Practical fixed time control method for state constraint nonlinear system

Abstract

The invention relates to a practical fixed time control method for a state constraint nonlinear system. The method comprises the following steps: step 1) establishing a dynamic equation and state constraint of a pure feedback constraint nonlinear system; 2) aiming at the state constraint, constructing an improved general barrier function; 3) based on the obstacle function, converting the constrained nonlinear system into an unconstrained system; 4) performing approximation to obtain an approximation form; 5) obtaining a coordinate transformation result; step 6) based on error dynamics, constructing an adaptive fixed time tracking control scheme and a parameter updating law control system; and 7) proving the stability of the system. Compared with the prior art, the method has the advantages of overcoming singularity in fixed time control, reducing calculation burden and the like.

Description

Technical Field

[0001] The invention relates to the field of adaptive control, and in particular to a practical fixed-time control method for a state-constrained nonlinear system. Background Art

[0002] In practical applications, autonomous systems such as quadcopters, robotic systems, and self-driving cars are subject to various constraints, including state and output constraints, for system specifications and safety considerations to ensure reliable and safe operation of the system. This is the well-known state / output constraint problem, which has been extensively studied based on barrier Lyapunov functions and control strategies based on barrier functions. The design idea of ​​the barrier Lyapunov function-based method is to convert the state constraints into new constraints on the virtual tracking error during the backstepping process, thereby indirectly satisfying the full state constraints, but it is necessary to check whether the virtual controller satisfies the new constraints, namely the feasibility conditions. The barrier function method circumvents the feasibility conditions, but there is still a lack of a general barrier function to remove the restrictions of the state lower limit being negative and the state upper limit being positive in existing studies, and to handle both constrained and unconstrained cases.

[0003] In addition to state / output constraints, inspired by engineering applications such as spacecraft systems, robotic arms, and ground vehicles, in order to improve the system convergence performance, it is necessary to ensure that the tracking error converges within a fixed time to achieve better transient performance and improve the robustness of the closed-loop system. In fixed-time control design, singularity problems may occur due to the fractional power derivatives in the virtual controller. Existing studies have eliminated singularities by constructing piecewise functions, but at the cost of designing more complex controls and complex Lyapunov stability analysis. Summary of the invention

[0004] The purpose of the present invention is to provide a practical fixed-time control method for a state-constrained nonlinear system in order to overcome the singularity in fixed-time control and reduce the computational burden. First, a universal barrier function is constructed to transform the state constraint problem into a problem of ensuring the boundedness of the system state after the transformation; at the same time, based on neural network and dynamic surface control technology, an adaptive tracking controller with a fixed-time convergence rate is designed for system uncertainty. The control design can ensure that the system always satisfies the state constraint and the tracking error converges to the neighborhood of zero within a fixed time.

[0005] The purpose of the present invention can be achieved by the following technical solutions:

[0006] A practical fixed-time control method for a state-constrained nonlinear system, the method comprising the following steps:

[0007] Step 1) establishing the dynamic equations and state constraints of the pure feedback constrained nonlinear system;

[0008] Step 2) constructing an improved universal barrier function for state constraints;

[0009] Step 3) based on the barrier function, transform the constrained nonlinear system into an unconstrained system;

[0010] Step 4) The nonlinear term of the unconstrained system is approximated by a radial basis function neural network to obtain an approximate form, and the nonlinear term in the obstacle function describing the unconstrained system is replaced by the approximate form;

[0011] Step 5) designing a fixed-time first-order filter, based on the universal barrier function and the fixed-time first-order filter, using a dynamic surface control method to perform coordinate transformation, and obtaining a coordinate transformation result;

[0012] Step 6) Based on the error dynamics, an adaptive fixed-time tracking control scheme and a parameter update law are constructed according to the coordinate transformation result, the barrier function and the fixed-time first-order filter, and the system is controlled based on the adaptive fixed-time tracking control scheme and the parameter update law, wherein the adaptive fixed-time tracking control scheme includes a virtual control input and a continuously differentiable neural adaptive control law with a fixed-time first-order filter;

[0013] Step 7) Prove the stability of the system controlled by the adaptive fixed-time tracking control scheme and the parameter update law.

