functions in multi-agent systems.

[0004] In the study of multi-agent systems, it is crucial to optimize the convergence speed to achieve consistent control, especially in engineering practice, where the goal is usually to achieve this goal within a limited or specific time frame. Fixed-time control methods have attracted attention for their fast convergence, high robustness, and precise control. Subsequently, many scholars have conducted research on this, especially fixed-time control of nonlinear multi-agent systems. In practical applications, a large initial control force is often required for fast convergence. At the same time, limiting the amplitude of the control input is particularly common in physical systems. Fixed-time controllers have achieved good results in these applications; however, experiments clearly reveal the huge initial control force. Therefore, despite the significant theoretical advantages of fixed-time control strategies, their application in engineering practice still faces various limitations and challenges, which require more in-depth research and solutions.

[0005] In practical engineering environments, systems often encounter state constraints due to external restrictions. Researchers have effectively addressed these challenges by integrating the Barrier Lyapunov function (BLF) into multi-agent systems, significantly advancing the development of this field. For example, a finite-time consistency control method using the Backstepping technique and BLF is developed for uncertain nonlinear MAS with full state constraints and actuator failures. This approach ensures that the performance of the MAS is maintained even in the event of actuator failures. An adaptive fuzzy control scheme for finite-time containment control in nonlinear multi-agent systems (MAS) is developed using an innovative nonlinear transformation function (NTF). The scheme ensures that the state is bounded within pre-set limits and achieves convergence of the follower output to the leader's convex hull without violating the state constraints. In addition, the state constraint problem of MAS with input delay and the application of BLF in solving the problem of delayed MAS are discussed. In summary, the combination of BLF and consistency control provides an effective method to solve the state constraints of MAS, but further research is still needed.

[0006] The study of fixed-time control in high-order nonlinear multi-agent systems, including full-state constraints and input saturation, is very limited. Summary of the invention

[0007] In view of the shortcomings of the prior art, the present invention provides a fixed-time adaptive control method for a high-order nonlinear multi-agent system. By designing a fixed-time adaptive control method based on a neural network, combined with the obstacle Lyapunov function and the backstepping method, high-precision and high-stability control can be achieved in the multi-agent system, ensuring that the states of all agents converge within the set time and remain within the predetermined constraints.

[0008] To achieve the above objectives, the present invention is implemented by the following technical solutions: a fixed-time adaptive control method for a high-order nonlinear multi-agent system, comprising the following steps:

[0009] Describe the state and model the topology of multi-agent systems;

[0010] The radial basis function neural network is used to approximate the unknown nonlinear functions in the system;

[0011] Approximate the input saturation characteristics by a smoothing function;

[0012] Combining the barrier Lyapunov function with the backstepping method, a control law that satisfies all state constraints is designed;

[0013] Using fixed-time control theory, design a fixed-time adaptive controller;

[0014] Design adaptive laws to estimate and compensate for disturbances;

[0015] A filter error compensation mechanism is designed to ensure that the filter error converges within a fixed time.

[0016] Preferably, the step of modeling the state description and topological structure of the multi-agent system and establishing the dynamic model of the agent comprises:

[0017] Using undirected graph Represent the topology of a multi-agent system;

[0018] Defining Vertex Sets and edge sets in Indicates that the nth agent can access the information of the mth agent;

[0019] Defining the adjacency matrix like Then a n,m =1, otherwise a n,m =0;

[0020] Define the in-degree matrix D = diag{d1,d2,…,d N} and Laplacian matrix L = DA.

[0021] Preferably, the step of using a radial basis function neural network to approximate an unknown nonlinear function in the system comprises:

[0022] Select the basis function S(Z) and the corresponding network structure;

[0023] Determine the optimal weight matrix W through network training * , so that the approximation error |δ(Z)| is minimized and satisfies the following relationship:

[0024] F(Z)=W * S(Z)+δ(Z)

[0025] Among them, δ(Z) satisfies |δ(Z)|≤δ * (Z) is the estimated error, W = [ω1, ω2, …, ω l ]∈R l is the weight vector, S(Z)=[S1(Z),S2(Z),…,S l (Z)] T is the basis function vector.

[0026] Preferably, the step of approximating the input saturation characteristic by a smoothing function comprises:

[0027] The hyperbolic tangent function is used to approximate the input saturation characteristic, which is expressed as:

[0028]

[0029] Among them, u Mi Indicates the input saturation threshold;

[0030] The input signal u i (w i ) is written as:

[0031]

[0032] Among them, σ(w i ) is the approximation error, σ(w i ) has an upper limit of

[0033] Preferably, the step of combining the barrier Lyapunov function with the backstepping method to design a control law that satisfies all state constraints comprises:

[0034] The choice barrier Lyapunov function form is:

[0035]

[0036] Among them, ε i,w is the system error, k i,bw is the preset constant value;

[0037] Design virtual control laws and actual control laws to ensure that the system status is within the preset safety range.

