Disturbance observation-based two-degree-of-freedom helicopter optimal tracking controller design method

Abstract

The invention discloses a two-degree-of-freedom helicopter optimal tracking controller design method based on disturbance observation, and relates to the technical field of two-degree-of-freedom helicopter control, and the method comprises the steps: considering the internal uncertainty and external disturbance of a system, and constructing a two-degree-of-freedom helicopter system equation; a pre-defined time nonlinear disturbance observer is introduced to estimate disturbance, coordinate transformation is constructed based on a dynamic surface control technology, a pre-defined time filter is introduced, a two-degree-of-freedom helicopter system model is decomposed into two stages of subsystems, and an optimal performance function is constructed for each subsystem. And designing an optimal virtual controller and a predefined time trajectory tracking controller. The controller designed by the invention can ensure that all signals of the whole two-degree-of-freedom helicopter are bounded within the preset time, can effectively improve the transient performance and the steady-state performance of the two-degree-of-freedom helicopter, and can ensure that the two-degree-of-freedom helicopter can still quickly and accurately track a reference trajectory within the preset time after being disturbed.

Description

Technical Field

[0001] The present invention relates to the technical field of two - degree - of - freedom helicopter control, and more specifically, to a design method for an optimal tracking controller of a two - degree - of - freedom helicopter based on disturbance observation. Background Art

[0002] As an important branch in the field of unmanned aerial vehicles, unmanned helicopters are widely used in fields such as transportation, marine supervision, and geological research due to their unique advantages such as vertical take - off and landing, low requirements for take - off environment, and hovering flight. It should be clear that the precise flight of a two - degree - of - freedom helicopter along a reference trajectory is the basis for completing the above - mentioned complex tasks. However, the two - degree - of - freedom helicopter is inevitably affected by its own strong coupling, uncertainty, and external unknown disturbances, making the design of its tracking controller more complex. If the expected tracking accuracy is not achieved, serious economic and safety accidents will occur. The invention patent application with the publication number CN115357005A designs an active fault - tolerant control method for the sensor faults of a two - degree - of - freedom helicopter, but does not consider the uncertainty of the two - degree - of - freedom helicopter. The invention patent application with the publication number CN 114578696A designs an adaptive neural network quantization fault - tolerant control method for a two - degree - of - freedom helicopter, but does not consider the transient performance of the system tracking error. In addition, both the invention patent applications with the publication numbers CN 115357005A and CN 114578696A are based on traditional asymptotic convergence, and the convergence time is affected by the initial state. The invention patent application with the publication number CN 118732509A designs a predefined - time tracking control method for a two - degree - of - freedom helicopter, but does not consider the influence of external disturbances and does not achieve optimal control design.

[0003] Therefore, there is an urgent need to design an optimal tracking controller to improve the transient and steady - state performance of a two - degree - of - freedom helicopter, and still accurately track the reference trajectory within a predetermined time in the presence of unknown disturbances. Summary of the Invention

[0004] The present invention aims to provide a design method for an optimal tracking controller of a two - degree - of - freedom helicopter based on disturbance observation, which can solve the above problems.

[0005] To solve the above problems, the technical solution adopted by the present invention is as follows:

[0006] A design method for an optimal tracking controller of a two - degree - of - freedom helicopter based on disturbance observation includes the following steps:

[0007] Step 1: Considering the internal uncertainty and external disturbance of the system, construct the system equation of the two - degree - of - freedom helicopter;

[0008] Step 2: Based on the two-degree-of-freedom helicopter system equation, introduce an auxiliary system, design a predefined-time nonlinear disturbance observer, and construct a Lyapunov function V for the estimation error of the auxiliary error o , and take the time derivative of the Lyapunov function V o to obtain such that the auxiliary error converges within a predefined time, and through inequality simplification, obtain the gain of the error function to be designed;

[0009] Step 3: Based on the two-degree-of-freedom helicopter system equation, combine the dynamic surface control method, introduce a predefined-time filter, construct a coordinate transformation equation including the output signal of the predefined-time filter, and decompose the two-degree-of-freedom helicopter system model into two-level subsystems according to the coordinate transformation equation;

