Self-adaptive fixed time tracking control method based on event triggering

Abstract

The invention provides a self-adaptive fixed time tracking control method based on event triggering, which is used for solving the technical problem that a virtual control law is easy to generate singularity in the traditional backstepping method design. The method comprises the following steps: establishing a mathematical model of a non-strict feedback nonlinear system with input time delay, and introducing an auxiliary system to counteract delay influence; an error dynamic system is obtained through coordinate transformation; according to system fixed time convergence, designing a piecewise function, introducing a backstepping method to design a controller and an adaptive law, using an RBFNN to approach an unknown function therein, designing a virtual control signal corresponding to each error signal, and obtaining an actual controller and an adaptive law; according to the actual self-adaptive controller, event triggering conditions are designed, when the event triggering moment is reached, the event triggering conditions containing input are triggered, and information transmission and response are carried out. An event triggering mechanism is designed, the number of times of control instruction transmission is reduced, communication resources are saved, and rapid convergence and safe operation of the system state are achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of tracking control, and in particular to an adaptive fixed-time tracking control method. Background Art

[0002] Nonlinear systems are widely used in industrial control, robotics, and other fields. Classical backstepping methods achieve tracking control by recursively designing virtual control laws, but this requires addressing the problems of unknown function approximation and input delay compensation. The barrier Lyapunov function (BLF) can handle state constraints, while fixed-time control (FTC) offers fast convergence and an event-triggered mechanism (ETM) reduces communication overhead. The barrier Lyapunov function (BLF) ensures that state variables remain within a predefined safe region.

[0003] Existing research uses fuzzy logic systems to handle input delays and state constraints, but has not achieved fixed-time convergence. Existing references have studied event-triggered control of multi-agent systems, but have not combined BLF with neural networks. None of the existing methods have simultaneously achieved the following in non-strict feedback systems:

[0004] 1. Input delay dynamic compensation;

[0005] 2. Multi-state constraint satisfaction;

[0006] 3. Fixed time convergence;

[0007] 4. Zeno behavior avoidance: Zeno behavior is the phenomenon where the controller triggers an infinite number of updates within a finite time.

[0008] 5. Neural network online learning.

[0009] For adaptive fixed-time tracking control of nonlinear systems with input delays and state constraints, existing control methods have the following drawbacks:

[0010] 1. It is impossible to simultaneously handle input delay compensation, state constraint satisfaction, and fixed-time convergence. That is, traditional methods cannot handle the coupling of input delay and state constraints within a fixed time. Fixed-time convergence means that the system state converges to the equilibrium point within a preset time, and the upper limit of the convergence time depends only on the controller parameters.

[0011] 2. The virtual control law in traditional backstepping design is prone to singularity problems;

[0012] 3. The event trigger mechanism is prone to Zeno behavior; the BLF design does not consider the introduction of the event trigger mechanism, which can easily lead to increased conservatism;

[0013] 4. The ability of neural networks to approximate unknown functions is not fully utilized, the adaptive ability of neural networks is not fully utilized, and precise mathematical models are relied upon.

[0014] Patent application number 202410776212.2 discloses an event-triggered quadrotor adaptive neural network trajectory tracking control method. The method includes a position loop control strategy and an attitude loop control strategy. The position loop control strategy employs a fixed-time sliding mode control law. The attitude loop control strategy employs an event trigger mechanism and an attitude loop controller. The adaptive update law employs an adaptive fixed-time virtual control law. This method effectively addresses existing technical issues, such as controller schemes that fail to consider convergence time, large errors in estimating unknown interference terms, the communication burden caused by continuous signal transmission from the UAV, and the complex design process of traditional event triggering mechanisms. Simulations and comparisons with finite-time control demonstrate the effectiveness and superiority of this control method in quadrotor trajectory tracking control, as well as its ability to reduce communication burden. However, this method also presents challenges, such as a failure to consider convergence time, large errors in estimating unknown interference terms, and high communication resource consumption. Summary of the Invention

[0015] Aiming at the technical problem that the virtual control law in the traditional backstepping design is prone to singularity, the present invention proposes an adaptive fixed-time tracking control method based on event triggering, which gives the upper bound of the convergence time, uses neural networks to approximate the terms containing unknown smooth functions and event triggering conditions, avoids the occurrence of Zeno behavior, transmits on demand, effectively saves communication resources, and realizes rapid convergence of system status and safe operation.