[0014] Furthermore, the dynamic equations and state constraints of the pure feedback nonlinear system are:

[0015]

[0016] y=x1,e=yy d

[0017] in, is the system status, are the control input and output, is the expected trajectory, is the tracking error, f i (·) is an unknown continuous nonlinear function, and Continuously differentiable with respect to u;

[0018] The state constraints are:

[0019]

[0020] Among them l i : and h i : For the pre-specified C 2 function.

[0021] Furthermore, the improved universal barrier function is: F: The function satisfies the following three properties:

[0022] Property 1: F(x i ,h i ,l i ) is C 2 Function, satisfying

[0023]

[0024] Property 2: F(·) with respect to x i The partial derivative of is strictly positive, that is,

[0025]

[0026] Property 3: For l i = -∞ and h i = +∞, that is, for x i Without constraints, F(·) satisfies

[0027]

[0028] Furthermore, the improved general barrier function is specifically:

[0029]

[0030] Furthermore, the approximate form is:

[0031]

[0032] Where, for i=1,…,n, x n+1 =0,ξ n+1 =0, is the ideal weight vector, N i is the number of nodes, ε i (X i ) is the approximate error, satisfying is a constant.

[0033] Furthermore, the unconstrained system is:

[0034]

[0035] in,

[0036]

[0037] Replace with an approximate form.

[0038] Furthermore, the coordinate transformation result is:

[0039] z1=ξ1-ξ d ,z i =ξ i -α if ,i=2,…,n

[0040] in,

[0041]

[0042] α if is the output of a fixed-time first-order filter.

[0043] Furthermore, the fixed-time first-order filter is:

[0044]

[0045] in is a fixed-time first-order filter, y i =α if -α i-1 is the filter error, α i-1 is the virtual control input to be designed, τ i1 >0,τ i2 >0 and ∈ if >0 is the control parameter.

[0046] Furthermore, the virtual control input and the continuously differentiable neural adaptive control law with a first-order filter at fixed time and the parameter update law are controlled as follows:

[0047]

[0048] Among them, α i represents the virtual control input, k i1 >0,k i2 >0,∈ i >0,δ i >0, γ>0, σ1>0 and σ2>0 are control parameters, represents the current parameter, φ i (X i )=||S i (X i )||+1,S i (X i ) represents the basis function vector, represents the parameter update law control, and u represents the continuously differentiable neural adaptive control law with a first-order filter with fixed time.

[0049] Furthermore, in step 7), it is proved that the upper bound of the convergence time after stabilization is:

[0050]

[0051] in, ρ1>0,ρ2>0 are normal numbers.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] (1) The present invention proposes an improved barrier function to handle a more general form of asymmetric time-varying constraints, that is, the sign of the constraint function can be arbitrary, that is, the restriction in existing studies that the lower limit of the state is negative and the upper limit is positive is cancelled. Such a barrier function is also a unified function because it can handle both constrained and unconstrained situations.

[0054] (2) In order to overcome the singularity in fixed-time control, a continuously differentiable neural adaptive control law with a first-order filter at fixed time is constructed by carefully designing the hyperbolic tangent function, which is different from the piecewise function method in existing research. It also adopts a simpler form and ensures the simplicity of the stability analysis procedure.

[0055] (3) In order to reduce the computational burden, the present invention designs a parameter update law with only one dimension for practical fixed-time tracking control, which avoids over-parameterization in the backstepping process. In practical fixed-time tracking control of high-order general nonlinear systems with state constraints, it is not easy to design such a low-cost adaptive method. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 is a flow chart of the method of the present invention;

[0057] Figure 2 is the change curve of the system state under constraints in the experiment;

[0058] Figure 3 It is the changing curve of tracking error and control input in the experiment;

[0059] Figure 4 It is the changing curve of the system state without constraints in the experiment. DETAILED DESCRIPTION

[0060] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0061] The present invention proposes a practical fixed-time control method for a state-constrained nonlinear system. The present invention combines neural network and dynamic surface control technology to provide an adaptive fixed-time tracking control scheme for a pure feedback system subject to asymmetric time-varying state constraints. The present invention needs to deal with a high-order nonlinear system with state constraints and uncertainties. The technical challenge lies in simultaneously dealing with the over-parameterization and singularity problems in the differential process of the virtual controller, as well as stabilizing a complex and time-varying closed-loop system. The method comprises the following steps:

[0062] Step 1) establishing the dynamic equations and state constraints of the pure feedback constrained nonlinear system;

[0063] Step 2) constructing an improved universal barrier function for state constraints;

[0064] Step 3) based on the barrier function, transform the constrained nonlinear system into an unconstrained system;

[0065] Step 4) The nonlinear term of the unconstrained system is approximated by a radial basis function neural network to obtain an approximate form, and the nonlinear term in the obstacle function describing the unconstrained system is replaced by the approximate form;

[0066] Step 5) designing a fixed-time first-order filter, based on the universal barrier function and the fixed-time first-order filter, using a dynamic surface control method to perform coordinate transformation, and obtaining a coordinate transformation result;

[0067] Step 6) Based on the error dynamics, an adaptive fixed-time tracking control scheme and a parameter update law are constructed according to the coordinate transformation result, the barrier function and the fixed-time first-order filter, and the system is controlled based on the adaptive fixed-time tracking control scheme and the parameter update law, wherein the adaptive fixed-time tracking control scheme includes a virtual control input and a continuously differentiable neural adaptive control law with a fixed-time first-order filter;

[0068] Step 7) Prove the stability of the system controlled by the adaptive fixed-time tracking control scheme and the parameter update law.

[0069] In step 1), the dynamic equation of the pure feedback nonlinear system is:

[0070]

[0071] y=x1,e=yy d

[0072] in is the system status, are the control input and output, is the expected trajectory, is the tracking error. i(·) is an unknown continuous nonlinear function, and Continuously differentiable with respect to u.

[0073] The state constraints of the system are

[0074]

[0075] Among them l i : and h i : For the pre-specified C 2 function.

[0076] The control objective is to design an adaptive control law for the system to track the desired trajectory y d , and ensure that the tracking error e converges to a small neighborhood of zero within a fixed time, while satisfying the state constraints during operation The formal description of the problem is as follows:

[0077] Problem 1: Considering a pure feedback nonlinear system and state constraints, design a continuous differentiable control law u so that for any initial condition The solution of the closed-loop system exists and is bounded, and satisfies the following properties:

[0078] 1) For all t ≥ 0;

[0079] 2) The equilibrium point e = 0 of the system is practically fixed-time stable, that is, the solution of the closed-loop system exists when t ≥ 0, and e = 0 is Lyapunov stable, there exists a constant ∈> 0 and a fixed time T> 0, and and d (0) irrelevant, making

[0080]

[0081] For a pure feedback system, by the middle value theorem, there exists α∈(0,1) such that

[0082]

[0083] in

[0084] In order to achieve the control objectives, the following three reasonable assumptions are made:

[0085] Assumption 1: Expected trajectory y d , and Bounded and continuous. There exists a condition that satisfies l i (t) <l d (t) <h d (t) <h i(t) is a time-varying function of d (t) and h d (t), so that

[0086] Assumption 2: Constraint function l i (t),h i (t) and its first and second time derivatives are continuous and bounded.

[0087] Assumption 3: g n The sign of (·) is known, and there is an unknown constant Make Without loss of generality, assume that g n (·)>0.

[0088] In step 2), in order to allow l i (t) and t i (t) is any i (t) <h i (t) is a time-varying function, covering both constrained and unconstrained cases, and a new universal barrier function F is proposed: Has the following properties.

[0089] Property 1: F(x i ,h i ,l i ) is C 2 Function, satisfying

[0090]

[0091] Property 2: F(·) with respect to x i The partial derivative of is strictly positive, that is,

[0092]

[0093] Property 3: For l i = -∞ and h i = +∞, that is, for x i Without constraints, F(·) satisfies

[0094]

[0095] A specific form of a general barrier function, using ξ i (t), i=1,…,n, expressed as

[0096]

[0097] It can be proved that i (t) satisfies properties 1-3. For any initial condition Ifi (t) is bounded, then for all t ≥ 0,

[0098] In step 3), based on the universal barrier function, the constrained nonlinear system is transformed into the following unconstrained system:

[0099]

[0100] in

[0101]

[0102] In step 4), for the nonlinear term F in the unconstrained system, i (·), i=1,…,n, radial basis function neural network is used for approximation:

[0103]

[0104] Where, for i=1,…,n, x n+1 =0,ξ n+1 =0, is the ideal weight vector, N I is the number of nodes, ε I (X I ) is the approximate error, satisfying For a constant is a basis function vector and S ij (X i )

[0105]

[0106] in, and i=1,…,n-1,j=1,…,N i is the center, is the width of the basis function. So

[0107]

[0108] in

[0109]

[0110] φ i (X i )=||S i (X i )||+1.

