Robust adaptive trajectory tracking control method based on nonlinear extended observer

Abstract

The invention discloses a robust adaptive trajectory tracking control method based on a nonlinear extended observer, and belongs to the technical field of intelligent control and nonlinear adaptive robustness. The method comprises the following steps: establishing a kinetic model of a nonlinear system and obtaining tracking error dynamics; constructing a mixed performance function; mapping the constrained trajectory tracking error into an equivalent unconstrained conversion variable; the nonlinear extended state observer performs real-time estimation and feed-forward compensation on system lumped disturbance; designing an adaptive evaluation neural network weight updating law and an optimal control strategy under the nominal system in combination with the nominal system; and constructing a time-varying gain mechanism with a preset time characteristic, and constructing a composite optimal controller. According to the method, it can be guaranteed that the disturbed nonlinear system converges the tracking error to the steady-state limit within the preset time, the whole process meets the preset transient performance envelope, and the trajectory tracking precision and robustness under complex disturbance are remarkably improved.

Description

Technical Field

[0001] This invention relates to a robust adaptive trajectory tracking control method based on a nonlinear extended observer, belonging to the field of intelligent control and nonlinear adaptive robust technology. Background Technology

[0002] With the rapid development of industrial automation and intelligent manufacturing, unprecedentedly stringent requirements have been placed on trajectory tracking control of nonlinear systems in mission-critical fields such as aerospace, precision machining, and minimally invasive surgery. These systems typically exhibit strong coupling, nonlinearity, and multivariable characteristics. Their control systems must not only achieve error convergence within a preset time but also ensure that the transient performance of the system state is strictly constrained throughout the entire process to avoid overshoot and collisions. However, such systems are often affected by the combined effects of model uncertainties, parameter perturbations, and external disturbances, making it extremely difficult to simultaneously meet the dual objectives of preset time convergence and preset performance constraints throughout the entire process.

[0003] Traditional methods (such as PID control and sliding mode control) can typically only guarantee asymptotic or exponential convergence, with convergence time depending on initial conditions, making it difficult to meet explicit cutoff time requirements. Preset performance control can constrain the transient and steady-state boundaries of the error, but it is generally based on an infinite time frame and cannot guarantee convergence within a finite time. Preset time control, while achieving convergence within a specified time, often ignores transient waveform constraints, easily leading to overshoot and requiring a large initial control energy.

[0004] Furthermore, in response to system lumped uncertainties, existing disturbance rejection techniques (such as linear extended state observers) are prone to phase lag and amplitude decay when dealing with rapid time-varying disturbances; conventional finite-time observers are difficult to balance convergence speed and noise suppression capability near the equilibrium point.

[0005] Therefore, for uncertain nonlinear systems under strong nonlinear disturbances, how to design a control method that can achieve high-precision convergence within a user-preset time, strictly follow the transient performance constraints throughout the process, and accurately compensate for time-varying disturbances in real time has become a key technical problem that urgently needs to be solved in the field of robust control. Summary of the Invention

[0006] To address the problems existing in the background technology, this invention provides a robust adaptive trajectory tracking control method based on a nonlinear extended observer.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: a robust adaptive trajectory tracking control method based on a nonlinear extended observer, the method comprising the following steps:

[0008] S1: Establish a dynamic model of the nonlinear system and obtain the tracking error dynamics by combining it with the reference signal;

[0009] S2: Construct a hybrid performance function that integrates preset time and preset performance attributes;

[0010] S3: The constrained trajectory tracking error is mapped to an equivalent unconstrained transformation variable through an error transformation mechanism;

[0011] S4: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure to perform real-time estimation and feedforward compensation of the system's lumped disturbance.

[0012] S5: Combining the weight update law of the adaptive evaluation neural network with the optimal control strategy under the nominal system;

[0013] S6: Construct a time-varying gain mechanism with preset time characteristics to build a composite optimal controller.