[0016] In order to achieve the above object, the technical solution of the present invention is implemented as follows: an adaptive fixed-time tracking control method based on event triggering, the steps of which are as follows:

[0017] Step 1: Establish a mathematical model of a non-strict feedback nonlinear system with input time lag and introduce an auxiliary system to offset the delay effect;

[0018] Step 2: Based on the mathematical model of the nonlinear system, the auxiliary system and the desired trajectory, the error dynamic system is obtained through coordinate transformation;

[0019] Step 3: Based on the fixed-time convergence of the system, a piecewise function is designed, and the backstepping method is introduced to design the controller and adaptive law: Based on the first error signal of the error dynamic system, a Lyapunov function V1 containing the adaptive law is constructed. The Lyapunov function is differentiated and the unknown function is approximated using RBFNN. The virtual control signal corresponding to each error signal is designed, and the derivative of the Lyapunov function V1 is determined.

[0020] Step 4: Construct the barrier Lyapunov function of the current error signal based on the previous error signal, and design the virtual control signal of the current error signal by combining the RBFNN and the corresponding piecewise function of the current error signal. Iterate the loop to obtain the virtual control signals of all error signals, and use the backstepping method to obtain the actual controller and adaptive law.

[0021] Step 5: According to the actual adaptive controller, the event trigger condition is designed based on the backstepping method and the system input threshold. The event trigger condition is used to control the actual adaptive controller. When the event trigger moment is reached, the event trigger condition containing the input is triggered. The strict feedback nonlinear system with input time lag is considered and combined with the auxiliary system with input time lag for information transmission and response.

[0022] Preferably, the mathematical model of the non-strict feedback nonlinear system with input time lag is:

[0023]

[0024] in, is the system state vector, x i+1 (t) is the system state at time i+1, is the system state x i The derivative of (t), is the system state x n (t), x1(t) is the first system state, and n is the total number of system states; Input variables for control; is the control output variable; f i (x(t)) is an unknown smooth function of the system state vector x(t); τ(t) is a known bounded function that satisfies represents the upper bound of the bounded function τ(t);

[0025] The control objectives of the non-strict feedback nonlinear system with input time lag include: 1) the closed-loop system state is fixed-time convergent; 2) the control output variable y(t) can be controlled by the reference signal y of the desired trajectory. r (t) Tracking and system status is within constraints:

[0026]

[0027] in, is a quadratically differentiable constraint function.

[0028] Preferably, the auxiliary system is:

[0029]

[0030] in, is the i+1th state variable of the auxiliary system, is the i-th state variable The derivative of q i For the control parameters, Represents the nth state variable The derivative of .

[0031] Preferably, the error dynamic system is:

[0032]

[0033] Among them, α i-1 is the i-1th virtual control signal, z i (t) is the i-th coordinate transformation state variable, z1(t) is the first coordinate transformation state variable, x1(t) is the first system state, x i (t) represents the i-th system state.

[0034] Preferably, the implementation method of step 3 is: constructing the Lyapunov function V1 according to the first coordinate transformation state variable z1 as

[0035]

[0036] Among them, the parameters Adaptive Law Represents the estimate of the uncertain parameter θ; uncertain parameter θ=max{‖W i ‖ 2 ,i=1,…,n}, represents a positive constant, W i represents the i-th weight vector;

[0037] Taking the derivative of the Lyapunov function V1, we get

[0038]

[0039] Among them, the unknown function Adaptive Law The derivative of is the derivative of the state variable z1, α1 is the first virtual control signal, and the control parameter q1>0.5, is the first state variable of the auxiliary system, f1 is the abbreviation of the first unknown smooth function f1(x(t)), represents the reference signal y r The derivative of (t);

[0040] Using RBFNN to approximate the unknown function Λ1, we can get

[0041] Among them, the intermediate variable W1 represents the weight vector for the unknown function Λ1, S1(Z1) is the basis function of the intermediate variable Z1, and δ1(Z1) is the approximation error of the intermediate variable Z1;

[0042] Combine have

[0043]

[0044] Where a1 is a positive constant, S1 is the abbreviation of the basis function S1(Z1);

[0045] There is a derivative of the Lyapunov function V1

[0046] have α1 and is the first virtual control signal;

[0047] When |z1|≥ε 10 When , the derivative of the Lyapunov function V1 is

[0048]

[0049] When |z1|<ε 10 , the derivative of the Lyapunov function V1 is

[0050]

[0051] Among them, the intermediate variable Preferably, the first virtual control signal α1 is designed according to the backstepping method as

[0052]

[0053] in, Represents a virtual control signal, Represents the piecewise function of the state variable z1. Design parameter K 11 ,K 12 ,ε1>0; piecewise function for

[0054]

[0055] Among them, the intermediate parameter ε 10 >0; segmentation parameter c j , j=1,…,n+1 is determined by the following formula

[0056]