[0038] Preferably, the step of designing a fixed-time adaptive controller using fixed-time control theory comprises:

[0039] Use fixed-time control theory to design a controller to ensure that the system is in a fixed time T max Inner convergence;

[0040] The design expression of the fixed-time adaptive controller is:

[0041]

[0042] in, is the design parameter, ε i,τ is the systematic error, k i,bτ is a preset constant value, Θ i,τ and Φ i,τ is the design parameter, is the disturbance term d i,τ The estimate, is the disturbance estimate corresponding to the mth agent.

[0043] Preferably, the step of designing an adaptive law to estimate and compensate for disturbances comprises:

[0044] Monitor system performance in real time and adjust control parameters dynamically;

[0045] Update the control parameters using the following expressions:

[0046]

[0047] in, and κ i,τ There are no parameters. is the parameter θ i,τ Estimates.

[0048] Preferably, the step of designing a filtering error compensation mechanism to ensure that the filtering error converges within a fixed time includes:

[0049] Evaluate the error between the system output and the expected output;

[0050] A dynamic filter is designed to adjust the compensation strategy to ensure that the filter error converges within a fixed time. The design expression of the filter error compensation mechanism is:

[0051]

[0052] in, is an estimated value, To compensate for the error.

[0053] Preferably, the method further comprises: analyzing the stability of the control system to ensure that the system converges within a fixed time, as follows:

[0054] The stability analysis is performed using the Lyapunov function, and the Lyapunov function is selected as:

[0055]

[0056] Combined with fixed time control theory, ensure that the system is in fixed time Internal convergence.

[0057] The present invention also provides a fixed-time adaptive control device for a high-order nonlinear multi-agent system, comprising:

[0058] The state description module is used to describe the state and model the topology of the multi-agent system;

[0059] A nonlinear function approximation module is used to approximate unknown nonlinear functions in the system using a radial basis function neural network;

[0060] An input saturation approximation module is used to approximate the input saturation characteristics through a smooth function;

[0061] Control law design module, which is used to combine the barrier Lyapunov function and backstepping method to design a control law that satisfies all state constraints;

[0062] Fixed-time control module, used to design fixed-time adaptive controller using fixed-time control theory;

[0063] The disturbance estimation and compensation module is used to design an adaptive law to estimate and compensate for disturbances;

[0064] A filter error compensation module is used to design a filter error compensation mechanism to ensure that the filter error converges within a fixed time;

[0065] The stability analysis module is used to analyze the stability of the control system and ensure that the system converges within a fixed time.

[0066] The present invention provides a fixed-time adaptive control method for a high-order nonlinear multi-agent system.

[0067] It has the following beneficial effects:

[0068] 1. The present invention designs a fixed-time adaptive control method based on neural network, combines the barrier Lyapunov function and backstepping method, and can achieve high-precision and high-stability control in a multi-agent system, ensuring that the states of all agents converge within a set time and remain within a predetermined constraint range.

[0069] 2. In the controller design process, the present invention avoids the "complexity explosion" problem in the traditional backstepping method by adopting dynamic surface control (DSC) and a first-order smooth mode differentiator, thereby simplifying the controller structure and improving the feasibility of practical application.

[0070] 3. The present invention estimates and compensates for the time-varying unknown disturbances in the system in real time by designing an adaptive law, which significantly enhances the robustness and anti-interference ability of the system to external interference and ensures the reliable operation of the control system.

[0071] 4. Aiming at the common input saturation problem in physical systems, the present invention approximates the input saturation characteristics through a smoothing function and designs a control law that adapts to input saturation, thereby effectively limiting the amplitude of the system input signal and preventing system instability caused by excessive input.

[0072] 5. The present invention introduces a novel filter error compensation mechanism in the controller design to ensure that the filter error converges within a fixed time, thereby improving the control accuracy and enhancing the performance of the control system.