[0014] Furthermore, step S1 includes the following steps:

[0015] S101: Establish the state space of the dynamic model of a continuous-time nonlinear system with external disturbances:

[0016] (1)

[0017] In formula (1):

[0018] For the system state variables in the dynamic model of a continuous-time nonlinear system;

[0019] System state variables The first derivative;

[0020] A nonlinear unknown function that describes the inherent dynamics of the internal state of a system;

[0021] A control input matrix that reflects the relationship between control inputs and system state;

[0022] For control input of continuous-time nonlinear systems;

[0023] The disturbance input matrix is ​​used to characterize the effect of external disturbances on the system state;

[0024] For an unknown bounded perturbation in a continuous-time nonlinear system;

[0025] S102: Given reference signal trajectory:

[0026] (2)

[0027] In formula (2):

[0028] The system state variables are reference signal dynamic models;

[0029] System state variables The first derivative;

[0030] A nonlinear function describing the reference signal system;

[0031] S103: Establish the dynamic equations of the tracking error system:

[0032] (3)

[0033] In formula (3):

[0034] This represents the original tracking state error.

[0035] Original tracking state error The first derivative.

[0036] Furthermore, the specific form of the hybrid performance function described in S2 is as follows:

[0037] (4)

[0038] In equation (4):

[0039] Original tracking state error The performance boundary function, for any time Meet the preset performance control conditions ,in, This is the system's initial time.

[0040] To constrain the performance boundary function of the original tracking state error magnitude The initial boundary values;

[0041] The performance boundary function that determines the final steady-state error accuracy of the system The steady-state boundary values ​​satisfy the following conditions: ;

[0042] It is a decay function;

[0043] The preset convergence time is independent of the initial conditions;

[0044] Performance boundary function The attenuation rate adjustment parameter;

[0045] For terminal smoothness parameters;

[0046] For irrational numbers An exponential function with base 0.

[0047] Furthermore, step S3 includes the following steps:

[0048] S301: Construction error conversion mechanism:

[0049] (5)

[0050] In equation (5):

[0051] These are the transformed unconstrained system state variables;

[0052] It is the inverse hyperbolic tangent function;

[0053] For irrational numbers The natural logarithm with base 1;

[0054] S302: Obtain the transformed unconstrained system:

[0055] (6)

[0056] In equation (6):

[0057] The first derivative of the state variables of the transformed unconstrained system;

[0058] Let be the nonlinear function of the transformed unconstrained system, where: Performance boundary function The first derivative, It is a hyperbolic cosine function. It is the hyperbolic tangent function;

[0059] This is the control input matrix for the transformed unconstrained system;

[0060] This is the perturbation input matrix of the transformed unconstrained system.

[0061] Furthermore, step S4 includes the following steps:

[0062] S401: Define nonlinear functions :

[0063] (7)

[0064] In equation (7):

[0065] The parameter is a non-linear power parameter.

[0066] For linear interval thresholds;

[0067] It is an absolute value function;

[0068] Standard symbolic functions;

[0069] S402: Define a lumped disturbance in a transformed system that includes unknown nonlinearities and unknown disturbances:

[0070] (8)

[0071] S403: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure:

[0072] (9)

[0073] In equation (9):

[0074] The transformed unconstrained system state variables The estimated value, For estimated value The first derivative;

[0075] For aggregated disturbance The estimated value, For estimated value The first derivative;

[0076] The nonlinear power parameters to be designed satisfy... and ;

[0077] All are nonlinear observer gains;

[0078] S404: Constructing a feedforward compensation term based on a nonlinear extended state observer:

[0079] (10)

[0080] In formula (10):

[0081] The control input matrix of the transformed unconstrained system The generalized inverse.

[0082] Furthermore, step S5 includes the following steps:

[0083] S501: Constructing the nominal system of the transformed unconstrained system:

[0084] (11)

[0085] In equation (11):

[0086] These are the state variables of the nominal system. State variables The first derivative;

[0087] The control input matrix of the nominal system;

[0088] For the control input of the nominal system;

[0089] S502: Constructing the value function and optimal value function in the finite time domain:

[0090] (12)

[0091] (13)

[0092] In equations (12)-(13):

[0093] State variables of the nominal system transpose;

[0094] For the control input of the nominal system transpose;

[0095] for The value function at time t, for The optimal value function at time t;

[0096] For terminal constraint functions;

[0097] The weight matrix represents the positive definite state.

