A quad-rotor unmanned aerial vehicle attitude control method based on an event triggering mechanism

Abstract

This invention relates to the field of unmanned aerial vehicles (UAVs). A quadrotor UAV attitude control method based on an event-triggered mechanism includes the following steps: Step 1, introducing a rotation matrix to describe the quadrotor UAV attitude; Step 2, providing the quadrotor UAV attitude dynamics system; Step 3, designing an angular velocity observer; Step 4, introducing an event-triggered mechanism; Step 5, constructing the system control input u. This invention introduces an event-triggered mechanism and designs a controller based on this mechanism.

Description

Technical Field

[0001] This invention relates to the field of unmanned aerial vehicles (UAVs), and more specifically to control methods. Background Technology

[0002] Quadrotor drones, as small unmanned aerial vehicles with vertical takeoff and landing capabilities, have demonstrated broad development potential in multiple fields due to their small size, high maneuverability, and long endurance. In recent years, with the continuous advancement of sensor technology, communication technology, and control theory, quadrotor drone systems are rapidly evolving from single-function to multi-functional integration. Currently, these drones are widely used in civilian and commercial scenarios such as agricultural and forestry plant protection, geographic surveying, logistics distribution, and emergency rescue, and are playing an increasingly important role in military reconnaissance, target location, and even collaborative operations.

[0003] Attitude control, as the core component of the flight control system of a quadcopter UAV, is fundamental to tasks such as trajectory tracking, and its performance directly determines the stability of the entire system. Attitude stabilization control, a key research direction, aims to enable the system state to converge quickly and smoothly and stabilize near the desired value. However, such systems generally suffer from inherent challenges such as strong coupling, strong time-varying characteristics, nonlinearity, and model uncertainty. In the design of practical control strategies, it is also necessary to comprehensively consider issues such as equipment performance limitations under complex external disturbances, limited onboard fuel capacity, and mission cycle time constraints. Furthermore, when facing complex situations such as multi-tasking parallelism, sudden failures, or extreme weather, the system exposes several other technical bottlenecks, bringing numerous complex challenges to related research.

[0004] In existing research on attitude control of quadrotor UAVs, time-triggered mechanisms are a traditional and widely adopted control strategy. This mechanism samples the system and updates control variables based on a preset fixed period (i.e., the control period), triggering the corresponding control behavior at the beginning of each period. To achieve better dynamic performance and control accuracy, traditional time-triggered mechanisms typically require short control periods, or even rely on high-frequency, near-real-time state monitoring and feedback. While this improves control performance, it also introduces problems such as high computational load, high communication resource consumption, high overall control cost, and excessive energy consumption. Summary of the Invention

[0005] The purpose of this invention is to provide an attitude control method for a quadcopter unmanned aerial vehicle based on an event-triggered mechanism to solve the above-mentioned technical problems.

[0006] The attitude control method for a quadcopter UAV based on an event-triggered mechanism is characterized by the fact that the system will only perform corresponding control actions when and only when the triggering condition is met. This method has advantages such as effectively reducing computational load, lowering energy consumption, and saving control costs. Specifically, it includes the following steps:

[0007] Step 1: Introduce a rotation matrix to describe the attitude of the quadcopter drone:

[0008] Let SO(3) be the set of rotation matrices, then we have

[0009] Step 2: Provide the attitude dynamics system of the quadcopter UAV:

[0010]

[0011] in, The inertia matrix represents the positive definite inertia matrix. Indicates the system's control input, It is the angular velocity defined in a fixed coordinate system. Indicates system interference;

[0012] Step 3: Design the angular velocity observer:

[0013]

[0014] in, These represent the estimated values ​​of angular velocity and attitude, respectively. The error between the true and estimated attitude values;

[0015] Step 4: Introduce an event triggering mechanism:

[0016] Assume the trigger times are t1, t2, ..., t i ..., the system control input u satisfies:

[0017]

[0018] The trigger condition is defined as follows:

[0019]

[0020] Where, constant Used to represent the state of the system, here e R Let e ​​be the attitude error vector. Ω This is the angular velocity error vector. Indicates error;

[0021] Step 5: Construct the system control input u:

[0022] u = -k R e R -k Ω e Ω +Ω×JΩ+Jα+ν+ks(t i )