[0073] 6. The method of the present invention introduces preset response parameters to enable the system error to converge in a timely manner, which is independent of the initial state of the system, thereby enhancing the flexibility and adaptability of the control strategy and being suitable for a variety of practical engineering application scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is a schematic diagram of the method flow of the present invention;

[0075] Figure 2 Schematic diagram of the experimental environment for simulation verification in Example 2 of the present invention;

[0076] Figure 3 This is a schematic diagram of the experimental system architecture in Example 2 of the present invention;

[0077] Figure 4 This is a schematic diagram of the communication topology structure in Embodiment 2 of the present invention;

[0078] Figure 5 is the output of the four agents and the leader y in Example 2 of the present invention r Schematic diagram of the trajectory;

[0079] Figure 6 A schematic diagram showing a comparison of tracking effects of four intelligent agents in Example 2 of the present invention;

[0080] Figure 7 Schematic diagram of tracking errors of an agent and a leader in Embodiment 2 of the present invention;

[0081] Figure 8 is a schematic diagram for comparing the tracking errors of four agents in Example 2 of the present invention;

[0082] Fig. 9 A schematic diagram of neighbor consensus error between leaders in Example 2 of the present invention;

[0083] Fig.10 Schematic diagram of comparison of neighbor consensus errors of four agents in Example 2 of the present invention;

[0084] Fig.11 Schematic diagram of control input for each agent in Embodiment 2 of the present invention;

[0085] Fig.12 This is a schematic diagram of the comparison between IS and NIS inputs in Example 2 of the present invention;

[0086] Fig.13 Schematic diagram of ITAE and RMSE of the intelligent agent in PAFTC and the proposed control scheme for the performance indicators in Example 2 of the present invention;

[0087] Fig.14 It is a schematic diagram of the device structure of the present invention.

[0088] Among them, 100, state description module; 200, nonlinear function approximation module; 300, input saturation approximation module; 400, control law design module; 500, fixed time control module; 600, disturbance estimation and compensation module; 700, filtering error compensation module; 800, stability analysis module. DETAILED DESCRIPTION

[0089] The following will be combined with the drawings in the specification of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0090] The present invention provides a fixed-time adaptive control method for a high-order nonlinear multi-agent system, which is specifically as follows:

[0091] S1. State description and topology modeling

[0092] First, it is necessary to describe the state and model the topology of the multi-agent system in order to lay the foundation for subsequent controller design.

[0093] In this section, we will use graph theory to describe the relationship between multiple agents. To represent the topological structure of a nonlinear multi-agent system. The graph consists of a vertex set and an edge set The edge from n to m in E indicates that the nth agent can access the information of the mth agent, identifying them as neighbors. The adjacency matrix, denoted as Assumptions Then a n,m =1; if Then a n,m = 0. The definition of the in-degree matrix is ​​D = diag = {d1, d2, ..., d N}, and the Laplacian matrix is ​​defined as L = DA. For the follower, if it can receive information from the leader, then b i >0; otherwise b i =0.

[0094] S2. System Description

[0095] Consider an n-order multi-agent system with a class of state constraints and input saturation. The system consists of N agents, each of which is described by the following dynamics: i = 1, 2, ..., N

[0096]

[0097] in, is the state vector. In addition, assume that each state of each agent is constrained to be within |ζ i,w | <k i,cw of the compact set, where k i,cw Indicates a preset constant value. i Represents the system output, is an unknown continuous function, D i,k (t) is a time-varying unknown disturbance, whose maximum value is u i (ω i ) represents an input signal with input saturation, where ω i is the input control signal. Input saturation u i (ω i ) is expressed as follows:

[0098]

[0099] Among them, u Mi Represents the system input u i the border.

[0100] System input u i The lack of smoothness means that backstepping is not directly applicable to controller design. Therefore, it is necessary to approximate the saturation characteristic with a smooth function as follows:

[0101]

[0102] In this way, i (ω i ) can be written as follows, where σ(ω i ) is the approximate error, and the upper limit is

[0103]

[0104] According to the mean value theorem, let the initial value be g(0) = 0, let Then g(ω i ) can be expressed as:

[0105]

[0106] According to the above expression, the tanh function and the control input ω i are all bounded; therefore, ξ(ω i ) is also bounded.

[0107] 0<ξ ζ ≤ξ(w i)≤1 (6)

[0108] Based on the above, system (1) can be converted into the following form:

[0109]

[0110] S3, radial basis function neural network approximation

[0111] RBFNN is used as a tool to approximate nonlinear unknown functions, and it has a good approximation effect. Its main principles can be summarized as follows. For a nonlinear continuous function F(Z):R q →R, where Ω Z is a compact set, δ is a positive constant, and there exists an RBFNN network W T S(Z) satisfies:

[0112] F(Z)=W * S(Z)+δ(Z) (8)

[0113] where δ(Z) satisfies |δ(Z)|≤δ * (Z) is the estimated error, W = [ω1, ω2, …, ω n ]∈R l represents the weight vector, S(Z)=[S1(Z),S2(Z),…,S l (Z)] T represents the basis function vector, and l>1 represents a neuron node. In particular, the Gaussian basis function has the following form:

[0114]

[0115] Among them, v i =[v i1 ,v i2 ,…,v in ]∈R q and π i are the center vector and width of the Gaussian basis function respectively.