[0098] The weight matrix is ​​the positive definite control input.

[0099] It is a minimum value function;

[0100] S503: Utilizing an evaluation neural network to evaluate the optimal value function Approximation:

[0101] (14)

[0102] In equation (14):

[0103] This is the weight vector used to evaluate the neural network between the hidden and output layers;

[0104] This represents the number of neurons in the hidden layer.

[0105] Weight vector transpose;

[0106] It is a time-varying activation function;

[0107] Remaining time;

[0108] This represents the approximation error of the neural network.

[0109] S504: Designing an approximate optimal control law for a nominal system :

[0110] (15)

[0111] In equation (15):

[0112] For positive definite control input weight matrix The inverse matrix;

[0113] The control input matrix of the nominal system transpose;

[0114] For partial derivatives transpose, For the time-varying activation function with respect to the state variables of the nominal system The partial derivatives;

[0115] To evaluate the weights of a neural network The estimated value;

[0116] S505: Design of a weight update law for evaluating neural networks based on gradient descent:

[0117] (16)

[0118] In equation (16):

[0119] For estimated value The first derivative;

[0120] The learning rate of the neural network;

[0121] For HJB residual error; HJB residual error transpose;

[0122] This represents the activation function value at the terminal moment; The activation function value at the terminal moment transpose;

[0123] For terminal constraint error; Terminal constraint error transpose; For estimated value transpose;

[0124] These are the weighting coefficients for the HJB residual term, terminal constraint term, and regularization term, respectively.

[0125] For time-varying activation functions with respect to time The partial derivatives;

[0126] For partial derivatives The transpose of .

[0127] Furthermore, step S6 includes the following steps:

[0128] S601: Construct a time-varying gain control term with preset time convergence characteristics. :

[0129] (17)

[0130] In equation (17):

[0131] The time-varying gain parameters to be designed;

[0132] S602: Feedforward compensation term combined with S404 Approximate optimal control law of S504 and the time-varying gain control term of S601 The final composite optimal controller is obtained:

[0133] (18)

[0134] Compared with the prior art, the beneficial effects of the present invention are:

[0135] 1. This invention employs a hybrid performance function and error transformation mechanism that integrates preset time and preset performance. By mapping the topology of the constrained tracking problem to an equivalent unconstrained stabilizing problem, it ensures that the system tracking error is within a safe envelope while forcing the tracking error to converge precisely within a user-specified time.

[0136] 2. This invention achieves fast and accurate estimation of the lumped uncertainty of the system by constructing a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure. The observer can ensure that the estimation error converges to near zero within a fixed time independent of the initial state, which significantly enhances the system's active compensation capability and robustness to unknown disturbances.

[0137] 3. This invention constructs a composite controller integrating disturbance feedforward compensation, online optimization, and time constraints. This controller achieves accurate estimation and real-time compensation of lumped disturbances through a nonlinear extended state observer; it utilizes an adaptive evaluation neural network to solve the finite-time optimal control problem online, achieving optimal allocation of control energy; and it introduces a time-varying gain mechanism to ensure strict convergence of the system state within a user-preset time. Attached Figure Description

[0138] Figure 1 This is a flowchart of the present invention;

[0139] Figure 2 This is a graph showing the angular position tracking of the two joints in Example 1;

[0140] Figure 3 This is a graph showing the tracking error of the two joints in Example 1 and their preset performance boundary curves;

[0141] Figure 4 This is a graph showing the estimation curve of the external disturbance torque by the nonlinear extended state observer of Example 1;

[0142] Figure 5 This is a perturbation estimation error curve of the nonlinear extended state observer in Example 1;

[0143] Figure 6 This is the evolution curve of the unconstrained variables after error transformation in Example 1;

[0144] Figure 7 This is a weight update curve of the adaptive evaluation neural network in Example 1;

[0145] Figure 8 This is a graph showing the change in control input torque of the two joints in Example 1. Detailed Implementation

[0146] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0147] A robust adaptive trajectory tracking control method based on a nonlinear extended observer is disclosed. Specifically, this method involves a robust adaptive trajectory tracking control method based on a nonlinear extended state observer under dual constraints of preset time and preset performance control. The method includes the following steps:

[0148] S1: Establish a dynamic model for a nonlinear system with external disturbances and unmodeled dynamics, and obtain the tracking error dynamics by combining the given reference signal;

[0149] S2: Construct a hybrid performance function that integrates preset time and preset performance attributes;

[0150] S3: The constrained trajectory tracking error is mapped to an equivalent unconstrained transformation variable through an error transformation mechanism;

[0151] S4: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure to perform real-time estimation and feedforward compensation of the system's lumped disturbance.