[0010] Step 4: Construct a first-level optimal performance function for the first-level subsystem, build a first-level Hamilton-Jacobi-Bellman equation according to the first-level optimal performance function, introduce an unknown but continuous first-level intermittent cost function, approximately estimate the first-level intermittent cost using a radial basis neural network, approximately solve the first-level Hamilton-Jacobi-Bellman equation through an actor-critic neural network, and obtain the actor-critic adaptive law and as well as the optimal virtual controller α, design the adaptive law Construct a Lyapunov function V1 for the first-level subsystem, and substitute α and into the time derivative of the Lyapunov function V1 to make the first-level subsystem tend to predefined-time stability, and through inequality simplification, obtain the value of the virtual controller gain to be designed for the first-level subsystem;

[0011] Step 5: For the second-level subsystem, construct a Lyapunov function V2, introduce a switching event-triggering mechanism, construct a second-level optimal performance function, build a second-level Hamilton-Jacobi-Bellman equation according to the second-level optimal performance function, introduce an unknown but continuous second-level intermittent cost function, approximately estimate the second-level intermittent cost using a radial basis neural network, approximately solve the second-level Hamilton-Jacobi-Bellman equation through an actor-critic neural network, and obtain the actor-critic-identification adaptive law and as well as the predefined-time optimal tracking controller and substitute them into the time derivative of the Lyapunov function V2 to make the second-level subsystem tend to predefined-time stability, and through inequality simplification, obtain the tracking controller gain to be designed for the second-level subsystem.

[0012] As a further description of the above technical solution, the two-degree-of-freedom helicopter system equation is:

[0013]

[0014] wherein: and respectively represent the pitch angle and yaw angle of the two-degree-of-freedom helicopter;

[0015] represents the pitch angle angular velocity of, represents the yaw angle angular velocity of;

[0016] represents the voltage of the pitch propeller motor, represents the voltage of the yaw propeller motor;

[0017] and are the gain matrices of the two-degree-of-freedom helicopter system model;

[0018] is a vector of unknown smooth nonlinear functions, d = [d1, d2] T is an unknown external disturbance;

[0019] [] T represents the transpose of [].

[0020] As a further description of the above technical solution, the auxiliary system is:

[0021]

[0022] wherein: represents the auxiliary system state vector, is the derivative with respect to time, represents the auxiliary error vector, and ρ is a positive constant to be designed.

[0023] As a further description of the above technical solution, the predefined time nonlinear disturbance observer is:

[0024]

[0025] wherein: is the estimated value of the disturbance of, is the estimated value of the auxiliary error vector z1(t) and is given by the following formula:

[0026]

[0027] wherein: is the time derivative of ,

[0028] represents the estimation error,

[0029] tanh(·) represents the hyperbolic tangent function, sig 1+γ (·) = [|·1| 1+γ sign(1), |·2| 1+γ sign(·2)] T , sign() represents the sign function; ω1, γ, c, and ω2 are control parameters to be designed, and γ ∈ (0, 1), T o is the designed predetermined time.

[0030] As a further description of the above technical solution, the predefined time filter is expressed as:

[0031]

[0032] where: T c is the designed predetermined time, is the time derivative of the output signal α f , represents the predefined time filter error, α f (0) = α(0) represents its initial value, α is the input signal of the predefined time filter, is the time derivative of α, and satisfies σ is an unknown positive constant; ξ, b > 0 are control parameters to be designed; Y is a positive time-varying parameter, is the estimated value of the parameter σ, |||| represents the 2-norm of the vector ·.

[0033] As a further description of the above technical solution, the execute-evaluate adaptive law and the optimal virtual controller α, and the adaptive law are respectively:

[0034]

[0035] where: and are respectively the weight estimation values of the execute neural network and the evaluate neural network; Φ s1 (X) is the basis function vector; and are parameters to be designed; e1(t) is the tracking error vector of the first-level subsystem; z1, z2 are control gain coefficients; To track the trajectory is the time derivative; ξ, b1, and γ are positive parameters to be designed; is the predefined time filter error; h, is a coefficient related to the design parameters.

[0036] As a further description of the above technical solution, the switching event trigger mechanism is as follows:

[0037]

[0038] where: is the actual execution signal of the i-th control input; is the ideal control signal updated at the trigger time ; are the k-th and (k + 1)-th trigger times of the i-th control input respectively, N + is a positive integer; υ i is the weight coefficient, 0 < υ i < 1; χ i (t) = μ i (t) - U i (t), is the trigger error; is the dynamic threshold; ζ1, ζ2, ζ3, H i , Λ i and l i are control parameters to be designed, represents the change rate of the control signal μ i (t) interval.