[0116] 4. Fixed-time stability theory

[0117] In order to ensure that the system is stable within a fixed time, the fixed-time stability theory is used for analysis and design.

[0118] Definition 1: For a nonlinear system where x∈R n is the system state, and f is a continuous nonlinear function satisfying f(0) = 0. If global finite-time stability is achieved, the convergence time is uniformly bounded for any initial state. Satisfy T≤T max , the system is called fixed-time stable.

[0119] Lemma 1: For a nonlinear system, if there exists a positive definite and radially unbounded function \(V(x):\mathbb{R}\) n \(\to\mathbb{R}\) such that and

[0120]

[0121] where \(\varphi\) a , \(\varphi\) b \(>0\), \(p > 1\) and \(0 < q < 1\), then the nonlinear system is said to be stable in fixed time, and the settling time \(T\leq T\) max \(=(1 / (\varphi\) a \((p - 1)))+(1 / (\varphi\) b \((1 - q)))\).

[0122] Lemma 2: Let \(\alpha>1\), for any \(x,y\geq0\) and \(\eta>0\), there is

[0123]

[0124] Lemma 3: For \(\lambda\in(0,1]\) and \(v\) i \(\in[0,+\infty)\), the following inequality holds

[0125]

[0126] Lemma 4: For any constant \(k\) i,bw \(>0\), if \(|\varepsilon\) i,w \(<k\) i,bv , \(\varepsilon\) i,w \(\in\mathbb{R}\) holds, then there is:

[0127]

[0128] Assumption 1: The output of the leader and its derivative are both bounded, denoted by \(y\) r \((t)\) and respectively.

[0129] Assumption 2: All state variables \(\zeta\) i,j \((j = 1,2,\cdots,n)\) are physically measurable and realizable, and these state variables can be used to design the control input.

[0130] Assumption 3: The minimum value of \(\xi(\omega\) i ) in Equation (6) is known, and the constant \(\xi\) ζ is known and accessible throughout the controller design process.

[0131] S5. Fixed-Time Adaptive Controller Design

[0132] In this step, for a class of multi-agent systems with full-state constraints and input saturation, the obstacle Lyapunov function is combined with the backstepping method, a neural network-based fixed-time adaptive controller is proposed, and its stability is analyzed.

[0133] For the i-th agent, the following coordinate transformation is performed:

[0134]

[0135] in is the virtual control law designed in step n-1, and b i represents positive fixed gain, ∑a ij =m i Then define a constant And define is θ i,1 The estimated value of .

[0136] Specifically, the designed virtual control law and actual control law are in the following forms:

[0137]

[0138] where p i,n ,q i,n , X i,n is a positive design parameter, for

[0139]

[0140] Adaptive Law The form is as follows:

[0141]

[0142] where μ i,τ and is a constant.

[0143] S6, Adaptive Law Design

[0144] Step 1: From formula (15), we can get:

[0145]

[0146] In the subsequent controller design, the following form of obstacle Lyapunov function will be used:

[0147]

[0148] where |ε i,w | <k i,bwis ε i,w restrictions, Combining formula (7), we can get

[0149]

[0150] in is a packaged unknown, and Using neural networks, the unknown It can be approximated as:

[0151]

[0152] where δ i,1 is the approximation error, and Then, for Applying Young's inequality, we get

[0153]

[0154]

[0155] Substituting (23-25) into (21), becomes:

[0156]

[0157] Bring in the previously designed virtual control law and adaptive law, can be rewritten as:

[0158]

[0159] in is a normal number.

[0160] Step s (s = 2, ..., n-1): Repeat the method similar to the first step and select the Lyapunov function of the following form:

[0161]

[0162] Then, taking the derivative of both sides of equation (28), we can obtain

[0163]

[0164] When examining the above equations, it is noted that the derivative of the previously designed virtual control law is required. In order to avoid the problem of "complexity explosion" caused by continuous derivation, the present invention uses a first-order sliding mode differentiator to approximate the virtual control law.