[0152] S5: Combining the weight update law of the adaptive evaluation neural network with the optimal control strategy under the nominal system;

[0153] S6: Construct a time-varying gain mechanism with preset time characteristics, and combine it with feedforward compensation term, optimal control strategy and time-varying gain mechanism to construct a composite optimal controller.

[0154] Furthermore, step S1 includes the following steps:

[0155] S101: Establish the state space of the dynamic model of a continuous-time nonlinear system with external disturbances:

[0156] (1)

[0157] In formula (1):

[0158] For the system state variables in the dynamic model of a continuous-time nonlinear system;

[0159] System state variables The first derivative;

[0160] A nonlinear unknown function that describes the inherent dynamics of the internal state of a system;

[0161] A control input matrix that reflects the relationship between control inputs and system state;

[0162] For control input of continuous-time nonlinear systems;

[0163] The disturbance input matrix is ​​used to characterize the effect of external disturbances on the system state;

[0164] For an unknown bounded perturbation in a continuous-time nonlinear system;

[0165] S102: Given reference signal trajectory:

[0166] (2)

[0167] In formula (2):

[0168] The system state variables are reference signal dynamic models;

[0169] System state variables The first derivative;

[0170] A nonlinear function describing the reference signal system;

[0171] S103: Establish the dynamic equations of the tracking error system:

[0172] (3)

[0173] In formula (3):

[0174] This represents the original tracking state error.

[0175] Original tracking state error The first derivative.

[0176] Furthermore, the specific form of the hybrid performance function described in S2 is as follows:

[0177] (4)

[0178] In equation (4):

[0179] Original tracking state error The performance boundary function, for any time Meet the preset performance control conditions ,in, This is the system's initial time.

[0180] To constrain the performance boundary function of the original tracking state error magnitude The initial boundary values;

[0181] The performance boundary function that determines the final steady-state error accuracy of the system The steady-state boundary values ​​satisfy the following conditions: ;

[0182] It is a decay function;

[0183] The preset convergence time is independent of the initial conditions;

[0184] Performance boundary function The attenuation rate adjustment parameter;

[0185] For terminal smoothness parameters;

[0186] For irrational numbers An exponential function with base 0.

[0187] Furthermore, step S3 includes the following steps:

[0188] S301: Construction error conversion mechanism:

[0189] (5)

[0190] In equation (5):

[0191] These are the transformed unconstrained system state variables;

[0192] It is the inverse hyperbolic tangent function;

[0193] For irrational numbers The natural logarithm with base 1;

[0194] S302: Obtain the transformed unconstrained system:

[0195] (6)

[0196] In equation (6):

[0197] The first derivative of the state variables of the transformed unconstrained system;

[0198] Let be the nonlinear function of the transformed unconstrained system, where: Performance boundary function The first derivative, It is a hyperbolic cosine function. It is the hyperbolic tangent function;

[0199] This is the control input matrix for the transformed unconstrained system;

[0200] This is the perturbation input matrix of the transformed unconstrained system.

[0201] Furthermore, step S4 includes the following steps:

[0202] S401: Define nonlinear functions :

[0203] (7)

[0204] In equation (7):

[0205] The parameter is a non-linear power parameter.

[0206] For linear interval thresholds;

[0207] It is an absolute value function;

[0208] Standard symbolic functions;

[0209] S402: Define a lumped disturbance in a transformed system that includes unknown nonlinearities and unknown disturbances:

[0210] (8)

[0211] S403: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure:

[0212] (9)

[0213] In equation (9):

[0214] The transformed unconstrained system state variables The estimated value, For estimated value The first derivative;

[0215] For aggregated disturbance The estimated value, For estimated value The first derivative;

[0216] The nonlinear power parameters to be designed satisfy... and To ensure fixed-time convergence characteristics;

[0217] All are nonlinear observer gains;

[0218] S404: Constructing a feedforward compensation term based on a nonlinear extended state observer:

[0219] (10)

[0220] In formula (10):

[0221] The control input matrix of the transformed unconstrained system The generalized inverse.