[0039] As a further description of the above technical solution, the predefined time-optimal tracking controller executes the execute-evaluate-identify adaptive law and are respectively:

[0040]

[0041] where: is the inverse matrix of the system gain matrix for decoupling the control input; z3, z4 are control gain coefficients; e2(t) is the tracking error of the second-level subsystem; b3 is the anti-saturation adjustment parameter; is the output of the execution neural network, is the weight estimate, is the input basis function vector; is the output of the identification neural network, is the weight estimate, is the input basis function vector; To evaluate the weight estimation value of the neural network; and are parameters to be designed.

[0042] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0043] 1) Different from the existing design method of asymptotically stable controllers, the predefined-time tracking controller designed based on the design method of the present invention can ensure that all signals of the entire two-degree-of-freedom helicopter system model are bounded in predefined time; the tracking error converges within a predefined time; effectively improves the transient performance and steady-state performance of the two-degree-of-freedom helicopter system.

[0044] 2) By designing a predefined-time nonlinear disturbance state observer, the adverse effects of unknown disturbances on the tracking performance of the two-degree-of-freedom helicopter are solved; through the predefined-time stability strategy; it is ensured that the observer accurately and quickly observes the unknown disturbance within a predetermined time.

[0045] 3) Introduce a reinforcement learning strategy to optimize the cost function, and establish an optimized predefined-time dynamic surface control framework with an execution-evaluation structure, which ensures that the two-degree-of-freedom helicopter system model can achieve approximate optimal control.

[0046] 4) Incorporate the switching event triggering mechanism into the design of the predefined-time tracking controller, so that the control signal switches under different triggering mechanisms, effectively avoiding the continuous update of the system control signal, reducing the update frequency of the control signal and the wear of the actuator, and ensuring that the two-degree-of-freedom helicopter can still accurately track the reference trajectory under the condition of intermittent update of the control signal.

[0047] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following specifically gives embodiments of the present invention and, in conjunction with the accompanying drawings, makes a detailed description as follows. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the embodiments. It should be understood that the following drawings only show some embodiments of the present invention, and therefore should not be regarded as limiting the scope. For those of ordinary skill in the art, without creative efforts, other related drawings can also be obtained based on these drawings.

[0049] Figure 1 is a schematic diagram of the model of the two-degree-of-freedom helicopter;

[0050] Figure 2 is the structure diagram of the two-degree-of-freedom helicopter tracking control method;

[0051] Figure 3It is the trajectory diagram of the angle tracking the expected angle of the pitch angle of the two-degree-of-freedom helicopter;

[0052] Figure 4 It is the trajectory diagram of the angle tracking the expected angle of the yaw angle of the two-degree-of-freedom helicopter;

[0053] Figure 5 It is the trajectory tracking response diagram of the angle error of the two-degree-of-freedom helicopter;

[0054] Figure 6 It is the trajectory diagram of the disturbance and disturbance estimation of the two-degree-of-freedom helicopter;

[0055] Figure 7 It is the control input curve diagram of the two-degree-of-freedom helicopter system;

[0056] Figure 8 It is the trajectory diagram of the weights of the execution-evaluation neural network of the two-degree-of-freedom helicopter system;

[0057] Figure 9 It is the trigger interval diagram of the control input of the two-degree-of-freedom helicopter system. Specific implementation manners

[0058] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention.

[0059] Please refer to Figure 1 and Figure 2 , the embodiments of the present invention provide a design method for a predefined time-optimal tracking controller of a two-degree-of-freedom helicopter based on a disturbance observer, including the following steps:

[0060] Step 1. According to Figure 1 the simplified diagram of the two-degree-of-freedom helicopter system model shown, construct the following dynamic model:

[0061]

[0062] In the formula: and respectively represent the pitch angle and yaw of the two-degree-of-freedom helicopter; and respectively represent the angular acceleration and angular velocity of the pitch angle ; and respectively represent the angular acceleration and angular velocity of the yaw angle ; represents the voltage of the pitch propeller motor; represents the voltage of the yaw propeller motor; and respectively represent the thrust gains of the pitch propeller acting on the pitch axis and the yaw axis; and respectively represent the thrust gains of the yaw propeller acting on the pitch axis and the yaw axis; is the total mass of the two-degree-of-freedom helicopter; is the acceleration due to gravity; sin(·) and cos() represent the sine function and the cosine function respectively; is the distance from the center of mass of the system to the fixed bracket point of the helicopter; and respectively represent the friction damping coefficients existing when the pitch angle and the yaw angle rotate; and respectively represent the moment of inertia of the pitch axis and the moment of inertia of the yaw axis.

