Two-degree-of-freedom helicopter safety controller design method based on zero-sum game

Abstract

The invention discloses a zero-sum game-based two-degree-of-freedom helicopter safety controller design method, which relates to the technical field of two-degree-of-freedom helicopter control, and comprises the steps of constructing a system equation, introducing an attack signal and a preset performance function, decomposing into two stages of subsystems, and designing a virtual controller, an adaptive law and a self-triggering mechanism. And solving a Hamiltonian-Jacobian-Exophone equation by using a zero-sum game theory, and finally verifying the stability of the system through a Lyapunov function. According to the invention, it can be ensured that the tracking error converges to the steady-state error interval within the predetermined time, and the transient and steady-state performance of the system is significantly improved; and meanwhile, a Nussbaum function and a zero-sum game strategy are introduced to improve the attack resistance, the communication burden and actuator abrasion are reduced through a self-triggering mechanism, and the sesame effect is avoided.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of two-degree-of-freedom helicopter control, and particularly relates to a two-degree-of-freedom helicopter safety controller design method based on zero-sum game. BACKGROUND

[0002] Unmanned helicopters have been widely used in transportation, ocean supervision, battlefield reconnaissance and other fields due to their unique advantages such as vertical take-off and landing, low environmental dependence and hovering capability. Two-degree-of-freedom helicopter, as a simplified model of unmanned helicopter, accurately flying along the reference trajectory is the basis for completing complex tasks. However, as a multi-input multi-output system, two-degree-of-freedom helicopter is inevitably affected by its strong coupling, uncertainty and external unknown disturbances. In addition, with the popularity of networked control systems, two-degree-of-freedom helicopters are vulnerable to network attacks such as deception attacks, which makes the design of tracking controllers more complex. If the expected tracking accuracy cannot be achieved, it may lead to serious economic and safety accidents.

[0003] In the prior art, an application with publication number CN 115357005 A proposes an active fault-tolerant control method for two-degree-of-freedom helicopter sensors, but does not consider the uncertainty of the system. An application with publication number CN 114578696 A designs an adaptive neural network quantization fault-tolerant control method, but ignores the transient performance optimization problem of system tracking error. An application with publication number CN 118732509 A proposes a predetermined time tracking control method, but does not involve the safety control problem of the system under attack. These prior arts have solved some problems to a certain extent, but still have significant deficiencies, such as insufficient attention to system transient performance, limited response to network attacks and failure to achieve accurate tracking control within a predefined time.

[0004] Therefore, it is urgent to design a safety controller that can achieve accurate tracking within a predefined time to improve the transient and steady-state performance of two-degree-of-freedom helicopters and still accurately track the reference trajectory under unknown attacks. This controller needs to consider the influence of system uncertainty, external disturbances and network attacks, while taking into account the update frequency of control signals and the wear and tear of actuators to ensure the efficient operation and long-term stability of the system. SUMMARY

[0005] The present application aims at the problem of insufficient trajectory tracking accuracy of existing two-degree-of-freedom helicopter control systems under complex environment caused by internal uncertainty, external disturbance and unknown attack, and proposes a two-degree-of-freedom helicopter predefined time safe tracking controller design method based on zero-sum game. The technical scheme realizes the goal of accurate trajectory tracking of the system within a predetermined time by introducing a predefined time preset performance function, a backstepping control method, a self-triggering mechanism and a zero-sum game theory, and has the ability to resist attacks.

[0006] To achieve the above object, the present application is implemented according to the following technical scheme:

[0007] The present application comprises the following steps:

[0008] S1: constructing a two-degree-of-freedom helicopter system equation, wherein the system equation contains the influence of internal uncertainty and external disturbance; the two-degree-of-freedom helicopter system equation is:

[0009]

[0010] In the formula: θ and φ represent the pitch angle and yaw angle of the two-degree-of-freedom helicopter respectively; represents the angular velocity of the pitch angle θ, represents the angular velocity of the yaw angle φ; u(t)=[V p ,V y ] T , V p represents the voltage of the pitch propeller motor, and V y represents the voltage of the yaw propeller motor; is an unknown smooth nonlinear function vector, d=[d1,d2] T is an external disturbance; [·] T represents the transpose of T ; and are gain matrices of the two-degree-of-freedom helicopter system model.