[0222] Furthermore, step S5 includes the following steps:

[0223] S501: Constructing the nominal system of the transformed unconstrained system:

[0224] (11)

[0225] In equation (11):

[0226] These are the state variables of the nominal system. State variables The first derivative;

[0227] The control input matrix of the nominal system;

[0228] For the control input of the nominal system;

[0229] S502: Constructing the value function and optimal value function in the finite time domain:

[0230] (12)

[0231] (13)

[0232] In equations (12)-(13):

[0233] State variables of the nominal system transpose;

[0234] For the control input of the nominal system transpose;

[0235] for The value function at time t, for The optimal value function at time t;

[0236] For terminal constraint functions;

[0237] The weight matrix represents the positive definite state.

[0238] The weight matrix is ​​the positive definite control input.

[0239] It is a minimum value function;

[0240] S503: Utilizing an evaluation neural network to evaluate the optimal value function Approximation:

[0241] (14)

[0242] In equation (14):

[0243] This is the weight vector used to evaluate the neural network between the hidden and output layers;

[0244] This represents the number of neurons in the hidden layer.

[0245] Weight vector transpose;

[0246] It is a time-varying activation function;

[0247] Remaining time;

[0248] This represents the approximation error of the neural network.

[0249] S504: Designing an approximate optimal control law for a nominal system :

[0250] (15)

[0251] In equation (15):

[0252] For positive definite control input weight matrix The inverse matrix;

[0253] The control input matrix of the nominal system transpose;

[0254] For partial derivatives transpose, For the time-varying activation function with respect to the state variables of the nominal system The partial derivatives;

[0255] To evaluate the weights of a neural network The estimated value;

[0256] S505: Design of a weight update law for evaluating neural networks based on gradient descent:

[0257] (16)

[0258] In equation (16):

[0259] For estimated value The first derivative;

[0260] The learning rate of the neural network;

[0261] For HJB residual error; HJB residual error transpose;

[0262] This represents the activation function value at the terminal moment; The activation function value at the terminal moment transpose;

[0263] For terminal constraint error; Terminal constraint error transpose; For estimated value transpose;

[0264] These are the weighting coefficients for the HJB residual term, terminal constraint term, and regularization term, respectively.

[0265] For time-varying activation functions with respect to time The partial derivatives;

[0266] For partial derivatives The transpose of .

[0267] Furthermore, step S6 includes the following steps:

[0268] S601: Construct a time-varying gain control term with preset time convergence characteristics. :

[0269] (17)

[0270] In equation (17):

[0271] The time-varying gain parameters to be designed;

[0272] S602: Feedforward compensation term combined with S404 Approximate optimal control law of S504 and the time-varying gain control term of S601 The final composite optimal controller is obtained:

[0273] (18)

[0274] Example 1:

[0275] This embodiment takes a dual-link robotic arm system as an example, and uses the method described in this invention to perform dual-link...

[0276] The robotic arm was simulated, and the model is as follows:

[0277]

[0278] in,

[0279]

[0280] In equation (19):

[0281] The angular positions of joint one and joint two;

[0282] The angular positions of joint one and joint two are respectively. The first derivative;

[0283] The angular positions of joint one and joint two are respectively. The second derivative;

[0284] Let the masses of connecting rod one and connecting rod two be the masses.

[0285] Let the lengths of link one and link two be given.

[0286] This represents the distance from the axis of each joint to the center of mass of the corresponding link;

[0287] Let be the moment of inertia of link one and link two about their center of mass;

[0288] This is the acceleration due to gravity.

[0289] The control input torque is applied to joint one and joint two;

[0290] For time-varying external perturbations of joint one and joint two;

[0291] These are the parameters for calculating centrifugal force;

[0292] Desired joint angle trajectory:

[0293]

[0294] Its initial state is , .