[0057] Among them, the segmentation parameter b1=1,

[0058] Preferably, the i-th virtual control law α is designed recursively and iteratively i , 2≤i≤n-1;

[0059] Construct the barrier Lyapunov function V i for

[0060] Design the virtual control signal as

[0061]

[0062] in, represents a positive constant, V i-1 represents the i-1th Lyapunov function, z i is the i-th coordinate transformation state variable, is the i-th virtual control signal, S i is the abbreviation of the i-th basis function, a i represents a positive constant;

[0063] Design parameter K i1 ,K i2 ,ε i >0, piecewise function for

[0064]

[0065] Step n: Construct the barrier Lyapunov function:

[0066] Design the virtual control signal as

[0067] Combining coordinate transformation, auxiliary system and RBFNN, the obstacle Lyapunov function V can be obtained n The derivative of

[0068]

[0069] Among them, V n-1 is the n-1th Lyapunov function, V n is the nth Lyapunov function, K j1 , K j2 、a j 、S j , σ n Both represent normal numbers.

[0070] Preferably, the actual adaptive controller ν(t) and the adaptive law are obtained for:

[0071]

[0072] Among them, ξ,ψ1,ι, κ is a positive constant.

[0073] Preferably, the design event triggering conditions are:

[0074]

[0075] Among them, t m+1 and t m denote time m+1 and time m respectively, m is a positive integer, inf denotes the lower bound; ψ2 is a positive constant; and ψ1>ψ2 / (1-ξ); function When t∈[t m ,t m+1 ), the control input vector u(t) remains unchanged and the event is not triggered When t = t m+1 , control the input vector u(t) to update to t m+1 The value of the moment;

[0076] because have Right now Derivatives of the actual adaptive controller is continuous, λ is a constant, satisfying have

[0077]

[0078] Arrangement available

[0079]

[0080] Among them, the constant

[0081] This effectively avoids Zeno behavior caused by the event triggering mechanism.

[0082] Preferably, a non-strict feedback nonlinear system with input time lag is considered and the practical adaptive controller ν(t) and the adaptive law are designed. And the event triggering conditions can ensure the following results:

[0083] 1) The system state will not violate the constraints;

[0084] 2) All signals in the closed-loop system are bounded;

[0085] 3) The tracking error is bounded and the barrier Lyapunov function V n Converges to the set at a fixed time interval And the convergence time satisfies Among them, Δ, χ1, μ1, and μ2 all represent control parameters;

[0086] 4): Avoids Zeno behavior caused by event triggering mechanism.

[0087] Compared with the prior art, the present invention has the following advantages: through coordinate transformation, event triggering mechanism and neural network approximation, it can achieve:

[0088] 1. Input delay compensation for non-strict feedback systems: Compensate for input delay through auxiliary systems, and transform non-strict feedback systems into cascade forms in combination with coordinate transformation to simplify control design. By integrating auxiliary systems with coordinate transformation, the effects of time lag can be compensated, and the controller can be designed using backstepping.

[0089] 2. Constraint satisfaction of multiple state variables;

[0090] 3. Convergence in fixed time;

[0091] 4. Actively avoid Zeno behavior; design an event trigger mechanism and embed the event trigger mechanism within a fixed time control framework to reduce the frequency of control signal updates, reduce energy consumption and communication overhead.

[0092] 5. Sparse update of control signals.

[0093] Adaptive neural network approximation: Use RBFNN to estimate unknown functions online, and combine it with adaptive laws to update weights to improve system robustness.

[0094] Multi-constraint collaborative processing: Through BLF and preset constraint technology, real-time constraint satisfaction and fixed-time convergence of multiple state variables are achieved.

[0095] The present invention solves the problem of fixed-time convergence of system state variables, thereby demonstrating the high efficiency of the present invention; solves the problem that traditional methods cannot handle the coupling of input delay and state constraints within a fixed time, further demonstrating the superiority of the present invention; and designs an event trigger mechanism to reduce the number of control instruction transmissions, "transmit on demand", and save communication resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0097] Figure 1 Flowchart of the present invention.

[0098] Figure 2 It is the simulation curve of the tracking performance of the present invention.

[0099] Figure 3 It is the changing trend curve of the state variable of the present invention.

[0100] Figure 4 This is the simulation curve of the tracking error of the present invention.

[0101] Figure 5 It is the changing trend curve of the adaptive law of the present invention.

[0102] Figure 6 This is the changing trend curve of the control input of the present invention.

[0103] Figure 7 This is a schematic diagram of the event-triggered execution interval of the present invention. DETAILED DESCRIPTION

[0104] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0105] like Figure 1 As shown in FIG, an adaptive fixed-time tracking control method based on event triggering, the steps are as follows:

[0106] Step 1: Establish a mathematical model of a non-strict feedback nonlinear system with input time lag and introduce an auxiliary system to offset the delay effect.