[0165]

[0166] Among them,i,sj and b i,s is the system state of the differentiator, γ i,y (j=0,1) is a positive design parameter. i,s0 and b i,s They are and Therefore, it can be deduced that

[0167]

[0168] Among them, d i,s represents the estimation error of the differentiator. In addition, a new filter compensation mechanism is developed to ensure that the estimation error d i,s Converges within a fixed time frame.

[0169]

[0170] in represents the compensation error, Yes i,s The estimated value of can be simplified to

[0171]

[0172] Estimating unknown functions using RBFNN for get:

[0173]

[0174] Where X i,s =[ζ i,1 ,…,ζ i,s ] T Substituting (34-35) into (33), we can obtain the following inequality:

[0175]

[0176] Substituting equations (16) and (18) into the above equation, we can obtain the following inequality:

[0177]

[0178] in

[0179] Step n: Define a Lyapunov function of the following form

[0180]

[0181] Similar to the above steps, V i,n Derivation and substitution get

[0182]

[0183] in Approach A radial basis function neural network (RBFNN) is used, where δ i,n Indicates positive approximation error And X i,n =[ζ i,1 ,…,ζ i,n ] T .for By applying Young's inequality, we can get the following results:

[0184]

[0185] Substitute equation (40-41) into The following inequality can be obtained:

[0186]

[0187] Substituting the designed virtual control rate (16) and adaptive rate (18) into equation (42), we get

[0188]

[0189] in

[0190] For the above multi-agent system with full-state constraints and input saturation, the virtual control law is approximated under the action of the first-order smooth mode differentiator, and the following adaptive law is designed to make the error converge to a fixed value time.

[0191]

[0192] For the above adaptive law, choose a Lyapunov function of the form

[0193]

[0194] S7. Stability analysis

[0195] After the controller is designed, a stability analysis is performed to theoretically verify the effectiveness of the proposed control scheme in achieving the set control objectives.

[0196] Proof: Combining (38) and (45), the Lyapunov function of the entire multi-agent system is selected as:

[0197]

[0198] According to equations (43), (44), and (46), we can get

[0199]

[0200] Using Lemma 2, Can be written as

[0201]

[0202] for as well as Using Young's inequality, we get:

[0203]

[0204] Will can be rewritten as follows:

[0205]

[0206] Here, using Young's inequality for the third term in the above formula, we can get the following form:

[0207]

[0208] Therefore, formula (51) is equivalent to:

[0209]

[0210] Combining equations (49) and (53), can be transformed into

[0211]

[0212] Similarly, for According to the above method, it can be transformed into

[0213]

[0214] for It can be rewritten as follows:

[0215]

[0216] By using Lemma 2, let l = 4 / 3, a = 4, V=oθ i,t , where O is a positive constant. We can get Let o 4 / 3 =3

[0217]

[0218] for Apply Young's inequality:

[0219]

[0220] Combining Eqs. (56), (57), and (58), is equivalent to:

[0221]

[0222] Similarly, for Performing the above processing can transform it into

[0223]

[0224] Combining Eqs. (48), (54), (55), (59), and (60), is equivalent to:

[0225]

[0226] where

[0227] According to Lemma 3 and Lemma 4, we can obtain

[0228]

[0229]

[0230] Substituting (62)-(65) into (61) gives:

[0231]

[0232] where φ a = min{p, γ, e}, and

[0233]

[0234] where Furthermore, substituting Eqs. (66) and (67) into Eq. (65), we get:

[0235]

[0236] where 0 < C < 1, choosing appropriate design parameters can ensure -Cφ b V 2 + Λ ≥ 0 and indicates that V is bounded.

[0237]

[0238] From Lemma 1, we can see that (69) shows that all signals of system (1) converge to the set At a fixed time T max =1 / ((1-C)φ b )+4 / φ a By choosing appropriate design parameters, and then based on the constraint of the barrier Lyapunov function, |ε i,w | <k i,bw It can be obtained, thereby indirectly realizing the constraint on the state. That is, the timing convergence of the error signal is realized.

[0239] Based on the above, it is proved that this method realizes the state constraint and input saturation control of the multi-agent system in a fixed time, ensuring the stability of the system and the timely convergence of the error.

[0240] In order to better understand the present invention, the above method is described in detail below in conjunction with specific embodiments.

