Unmanned vehicle formation obstacle avoidance control method based on hierarchical control

Abstract

The invention relates to an unmanned vehicle formation obstacle avoidance control method based on hierarchical control, and the method comprises the steps: building i unmanned vehicle hierarchical control system models, including building a kinematic equation and a kinetic equation, and defining a control target; the adaptive neural network processes uncertain items; obstacle avoidance is realized based on artificial field potential, and the method comprises the steps of introducing a potential field function to solve corresponding repulsive force; and designing a non-fragile sliding mode controller, including designing a kinematics control law and a dynamics control law, so as to realize unmanned vehicle formation obstacle avoidance. The method solves the problem of gain perturbation of the controller possibly caused by various factors under hierarchical control, adopts hierarchical control system modeling, is clearer in control logic compared with a traditional state space expression form, has more practical physical significance, and is more favorable for deploying theories to real objects.

Description

Technical Field

[0001] This invention relates to the field of unmanned vehicle control technology, and specifically to an unmanned vehicle platooning obstacle avoidance control method based on hierarchical control. Background Technology

[0002] In the field of unmanned vehicle (UAV) platooning control, several problems remain to be solved in practical applications: some control methods based on consistent state-space expressions only achieve position control of the UAVs, not attitude control. Their control logic is less rigorous and comprehensive than hierarchical control methods, and their system modeling lacks physical meaning, making practical verification difficult. Furthermore, some similar solutions fail to adequately consider controller gain perturbations caused by various factors, including actuator degradation, numerical calculation errors, component aging, and the need for parameter readjustment during use. This affects the stability of control performance. In addition, current research on hierarchical control theories is largely limited to networked control problems, such as network attacks and communication resource waste, rather than focusing on obstacle avoidance—problems more directly related to practical applications. This prevents the advantages of hierarchical control from being fully realized in real-world scenarios. These problems hinder the further application and development of UAV platooning control technology in search and rescue, military reconnaissance, and automated production lines. Therefore, a more comprehensive control method is urgently needed to overcome the bottlenecks of existing technologies. Summary of the Invention

[0003] The purpose of this invention is to provide an obstacle avoidance control method for unmanned vehicles based on hierarchical control, which can solve the controller gain perturbation phenomenon that may be caused by various factors under hierarchical control, and use a neural network adaptive strategy to solve the uncertainty in the modeling of hierarchical control system, and use an artificial force field method to solve the obstacle avoidance problem under hierarchical control of multiple unmanned vehicles, which is simple and effective.

[0004] This invention provides a hierarchical control-based obstacle avoidance control method for unmanned vehicle platooning, comprising:

[0005] Construct the hierarchical control system model for the i-th unmanned vehicle, including establishing kinematic and dynamic equations and defining the control objectives;

[0006] Adaptive neural networks handle uncertainties;

[0007] Obstacle avoidance based on artificial potential fields includes introducing a potential field function to calculate the corresponding repulsive force; and

[0008] Design non-fragile sliding mode controllers, including designing kinematic and dynamic control laws, to achieve obstacle avoidance in autonomous vehicle formations.

[0009] Furthermore, before constructing the hierarchical control system model for unmanned vehicles, the number of followers and leaders of unmanned vehicles in the platoon is determined, and the communication topology of the multi-unmanned vehicle system is determined to determine the corresponding adjacency matrix.

[0010] Furthermore, the construction of the i-th hierarchical control system model for the autonomous vehicle includes:

[0011] Establish the kinematic equations:

[0012]

[0013] Among them, x i (t), y i (t) represents the x and y coordinates of the position of the i-th unmanned vehicle in the Earth coordinate system as a function of time t, where the unmanned vehicle acts as a follower. Let x be the x-coordinate of the i-th autonomous vehicle in the Earth coordinate system. i (t) and ordinate y i (t) is the derivative with respect to time t, used to update the x and y coordinates of the autonomous vehicle's position; ψ i (t) is the yaw angle, which is used to determine the forward direction of the unmanned vehicle. ψ i (t)=atan2(y i (t),x i (t)), where atan2 is a mathematical function that returns y in radians. i (t) / x i The arctangent of (t), Represents ψ i The derivative of (t) with respect to time t is used to update the yaw angle of the autonomous vehicle; v i (t) is the linear velocity, used to represent the forward speed of the unmanned vehicle;

[0014] Establish the dynamic equations:

[0015]

[0016] Where, ω i (t) represents the angular velocity, which indicates the number of radians the autonomous vehicle rotates per unit time. Let be the moment of inertia of a single autonomous vehicle; This refers to the uncertainties in the linear and angular velocity directions of the system model; r i h is the radius of the wheels of a single autonomous vehicle. i It is half the distance between the two drive wheels of a single autonomous vehicle; The input torque is applied to the drive wheels of the autonomous vehicle; For external disturbances, satisfy in and A positive number indicates the upper bound of the external disturbance.