[0107] System modeling: Consider the following mathematical model of a non-strict feedback nonlinear system with input time lag:

[0108]

[0109] in, is the system state vector, x i+1 (t) is the system state at the i+1th system state (variable) at time t, is the system state x i The derivative of (t), is the system state x n (t), x1(t) is the first system state, and n is the total number of system states; Input variables for control; is the control output variable; f i (x(t)) is an unknown smooth function of the system state vector x(t); τ(t) is a known bounded function that satisfies in, Represents the upper bound of the bounded function τ(t).

[0110] Control objectives: 1) The closed-loop system state converges in a fixed time.

[0111] 2) The control output variable y(t) can be replaced by the reference signal y of the desired trajectory r (t) Tracking and system status are within the preset (constraint) conditions as follows:

[0112]

[0113] in, is a quadratically differentiable constraint function. In this case, the system state x i (t) is always under this constraint. In particular, when the constraint function This constraint is transformed into a symmetric constraint. For the above system (1), the problem containing symmetric constraints can be solved. and When is a constant, this constraint becomes a static constraint, which can solve problems with static constraints. In short, when facing different constraint problems, adjust and value, and solve the problem flexibly.

[0114] The following important theorems are given:

[0115] Lemma 1 If there exists a continuous positive definite function V(x) such that

[0116]

[0117] Among them, the control parameters μ1>0, μ2>0, 0<β1<1, β2>1, Δ>0, is the derivative of the positive definite function V(x). When 0<Δ<min{(1-χ)μ1,(1-χ)μ2} and the control parameter 0<χ<1,

[0118]

[0119] Then the equilibrium point x(t) = 0 of the nonlinear system is fixed-time stable, and the rest time T(x0) satisfies the following formula

[0120]

[0121] Lemma 2 For any real number and a positive constant k p and |z|≤k p ,have

[0122]

[0123] Lemma 3 For any real number ε>0, the following equation holds true

[0124]

[0125] Lemma 4 For any real number b∈[0,1], there is

[0126]

[0127] Lemma 5 For any real number x a ≥0(a=1,…,m), there is

[0128]

[0129] Lemma 6 For any integers s1, s2, s3 > 0, have

[0130]

[0131] Lemma 7 (Young's inequality) For any real number ∈>0, there is

[0132]

[0133] Among them, (m-1)(n-1)=1 and m,n>1.

[0134] Lemma 8 For any real number ρ>0, there is

[0135]

[0136] Lemma 9 (Cauchy-Schwarz inequality) For any real number sequence a1, a2, ..., a n and b1,b2,...,b n ,have

[0137]

[0138] Among them, if and only if a i and b i The linear correlation equality holds, that is, there exists a constant k such that a i =kb i holds for all i.

[0139] Based on the excellent properties of RBFNN (Radial Basis Function Neural Network), the present invention uses RBFNN to approximate unknown continuous functions. Any unknown continuous function Λ(Z), It can be expressed as

[0140]

[0141] in, Represents the input vector of the input layer, z1,z2,...,z n represents the n input components of the input vector Z, Ω z represents a compact set; is the weight vector, w1,w2,...,w γ They represent the γ weight vectors of the weight vector W, and the weight vector is defined as

[0142]

[0143] Where sup represents the supremum, and F(Z) represents the unknown function containing the variable Z. Basis function vector Basis function s j (Z)(j=1,2,...,γ) Gaussian function is usually selected as the basis function, that is,

[0144]

[0145] in, represents the basis function center, § j Represents the width of the Gaussian function; γ is the number of network nodes; δ(Z) is the approximation error. You can select appropriate parameters according to your needs. γ=6. An auxiliary system is introduced to offset the impact of input delay. The auxiliary system is designed as follows:

[0146]

[0147] in, is the i+1th state variable of the auxiliary system, is the i-th state variable The derivative of q i is the control parameter, Represents the nth state variable The derivative of q1>0.5,q i >1, (i=1,…,n). If the system is the equation (1), there is no time delay, then τ=0; when the auxiliary system is the equation (2) When the state variables of the auxiliary system Will tend to zero. For the convenience of writing, some of the following formulas will be abbreviated, such as xi (t),z i (t) will be abbreviated as x i ,z i .

[0148] If there is no time delay in the auxiliary system, then τ=0; when When the variable will approach zero.

[0149] Step 2: According to the mathematical model of the nonlinear system, the auxiliary system and the expected trajectory, the error dynamic system is obtained through coordinate transformation.