[0011] The deception attack suffered by the two-degree-of-freedom helicopter system can be expressed as:

[0012]

[0013] In the formula, j=1,2; represents the damaged state after the deception attack, represents the state-related sensor attack, and satisfies λ j (t) is an unknown time-varying sensor attack signal, β j (t)=(1+λ j (t)); ua (t) represents the actuator attack signal, u c (t) represents the designed safety controller.

[0014] S2: Based on the system equations, unknown sensor and actuator attack signals are introduced, and according to the state after the attack, a predefined time-preset performance function is introduced to constrain the tracking error and perform equivalent error transformation; the predefined time-preset performance function is:

[0015]

[0016] In the formula: T represents a constant. c The convergence time is represented by p, where p > 0 is a constant representing the convergence rate. Construct the error transformation function. Tracking error subject to equality constraints Converted to an unconstrained equivalent error z1(t)=[z 11 (t),z 12 (t)] T .

[0017] S3: Based on the equivalent error and backstep control method, the two-degree-of-freedom helicopter system model is decomposed into two-level subsystems;

[0018] S4: For the system state of the first-level subsystem of the two-degree-of-freedom helicopter system model after the attack, design a virtual controller and a first-level adaptive law using the Nussbaum function, and construct a first-level Lyapunov function. Calculate the time derivative of the first-level Lyapunov function. The virtual controller is represented by α, and the first-level adaptive law is represented by... Virtual controller α and adaptive law They are respectively:

[0019]

[0020] In the formula: This is an estimate of parameter θ1. To track the desired trajectory Time derivative, N(ξ1) represents the Nussbaum function. sec(·) represents the secant function. tanh(·) denotes the hyperbolic tangent function, sig 1+γ (z1)=[|z 11 | 1+γ sgn(z 11 ),|z 12 | 1+γ sgn(z12 T sgn(·) represents a sign function, γ, and h1 are control parameters to be designed, and γ ∈ (0, 1), T c is a predetermined time to be designed, is a fuzzy basis function vector, and ||z1|| represents a length of the vector z1.

[0021] S5: introducing a self-triggering mechanism for the system state of the second-level subsystem of the attacked two-degree-of-freedom helicopter system model, designing a performance function according to the zero-sum game theory, constructing a Hamilton-Jacobi-Isaacs (H-J-I) equation according to the performance function, approximately solving the H-J-I equation by executing-critic fuzzy logic systems, and obtaining two participants of the zero-sum game, an executing-critic learning law, and a second-level adaptive law by combining the Nussbaum function, and constructing a second-level Lyapunov function and taking a time derivative of the second-level Lyapunov function; the self-triggering mechanism is:

[0022]

[0023] wherein: u i (t) is an actual execution signal of the i th control input; is an updated ideal control signal at a triggering time; is an updated ideal control signal at a triggering time; are the k th and k+1 th triggering times of the i th control input, respectively, and N + is a positive integer; ρ i is a weight coefficient, and 0 < ρ i < 1; and are control parameters to be designed, represents a change rate of the control signal u i (t) interval.

[0024] S6: constructing a whole Lyapunov function of the two-degree-of-freedom helicopter system model, taking a time derivative of the whole Lyapunov function, making the two-degree-of-freedom helicopter system tend to be stable in a predefined time, and obtaining control gains to be designed of the system by inequality simplification. The two participants of the zero-sum game are μ c and μ a , the executing-critic learning law is the second-level adaptive law includes and Specifically,

[0025]

[0026] wherein: ​ is the inverse matrix of the system gain matrix , c3>c2>0 is the control gain, N(ξ2) is a Nussbaum function, b2>0, b3>0, and is the weight estimation, is the input is the basis function vector, is the input is the basis function vector, is the estimated value of the parameter Λ, is the estimated value of the parameter θ2, h2 is the control parameter with design.