[0241] Embodiment 1:

[0242] Please see attached Figure 1 The embodiment of the present invention provides a fixed-time adaptive control method for a high-order nonlinear multi-agent system, comprising the following steps:

[0243] S1. State description and topological structure modeling of multi-agent systems

[0244] First, the state of the multi-agent system is described and its topological structure model is established. Specifically, it is assumed that the system consists of N agents, each of which has a dynamic equation of the following form:

[0245]

[0246] S2. Use radial basis function neural network to approximate unknown nonlinear functions in the system

[0247] In order to approximate the unknown nonlinear function in the system, a radial basis function neural network (RBFNN) is used. Specifically, a Gaussian basis function is selected, which is in the form of:

[0248]

[0249] Among them, v i is the center vector, π i is the width. Through network training, the weight vector W is adjusted so that the nonlinear function F(Z) satisfies:

[0250] F(Z)=W * S(Z)+δ(Z)

[0251] Where δ(Z) is the approximation error.

[0252] S3, approximating input saturation characteristics through smoothing functions

[0253] Since the system input u i Lack of smoothness requires a smoothing function to approximate the input saturation characteristics. Specifically, the smoothing function g(w i ):

[0254]

[0255] The input signal u i (ω i ) is expressed as:

[0256] u i (w i )=g(w i )+σ(w i )

[0257] Among them, σ(w i ) is the approximate error.

[0258] S4. Design of control law by combining obstacle Lyapunov function and backstepping method

[0259] Combining the barrier Lyapunov function with the backstepping method, a control law that satisfies all state constraints is designed. Specifically, the barrier Lyapunov function of the following form is selected:

[0260]

[0261] Design virtual control laws and actual control laws to ensure that the system status is within the preset safety range.

[0262] S5. Design of adaptive controller using fixed-time control theory

[0263] Using the fixed-time control theory, a fixed-time adaptive controller is designed to ensure that the system error converges within a fixed time. Specifically, the design expression of the fixed-time adaptive controller is:

[0264]

[0265] S6. Design adaptive laws to estimate and compensate for disturbances

[0266] Design adaptive laws to estimate and compensate for disturbances within the system and improve the robustness of the system. Specifically, dynamically adjust control parameters to compensate for uncertainties and external disturbances within the system:

[0267]

[0268] S7. Design a filter error compensation mechanism to ensure that the filter error converges within a fixed time.

[0269] In order to ensure that the filter error converges within a fixed time, a filter error compensation mechanism is designed. Specifically, the filter compensation error is designed to ensure that the filter error converges within a fixed time:

[0270]

[0271] in represents the compensation error, Yes i,s The estimated value of .

[0272] S8. Stability analysis

[0273] The stability of the control system is analyzed using Lyapunov stability theory. The Lyapunov function is selected as:

[0274]

[0275] Combined with the fixed time control theory, ensure that the system is within the fixed time T max =1 / ((1-C)φ b )+4 / φ a Internal convergence.

[0276] Through the above steps, the present invention provides an efficient fixed-time adaptive control method for high-order nonlinear multi-agent systems, which can effectively solve technical problems such as complexity explosion, unknown nonlinear function approximation, input saturation, state constraints, fixed-time convergence, disturbance estimation and compensation, and filtering error compensation in the system, thereby improving the control performance and stability of the system.

[0277] Embodiment 2:

[0278] Experimental verification

[0279] This paper experimentally verifies the proposed control scheme on the StarSim semi-physical simulation experimental platform. Figure 2 and Figure 3 As shown. Contains the following modules:

[0280] NIPXIe-1071: MT RCP (Rapid Control Prototype) is a PXIe base produced by NI, which is used for various measurement and control tasks and acts as a controller in this embodiment. The experimental equipment is equipped with a Kintex-7 325T FPGA@Xilinx and also has 16 analog I / O channels with a transmission rate of 1MS / s for collecting and outputting analog signals, making it highly accurate and flexible in the control process.

[0281] NIPXIe-1082: MT real-time simulator (RTS) uses Kintex7325T FPGA chip, has 16-bit synchronous analog I / O port, data transmission rate is 1MS / s, and acts as the hardware model of the intelligent agent in this embodiment to facilitate real-time control and simulation.

[0282] Adapter board: used to implement the signal connection between the fast controller and the real-time simulator.

[0283] Main control computer: Use StarSim RCP and StarSim HIL software to download the controller algorithm and multi-agent system model to the fast controller and real-time simulator respectively. The simulator receives the control signal sent by the controller, calculates the response signal of the multi-agent system in real time, and outputs it to the control side (RCP) to further correct the control signal. All signals can be displayed in real time on the main control computer screen.

[0284] The simulation research object of this embodiment is a nonlinear multi-agent system composed of 4 agents. The given tracking signal is a virtual leader. The communication topology between agents is as follows: Figure 4 As shown. The dynamic model of the i-th (i=1,2,3,4) agent is defined as follows:

[0285]

[0286] in Disturbance term D i,1 (t) = 0.27sin(t) and D i,2 (t) = 0.28sin(t). The designed tracking signal is y r =sin(0.5t), where only agent No. 2 can receive the leader's information.