[0150] The mathematical model of the nonlinear system is converted into an error dynamic system through coordinate transformation:

[0151]

[0152] Among them, α i-1 is the virtual control rate. i (t) is the i-th coordinate transformation state variable, z1(t) is the first coordinate transformation state variable, x1(t) is the first system state variable, x i (t) represents the i-th system state variable.

[0153] Combining equations (2) and (3), the influence of input delay (time lag) is compensated; the controller and adaptive law of equation (1) are designed using the backstepping method.

[0154] Step 3: Based on the system's fixed-time convergence and incorporating existing research findings, a piecewise function is designed and introduced into the backstepping controller and adaptive law design process, ensuring that the system under consideration converges in fixed time. A Lyapunov function V1 containing the adaptive law is constructed based on the first error signal of the error dynamics system. The Lyapunov function is differentiated and the unknown function is approximated using an RBFNN. A virtual control signal corresponding to each error signal is designed, and the derivative of the Lyapunov function V1 is determined.

[0155] Step 1: Construct Lyapunov function V1 based on z1:

[0156]

[0157] Among them, the parameters Adaptive Law Represents the estimate of the uncertain parameter θ; uncertain parameter θ=max{‖W i ‖ 2 ,i=1,…,n}, and W idenote positive constants and weight vectors respectively.

[0158] Taking the derivative of the Lyapunov function V1, we get

[0159]

[0160] Among them, the unknown function Adaptive Law The derivative of is the derivative of the state variable z1, α1 is the first virtual control signal, parameter q1>0.5 and is the first state variable of the auxiliary system. Represent the unknown smooth function f i Abbreviation of (x(t)) and reference signal y r The derivative of (t).

[0161] Then, using RBF neural network (RBFNN) to approximate the unknown function Λ1, we can get

[0162]

[0163] Among them, the intermediate variable W1 represents the weight vector for the unknown function Λ1. S1(Z1) is the basis function of the intermediate variable Z1, and δ1(Z1) is the approximation error of the intermediate variable Z1.

[0164] Combine Formula (6) and Lemma 7, we have

[0165]

[0166] Where a1 is a positive constant and S1 is the abbreviation of the basis function S1(Z1).

[0167] Substituting formula (7) into formula (5), we have

[0168]

[0169] According to the backstepping method, the virtual control signal (virtual control rate) α1 is designed as

[0170]

[0171] in, They represent the virtual control signal and the piecewise function with the subscript z1 as the state variable. Design parameter K 11 ,K 12 ,ε1>0; piecewise function is defined as

[0172]

[0173] Among them, the intermediate parameter ε 10 >0; segmentation parameter c j , (j=1,…,n+1) is determined by the following formula

[0174]

[0175] Among them, the segmentation parameter b1=1, When n=2, we have

[0176]

[0177] Combined with Lemma 3, we have

[0178]

[0179] Based on formula (11), when |z1|≥ε 10 When , substitute equations (9)-(13) into equation (8), we can get

[0180]

[0181] When |z1|<ε 10 ,have

[0182]

[0183] Among them, the intermediate variable

[0184] Compared with formula (14) and formula (15), formula (15) has an additional bounded term In this case, the following steps will be ignored.

[0185] For existing research work, although these results have successfully solved the singularity problem and robustness problem, they cannot avoid the occurrence of jitter problem. The present invention introduces a series of piecewise functions, which successfully avoids the occurrence of singularity problem and ensures the system state convergence in fixed time.

[0186] Step 4: Construct the barrier Lyapunov function of the current error signal based on the previous error signal, design the virtual control signal of the current error signal by combining RBFNN and the corresponding piecewise function of the current error signal, obtain the virtual control signals of all error signals through cyclic iteration, and use the backstepping method to obtain the actual controller and adaptive law.

[0187] Step i (2≤i≤n-1):

[0188] Combining BLF, coordinate transformation and RBFNN, recursively iteratively design virtual control law α i , ensuring that the state constraints are satisfied.

[0189] Construct the barrier Lyapunov function V i for

[0190]

[0191] Design the virtual control signal as

[0192]

[0193] in, represents a positive constant, V i-1 represents the i-1th Lyapunov function, z i is the i-th coordinate transformation state variable, is the i-th virtual control signal, S i is the abbreviation of basis function, a i Represents a positive constant. Design parameter K i1 ,K i2 ,ε i >0, piecewise function Defined as

[0194]

[0195] Among them, the segmentation parameter c j Similar to formula (12).

[0196] Step n:

[0197] Construct the following obstacle Lyapunov function:

[0198]

[0199] Design the virtual control signal as

[0200]

[0201] Combining coordinate transformation, auxiliary system, RBFNN and Lemma 7 and Lemma 3, we can derive Equation (19) as

[0202]

[0203] Among them, V n-1 is the n-1th Lyapunov function, V n is the nth Lyapunov function, K j1 , K j2 、a j 、S j , σ n Both represent normal numbers.