[0017] Furthermore, the control objective includes a geometric control objective and a dynamic control objective, wherein the geometric control objective is:

[0018]

[0019] The objective of the dynamic control is:

[0020]

[0021] Where, p i (t)=[x i (t),y i (t)] T Let p0(θ(t)) represent the position of the i-th autonomous vehicle, where p0(θ(t)) = [x0(θ(t)), y0(θ(t))]. T The virtual leader represents the position of the virtual leader, which is not a real autonomous vehicle but serves to provide a reference path for the autonomous vehicle to follow. θ(t) represents the path variable, used to describe the positional changes of the virtual leader, satisfying the expression... in Let θ(t) represent the derivative of θ(t) with respect to time t, used to update θ(t); v0(t) represents the forward speed of the virtual leader; χ(t) represents the update law, used to enable the autonomous vehicle to better follow the reference path provided by the virtual leader; q i0 (t) represents the expected distance between the i-th autonomous vehicle and the virtual leader (the virtual leader is marked with subscript 0); Representing a very small positive number, the significance of the geometric control objective is that, as time increases, the difference between the positional distance between the vehicles in the autonomous vehicle platoon and the set desired positional distance can be controlled within a very small range; Representing a very small positive number, the meaning of the dynamic control objective is to control the difference between the speed of the autonomous vehicle and the desired speed within a very small range as time increases.

[0022] Furthermore, adaptive neural networks handle uncertainties including:

[0023] An adaptive neural network is used to approximate the uncertainty term. and Its expression is:

[0024]

[0025] Among them, c i (t)=[v i (t),w(t)] T ε is the input vector of the neural network; iv (t), ε iw (t) represents the estimation error of the neural network, and satisfies Two unknown, very small positive numbers are used to illustrate that the estimation error of the neural network is bounded; σ(·) is the basis function; The ideal weight matrix;

[0026] The form of σ(·) is as follows:

[0027]

[0028] Where, r m x is a positive number, set to 1, where m represents the number of neurons; m This is the input acting on each neuron;

[0029] For ideal weights The estimation is performed, and the corresponding estimation error is:

[0030]

[0031] in, To and The estimate;

[0032] Therefore, the estimation error of the uncertainty function using neural network estimation is:

[0033] in,

[0034] That is, the uncertainty term and The estimation error is:

[0035]

[0036] Furthermore, obstacle avoidance based on artificial field potential includes:

[0037] Define the distance z between the autonomous vehicle and the obstacle. il (t) is:

[0038] z il (t)=p i (t)―O l

[0039] Where, p i(t)=[x i (t),y i (t)] T Let o be the position coordinates of the unmanned vehicle i. l Let l be the position vector of obstacle l;

[0040] The force field function is:

[0041]

[0042] Among them, Ψ l For ||z il (t)|| is the symbolic representation of the power field function of the independent variable;

[0043] The repulsive force is calculated using the force field function:

[0044]

[0045] When the autonomous vehicle enters the set range of the force field function, the obstacle begins to generate the repulsive force on the autonomous vehicle.

[0046] Furthermore, the design of kinematic control laws includes:

[0047] The positional error between autonomous vehicles is defined as:

[0048]

[0049] Differentiating with respect to time t, we get:

[0050]

[0051] Where, p j (t)=[x j (t),y j (t)] T Let q be the position of the j-th driverless car; ij (t) represents the position offset of the ideal formation; a ij The elements in the adjacency matrix represent whether there is communication interaction between autonomous vehicles. ij =1 indicates that driverless car j has a channel to transmit information to driverless car i, when a ij = 0 indicates that there is no channel for autonomous vehicle j to transmit information to autonomous vehicle i. The adjacency matrix is ​​a matrix used in graph theory to represent the adjacency relationship between nodes; therefore, in this invention, element a from the matrix is ​​used. ij This indicates whether there is a channel for information transmission between driverless vehicles;

[0052]

[0053] Where N represents the number of autonomous vehicles following the virtual leader in the entire autonomous vehicle platoon; This represents the partial derivative of p0(θ(t)) with respect to θ(t); α ei (t) represents the kinematic tracking error; Let v represent the expected linear velocity of driverless car i. i (t) represents the actual linear velocity of driverless vehicle i, and col(·) represents a column vector; Let ψ represent the expected heading angle of driverless vehicle i. i (t) represents the actual heading angle of the unmanned vehicle;

[0054] Based on the aforementioned position error, a virtual control law is derived:

[0055]

[0056] in, Indicates that for a si Find the inverse, K i4 ,i=1,2,3,…,N represents the gain of the kinematic controller to be designed, which is a two-dimensional positive definite matrix, and its value range can be determined through subsequent stability analysis.