[0027] The beneficial effects of the present application are:

[0028] 1) Different from the existing asymptotic stability controller design method and the traditional pre-defined performance controller design method, the pre-defined time pre-defined performance tracking controller designed based on the design method of the present application can ensure that all signals of the entire two-degree-of-freedom helicopter attitude system are pre-defined time bounded; the tracking error converges to the pre-defined steady-state error interval within the pre-defined time, and will never violate the boundary of the region; the transient performance and the steady-state performance of the two-degree-of-freedom helicopter system are effectively improved.

[0029] 2) By introducing the Nussbaum function, the adverse effects of the sensor attack with unknown attack direction on the tracking performance of the two-degree-of-freedom helicopter are solved; for the actuator attack, a control method based on the zero-sum game theory is adopted, a reinforcement learning strategy is introduced, and a pre-defined time safe tracking controller with an actuator-evaluator structure is established.

[0030] 3) The self-triggering mechanism is incorporated into the design of the pre-defined time pre-defined performance tracking controller, so that the control signal is strictly updated intermittently according to the pre-planned triggering condition, the continuous updating of the system control signal is effectively avoided, the updating frequency of the control signal and the wear of the actuator are reduced, and under the condition of intermittent updating of the control signal, the two-degree-of-freedom helicopter can still accurately track the reference trajectory. BRIEF DESCRIPTION OF DRAWINGS

[0031] Figure 1 is a model diagram of a two-degree-of-freedom helicopter;

[0032] Figure 2 is an adaptive safe control design structure diagram of a two-degree-of-freedom helicopter;

[0033] Figure 3 is a trajectory diagram of the angle tracking expected angle of the pitch angle of a two-degree-of-freedom helicopter;

[0034] Figure 4 It is a trajectory diagram of a two-degree-of-freedom helicopter tracking the desired angle from the yaw angle.

[0035] Figure 5 This is a trajectory tracking response diagram for a two-degree-of-freedom helicopter with angular error.

[0036] Figure 6 This is a control input curve diagram of a two-degree-of-freedom helicopter system;

[0037] Figure 7 This is the execution-evaluation weighted trajectory diagram of a two-degree-of-freedom helicopter system;

[0038] Figure 8 This is a trajectory diagram of a two-degree-of-freedom helicopter with adaptive law.

[0039] Figure 9 This is a diagram showing the trigger intervals for the control inputs of a two-degree-of-freedom helicopter system. Detailed Implementation

[0040] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. The illustrative embodiments and descriptions herein are used to explain the present invention, but are not intended to limit the present invention.

[0041] Reference Figure 1 and Figure 2 This invention provides a design method for a two-degree-of-freedom helicopter predefined time safety tracking controller based on zero-sum game theory, comprising the following steps:

[0042] Step 1, according to Figure 1 The simplified diagram of the two-degree-of-freedom helicopter system shown is used to construct the following dynamic model:

[0043]

[0044] In the formula: θ and φ represent the pitch angle and yaw angle of a two-degree-of-freedom helicopter, respectively; and These represent the angular acceleration and angular velocity at the pitch angle θ, respectively. and F represents the angular acceleration and angular velocity at the yaw angle φ, respectively; p This indicates the voltage V of the pitch propeller motor. p The effect of r p The thrust, and satisfying F p r p =K pp V p F y This indicates the voltage V of the yaw propeller motor. y The effect of r y The thrust, and satisfying Fy r y yy V y pp yp py yy M is the total mass of the two-degree-of-freedom helicopter; g is the gravity acceleration; sin(·) and cos(·) represent the sine function and the cosine function, respectively; l a is the distance from the system mass center to the helicopter fixed support point; and represent the friction damping coefficients when the pitch angle and the yaw angle rotate, respectively; and represent the moment of inertia of the pitch axis and the yaw axis, respectively.