[0204] Substituting Equation (20) into (21) and using Lemma (3), we can obtain the actual adaptive controller ν(t) and the adaptive law as follows:

[0205]

[0206] Among them, ξ,ψ1,ι, κ is a positive constant.

[0207] Step 5: According to the actual adaptive controller, the event trigger condition is designed based on the backstepping method and the system input threshold. The event trigger condition is used to control the actual adaptive controller. When the event trigger moment is reached, the event trigger condition containing the input is triggered. The strict feedback nonlinear system with input time lag is considered and combined with the auxiliary system with input time lag for information transmission and response.

[0208] When time m+1 is the event triggering time, the designed event triggering condition includes input time lag, and then the strict feedback nonlinear system under consideration also has input time lag. Design event triggering condition:

[0209]

[0210] Among them, t m+1 , t m They represent time m+1 and time m respectively, and inf represents the infimum.

[0211] By adjusting the parameters ξ and ψ2 involved in the event triggering conditions to balance the control accuracy and communication load, the Zeno phenomenon can be avoided. Where ψ2 is a positive constant; and ψ1>ψ2 / (1-ξ); a function specified for the convenience of writing proofs t m is the update time, m is a positive integer. From this formula, it is easy to know that: when t∈[t m ,t m+1 ), the control input vector u(t) remains unchanged; when t=t m+1 , the control input vector u(t) will be updated to t m+1 The value at time t∈[t m ,t m+1 ), the event is not triggered when

[0212] Stability analysis: Consider the non-strict feedback nonlinear system with time-varying input delay (Eq. (1)), design a practical adaptive controller and adaptive law as Eq. (22) and event triggering conditions as Eq. (23). The proposed strategy can ensure the following results:

[0213] 1): The system state will not violate the constraints;

[0214] 2): All signals in the closed-loop system are bounded;

[0215] 3): Tracking error is bounded and the barrier Lyapunov function V n Converges to the set at a fixed time interval And the convergence time satisfies Among them, Δ, χ1, μ1, and μ2 all represent control parameters.

[0216] 4): Avoids Zeno behavior caused by event triggering mechanism.

[0217] Proof: According to formula (23), t∈[t m ,t m+1 )

[0218] |ν(t)-u(t)|<ξ|u(t)|+ψ2. (24)

[0219] Then, the adaptive controller ν(t) can be written as

[0220] ν(t)=(1+ξη1(t))u(t)+ψ2η2(t), (25)

[0221] Where |η1(t)|,|η2(t)|≤1. η1(t) and η2(t) represent adjustment functions, converting Equation (24) into Equation (25).

[0222] Based on formula (25), the control input vector can be obtained

[0223]

[0224] Substituting equation (26) into equation (21), we can obtain the Lyapunov function V n The derivative of

[0225]

[0226] Combining equations (22) and (26), the intermediate term is

[0227]

[0228] Based on Lemma 8 and have

[0229]

[0230] Substituting equations (23) and (29) into equation (21), we have

[0231]

[0232] According to Lemma 7, we have

[0233]

[0234] Substituting formula (31) into formula (30), we have

[0235]

[0236] Based on Lemma 6 and s1=1-s2, have

[0237]

[0238] According to formula (33) and Formula (32) can be organized as

[0239]

[0240] Based on Lemma 4, Lemma 5 and Lemma 7, Equation (34) can be written as

[0241]

[0242] Among them, the intermediate variable

[0243]

[0244] Based on Lemma 4 and Lemma 5, Equation (35) can be organized as

[0245]

[0246] in

[0247] According to formula (36), we can know that the Lyapunov function V n Bounded Right now Given the state constraints: Then z i ,θ, Bounded. From equation (36), the error signal When the state vector x i When bounded, Bounded. In the easy-to-proven auxiliary system It is bounded, that is, all signals in the closed-loop system are bounded.

[0248] Formula (36) is transformed into when have According to Lemma 1, the Lyapunov function V n Converge to the set And the fixed convergence time satisfies

[0249] because have Right now From formula (22), we know is continuous. λ is a constant that satisfies Therefore, there is

[0250]

[0251] because Bring in this specified function respectively Combining with formula (24), we can get the second equal sign.

[0252] Formula (37) can be obtained by rearranging

[0253]

[0254] in

[0255] Finally, Equation (38) effectively avoids the Zeno behavior caused by the event triggering mechanism.