[0023] in

[0024] Preferably, when there is no system interference, the constant c in the system control input u satisfies the following equation:

[0025]

[0026] Preferably, when dynamic disturbances exist, the constant c in the system control input u satisfies the following equation:

[0027]

[0028] Beneficial effects: 1. This invention incorporates an external disturbance component (Δ) into the attitude dynamics system of a quadcopter UAV, which can more realistically reflect the real-world problems faced by quadcopter UAVs, such as air disturbances and measurement noise; 2. The angular velocity observer designed in this invention can acquire angular velocity and attitude estimates, overcoming the difficulty in timely and accurate acquisition of angular velocity in real-world quadcopter UAVs; 3. This invention introduces an event triggering mechanism into the attitude control of quadcopter UAVs, which can effectively reduce system energy consumption, save operating costs, and reduce computational load, with broad application prospects; 4. The solution of this invention is adaptable to various task requirements, and the attitude control target can be achieved simply by modifying the event triggering conditions. Attached Figure Description

[0029] Figure 1 This is a schematic diagram of the steps of the present invention;

[0030] Figure 2 When there is no system interference, k = 3, β = 1, attitude error vector e R Simulation diagram;

[0031] Figure 3 When there is no system interference, k = 3, β = 1, angular velocity error vector e Ω Simulation diagram;

[0032] Figure 4 When there is no system interference, k = 5, β = 2, attitude error vector e R Simulation diagram;

[0033] Figure 5 When there is no system interference, k = 5, β = 2, angular velocity error vector e Ω Simulation diagram;

[0034] Figure 6 When there is no system interference, k = 1, β = 0.5, attitude error vector e R Simulation diagram;

[0035] Figure 7 When there is no system interference, k = 1, β = 0.5, angular velocity error vector e Ω Simulation diagram;

[0036] Figure 8 For the existence of dynamic disturbance Δ=3e A hour, k = 3, β = 1, attitude error vector e R Simulation diagram;

[0037] Figure 9 For the existence of dynamic disturbance Δ=3e A hour, k = 3, β = 1, angular velocity error vector e Ω Simulation diagram;

[0038] Figure 10 For the existence of dynamic disturbance Δ=2e A hour, k = 2, β = 2, attitude error vector e R Simulation diagram;

[0039] Figure 11 For the existence of dynamic disturbance Δ=2e A hour, k = 2, β = 2, angular velocity error vector e Ω Simulation diagram;

[0040] Figure 12 For the existence of dynamic disturbance Δ=e A hour, k = 1, β = 0.5, attitude error vector e R Simulation diagram;

[0041] Figure 13 For the existence of dynamic disturbance Δ=e A hour, k = 1, β = 0.5, angular velocity error vector e Ω Simulation diagram;

[0042] Figure 14 When there is no system interference and no event triggering mechanism is introduced, the attitude error vector e R Simulation diagram;

[0043] Figure 15 When there is no system interference and no event triggering mechanism is introduced, the angular velocity error vector e Ω Simulation diagram;

[0044] Figure 16 For the existence of dynamic disturbance Δ=3e A Furthermore, without an event-triggered mechanism, the attitude error vector eR Simulation diagram;

[0045] Figure 17 For the existence of dynamic disturbance Δ=3e A Furthermore, without an event-triggered mechanism, the angular velocity error vector e Ω The simulation diagram. Detailed Implementation

[0046] To make the technical means, creative features, objectives and effects of this invention easier to understand, the invention will be further described below with reference to specific illustrations.

[0047] This invention relates to the following mathematical symbols:

[0048]

[0049] For an event-triggered mechanism-based attitude control method for quadrotor UAVs, see [link to relevant documentation]. Figure 1 It includes the following steps:

[0050] Step 1: Introduce a rotation matrix to describe the attitude of the quadcopter UAV;

[0051] Step 2: Provide the attitude dynamics system of the quadcopter UAV;

[0052] Step 3: Design an angular velocity observer;

[0053] Step 4: Introduce an event triggering mechanism;

[0054] Step 5: Construct the system control input u.

[0055] The details are as follows:

[0056] Step 1. Introduce the rotation matrix R to describe the attitude of the quadcopter UAV.

[0057] This invention selects rotation matrices to describe the attitude information of a quadcopter UAV. Let SO(3) be the set of rotation matrices, then...