[0045] The detailed system parameters are shown in Table 1:

[0046] Table 1: Two-degree-of-freedom helicopter system model parameters

[0047]

[0048] Definition

[0049] u(t)=[u1(t),u2(t)] T p y T , [] T represent the transpose of [·], and further, the two-degree-of-freedom helicopter system model is expressed as:

[0050]

[0051] In the formula: is an unknown smooth nonlinear function vector, d=[d1,d2] T is an external disturbance; and are gain matrices of the two-degree-of-freedom helicopter system model, and are respectively:

[0052]

[0053] Step 2, based on the system equation obtained in step 1, the following deception attack signal is introduced:

[0054]

[0055] ​​​​​​​​​​​​In the formula; j = 1, 2; This indicates the damaged state following a deception attack. This indicates a state-dependent sensor attack, and satisfies λ j (t) represents the unknown time-varying sensor attack signal, β j (t)=(1+λ j (t)); u a (t) represents the actuator attack signal, u c (t) represents the designed safety controller.

[0056] Furthermore, a predefined time-preset performance function is constructed to constrain the tracking error. in For the desired tracking trajectory, the predefined time-preset performance function ζ(t) is expressed as:

[0057]

[0058] In the formula: T represents a constant. c The convergence time is represented by p, where p > 0 is a constant representing the convergence rate. Construct the error transformation function. Tracking error subject to equality constraints Converted to an unconstrained equivalent error z1(t)=[z 11 (t),z 12 (t)] T .

[0059] Step 3: Based on the equivalent error z1(t) obtained in Step 2 and the backstepping method, construct the following coordinate transformation:

[0060]

[0061] In the formula: Representing the tracking error, α = [α1, α2] T As a virtual controller, the two-degree-of-freedom helicopter system model is decomposed into two-level subsystems, and an error variable z1(t) is introduced. These are used as tracking error variables for the first-level subsystem and the second-level subsystem, respectively.

[0062] Step 4: Based on the coordinate transformation obtained in Step 3, construct the Lyapunov function V1 for the first-level subsystem of the two-degree-of-freedom helicopter attitude system:

[0063]

[0064] In the formula: sec() represents the secant function. This indicates the estimation error.

[0065] Furthermore, by differentiating the Lyapunov function V1 with respect to time, we obtain:

[0066]

[0067] In the formula: Using Young's inequality, we can obtain:

[0068]

[0069] In the formula: This represents the upper bound of the attack signal β1. β 2 represents the lower bound of the attack signal β2. Furthermore, we have...

[0070]

[0071] In the formula: And approximate it using a fuzzy logic system. The specific form is as follows:

[0072]

[0073] In the formula: W1 is the weight matrix,

[0074] Indicates that the input is The basis function vectors, ∈1, represent the approximation error of the fuzzy logic system. Definition We have It is a normal number.

[0075] Furthermore, to ensure that the first-stage subsystem of the two-degree-of-freedom helicopter system model tends to predefined time stability, the following virtual controller and adaptive law are designed.

[0076]

[0077] In the formula: This is an estimate of parameter θ1. To track the desired trajectory Time derivative, N(ξ1) represents the Nussbaum function. sec(·) represents the secant function. tanh() represents the hyperbolic tangent function, sig 1+γ (z1)=[|z 11 | 1+γ sgn(z 11 ),|z 12 | 1+γsgn(z 12 ] T , sgn(·) represents the sign function, γ, and h1 are control parameters to be designed, and γ ∈ (0, 1), T c is the predetermined time of design, is the fuzzy basis function vector, and ||z1|| represents the length of the vector z1.

[0078] Definition

[0079]

[0080] Then the designed virtual controller α, adaptive law is brought into formula (9), and the inequality is simplified to obtain:

[0081]

[0082] In the formula:

[0083] In the process of inequality simplification, the value of the first subsystem with the designed virtual controller gain is obtained.