[0063] The detailed parameters of the two-degree-of-freedom helicopter system model are shown in Table 1.

[0064] Table 1

[0065]

[0066] Define [·] T represents the transpose of [], and further, the two-degree-of-freedom helicopter system model is expressed as:

[0067]

[0068] In the formula: is the vector of unknown smooth non-linear functions, d = [d1, d2] T is the unknown external disturbance, and are the gain matrices of the two-degree-of-freedom helicopter system model, which are respectively:

[0069]

[0070] Step 2: Based on the two-degree-of-freedom helicopter system equation obtained in Step 1, introduce the following auxiliary system:

[0071]

[0072] In the formula: represents the auxiliary error vector, represents the auxiliary system state vector, is the derivative with respect to time, and ρ is a positive constant to be designed.

[0073] Furthermore, construct the following predefined-time non-linear disturbance observer:

[0074]

[0075] Wherein: is the disturbance estimate value, is the time derivative of the vector z1(t), is the estimated value of the vector z1(t) and is given by:

[0076]

[0077] Wherein: is time derivative,

[0078] represents the estimation error,

[0079] tanh(·) represents the hyperbolic tangent function, sig 1+γ (·) = [|·1| 1+γ sign(·1), |·2| 1+γ sign(·2)] T , sign(·) represents the sign function, ω1, γ, c, and ω2 are control parameters to be designed, and γ ∈ (0, 1), T o is the designed predefined time.

[0080] Furthermore, a Lyapunov function V is constructed for the estimation error of the auxiliary error vector o

[0081]

[0082] Taking the time derivative of the Lyapunov function V o yields:

[0083]

[0084] Through inequality simplification, we get:

[0085]

[0086] According to the predefined time stability strategy, when the time t ≥ T o , the error Furthermore, we get:

[0087]

[0088] Thus, when t ≥ T o , the disturbance estimation error can be adjusted to 0.

[0089] Step 3: Based on the two-degree-of-freedom helicopter system equation obtained in Step 1, combined with the dynamic surface control technology, construct the following coordinate transformation:

[0090]

[0091] where: e1(t) = [e 11 (t), e 12 (t)] T and e2(t) = [e 21 (t), e 22 (t)] T represent the tracking errors, is the desired tracking trajectory, α f = [α f1 , α f2 T is the output signal of the predefined time filter, and the predefined time filter is expressed as follows:

[0092]

[0093] where: T c is the designed predefined time, is the time derivative of α f , represents the predefined time filter error, α f (0) = α(0) represents its initial value, α = [α1, α2] T is the input signal of the predefined time filter, is the time derivative of α, and satisfies σ is an unknown positive constant, ξ, b > 0 are control parameters to be designed, Y is a positive time-varying parameter, is the estimated value of the parameter σ, ||·|| represents the 2-norm of the vector ·.

[0094] Decompose the two-degree-of-freedom helicopter system model into two-level subsystems, and introduce the error variables e1(t) and e2(t) as the tracking error variables of the first-level subsystem and the second-level subsystem respectively.

[0095] Step 4: Based on the coordinate transformation obtained in Step 3, construct the first-level optimal performance function for the first-level subsystem of the two-degree-of-freedom helicopter system model

[0096]

[0097] where: Ω x represents the set of all allowable control inputs, α * ​(e1) is the ideal optimal virtual controller.

[0098] Furthermore, according to the first-level optimal performance function the following first-level HJB (Hamilton-Jacobi-Bellman) equation is constructed:

[0099]

[0100] In the formula: denotes the partial derivative of e1(t), and is the derivative of e1(t) with respect to time. To achieve the optimal control objective, the following first-level intermittent cost function is introduced

[0101]

[0102] In the formula: is the desired tracking trajectory the derivative with respect to time, and b1>0 is the parameter to be designed.