[0058] For any aircraft attitude, there can be a uniquely determined rotation matrix R∈SO(3) to represent it.

[0059] Step 1.1. Define two coordinate systems: a fixed coordinate system and an inertial coordinate system, both of which satisfy the right-hand rule.

[0060] A fixed coordinate system is usually established on the ground and serves as the unified reference coordinate system for the entire system. An inertial coordinate system is often used to describe the position and attitude of an object in a fixed coordinate system. By rotating one coordinate system by a certain angle, it can coincide with another coordinate system. Common rotation angles include roll angle φ, pitch angle θ, and yaw angle ψ. That is, starting from the fixed coordinate system, rotating ψ around the Z-axis, rotating θ around the Y-axis, and finally rotating φ around the X-axis will result in coinciding with the inertial coordinate system.

[0061] Step 1.2. Let the fixed coordinate system be represented as... Inertial coordinate system is represented as The rotation matrix from the fixed coordinate system to the inertial coordinate system is expressed as: satisfy:

[0062]

[0063] Where R φ R θ and R ψ The rotation matrices representing the roll angle, pitch angle, and yaw angle are shown below:

[0064]

[0065] Then the rotation matrix The expression can be simplified as follows:

[0066]

[0067] For ease of understanding, the rotation matrix will be discussed later. Abbreviated as R, it satisfies the following equation:

[0068] RR T =R T R = I 3×3 ,

[0069] det(R) = 1

[0070] Let SO(3) be the set of rotation matrices, i.e.

[0071] For any aircraft attitude, there can be a uniquely determined rotation matrix R∈SO(3) to represent it.

[0072] Step 2. Give the attitude dynamics system of the quadcopter UAV.

[0073] Using Newton's second law of motion and Euler's equations, the attitude dynamics system of the quadcopter UAV is obtained:

[0074]

[0075] in, The inertia matrix represents the positive definite inertia matrix. Indicates the system's control input, It is the angular velocity defined in a fixed coordinate system. This indicates system interference.

[0076] Step 2.1. Make the following assumptions:

[0077] 1) Ignoring factors such as the Earth's curvature and rotation, assume that the quadcopter drone moves within a plane;

[0078] 2) Treat the quadcopter drone as a rigid body, that is, do not consider its shape, size, mass, and deformation during motion;

[0079] 3) The origin of the inertial coordinate system coincides with that of the fixed coordinate system and is located at the center of the rigid body;

[0080] 4) The principal axes of the inertial coordinate system and the fixed coordinate system coincide, and the direction of the coordinate axes is determined by the task currently being performed by the rigid body.

[0081] Step 2.2. Introduce the attitude dynamics system of the quadcopter UAV.

[0082] When disturbances are present, according to Newton's second law of motion and Euler's equations, the attitude dynamics system of the quadcopter UAV is as follows:

[0083]

[0084] in, The inertia matrix represents the positive definite inertia matrix. Indicates the system's control input, It is the angular velocity defined in a fixed coordinate system. Indicates system interference. Mapping Transform a three-dimensional vector into a 3×3 skew-symmetric matrix. For any vector... Satisfy the following formula:

[0085]

[0086] At the same time, the inverse mapping of mapping (.)^ is defined as

[0087] Step 3. Design an angular velocity observer.

[0088] The angular velocity observer is designed as follows:

[0089]

[0090] in, These represent the estimated values ​​of angular velocity and attitude, respectively. This represents the error between the true and estimated attitude values.

[0091] Step 3.1. Introduce the dynamic equations for angular velocity and attitude estimates.

[0092] Design an angular velocity observer to obtain estimates of angular velocity and attitude, let Let represent the estimated values ​​of angular velocity and attitude, respectively. The estimated values ​​of angular velocity and attitude also satisfy the rigid body dynamics equations, i.e.:

[0093]

[0094] Since the magnitude of the control input is usually finite, it can be assumed that the angular velocity and its first derivative are bounded, i.e., that there exist positive values. Make:

[0095]

[0096] Step 3.2. Construct the error model.

[0097] Let the error between the true and estimated attitude values ​​be Q∈SO(3), that is... when When, Q = I 3×3 .