[0111] In step 5), in order to avoid repeatedly obtaining the virtual controller α i-1,i=2,…,n derivatives make the design too complicated, so the dynamic surface control method is adopted, and the following coordinate transformation is further performed:

[0112] z1=ζ1-ζ d ,z i =ξ i -α if ,i=2,…,n

[0113] in α if is the output of the following fixed-time first-order filter:

[0114]

[0115] where y i =α if -α i-1 is the filter error, α i-1 is the virtual control input to be designed, τ i1 >0,τ i2 >0 and ∈ if >0 is the control parameter.

[0116] In step 6), the adaptive fixed time tracking control scheme is designed as follows:

[0117]

[0118] And the parameter update law

[0119]

[0120] Among them, k i1 >0,k i2 >0,∈ i >0,δ i >0, γ>0, σ1>0 and σ2>0 are control parameters.

[0121] In step 7), the stability of the system is proved based on the Lyapunov theory, and the upper bound of the convergence time is obtained. The proof process is as follows:

[0122] (1) Using the backstepping method, which includes n steps, consider the first Lyapunov function as The i-th Lyapunov function is The nth Lyapunov function is And the overall Lyapunov function is Derived and scaled using lemmas such as Young's inequality, substituted into the virtual control law and the actual control law for calculation, and finally obtained

[0123]

[0124] where Θ is a positive constant,

[0125] ρ1=min{2 μ k i1 ,(2 g n ) μ k n1 ,σ1,2 μ τ i1},i=1,…,n-1

[0126] ρ2=min{4k i2 ,(2 g n ) 2 k n2 ,σ2,4τ i2}.

[0127] (2) Prove that all signals in the closed-loop system are bounded. Prove that ξ i (t) is bounded, then for all t≥0.

[0128] (3) According to the fixed time stability theorem, the tracking error e(t) is <T<T max converges to a small neighborhood of zero, T max satisfy

[0129]

[0130] in ρ1>0,ρ2>0 are normal numbers.

[0131] The present invention proposes an improved barrier function to handle a more general form of asymmetric time-varying constraints, that is, the sign of the constraint function can be arbitrary, that is, the restriction that the lower limit of the state is negative and the upper limit is positive in the existing research is cancelled. Such a barrier function is also a unified function because it can handle both constrained and unconstrained situations.

[0132] In order to overcome the singularity in fixed-time control, a continuously differentiable neural adaptive control law with a first-order filter at fixed time is constructed by carefully designing the hyperbolic tangent function, which is different from the piecewise function method in existing research. It also adopts a simpler form and ensures the simplicity of the stability analysis procedure.

[0133] In order to reduce the computational burden, the present invention designs a parameter update law with only one dimension for practical fixed-time tracking control, avoiding over-parameterization in the backstepping process. In practical fixed-time tracking control of high-order general nonlinear systems with state constraints, it is not easy to design such a low-cost adaptive method.

[0134] The control flow chart of the present invention is as follows: Figure 1 shown.

[0135] The present invention discloses a practical fixed-time control method for a state-constrained nonlinear system based on a universal barrier function. A universal barrier function is constructed for a state-constrained pure feedback nonlinear system; the original constraint system is transformed into an unconstrained system according to the universal barrier function; and an adaptive tracking controller with a fixed-time convergence property is designed based on a neural network and a dynamic surface control technology for system uncertainty. The present invention can ensure that the system always satisfies the state constraint and the tracking error converges to a neighborhood of zero within a fixed time. It is also a low-cost design based on a neural network, because only one parameter update rule needs to be designed, thereby greatly reducing the dimension of the closed-loop system.