[0084] Step 5, the second subsystem of the two-degree-of-freedom helicopter system model is:

[0085]

[0086] In the formula: represents the derivative of the virtual signal α with respect to time, in order to reduce the communication resources, the following self-triggering mechanism is introduced:

[0087]

[0088] In the formula: u i (t) is the actual execution signal of the i-th control input; is the updated ideal control signal at the triggering time ; are the k-th and k+1-th triggering times of the i-th control input, respectively, and N + is a positive integer; ρ i is a weight coefficient, and 0 < ρ i < 1; and are control parameters to be designed, represents the change rate of the control signal u i (t) interval.

[0089] The time-varying functions w 1i (t) and w 2i (t) are constructed, and satisfy |w 1i(t)|≤1 and |w 2i (t)|≤1, based on formula (15), for We can obtain:

[0090]

[0091] In the formula:

[0092] For the second-level subsystem of a two-degree-of-freedom helicopter system model, attack signals and control signals are considered as two players. To balance player u... c and u a The need between the participants ensures system stability. Therefore, the cost strategies of the two participants are defined as follows:

[0093]

[0094] In the formula: and The optimal control strategy to achieve Nash equilibrium is constructed, where c1 > 0 and c3 > c2 > 0 are constants to be designed.

[0095] Furthermore, based on Leibniz's formula and differential calculus, the Bellman equation can be obtained, expressed in terms of Hamiltonian functions as follows:

[0096]

[0097] In the formula: express right The partial derivative, for The derivative with respect to time. Then, using the stability condition. and We can obtain:

[0098]

[0099] In the formula:

[0100] Furthermore, we have

[0101]

[0102]

[0103] In the formula, the auxiliary cost function for:

[0104]

[0105] In the formula: B -1is the inverse matrix of the system gain matrix B, c3>c2>0 are control gains, N(ξ2) is a Nussbaum function, b2>0, b3>0, is the input basis function vector, is the estimate of the parameter Λ, is the estimate of the parameter θ2, h2 is a control parameter to be designed.

[0106] Since is unknown but continuous, we choose a fuzzy logic system to approximate the estimate:

[0107]

[0108] where: is the ideal weight, is the input basis function vector, is the approximation error vector.

[0109] Further, by performing a critic structure to approximate the HJI equation, the performing-critic adaptive law and and the zero-sum game player μ c and μ a are obtained:

[0110]

[0111]

[0112] where: and are the estimates of κ a1 and κ c1 are parameters to be designed.

[0113] For the second level subsystem of the two-degree-of-freedom helicopter system model, a Lyapunov function V2 is constructed:

[0114]

[0115] where: and tr(·) represents the trace of a matrix,

[0116] Further, by taking the derivative of the Lyapunov function V2 with respect to time, we obtain:

[0117] ​​

[0118] make Using fuzzy logic systems, we can obtain:

[0119]

[0120] In the formula: W2 is the weight matrix, Indicates that the input is The basis function vectors, ∈2, represent the approximation error of the fuzzy logic system. Definition We have It is a normal number.

[0121] To ensure that the second-stage subsystem of the two-degree-of-freedom helicopter system model tends to predefined time stability, the following adaptive law is designed. and Simultaneously define for:

[0122]

[0123] The designed zero-sum game participants μ c and μ a Adaptive law and Execution-evaluation adaptive law and Substituting into formula (30), we can simplify using inequalities to obtain:

[0124]

[0125] In the formula:

[0126] g2(t)=β2τ1.

[0127] The value of the virtual controller gain is obtained during the inequality simplification process.

[0128] Step 6: Verify the results obtained in Step 5. Construct the overall Lyapunov function V = V1 + V2 for the two-degree-of-freedom helicopter system model, and take its derivative with respect to time to obtain:

[0129]

[0130] In the formula: Representation matrix The minimum eigenvalue, k = min{k1,k2}.