[0256] The present invention verifies the effectiveness of the proposed control method through simulation experiments. The experimental design includes the following steps:

[0257] 1. System model: Select a non-strict feedback nonlinear system as the research object, and its dynamic equation is as follows:

[0258]

[0259] where f1(x(t))=1+sin(x1(t))x2(t), y r =sin(t),τ(t)=0.8|sin(t)|.

[0260] 2. Initial conditions: Set the initial conditions to

[0261] 3. Parameter setting: K ij =20(i,j=1,2),r=0.01,ε1=ε2=ε 10 =ε 20 =1,a1=a2=1,q1=1,q2=2,

[0262] ψ1=4, ψ2=0.1, ι=12, κ=1, ξ=0.3,

[0263] Results and Analysis:

[0264] 1. Tracking performance: All signals in the closed-loop system are bounded. According to formula (36), the system state x1 and the reference signal y can be obtained.r and constraints The simulation results show that Figure 2 As shown, according to Figure 2 It can be seen that the proposed method can effectively track the reference signal y r , and the state variables x1 and x2 always remain within the predetermined boundaries.

[0265] 2. State variable: Get the state variable x2 constraint L x2 ,H x2 The changing trend of Figure 3 As shown, Figure 3 This shows that the system remains stable in the presence of input delay.

[0266] 3. Tracking error: The error z1 simulation results can be shown as follows Figure 4 As shown, it shows that the tracking error z1 tends to be bounded in a fixed time.

[0267] 4. Adaptive parameters: Adaptive law can be obtained The changing trends, such as Figure 5 As shown, Figure 5 It is shown that the parameter estimates remain bounded during the control process.

[0268] 5. Control input: Through the two sentences above formula (37) and the simulation results, the changing trends of the control inputs u(t) and u(t-τ) can be obtained as follows: Figure 6 As shown, Figure 6 It shows that the control inputs u(t) and u(t-τ) can be bounded in a fixed time.

[0269] 6. Event trigger interval: The interval time of event trigger can be obtained by simulation through formula (38) as follows: Figure 7 As shown, Figure 7 It shows that the designed event trigger mechanism can effectively reduce the number of control signal updates while ensuring control performance and avoid Zeno behavior.

[0270] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. An adaptive fixed-time tracking control method based on event triggering, characterized in that: The steps are as follows: Step 1: Establish a mathematical model of a non-strict feedback nonlinear system with input time lag and introduce an auxiliary system to offset the delay effect; Step 2: Based on the mathematical model of the nonlinear system, the auxiliary system and the desired trajectory, the error dynamic system is obtained through coordinate transformation; Step 3: Based on the fixed-time convergence of the system, a piecewise function is designed, and the backstepping method is introduced to design the controller and adaptive law: Based on the first error signal of the error dynamic system, a Lyapunov function V1 containing the adaptive law is constructed. The Lyapunov function is differentiated and the unknown function is approximated using RBFNN. The virtual control signal corresponding to each error signal is designed, and the derivative of the Lyapunov function V1 is determined. Step 4: Construct the barrier Lyapunov function of the current error signal based on the previous error signal, and design the virtual control signal of the current error signal by combining the RBFNN and the corresponding piecewise function of the current error signal. Iterate the loop to obtain the virtual control signals of all error signals, and use the backstepping method to obtain the actual controller and adaptive law. Step 5: According to the actual adaptive controller, the event trigger condition is designed based on the backstepping method and the system input threshold. The event trigger condition is used to control the actual adaptive controller. When the event trigger moment is reached, the event trigger condition containing the input is triggered. The strict feedback nonlinear system with input time lag is considered and combined with the auxiliary system with input time lag for information transmission and response.

2. The event-triggered adaptive fixed-time tracking control method according to claim 1, characterized in that: The mathematical model of the non-strict feedback nonlinear system with input time lag is: in, is the system state vector, x i+1 (t) is the system state at time i+1, is the system state x i The derivative of (t), is the system state x n (t), x1(t) is the first system state, and n is the total number of system states; Input variables for control; is the control output variable; f i (x(t)) is an unknown smooth function of the system state vector x(t); τ(t) is a known bounded function that satisfies represents the upper bound of the bounded function τ(t); The control objectives of the non-strict feedback nonlinear system with input time lag include: 1) the closed-loop system state is fixed-time convergent; 2) the control output variable y(t) can be controlled by the reference signal y of the desired trajectory. r (t) Tracking and system status is within constraints: in, is a quadratically differentiable constraint function.

3. The event-triggered adaptive fixed-time tracking control method according to claim 2, characterized in that: The auxiliary system is: in, is the i+1th state variable of the auxiliary system, is the i-th state variable The derivative of q i For the control parameters, Represents the nth state variable The derivative of .