[0057] Furthermore, based on the virtual control law, the corresponding desired linear velocity and desired heading angle are obtained:

[0058]

[0059] Where, ||α i (t)|| represents the virtual control law α i (t) Find the norm; k is an integer, k = 1, 2, 3, ...; With y i (t) and x i (t) has the same meaning, representing the ordinate and abscissa of the position of the unmanned vehicle i, respectively;

[0060] The update law is obtained:

[0061]

[0062] in, λ1>0 and λ2>0 are the derivatives of χ(t) with respect to time t, used to update χ(t); λ1>0 and λ2>0 are the adjustment parameters of the update law, and the specific values ​​are adjusted according to the actual effect.

[0063] Furthermore, the design of the dynamic control law includes:

[0064] The sliding mode function is:

[0065] S iv (t)=K i1 v ie(t)

[0066] S iw (t)=K i2 w ie (t)

[0067] in, This represents the difference between the actual linear velocity and the desired linear velocity of driverless vehicle i. K represents the difference between the actual angular velocity and the desired angular velocity of driverless vehicle i. i1 and K i2 For controller gain;

[0068] The controller gain perturbation is as follows:

[0069]

[0070] Where ΔK i1 (t), ΔK i2 (t) represents possible controller gain perturbations; Let be a random variable, following a Bernoulli distribution. ΔK i1 (t), ΔK i2 (t) is:

[0071] ΔK i1 (t)=μ i1 m i (t)μ i2

[0072] ΔK i2 (t)=μ i1 m i (t)μ i2

[0073] Where, m i (t) is a time-varying constant, satisfying (m i (t)) 2 <1;μ i1 =0.6, μ i2 =0.5. The time-varying constant can be set to m. i (t) = 0.5sin(t).

[0074] Furthermore, designing a non-fragile sliding mode controller also includes:

[0075]

[0076]

[0077] in and Let be the derivatives of the desired linear velocity and the desired angular velocity with respect to time t, respectively; sgn(Siv (t)) and sgn(S iw (t) represents the values ​​of S respectively. iv (t) and S iw (t) takes the sign function sgn(·); where, and If it is a positive number, it satisfies and External disturbances and The upper bound when external disturbances are considered. and When set to 0.5sin(t) and 0.6sin(t) respectively, It can be set to 0.51. It can be set to 0.61; K i1 and K i2 The range of values ​​for the controller gain is obtained through Lyapunov reachability and stability analysis; the specific values ​​need to be adjusted according to the specific situation. and Used to handle external interference and The adverse effects on the system; and Used to handle uncertainties in the system; and |μ i2 ||w ie (t)|sgn(S iw (t) is used to handle controller gain perturbations generated on the controller. and

[0078] The desired angular velocity is designed as follows:

[0079]

[0080] in K represents the difference between the actual heading angle and the desired heading angle of driverless vehicle i; i3 The gain of the controller to be designed can be obtained through Lyapunov stability analysis, and the specific value needs to be adjusted according to the actual situation.

[0081] The obstacle avoidance items are:

[0082]

[0083] Among them, U R Let β be the sum of the repulsive forces generated by k obstacles.l >0 represents the obstacle avoidance coefficient, which is adjusted according to the actual obstacle avoidance effect; l represents the l-th obstacle, and the role of this item in the above non-fragile sliding mode controller is to control the unmanned vehicle to avoid obstacles.

[0084] Furthermore, it also includes reachability analysis and stability analysis to prove that the tracking error system of the multi-unmanned vehicle platooning system enters the sliding diaphragm domain and stabilizes within a finite time, and obtains the inequality conditions for the gain of the controller to be designed.