[0131] According to formula (36), a virtual controller α is designed, and an adaptive law is employed. and Performing-judging self-adaptive law And Zero-sum game participant mu c And mu a Guaranteeing that all signals of the two-degree-of-freedom helicopter closed-loop system are bounded in a predefined time, and the upper bound of the stable time of the two-degree-of-freedom helicopter closed-loop system is The stable error of the two-degree-of-freedom helicopter closed-loop system can be converged into a small enough area, that is:

[0132]

[0133] In addition, for the self-triggering mechanism (15), there are:

[0134]

[0135] In the formula: is a constant, obtained from the initial condition Obtained Then further obtained:

[0136]

[0137] Therefore, the time interval of the two triggers Since And Therefore The Zeno effect is effectively avoided.

[0138] In order to explain the method control effect of the application in detail, the following simulation experiments will be carried out in MATLAB, and the reference trajectory is set to The unknown disturbance is set to The initial condition of the system is set to The selected controller gain is shown in Table 2.

[0139] Table 2: Controller gain

[0140]

[0141]

[0142] The following results are obtained through MATLAB simulation experiments, Figure 3 is the trajectory diagram of the angle tracking expected angle of the pitch angle of the two-degree-of-freedom helicopter, from which it can be seen that the actual pitch angle of the two-degree-of-freedom helicopter system can track the expected reference pitch angle within a predetermined time; Figure 4 is the trajectory diagram of the angle tracking expected angle of the yaw angle of the two-degree-of-freedom helicopter, from which it can be seen that the actual yaw angle of the two-degree-of-freedom helicopter system can track the expected reference yaw angle within a predetermined time; Figure 5The figure shows the angle error trajectory response of the two-degree-of-freedom helicopter under the preset function, and it can be seen from the figure that the angle error of the two-degree-of-freedom helicopter is completely constrained in a given range, and converges to the neighborhood of zero within a predetermined time; Figure 6 The figure is a control input curve of the two-degree-of-freedom helicopter system, and it can be seen from the simulation figure that the tracking controller of the application effectively avoids continuous updating of the control signal, relieves the burden of network communication, and reduces the wear of the actuator; Figure 7 The figure is an execution-judgment weight trajectory of the two-degree-of-freedom helicopter system, and it can be seen from the figure that the execution weight of the two-degree-of-freedom helicopter system converges to 0.545 within a predetermined time, and the judgment weight converges to 0.273 within a predetermined time; Figure 8 The figure is an adaptive law trajectory of the two-degree-of-freedom helicopter, and it can be seen from the figure that the adaptive law is positive and changes with time; Figure 9 The figure is a trigger interval of the control input of the two-degree-of-freedom helicopter system, and it can be seen from the figure that the Zeno effect is effectively avoided. The simulation results show that the predefined time tracking controller design method of the application ensures the predefined time stability of the closed-loop system of the two-degree-of-freedom helicopter; the tracking error converges within a given time, effectively improves the transient performance and steady-state performance of the two-degree-of-freedom helicopter system, and the two-degree-of-freedom helicopter can still accurately track the reference trajectory under unknown deception attacks.

[0143] The technical scheme of the application is not limited to the above specific embodiments, and any technical modification made according to the technical scheme of the application falls within the protection scope of the application.