4. The event-triggered adaptive fixed-time tracking control method according to claim 3, characterized in that: The error dynamic system is: Among them, α i-1 is the i-1th virtual control signal, z i (t) is the i-th coordinate transformation state variable, z1(t) is the first coordinate transformation state variable, x1(t) is the first system state, x i (t) represents the i-th system state.

5. The event-triggered adaptive fixed-time tracking control method according to claim 4, characterized in that: The implementation method of step 3 is: construct the Lyapunov function V1 according to the first coordinate transformation state variable z1: Among them, the parameters Adaptive Law Represents the estimate of the uncertain parameter θ; uncertain parameter θ=max{‖W i ‖ 2 ,i=1,…,n}, k p1 represents a positive constant, W i represents the i-th weight vector; Taking the derivative of the Lyapunov function V1, we get Among them, the unknown function Adaptive Law The derivative of is the derivative of the state variable z1, α1 is the first virtual control signal, and the control parameter q1>0.5, is the first state variable of the auxiliary system, f1 is the abbreviation of the first unknown smooth function f1(x(t)), represents the reference signal y r The derivative of (t); Using RBFNN to approximate the unknown function Λ1, we can get Λ1(Z1)=W1 T S1(Z1)+δ1(Z1), Among them, the intermediate variable W1 represents the weight vector for the unknown function Λ1, S1(Z1) is the basis function of the intermediate variable Z1, and δ1(Z1) is the approximation error of the intermediate variable Z1; Combine have Where a1 is a positive constant, S1 is the abbreviation of the basis function S1(Z1); There is a derivative of the Lyapunov function V1 have α1 and is the first virtual control signal; When |z1|≥ε 10 When , the derivative of the Lyapunov function V1 is When |z1|<ε 10 , the derivative of the Lyapunov function V1 is Among them, the intermediate variable 6. The event-triggered adaptive fixed-time tracking control method according to claim 5, characterized in that: According to the backstepping method, the first virtual control signal α1 is designed as in, Represents a virtual control signal, Represents a piecewise function of the state variable z1. Design parameter K 11 ,K 12 ,ε1>0; piecewise function for Among them, the intermediate parameter ε 10 >0; segmentation parameter c j , j=1,…,n+1 is determined by the following formula Among them, the segmentation parameter b1=1, 7. The event-triggered adaptive fixed-time tracking control method according to claim 6, characterized in that: Recursive iterative design of the i-th virtual control law α i , 2≤i≤n-1; Construct the barrier Lyapunov function V i for Design the virtual control signal as in, represents a positive constant, V i-1 represents the i-1th Lyapunov function, z i is the i-th coordinate transformation state variable, is the i-th virtual control signal, S i is the abbreviation of the i-th basis function, a i represents a positive constant; Design parameter K i1 ,K i2 ,ε i >0, piecewise function for Step n: Construct the barrier Lyapunov function: Design the virtual control signal as Combining coordinate transformation, auxiliary system and RBFNN, the obstacle Lyapunov function V can be obtained n The derivative of Among them, V n-1 is the n-1th Lyapunov function, V n is the nth Lyapunov function, K j1 , K j2 、a j 、S j , σ n Both represent normal numbers.

8. The event-triggered adaptive fixed-time tracking control method according to claim 7, characterized in that: Get the actual adaptive controller ν(t) and the adaptive law for: Among them, ξ,ψ1,ι, κ is a positive constant.

9. The adaptive fixed-time tracking control method based on event triggering according to any one of claims 5 to 8, characterized in that: The design event trigger conditions are: Among them, t m+1 and t m denote time m+1 and time m respectively, m is a positive integer, inf denotes the lower bound; ψ2 is a positive constant; and ψ1>ψ2 / (1-ξ); function When t∈[t m ,t m+1 ), the control input vector u(t) remains unchanged and the event is not triggered When t = t m+1 , control the input vector u(t) to update to t m+1 The value of the moment; because have Right now Derivatives of the actual adaptive controller is continuous, λ is a constant, satisfying have Arrangement available Among them, the constant This effectively avoids Zeno behavior caused by the event triggering mechanism.

10. The event-triggered adaptive fixed-time tracking control method according to claim 9, characterized in that: Considering a non-strict feedback nonlinear system with input time lag, design a practical adaptive controller ν(t) and an adaptive law And the event triggering conditions can ensure the following results: 1) The system state will not violate the constraints; 2) All signals in the closed-loop system are bounded; 3) The tracking error is bounded and the barrier Lyapunov function V n Converges to the set at a fixed time interval And the convergence time satisfies Among them, Δ, χ1, μ1, and μ2 all represent control parameters; 4): Avoids Zeno behavior caused by event triggering mechanism.

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