[0136] Considering a chemical reaction dynamics model called the Brussels model, which is one of the most popular nonlinear oscillation models of chemical kinetics, the steps of the present invention are:

[0137] Step 1) establishing the dynamic equations and state constraints of the pure feedback nonlinear system;

[0138] Consider the following Brussels model in the form of a second-order pure feedback system:

[0139]

[0140] Among them, x1 and x2 represent the concentrations of two chemical reaction intermediates, and C, D>0 are system parameters describing the supply of reservoir chemicals. In order to achieve the ideal chemical synthesis reaction efficiency, the concentration x1 must track the reference signal y d (t) and is restricted to a dynamic range, denoted by (l1(t),h1(t)), and the same is true for x2.

[0141] The state constraints of the system are

[0142]

[0143] Among them l i : and h i : For the pre-specified C 2 function.

[0144] Step 2) construct a general barrier function for the state constraints;

[0145] In order to allow i (t) and h i (t) is any i (t) <h i (t) is a time-varying function, covering both constrained and unconstrained cases, and constructing a new universal barrier function ξ i (t):

[0146]

[0147] Step 3) Based on the barrier function transformation, an unconstrained system is obtained;

[0148] Based on the barrier function in the previous step, the Brussels nonlinear system is transformed into the following unconstrained system:

[0149]

[0150] in

[0151]

[0152] Step 4) Processing unknown nonlinear terms based on neural network;

[0153] For the nonlinear term F in the previous step i (·), i=1,2, radial basis function neural network is used for approximation:

[0154]

[0155] in, is the ideal weight vector, N i is the number of nodes, ε i (X i ) is the approximate error, satisfying For a constant is a basis function vector and S ij (X i )

[0156]

[0157] in, and j=1,…,N i is the center, is the width of the basis function. So

[0158]

[0159] in

[0160]

[0161] φ i (X i )=||S i (X i )||+1.

[0162] Step 5) define coordinate transformation and design fixed time filter;

[0163] In order to avoid repeatedly calculating the derivative of the virtual controller α1, which would make the design too complicated, the dynamic surface control method was adopted, and the following coordinate transformation was further performed:

[0164] z1=ξ1-ξ d ,z2=ξ2-α 2f

[0165] in α if is the output of the following fixed-time first-order filter:

[0166]

[0167] Where y2 = α 2f -α1 is the filter error, α1 is the virtual control input to be designed, τ 21 >0,τ 22 >0 and ∈ 2f >0 is the control parameter.

[0168] Step 6) Design a control scheme based on error dynamics:

[0169]

[0170] And the parameter update law

[0171]

[0172] Among them, k i1 >0,k i2 >0,∈ i >0,δ i >0, i=1,2,γ>0,σ1>0 andσ2>0 are control parameters.

[0173] Step 7) Prove the stability of the system based on Lyapunov theory and obtain the upper bound of the convergence time.

[0174] Based on Lyapunov theory, it is proved For all t ≥ 0, and the tracking error e(t) is fixed at time 0 <T<T max converges to a small neighborhood of zero, T max satisfy

[0175]

[0176] in ρ1>0,ρ2>0 are normal numbers.

[0177] Finally, this embodiment performs simulation verification of state-constrained Brussels model tracking control.

[0178] Tracking signal is y d (t) = 0.5sin(0.5t). The constraint function is:

[0179] l1(t)=y d -(0.25e -0.3t +0.05),h1(t)=y d +(0.15e -0.3t +0.05),

[0180] l2(t)=sin(0.5t)-1,h2(t)=0.6sin(0.5t)+0.4.

[0181] The initial state is selected as

[0182]

[0183] satisfy Design an adaptive timing tracking control scheme based on neural network, and the parameters are selected as follows:

[0184] μ=0.75,N1=N2=10,μ 1j =03,μ 2j =02,ψ 1j =ψ 1j =3,j=1,…,10

[0185] k 11 =k 12 =1.5,∈1=0.001,δ1=0.5,τ 21 =τ 22 =6,∈ 2f =0.0001,

[0186] k 21 =k 22 =1,∈2=0.001,δ2=0.5,γ=0.1,σ1=0.1,σ2=4.

[0187] The simulation results are as follows Figures 2 to 4 shown. Figure 2 This indicates that the state constraints are not violated. Figure 3 It indicates that |e(t)|≤0.055, for t≥6.8s, fixed time tracking is achieved.