[0085] Further, accessibility analysis:

[0086] Choosing Lyapunov functions Taking its infinitesimal operator yields:

[0087]

[0088] in

[0089] tr(·) represents the trace of an expression within parentheses; This represents the expectation of the content within the parentheses; Δt represents the time interval; min{·} represents the minimum value of the content within the parentheses; through the above derivation process, the linear velocity error v is obtained. ie (t) will converge to the sliding region. Where τ1∈(0,1) is an adjustable parameter; and the controller gain K is obtained simultaneously. i1 The range of values ​​for needs to satisfy the inequality

[0090] Choosing Lyapunov functions Taking its infinitesimal operator yields:

[0091]

[0092] in

[0093] The above derivation yields the following:

[0094] angular velocity error w ie (t) will converge to the sliding region. Where τ2∈(0,1) is an adjustable parameter; and the controller gain K is obtained simultaneously. i2 The range of values ​​for needs to satisfy the inequality as well as

[0095] Stability analysis:

[0096] Choosing Lyapunov functions

[0097]

[0098] Taking its infinitesimal operator yields:

[0099]

[0100]

[0101] in

[0102] Based on the above derivation, we can conclude that:

[0103] The dynamic subsystem is stable and the dynamic error will be resolved in a finite time. The convergence is achieved, where t0 is an arbitrary initial time; simultaneously, the controller gain K is derived. i1 The range of values ​​needs to satisfy Controller gain K i3 The value range needs to satisfy 2K i3 -1>0; Controller gain K i2 The range of values ​​needs to satisfy

[0104] Based on Lyapunov stability analysis, the adaptive law design for neural network weights is determined as follows:

[0105]

[0106] Among them, γ1 and γ2 are adaptive law parameters, γ1>0 and γ2>0. Both values ​​need to be adjusted according to the actual situation, and there is no specific theoretical derivation method. and They represent as well as The derivative with respect to time t is used to estimate the neural network weights using the above formula. and Update;

[0107] Choosing Lyapunov functions Taking its derivative with respect to time t, we get:

[0108]

[0109] make

[0110]

[0111] in

[0112]

[0113] K i4The values ​​of i = 1, 2, 3, ..., N represent the kinematic controller gain to be designed, which is a two-dimensional positive definite matrix. This gain is obtained by solving the matrix inequality D < 0. Both values ​​need to be adjusted based on the actual situation, a 10 This indicates whether the virtual leader has a channel to transmit information to vehicle number 1, and so on. N0 This indicates whether a virtual leader is transmitting information to vehicle N; for The upper bound of the norm, It is a positive number, and the specific value needs to be determined according to the actual situation; I represents the identity matrix, P 11 , P 21 , P N1 All are positive definite matrices, solved together using the matrix inequality D < 0; G3 = col{z(t), χ(t)}, where col{·} represents a column vector; z(t) = col{z1(t), ..., z N (t)};when hour, This indicates that the kinematic subsystem is stable and the kinematic error converges; where ||·|| represents the norm of the expression within the symbol; λ max (·) indicates taking the largest eigenvalue of the expression within the parentheses; P1=diag{P 11 ,P 21 ,…,P N1}, diag{·} denotes a diagonal matrix; a s =diag{a s1 ,a s2 ,…,a sN};α e (t)=col{α e1 (t),α e2 (t),…,α eN (t)};

[0114] The controller gain K can be obtained through reachability analysis and stability analysis. i1 The range of values ​​for needs to satisfy the inequality Controller gain K i2 The range of values ​​for needs to satisfy the inequality Controller gain K i3 The value range needs to satisfy 2K i3 -1>0; Controller gain K i4 The value is obtained by solving the matrix inequality D<0; the specific value can be adjusted according to actual needs.

[0115] The present invention has at least the following beneficial effects: 1) The present invention solves the controller gain perturbation phenomenon that may be caused by various factors under hierarchical control. The system modeling of hierarchical control is clearer in terms of control logic and has more practical physical meaning than the traditional state-space expression form, and is also more conducive to the deployment of theory to physical objects; 2) The present invention uses a neural network adaptive strategy to solve the uncertainty in the modeling of hierarchical control system, which can effectively offset the impact of modeling uncertainty on formation; 3) The present invention adopts the artificial force field method, which has low computational load, is easy to understand, and is easier to integrate into the design of controller. This method has also been widely used in robot and autonomous driving systems, which can achieve obstacle avoidance and plan safe and effective paths in complex environments, solving the obstacle avoidance problem under hierarchical control of multiple unmanned vehicles, with simple implementation and significant effect; 4) In the controller design of the present invention, the kinematic part introduces virtual control law and update law for leader path, which has better control effect and more compact control logic compared with traditional single controller, with progressive layers. The dynamic part considers the impact of controller gain perturbation and adopts sliding mode control to offset the adverse effects of external disturbances on the system, achieving rapid convergence of the system. Attached Figure Description

[0116] To further illustrate the above and other advantages and features of the various embodiments of the present invention, a more specific description of the embodiments of the invention will be presented with reference to the accompanying drawings. It is to be understood that these drawings depict only typical embodiments of the invention and are therefore not intended to limit its scope. In the drawings, identical or corresponding parts will be indicated by identical or similar reference numerals for clarity.