[0295] Apply time-varying external perturbations to the joints :

[0296]

[0297] The parameters of the dual-link robotic arm system are designed as follows:

[0298] , , , , , , , , .

[0299] The tracking error vector is defined as follows:

[0300]

[0301] The corresponding speed error is defined as ;

[0302] The initial joint positions and velocities of the system are set as follows: , .

[0303] Hybrid performance function The parameters are designed as follows:

[0304] , , , , , .

[0305] The parameters for the nonlinear extended state observer are set as follows:

[0306] , , , , , , .

[0307] The parameter settings for the composite optimal controller and neural network are as follows:

[0308] The time-varying gain parameter is Weight matrix , The number of neurons in the hidden layer is The initial weights are , , , The activation function is Neural network learning rate .

[0309] Control effect of the dual-link robotic arm:

[0310] Figure 2 and Figure 3 The angular position tracking curves of the two joints of the dual-link robotic arm and the corresponding tracking error versus preset performance boundary curves are presented respectively. Figure 2 The actual joint trajectory can be displayed to track the desired reference trajectory with high precision. Figure 3 The tracking errors of the two joints were strictly kept within a preset performance envelope throughout the entire process and converged to a minimum steady-state boundary within a preset time. Experimental results verified the effectiveness of this invention in achieving the dual control objectives of preset time convergence and transient performance constraints throughout the process.

[0311] Figure 4 and Figure 5The estimation curves of the lumped disturbance by the nonlinear extended state observer and the corresponding disturbance estimation error curves are presented respectively. It can be seen that even in the presence of fast time-varying harmonic disturbances, the proposed observer can still achieve accurate estimation of the disturbance, and the estimation error can converge to near zero within a fixed time independent of the initial state, verifying the superiority and strong robustness of the proposed method in terms of disturbance resistance.

[0312] Figure 6 , Figure 7 and Figure 8 The evolution curves of the unconstrained variables after error transformation, the weight update curve of the adaptive evaluation neural network, and the control input torque variation curve of the composite controller are presented respectively. As can be seen from the figures, under the design of the composite controller, the transformed variables are stable within a preset time and the weights of the evaluation neural network converge rapidly.

[0313] Comprehensive experimental results show that, under the premise of strictly satisfying the dual constraints of preset time convergence and preset performance envelope, the present invention can effectively suppress the influence of strong external disturbances and achieve high-precision and robust trajectory tracking control.

[0314] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0315] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A robust adaptive trajectory tracking control method based on a nonlinear extended observer, characterized in that: The method includes the following steps: S1: Establish a dynamic model of the nonlinear system and obtain the tracking error dynamics by combining it with the reference signal; S2: Construct a hybrid performance function that integrates preset time and preset performance attributes; S3: The constrained trajectory tracking error is mapped to an equivalent unconstrained transformation variable through an error transformation mechanism; S4: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure to perform real-time estimation and feedforward compensation of the system's lumped disturbance. S5: Combining the weight update law of the adaptive evaluation neural network with the optimal control strategy under the nominal system; S6: Construct a time-varying gain mechanism with preset time characteristics to build a composite optimal controller.

2. The robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 1, characterized in that: S1 includes the following steps: S101: Establish the state space of the dynamic model of a continuous-time nonlinear system with external disturbances: (1) In formula (1): For the system state variables in the dynamic model of a continuous-time nonlinear system; System state variables The first derivative; A nonlinear unknown function that describes the inherent dynamics of the internal state of a system; A control input matrix that reflects the relationship between control inputs and system state; For control input of continuous-time nonlinear systems; The disturbance input matrix is ​​used to characterize the effect of external disturbances on the system state; For an unknown bounded perturbation in a continuous-time nonlinear system; S102: Given reference signal trajectory: (2) In formula (2): The system state variables are reference signal dynamic models; System state variables The first derivative; A nonlinear function describing the reference signal system; S103: Establish the dynamic equations of the tracking error system: (3) In formula (3): This represents the original tracking state error. Original tracking state error The first derivative.