[0188] Figure 4 It shows that the proposed control scheme can handle the unconstrained situation at the same time.

[0189] The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A practical fixed-time control method for state-constrained nonlinear systems, characterized in that: The method comprises the following steps: Step 1) establishing the dynamic equations and state constraints of the pure feedback constrained nonlinear system; Step 2) constructing an improved universal barrier function for state constraints; Step 3) based on the barrier function, transform the constrained nonlinear system into an unconstrained system; Step 4) The nonlinear term of the unconstrained system is approximated by a radial basis function neural network to obtain an approximate form, and the nonlinear term in the obstacle function describing the unconstrained system is replaced by the approximate form; Step 5) designing a fixed-time first-order filter, based on the universal barrier function and the fixed-time first-order filter, using a dynamic surface control method to perform coordinate transformation, and obtaining a coordinate transformation result; Step 6) Based on the error dynamics, an adaptive fixed-time tracking control scheme and a parameter update law are constructed according to the coordinate transformation result, the barrier function and the fixed-time first-order filter, and the system is controlled based on the adaptive fixed-time tracking control scheme and the parameter update law, wherein the adaptive fixed-time tracking control scheme includes a virtual control input and a continuously differentiable neural adaptive control law with a fixed-time first-order filter; Step 7) Prove the stability of the system controlled by the adaptive fixed-time tracking control scheme and the parameter update law.

2. A practical fixed-time control method for a state-constrained nonlinear system according to claim 1, characterized in that: The dynamic equations and state constraints of the pure feedback nonlinear system are: y=x1,e=yy d in, is the system status, are the control input and output, is the expected trajectory, is the tracking error, f i (·) is an unknown continuous nonlinear function, and Continuously differentiable with respect to u; The state constraints are: in and h i : For the pre-specified C 2 function.

3. A practical fixed-time control method for a state-constrained nonlinear system according to claim 2, characterized in that: The improved universal barrier function is: The function satisfies the following three properties: Property 1: F(x i ,h i , l i ) is C 2 Function, satisfying Property 2: F(·) with respect to x i The partial derivative of is strictly positive, that is, Property 3: For l i = -∞ and h i = +∞, that is, for x i Without constraints, F(·) satisfies 4. A practical fixed-time control method for a state-constrained nonlinear system according to claim 3, characterized in that: The improved general barrier function is as follows:

5. A practical fixed-time control method for a state-constrained nonlinear system according to claim 1, characterized in that: The approximate form is: F i (X i )=W i T S i (X i )+ε i (X i ) Where, for i=1,…,n, x n+1 =0,ξ n+1 =0, is the ideal weight vector, Ni is the number of nodes, ε i (X i ) is the approximate error, satisfying is a constant.

6. A practical fixed-time control method for a state-constrained nonlinear system according to claim 5, characterized in that: The unconstrained system is: in, Replace with an approximate form.

7. A practical fixed-time control method for a state-constrained nonlinear system according to claim 1, characterized in that: The coordinate transformation result is: z1=ξ1-ξ d ,z i =ξ i -a if ,i=2,…,n in, α if is the output of a fixed-time first-order filter.

8. A practical fixed-time control method for a state-constrained nonlinear system according to claim 7, characterized in that: The fixed-time first-order filter is: in is a fixed-time first-order filter, y i =α if -α i-1 is the filter error, α i-1 is the virtual control input to be designed, τ i1 >0,τ i2 >0 and∈ if >0 is the control parameter.

9. A practical fixed-time control method for a state-constrained nonlinear system according to claim 8, characterized in that: The virtual control input and the continuously differentiable neural adaptive control law with a first-order filter with fixed time and the parameter update law are controlled as: Among them, α i represents the virtual control input, k i1 >0,k i2 >0,∈ i >0,δ i >0, γ>0, σ1>0 and σ2>0 are control parameters. represents the current parameter, φ i (X i )=||S i (X i )||+1,S i (X i ) represents the basis function vector, represents the parameter update law control, and u represents the continuously differentiable neural adaptive control law with a first-order filter with fixed time.

10. A practical fixed-time control method for a state-constrained nonlinear system according to claim 1, characterized in that: In step 7), it is proved that the upper bound of the convergence time after stabilization is: in, ρ1>0,ρ2>0 are normal numbers.

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