[0098] Define the attitude error function Ψ and the attitude error vector e. R Angular velocity error vector e Ω They are respectively:

[0099]

[0100] Where G = diag(g1, g2, g3), It is a positive real number.

[0101] Since angular velocity and its first derivative are bounded, it is easy to obtain

[0102] Attitude error function Ψ, attitude error vector e R and angular velocity error vector e Ω It has the following properties:

[0103] Property 1:

[0104] Property 2:

[0105] Property 3: in, satisfy:

[0106] Property 4: in

[0107] Property 5: Suppose there exist Ψ≤ψ≤n1 such that:

[0108] Where, n1 = min{g1 + g2, g2 + g3, g3 + g1}, n2 = max{(g1 - g2)} 2 (g2-g3) 2 (g3-g1) 2},

[0109] n3 = max{(g1 + g2)} 2 (g2+g3) 2 (g3+g1) 2},n4=max{g1+g2,g2+g3,g3+g1},

[0110] n5=min{(g1+g2) 2 (g2+g3) 2 (g3+g1) 2}

[0111] Property 6: ||e R ||≤B2, where,

[0112] Attitude stability control aims to make the values ​​of the attitude error function and angular velocity error function approach zero, that is:

[0113]

[0114] Step 3.3. Design of the angular velocity observer.

[0115] Design an angular velocity observer, that is, give the dynamic equations for the estimated angular velocity:

[0116]

[0117] Step 4. Introduce an event triggering mechanism.

[0118] Event-triggered mechanisms involve pre-defined conditions (i.e., trigger conditions), where the system will only perform corresponding control actions if and only if these conditions are met. Experiments have shown that event-triggered mechanisms can fulfill control requirements while significantly reducing system communication frequency, offering advantages such as effectively reducing computational load, lowering energy consumption, and saving control costs.

[0119] Assume the trigger times are t1, t2, ..., t i ..., the system control input u satisfies:

[0120]

[0121] The trigger condition is defined as follows:

[0122]

[0123] Here, constant Used to describe the system state; Used to represent error.

[0124] Step 5. Construct the system control input u.

[0125] For systems (1) and (2), design an angular velocity observer (3) and construct the system control input u as follows:

[0126] u = -k R e R -k Ω e Ω +Ω×JΩ+Jα+ν+ks(t i )----------------------(4)

[0127] in

[0128] Case 1. When there is no system interference, the system control input u is set to meet the conditions so that the attitude dynamics system (1), (2), (4) of the quadcopter UAV is exponentially stable, thereby completing the stable control of the attitude of the quadcopter UAV.

[0129] Assume that the attitude control system of the quadcopter UAV is in an ideal state with no external interference, i.e., Δ = 0.

[0130] When there exists a positive constant c that satisfies the following equation (*),

[0131]

[0132] It can be proven that the attitude dynamics system (1), (2), and (4) of the quadcopter UAV is exponentially stable, thus achieving stable control of the attitude of the quadcopter UAV.

[0133] The proof is as follows:

[0134] Consider Lyapunov functions

[0135] From matrix theory, we can obtain

[0136]

[0137] cλ m ||e Ω ||≤ce Ω T Je R ≤cλ M||e Ω ||

[0138] From property 5, we can obtain:

[0139]

[0140] Therefore

[0141] That is, satisfying:

[0142] ξ T W1ξ≤V≤ξ T W2ξ·

[0143] Where ξ=(||e Ω ||,||e R ||) T .matrix They are respectively:

[0144]

[0145] Since Δ=0, substituting equation (4) into property 4, we get:

[0146]

[0147] Taking the first derivative of the Lyapunov function V, we get:

[0148]

[0149]

[0150] From the definitions of ν and M, we get:

[0151]

[0152] From the definitions of s and e, we get:

[0153] s(t i ) = e + s = e + e Ω +βe R

[0154] but:

[0155] And because The above expression can then be simplified to:

[0156]

[0157] Substituting the above result into the first derivative of the Lyapunov function V, we get:

[0158]

[0159] Among them, matrix W3 satisfies the following equation:

[0160]

[0161] In this step, under the condition of constant c, matrix W3 is positive definite. It can be proved by Lyapunov stability theory that the attitude dynamics system (1), (2), and (4) of the quadcopter UAV is exponentially stable, thus completing the stable control of the attitude of the quadcopter UAV.