3. The robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 2, characterized in that: The specific form of the hybrid performance function described in S2 is as follows: (4) In equation (4): Original tracking state error The performance boundary function, for any time Meet the preset performance control conditions ,in, This is the system's initial time. To constrain the performance boundary function of the original tracking state error magnitude The initial boundary values; The performance boundary function that determines the final steady-state error accuracy of the system The steady-state boundary values ​​satisfy the following conditions: ; It is a decay function; The preset convergence time is independent of the initial conditions; Performance boundary function The attenuation rate adjustment parameter; For terminal smoothness parameters; For irrational numbers An exponential function with base 0.

4. The robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 3, characterized in that: S3 includes the following steps: S301: Construction error conversion mechanism: (5) In equation (5): These are the transformed unconstrained system state variables; It is the inverse hyperbolic tangent function; For irrational numbers The natural logarithm with base 1; S302: Obtain the transformed unconstrained system: (6) In equation (6): The first derivative of the state variables of the transformed unconstrained system; Let be the nonlinear function of the transformed unconstrained system, where: Performance boundary function The first derivative, It is a hyperbolic cosine function. It is the hyperbolic tangent function; This is the control input matrix for the transformed unconstrained system; This is the perturbation input matrix of the transformed unconstrained system.

5. The robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 4, characterized in that: S4 includes the following steps: S401: Define nonlinear functions : (7) In equation (7): The parameter is a non-linear power parameter. For linear interval thresholds; It is an absolute value function; Standard symbolic functions; S402: Define a lumped disturbance in a transformed system that includes unknown nonlinearities and unknown disturbances: (8) S403: Design a fixed-time convergent nonlinear extended state observer with a multi-power composite feedback structure: (9) In equation (9): For the transformed unconstrained system state variables The estimated value, For estimated value The first derivative; For aggregated disturbance The estimated value, For estimated value The first derivative; The nonlinear power parameters to be designed satisfy... and ; All are nonlinear observer gains; S404: Constructing a feedforward compensation term based on a nonlinear extended state observer: (10) In formula (10): The control input matrix of the transformed unconstrained system The generalized inverse.

6. The robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 5, characterized in that: S5 includes the following steps: S501: Constructing the nominal system of the transformed unconstrained system: (11) In equation (11): These are the state variables of the nominal system. State variables The first derivative; The control input matrix of the nominal system; For the control input of the nominal system; S502: Constructing the value function and optimal value function in the finite time domain: (12) (13) In equations (12)-(13): State variables of the nominal system transpose; For the control input of the nominal system transpose; for The value function at time t, for The optimal value function at time t; For terminal constraint functions; The weight matrix represents the positive definite state. The weight matrix is ​​the positive definite control input. It is a minimum value function; S503: Utilizing an evaluation neural network to evaluate the optimal value function Approximation: (14) In equation (14): This is the weight vector used to evaluate the neural network between the hidden and output layers; This represents the number of neurons in the hidden layer. Weight vector transpose; It is a time-varying activation function; Remaining time; This represents the approximation error of the neural network. S504: Designing an approximate optimal control law for a nominal system : (15) In equation (15): For positive definite control input weight matrix The inverse matrix; The control input matrix of the nominal system transpose; partial derivatives transpose, For the time-varying activation function with respect to the state variables of the nominal system The partial derivatives; To evaluate the weights of a neural network The estimated value; S505: Design of a weight update law for evaluating neural networks based on gradient descent: (16) In equation (16): For estimated value The first derivative; The learning rate of the neural network; This refers to the HJB residual error. HJB residual error transpose; This is the activation function value at the terminal moment; The activation function value at the terminal moment transpose; For terminal constraint error; Terminal constraint error transpose; For estimated value transpose; These are the weighting coefficients for the HJB residual term, terminal constraint term, and regularization term, respectively. For time-varying activation functions with respect to time The partial derivatives; partial derivatives The transpose of .

7. A robust adaptive trajectory tracking control method based on a nonlinear extended observer according to claim 6, characterized in that: S6 includes the following steps: S601: Construct a time-varying gain control term with preset time convergence characteristics. : (17) In equation (17): The time-varying gain parameters to be designed; S602: Feedforward compensation term combined with S404 Approximate optimal control law of S504 and the time-varying gain control term of S601 The final composite optimal controller is obtained: (18)。

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