[0287] according to Figure 4 As shown, its adjacency matrix and Laplacian matrix are as follows:

[0288]

[0289] The initial condition is ζ(0) = [0, 0.1, 0.1, 0.2, 0.1, 0.2, 0.1, 0.3] T ,θ(0)=[0.1,0,0.15,0,0.1,0.1,0.05,0] T .

[0290] The main design parameters are c i,1 =c i,2 =80,q i,1=q i,2 =80,a i,1 =a i,2 =1,γ i,1 =γ i,2 =5, k i,b1 =k i,b2 =0.1, b i,1 =b i,2 =0.5,β i,1 =β i,2 =0.5,o i,1 =o i,2 =1,m i1 =m i,2 =0.5, n i,1 =n i,2 =0.5, R i20 =R i21 =0.1. From the values ​​of J, X, and C, we can get T max =1.4. For agent 1, the first RBF neural network structure used for approximation is 3-9-1, and the network input is ζ = [ζ 11 ,ζ 21 ,ζ 22 ] T , the centers of the Gaussian basis functions are uniformly distributed in the range [-8,8], and π 1,1 =2. The second RBF neural network structure used for approximation is 2-9-1, and the network input is ζ = [ζ 21 ,ζ 22 ] T , the centers of the Gaussian basis functions are uniformly distributed in the range [-8,8], and π 1,2 =2.

[0291] The simulation results of the adaptive fixed time control scheme proposed by the present invention are as follows: Figure 5-13 . Figure 5 The actual and expected output response curves of the system are shown, indicating that each agent y i Can effectively track the input signal y r . Figure 6 The tracking trajectories of the proposed control scheme and pure adaptive fixed-time control (PAFTC) are compared, demonstrating the superior performance of the former. Figure 7 The tracking error between each agent and the leader is depicted and compared with PAFTC. Figure 8 . Fig. 9 is the neighbor consistency error. As can be seen from the figure, due to the use of the barrier Lyapunov function, the system's neighbor consistency error e i,1 is restricted to the given range [-k i,b1 ,k i,b1 ,], the state constraint value of the system is not violated, Fig.10 The neighbor consensus error is compared with PAFTC in the present invention, and the neighbor consensus error of the controller proposed in the present invention is significantly smaller than that of PAFTC. Figure 11-12 are the control input u of each agent considering the input saturation condition. i and the control input u without considering input saturation (NIS) i It can be seen from the figure that under the control scheme designed by the present invention, the multi-agent system still maintains acceptable tracking performance under the conditions of external disturbance and input saturation, and the input signal is also limited to the required range. In order to highlight the advantages of the proposed control scheme, the effects of the time integral and absolute error (ITAE) and root mean square error (RMSE) of each agent under the proposed control scheme and PAFTC are also compared, as well as the specific control scheme. The results are shown in Fig.13 As shown in the figure, it can be seen that under the control scheme designed by the present invention, in the presence of external disturbances and input saturation, the multi-agent system still maintains acceptable tracking performance, and the input signal is limited to the required range. Figure 5-13 It can be seen that the proposed control scheme has excellent convergence speed and small error fluctuation amplitude, indicating that the system is fixed-time stable, verifying the effectiveness of the design scheme.

[0292] The embodiment of the fixed-time adaptive control device for a high-order nonlinear multi-agent system described below and the embodiment of the fixed-time adaptive control method for a high-order nonlinear multi-agent system described above can be referred to each other.

[0293] Please see attached Fig.14 The present invention also provides a fixed-time adaptive control device for a high-order nonlinear multi-agent system, comprising:

[0294] A state description module 100 is used to describe the state and model the topology of the multi-agent system;

[0295] A nonlinear function approximation module 200 is used to approximate an unknown nonlinear function in the system using a radial basis function neural network;

[0296] An input saturation approximation module 300, for approximating input saturation characteristics by a smoothing function;

[0297] A control law design module 400, for designing a control law satisfying all state constraints by combining the barrier Lyapunov function and the backstepping method;

[0298] A fixed time control module 500 is used to design a fixed time adaptive controller using fixed time control theory;

[0299] The disturbance estimation and compensation module 600 is used to design an adaptive law to estimate and compensate for disturbances;

[0300] A filter error compensation module 700 is used to design a filter error compensation mechanism to ensure that the filter error converges within a fixed time;

[0301] The stability analysis module 800 is used to analyze the stability of the control system to ensure that the system converges within a fixed time.