Claims

1. A method for safety controller design of a two-degree-of-freedom helicopter based on zero-sum game, characterized in that, The method comprises the following steps: S1: constructing a two-degree-of-freedom helicopter system equation, which contains the influence of internal uncertainty and external disturbance; S2: based on the system equation, introducing unknown sensor and actuator attack signals, and according to the state after the attack, introducing a predefined time preset performance function to constrain the tracking error, and performing equivalent error transformation; S3: based on the equivalent error and backstepping control method, the two-degree-of-freedom helicopter system model is decomposed into two levels of subsystems; S4: for the system state of the first level subsystem of the attacked two-degree-of-freedom helicopter system model, a virtual controller and a first level adaptive law are designed in combination with a Nussbaum function, and a first level Lyapunov function is constructed, and the time derivative of the first level Lyapunov function is calculated; S5: for the system state of the second level subsystem of the attacked two-degree-of-freedom helicopter system model, a self-triggering mechanism is introduced, a performance function is designed according to the zero-sum game theory, a Hamilton-Jacobi-Isaacs (H-J-I) equation is constructed according to the performance function, the H-J-I equation is approximately solved by an execution-evaluation fuzzy logic system, and two participants of the zero-sum game, an execution-evaluation learning law and a second level adaptive law are obtained by combining the Nussbaum function, and a second level Lyapunov function is constructed, and the time derivative of the second level Lyapunov function is calculated; S6: constructing a whole Lyapunov function of the two-degree-of-freedom helicopter system model, and calculating the time derivative of the whole Lyapunov function, so that the two-degree-of-freedom helicopter system tends to be pre-defined time stable, and through inequality simplification, the control gain to be designed of the system is obtained.

2. The method of claim 1, wherein: The two-degree-of-freedom helicopter system equation is: where: θ and φ represent the pitch and yaw angles of the two-degree-of-freedom helicopter, respectively; represents the angular velocity of the pitch angle θ, represents the angular velocity of the yaw angle φ; u(t) = [V p , y ] T , V p represents the voltage of the pitch propeller motor, V y represents the voltage of the yaw propeller motor; is an unknown smooth nonlinear function vector, d = [d1, d2] T is an external disturbance; [·] T represents the transpose of T ; and are the gain matrices of the two-degree-of-freedom helicopter system model.

3. The method of claim 1, wherein: The attack on the two-degree-of-freedom helicopter system model can be expressed as: In the formula, j = 1,2; represents the compromised state after a deception attack, represents a state-dependent sensor attack, and satisfies λ j (t) is an unknown time-varying sensor attack signal, β j (t) = (1 + λ j (t)); u a (t) represents an actuator attack signal, u c (t) represents the designed secure control controller.

4. The method of claim 1, wherein: The predefined time preset performance function is: where: represents a constant, T c represents a convergence time, p > 0 is a constant representing a convergence rate. The error transfer function is constructed The tracking error subject to the equality constraint is converted to an equivalent error z1(t) = [z 11 (t), z 12 (t)] T .

5. The method of claim 1, wherein: The virtual controller is denoted as a, the first order adaptive law is denoted as Virtual controller a and adaptive law respectively: where: is an estimate of the parameter θ1, is the time derivative of the desired tracking trajectory N(ξ1) denotes a Nussbaum function, sec(·) denotes the secant function, tanh() denotes the hyperbolic tangent function, sig 1+γ (z1) = [|z 11 1+γ sgn(z 11 12 1+γ 12 T sgn() represents the sign function, γ, and h1 are control parameters to be designed, and γ ∈ (0, 1), T c is a predetermined time to be designed, is a fuzzy basis function vector, and ||z1|| denotes the length of the vector z1.​​​​​​ 6. The method of claim 1, wherein: The self-triggering mechanism is: wherein: u i (t) is the actual execution signal of the i-th control input; is the ideal control signal updated at the triggering moment ; respectively the k-th and the k+1-th triggering moment of the i-th control input, N + is a positive integer; p i is a weight coefficient, 0 < p i < 1; and l i is the control parameter to be designed, indicates the rate of change of the control signal u i (t) interval.

7. The method of claim 1, wherein: The two participants of the zero-sum game are μ c and μ a , the execution-judgment learning law is respectively The second level adaptive law includes and Specifically: wherein: is an inverse matrix of a system gain matrix is a control gain, N(ξ2) is a Nussbaum function, b2>0, b3>0, and is a weight estimate, is a basis function vector with input is a basis function vector with input is a basis function vector with input is a basis function vector with input is an estimate of the parameter Λ, is an estimate of the parameter θ2, h2is a control parameter with band design.

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