[0103] Since the first-level intermittent cost function is unknown but continuous, we choose a radial basis neural network to approximately estimate the first-level intermittent cost, that is:

[0104]

[0105] In the formula: is the ideal weight, diag{} is a diagonal matrix, and Φ s1 (X)=[Φ s11 (X),Φ s12 (X)] T represents the basis function vector with input ε s1 (X) is the approximation error vector and satisfies

[0106] Since the ideal weight is unknown, in order to obtain the optimal virtual controller, an actor-critic neural network is introduced for online learning. Furthermore, the first-level HJB equation is approximately solved by the actor-critic neural network to obtain the actor-critic adaptive law and as well as the optimal virtual controller α:

[0107]

[0108] In the formula: and are the estimated values of and are the parameters to be designed.

[0109] Construct the Lyapunov function V1 for the first - stage subsystem of the two - degree - of - freedom helicopter system model:

[0110]

[0111] Where: and tr{·} represents the trace of the matrix.

[0112] Furthermore, take the derivative of the Lyapunov function V1 with respect to time, and we can get:

[0113]

[0114] Where: is the time derivative of α f .

[0115] To ensure that the first - stage subsystem of the two - degree - of - freedom helicopter system model tends to predefined - time stability, design the following adaptive law

[0116]

[0117] Where:

[0118] Bring the designed optimal virtual controller α, the adaptive law the execute - evaluate adaptive law and into formula (20), and through inequality simplification, we can get:

[0119]

[0120] Where:

[0121] During the process of inequality simplification, the value of the virtual controller gain to be designed for the first - stage subsystem is obtained.

[0122] Step 5: The second - stage subsystem of the two - degree - of - freedom helicopter system model is

[0123]

[0124] Where: is the derivative of α f with respect to time. For the non - linear term introduce the identification network estimation

[0125]

[0126] In the formula: is the ideal weight, represents the basis function vector with the input being , is the approximation error vector and satisfies

[0127] Introduce the following switching event-triggering mechanism:

[0128]

[0129] In the formula: N + is a positive integer, 0 < υ i < 1, ζ1, ζ2, ζ3, H i , Λ i and l i are control parameters to be designed, υ i |μ i (t)| + Λ i represents the control signal gap starting from two consecutive triggering moments, represents the change rate of the interval of the control signal μ i (t).

[0130] If μ i (t) is designed as:

[0131]

[0132] In the formula: is the optimal tracking controller to be designed, b2 > 0,

[0133] Select the time-varying function such that:

[0134]

[0135] If then μ i (t) is designed as:

[0136]

[0137] In the formula: ζ4 ≥ ζ1 + ζ3, select the time-varying function such that:

[0138]

[0139] Construct the second-stage optimal performance function for the second-stage subsystem of the two-degree-of-freedom helicopter system model

[0140]

[0141] In the formula: is the ideal optimal tracking controller.

[0142] Furthermore, according to the second-level optimal performance function construct the following second-level HJB equation:

[0143]

[0144] In the formula: denotes the partial derivative of e2(t), and is the derivative of e2(t) with respect to time. To achieve the optimal control objective, introduce the following second-level intermittent cost function

[0145]

[0146] In the formula: b3 > 0 is the parameter to be designed.

[0147] Since the second-level intermittent cost function is unknown but continuous, we choose a radial basis neural network to approximately estimate the second-level intermittent cost, that is:

[0148]

[0149] In the formula: is the ideal weight,

[0150] denotes that the input is the basis function vector,

[0151] is the approximation error vector and satisfies

[0152] Since the ideal weight is unknown, to obtain the optimal tracking controller, introduce an actor-critic neural network for online learning. Furthermore, approximately solve the second-level HJB equation through the actor-critic neural network to obtain the actor-critic-identification adaptive law and as well as the predefined time-optimal tracking controller

[0153]

[0154]

[0155] In the formula: and are Estimated value and are parameters to be designed

[0156] Construct the Lyapunov function V2 for the second - level subsystem of the two - degree - of - freedom helicopter system model

[0157]

[0158] Furthermore, taking the derivative of the Lyapunov function V2 with respect to time, we can obtain

[0159]

[0160] Bring the designed predefined time - optimal tracking controller Execute - evaluate - identify adaptive law and Substitute them into formula (40), and through inequality simplification, we can obtain

[0161]