[0302] The device of this embodiment can be used to execute the above method embodiment, and its principles and technical effects are similar, which will not be repeated here.

[0303] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A fixed-time adaptive control method for a high-order nonlinear multi-agent system, characterized in that: The following steps are involved: Describe the state and model the topology of multi-agent systems; Approximate the input saturation characteristics by a smoothing function; The radial basis function neural network is used to approximate the unknown nonlinear functions in the system; Combining the barrier Lyapunov function with the backstepping method, a control law that satisfies all state constraints is designed; Using fixed-time control theory, design a fixed-time adaptive controller; Design adaptive laws to estimate and compensate for disturbances; Design a filter error compensation mechanism to ensure that the filter error converges within a fixed time; The steps of designing a fixed-time adaptive controller using fixed-time control theory include: Use fixed-time control theory to design a controller to ensure that the system runs in fixed time. Inner convergence; The design expression of the fixed-time adaptive controller is: ， in, is the design parameter, is the external disturbance error of the system, is a preset constant value, and is the design parameter, is the disturbance term Estimates.

2. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 1, characterized in that: The steps of modeling the state description and topological structure of the multi-agent system and establishing the dynamic model of the agent include: Using undirected graph Represent the topology of a multi-agent system; Defining Vertex Sets and edge sets ,in Indicates Agents can access Information about an agent; Defining the adjacency matrix ,like ,but ,otherwise ; Define the in-degree matrix and the Laplacian matrix .

3. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 2, characterized in that: The step of using the radial basis function neural network to approximate the unknown nonlinear function in the system comprises: Choosing Basis Functions and the corresponding network structure; Determine the optimal weight matrix through network training , so that the approximation error Minimum, satisfying the following relationship: ， in, satisfy The estimated error is the weight vector, is the input control signal; is the basis function vector.

4. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 3, characterized in that: The step of approximating the input saturation characteristic by a smoothing function comprises: The hyperbolic tangent function is used to approximate the input saturation characteristic, which is expressed as: ， in, represents the input saturation threshold, Indicates the input control signal; The input signal Written as: ， in, is the approximate error, The upper limit is .

5. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 4, characterized in that: The steps of combining the barrier Lyapunov function with the backstepping method to design a control law that satisfies all state constraints include: The choice barrier Lyapunov function is in the form of: ， in, is the control error of the system, is the preset constant value; Design virtual control laws and actual control laws to ensure that the system status is within the preset safety range.

6. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 5, characterized in that: The step of designing an adaptive law to estimate and compensate for disturbances comprises: Monitor system performance in real time and adjust control parameters dynamically; Update the control parameters using the following expressions: ， in, and is the design parameter, It is a parameter The estimate, and is a constant.

7. A fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 6, characterized in that: The step of designing a filtering error compensation mechanism to ensure that the filtering error converges within a fixed time includes: Evaluate the error between the system output and the expected output; A dynamic filter is designed to adjust the compensation strategy to ensure that the filter error converges within a fixed time. The design expression of the filter error compensation mechanism is: ， in, is an estimated value, To compensate for the error.

8. The fixed-time adaptive control method for a high-order nonlinear multi-agent system according to claim 7, characterized in that: The method further includes: analyzing the stability of the control system to ensure that the system converges within a fixed time, as follows: The stability analysis is performed using the Lyapunov function, and the Lyapunov function is selected as: ， in, is a component of the Lyapunov function, is the error term, Is with the agent The state, disturbance estimate, and time Parameters related to the dynamic characteristics of Combined with fixed time control theory, ensure that the system is in fixed time Inner convergence, where represents the constant related to error convergence, represents the constant related to the error and disturbance terms, is a constant.

9. A fixed-time adaptive control device for a high-order nonlinear multi-agent system, used to implement the method described in any one of claims 1 to 8, characterized in that: include: The state description module is used to describe the state and model the topology of the multi-agent system; A nonlinear function approximation module is used to approximate unknown nonlinear functions in the system using a radial basis function neural network; An input saturation approximation module is used to approximate the input saturation characteristics through a smooth function; Control law design module, which is used to combine the barrier Lyapunov function and backstepping method to design a control law that satisfies all state constraints; Fixed-time control module, used to design fixed-time adaptive controller using fixed-time control theory; The disturbance estimation and compensation module is used to design adaptive laws to estimate and compensate disturbances; A filter error compensation module is used to design a filter error compensation mechanism to ensure that the filter error converges within a fixed time; The stability analysis module is used to analyze the stability of the control system and ensure that the system converges within a fixed time.

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