[0162] Case 2. When dynamic disturbances exist, the system control input u is set to meet certain conditions so that the attitude dynamics system (1), (2), and (4) of the quadcopter UAV is exponentially stable, thereby achieving stable control of the attitude of the quadcopter UAV.

[0163] Assume the attitude control system of a quadcopter UAV is under dynamic disturbance. Define vectors. in For system disturbances, we assume that they are related to vector e. A Linear dependence, that is: there exists a positive number k such that Δ = ke A .

[0164] When the constant c in the system control input u satisfies the following equation (**),

[0165]

[0166] It can be proven that the attitude dynamics system (1), (2), and (4) of the quadcopter UAV is exponentially stable, thereby achieving stable control of the attitude of the quadcopter UAV.

[0167] The proof is as follows:

[0168] Consider Lyapunov functions

[0169] From matrix theory, we can obtain

[0170]

[0171] cλ m ||e Ω ||≤ce Ω T Je R ≤cλ M ||e Ω ||

[0172] From property 5, we can obtain:

[0173]

[0174] Therefore

[0175] That is, satisfying:

[0176] ξ T W1ξ≤V≤ξ T W2ξ

[0177] Where ξ=(||e Ω ||,||e R ||) T .matrix They are respectively:

[0178]

[0179] Based on the condition of constant c in this step, we know that matrices W1 and W2 are positive definite.

[0180] Substituting equation (4) into property 4, we get:

[0181]

[0182] Taking the first derivative of the Lyapunov function, we get:

[0183]

[0184] From the definition of Δ, we get:

[0185]

[0186] Substituting the above equation into... have to:

[0187]

[0188] in,

[0189]

[0190] In this step, given the constant c, matrix W4 is positive definite; therefore, the angular velocity error function and attitude error function converge exponentially to 0, i.e., (e Ω ,e R )→(0,0), thereby completing the stable control of the attitude of the quadcopter drone.

[0191] Simulation Experiment

[0192] Simulation experiments were conducted to test the stability of the attitude system of a quadcopter UAV under an event-triggered mechanism, thus visually demonstrating the performance of the proposed controller through numerical simulation.

[0193] The parameter values ​​are: R(0) = I 3×3 , J(0)=diag(1,2,3), G=diag(1,1,1), Ω=(1,3,2),kR =k Ω =5, ε=δ=3, c=0.02. In the simulation diagram below, the red, blue, and green lines represent the first, second, and third quantities of the error vector, respectively, for example: e Ω The i-th quantity.

[0194] Case 1. Simulation experiment on the attitude stability of quadcopter UAV without system interference.

[0195] ① Take k=3, β=1, simulation graph as follows Figure 2 , Figure 3 As shown.

[0196] ② Take k=5, β=2, simulation graph as follows Figure 4 , Figure 5 As shown.

[0197] ③ Take k=1, β=0.5, simulation graph as follows Figure 6 , Figure 7 As shown.

[0198] As can be seen from the figure, the error vector exponent approaches 0, which shows that the controller u under our designed event-triggered mechanism can effectively ensure the stability of the system and thus complete the control of the attitude of the quadcopter UAV.

[0199] Case 2. When dynamic interference exists, conduct a simulation experiment on the stability of the attitude of the quadcopter UAV.

[0200] ①Take Δ=3e A , k=3, β=1, simulation graph as follows Figure 8 , Figure 9 As shown.

[0201] ②Take Δ=2e A , k=2, β=2, simulation graph as follows Figure 10 , Figure 11 As shown.

[0202] ③Take Δ = e A , k=1, β=0.5, simulation graph as follows Figure 12 , Figure 13 As shown.

[0203] As can be seen from the figure, the error vector exponent approaches 0, which shows that the controller u under our designed event-triggered mechanism can effectively ensure the stability of the system and thus complete the control of the attitude of the quadcopter UAV.

[0204] Analyze the impact of event triggering mechanisms on system convergence speed:

[0205] First, the stability of the attitude of the quadcopter UAV without an event triggering mechanism is simulated. To intuitively demonstrate the impact of the event triggering mechanism on the system convergence speed, the parameters are selected as above.

[0206] ① The attitude stability of a quadcopter UAV without system interference and without an event-triggered mechanism is shown in the simulation diagram. Figure 14 , Figure 15 As shown.