[0162] Wherein

[0163] For the second - level subsystem of the two - degree - of - freedom helicopter system model, adopt the designed predefined time - optimal tracking controller Execute - evaluate - identify adaptive law and Make the second - level subsystem of the two - degree - of - freedom helicopter system model tend to be predefined - time stable, and obtain the tracking controller gain to be designed for the second - level subsystem during the inequality simplification process

[0164] Step 6: Verify the result obtained in step 5. Construct the overall Lyapunov function V = V1 + V2 for the two - degree - of - freedom helicopter system model, and take the derivative of it with respect to time to obtain

[0165]

[0166] Wherein

[0167] According to formula (42), adopt the designed optimal virtual controller α, adaptive law Execute - evaluate adaptive law and Predefined time - optimal tracking controller Execute - evaluate - identify adaptive law and Ensure that all signals of the two - degree - of - freedom helicopter closed - loop system are predefined - time bounded, and the upper bound of its stable time Its steady-state error can be converged to a sufficiently small region, i.e.:

[0168]

[0169] In addition, for the switching event-triggering mechanism (26), there is:

[0170]

[0171] In the formula: is a constant, obtained from the initial condition to get

[0172] Then further obtain:

[0173]

[0174] Therefore, the time interval between two triggers Since and So Effectively avoids the Zeno effect.

[0175] To illustrate the control effect of the method of the present invention in detail, a simulation experiment will be carried out in MATLAB next. The reference trajectory is set to The unknown disturbance is set to

[0176] The system initial conditions are set to

[0177] The selected controller gains are shown in Table II.

[0178] Table II

[0179]

[0180] The following results are obtained through the MATLAB simulation experiment, Figure 3 is the trajectory diagram of the angle of the pitch angle of the two-degree-of-freedom helicopter tracking the desired angle; Figure 4 is the trajectory diagram of the angle of the yaw angle of the two-degree-of-freedom helicopter tracking the desired angle; Figure 5 is the trajectory tracking response diagram of the angle error of the two-degree-of-freedom helicopter; Figure 6 is the trajectory diagram of the disturbance and disturbance estimation of the two-degree-of-freedom helicopter; Figure 7 is the control input curve diagram of the two-degree-of-freedom helicopter system. It can be seen from the simulation diagram that the invented tracking controller effectively avoids the continuous update of the control signal, alleviates the burden of network communication, and reduces the wear of the actuator; Figure 8 is the trajectory diagram of the weights of the execution-evaluation neural network of the two-degree-of-freedom helicopter system;Figure 9 is the trigger interval of the control input of the two-degree-of-freedom helicopter system, and the Zeno effect is effectively avoided. From the simulation experiment results, it is obtained that the predefined-time tracking controller design method proposed by the present invention ensures the predefined-time stability of the two-degree-of-freedom helicopter closed-loop system; the tracking error converges within a given time, effectively improving the transient performance and steady-state performance of the two-degree-of-freedom helicopter system, and in the case of being subjected to unknown disturbances, the two-degree-of-freedom helicopter can still accurately track the reference trajectory.

[0181] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for designing an optimal tracking controller for a two-degree-of-freedom helicopter based on disturbance observation, characterized in that: The following steps are involved: Step 1: Considering the internal uncertainty and external disturbance of the system, construct the two-degree-of-freedom helicopter system equation; Step 2: Based on the two-degree-of-freedom helicopter system equation, introduce the auxiliary system, design the predefined time nonlinear disturbance observer, and construct the Lyapunov function V for the estimated error of the auxiliary error. o , for the Lyapunov function V o The time derivative of The auxiliary error is made to converge within a predetermined time, and the error function gain to be designed is obtained by simplifying the inequality; Step 3: Based on the two-degree-of-freedom helicopter system equation and in combination with the dynamic surface control method, a predefined time filter is introduced to construct a coordinate transformation equation including the output signal of the predefined time filter, and the two-degree-of-freedom helicopter system model is decomposed into two-level subsystems according to the coordinate transformation equation; Step 4: Construct the first-level optimal performance function for the first-level subsystem, construct the first-level Hamilton-Jacobi-Bellman equation based on the first-level optimal performance function, introduce the unknown but continuous first-level intermittent cost function, use the radial basis neural network to approximate the first-level intermittent cost, and use the execution-evaluation neural network to approximately solve the first-level Hamilton-Jacobi-Bellman equation to obtain the execution-evaluation adaptive law and and the optimal virtual controller α, design the adaptive law Construct Lyapunov function V1 for the first-level subsystem. α and Substitute the time derivative of the Lyapunov function V1 Make the first-level subsystem stable within a predefined time, and obtain the value of the virtual controller gain to be designed for the first-level subsystem by simplifying the inequality; Step 5: For the second-level subsystem, construct the Lyapunov function V2, introduce the switching event trigger mechanism, construct the second-level optimal performance function, construct the second-level Hamilton-Jacobi-Bellman equation based on the second-level optimal performance function, introduce the unknown but continuous second-level intermittent cost function, use the radial basis neural network to approximate the second-level intermittent cost, and use the execution-evaluation neural network to approximately solve the second-level Hamilton-Jacobi-Bellman equation to obtain the execution-evaluation-identification adaptive law. and and the optimal tracking controller for a predefined time And substitute the time derivative of the Lyapunov function V2 The second-level subsystem is made to be stable within a predefined time, and the tracking controller gain to be designed for the second-level subsystem is obtained by simplifying the inequality.