[0207] ② When dynamic interference exists, take Δ = 3e A The stability of the attitude of a quadcopter drone without an event-triggered mechanism is shown in the simulation diagram. Figure 16 , Figure 17 As shown.

[0208] Secondly, the impact of the event triggering mechanism on the system convergence speed is analyzed through simulation diagrams.

[0209] Scenario 1. The impact of event triggering mechanisms on system stability when there is no system interference.

[0210] Compared to e without an event-triggered mechanism R Simulation diagram ( Figure 14 ) and the introduction of event-triggered mechanisms in e R Simulation diagram ( Figure 2 , Figure 4 , Figure 6 ), and e without an event triggering mechanism Ω Simulation diagram ( Figure 15 ) and the introduction of event-triggered mechanisms in e Ω Simulation diagram ( Figure 3 , Figure 5 , Figure 7 Eight simulation images show that the angular velocity error vector e Ω and attitude error vector e R Ultimately, all values ​​tend to zero, indicating system stability. However, comparing these eight graphs reveals that the system under a controller without an event-triggered mechanism stabilizes quickly (around 50 seconds), while the system under a controller with an event-triggered mechanism stabilizes around 400, 500, and 350 seconds. This shows that introducing an event-triggered mechanism into the attitude control system of a quadcopter UAV can effectively reduce system power consumption, but it also prolongs the time it takes for the system to reach stability. Therefore, in practical applications, it is necessary to determine, based on different task requirements, whether to constantly update system signals, use an event-triggered mechanism to control signal updates, or adjust the event triggering conditions within the event-triggered mechanism.

[0211] Case 2. When dynamic disturbances exist, with Δ = 3e A For example, the impact of event triggering mechanisms on system stability.

[0212] Compared to e without an event-triggered mechanism R Simulation diagram ( Figure 16 ) and the introduction of event-triggered mechanisms in e R Simulation diagram ( Figure 8 ), and e without an event triggering mechanism Ω Simulation diagram ( Figure 17 ) and the introduction of event-triggered mechanisms in e Ω Simulation diagram ( Figure 9 Four simulation images show that the angular velocity error vector e Ω and attitude error vector e R Ultimately, all values ​​tend to zero, indicating system stability. However, comparing these four graphs reveals that the system under the controller without an event-triggered mechanism reaches a stable state around 200 seconds, while the system under the controller with an event-triggered mechanism takes around 900 seconds to stabilize. This shows that introducing an event-triggered mechanism into the attitude control system of a quadcopter drone can effectively reduce system power consumption, but it also prolongs the time it takes for the system to reach stability. Therefore, in practical applications, it is necessary to determine, based on different task requirements, whether to continuously update system signals, use an event-triggered mechanism to control signal updates, or adjust the event triggering conditions within the event-triggered mechanism.

[0213] The foregoing has shown and described the basic principles and main features of the present invention, as well as its advantages. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A quadcopter UAV attitude control method based on an event-triggered mechanism, characterized in that, Includes the following steps: Step 1: Introduce a rotation matrix to describe the attitude of the quadcopter drone: Let SO(3) be the set of rotation matrices, then we have Step 2: Provide the attitude dynamics system of the quadcopter UAV: in, The inertia matrix represents the positive definite inertia matrix. Indicates the system's control input, It is the angular velocity defined in a fixed coordinate system. Indicates system interference; Step 3: Design the angular velocity observer: The dynamic equation for the estimated angular velocity is as follows in, These represent the estimated values ​​of angular velocity and attitude, respectively. The error between the true and estimated attitude values; Step 4: Introduce an event triggering mechanism: Assume the trigger times are t1, t2, ..., t i ..., the system control input u satisfies: The trigger condition is defined as follows: Here, constant Used to describe the system state; Used to represent error. Step 5: Construct the system control input u: u6-k R e R -k Ω e Ω +Ω×JΩ+Jα+ν+ks(t i ) in 2. The quadrotor UAV attitude control method based on an event-triggered mechanism according to claim 1, characterized in that, When there is no system interference, the constant c in the system control input u satisfies the following equation:

3. The quadrotor UAV attitude control method based on an event-triggered mechanism according to claim 1, characterized in that, When dynamic disturbances exist, the constant c in the system control input u satisfies the following equation:

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