2. The method according to claim 1, characterized in that: The system equation of a two-degree-of-freedom helicopter is: Where: and Respectively represent the pitch angle and yaw angle of a two-degree-of-freedom helicopter; Represents the pitch angle The angular velocity of Represents the yaw angle Angular velocity; represents the voltage of the pitch propeller motor, Represents the voltage of the yaw propeller motor; and is the gain matrix of the two-degree-of-freedom helicopter system model; is the unknown smooth nonlinear function vector, d=[d1,d2] T is an unknown external disturbance; [·] T represents the transpose of [·].

3. The method according to claim 2, characterized in that The auxiliary systems are: Where: represents the auxiliary system state vector, for The time derivative, represents the auxiliary error vector, and ρ is a positive constant to be designed.

4. The method according to claim 3, characterized in that: The predefined time nonlinear disturbance observer is: Where: For disturbance The estimated value of is an estimate of the auxiliary error vector z1(t) and is given by: Where: for The time derivative of represents the estimation error, tanh(·) represents the hyperbolic tangent function, sign(·) represents the sign function; ω1,γ, and ω2 are the control parameters to be designed, and γ∈(0,1), T o Scheduled time for design.

5. The method according to claim 4, characterized in that The predefined time filter is expressed as: Where: T c The scheduled time for design, The output signal α f The time derivative of represents the predefined time filter error, α f (0) = α(0) represents its initial value, α is the input signal of the predefined time filter, is the time derivative of α and satisfies σ is an unknown positive constant; ξ, b>0 is the control parameter to be designed; Y is a positive time-varying parameter, is the estimated value of parameter σ, and |||| represents the 2-norm of vector ·.

6. The method according to claim 5, characterized in that Execution-Evaluation Adaptive Law and The optimal virtual controller α, and the adaptive law They are: Where: and are the weight estimates of the execution neural network and the evaluation neural network respectively; Φ s1 (X) is the basis function vector; and are the parameters to be designed; e1(t) is the tracking error vector of the first-level subsystem; z1, z2 are the control gain coefficients; To track the trajectory The time derivative of ξ, b1 and γ are the positive parameters to be designed; is the predefined time filter error; h, are coefficients related to the design parameters.

7. The method according to claim 6, characterized in that The switching event triggering mechanism is: Where: is the actual execution signal of the ith control input; For the triggering moment Updated ideal control signal; are the kth and k+1th triggering moments of the ith control input, respectively, N + is a positive integer; i is the weight coefficient, 0<υ i <1; is the trigger error; is the dynamic threshold; ζ1, ζ2, ζ3, H i , Λ i and l i is the control parameter to be designed, Indicates the control signal μ i (t) The rate of change of the interval.

8. The method according to claim 7, characterized in that Predefined time optimal tracking controller Execution-Evaluation-Identification Adaptive Law and They are: Where: is the system gain matrix The inverse matrix of is used to decouple the control input; z3, z4 are control gain coefficients; e2(t) is the tracking error of the second-level subsystem; b3 is the anti-saturation adjustment parameter; To execute the output of the neural network, is the weight estimate, For input The basis function vector of ; To identify the output of the neural network, is the weight estimate, For input The basis function vector of ; To evaluate the weight estimates of the neural network; and are the parameters to be designed.

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