[0117] Figure 1 The diagram illustrates the communication topology of four driverless vehicles led by a virtual leader in some embodiments of the present invention.

[0118] Figure 2 This diagram illustrates a multi-driver system guided by a virtual leader in some embodiments of the present invention.

[0119] Figure 3 The diagram shows a schematic of the artificial field potential structure with two obstacles in some embodiments of the present invention;

[0120] Figure 4 The diagram shows an unmanned vehicle platoon obstacle avoidance schematic in some embodiments of the present invention;

[0121] Figure 5 The diagram shows the horizontal axis error of unmanned vehicle formation in some embodiments of the present invention;

[0122] Figure 6 The diagram shows the longitudinal coordinate error of unmanned vehicle formation in some embodiments of the present invention;

[0123] Figure label:

[0124] This represents the maximum distance at which the l-th obstacle can generate a repulsive force. d l This represents the minimum separation distance when approaching obstacle l, and is greater than... Greater than d l When the distance between the autonomous vehicle and the obstacle is less than d l At that time, the repulsive force will approach infinity, forcing the driverless car to avoid the obstacle. rep1 F represents the repulsive force exerted by obstacle 1 on the driverless car. rep2 F represents the repulsive force exerted by obstacle 2 on the driverless car. s F represents the resultant force of the repulsive forces exerted by obstacle 1 and obstacle 2 on the unmanned vehicle. att This indicates the attractiveness of the target point to the autonomous vehicle. Detailed Implementation

[0125] It should be noted that the components in the accompanying drawings may be shown exaggerated for illustrative purposes and may not be to scale.

[0126] In this invention, the various embodiments are merely intended to illustrate the solutions of the invention and should not be construed as limiting.

[0127] In this invention, unless otherwise specified, the quantifiers “a” and “one” do not exclude scenarios involving multiple elements.

[0128] It should also be noted that, in the embodiments of the present invention, only a portion of the parts or components may be shown for clarity and simplicity. However, those skilled in the art will understand that, under the teachings of the present invention, the required parts or components can be added as needed for specific scenarios.

[0129] It should also be noted that within the scope of this invention, the terms "same", "equal", and "equal to" do not mean that the two values ​​are absolutely equal, but allow for a certain reasonable error. In other words, the terms also cover "substantially the same", "substantially equal", and "substantially equal to".

[0130] It should also be noted that in the description of this invention, the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not explicitly or implicitly suggest that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0131] Furthermore, the embodiments of the present invention describe the process steps in a specific order. However, this is only for the convenience of distinguishing each step, and is not a limitation on the order of each step. In different embodiments of the present invention, the order of each step can be adjusted according to the process.

[0132] The following embodiments provide a hierarchical control-based obstacle avoidance control method for unmanned vehicle platooning, including:

[0133] Determine the number of followers and leaders of autonomous vehicles in a platoon, and determine the communication topology of the multi-autonomous vehicle system to determine the corresponding adjacency matrix;

[0134] Constructing the i-th hierarchical control system model for the autonomous vehicle includes establishing kinematic and dynamic equations and defining the control objective, specifically:

[0135] Establish the kinematic equations:

[0136]

[0137] Among them, x i (t), y i (t) represents the x and y coordinates of the i-th autonomous vehicle's position in the Earth coordinate system as a function of time t, where the autonomous vehicle acts as a follower. Let x be the x-coordinate of the i-th autonomous vehicle in the Earth coordinate system. i (t) and ordinate y i (t) is the derivative with respect to time t, used to update the x and y coordinates of the autonomous vehicle's position; ψ i (t) is the yaw angle, which is used to determine the forward direction of the unmanned vehicle. ψ i (t)=atan2(y i (t),x i (t)), where atan2 is a mathematical function that returns y in radians. i (t) / x i The arctangent of (t), (t) represents ψ i The derivative of (t) with respect to time t is used to update the yaw angle of the autonomous vehicle; v i (t) is the linear velocity, used to represent the forward speed of the unmanned vehicle;

[0138] Establish the dynamic equations:

[0139]

[0140] Where, ω i (t) represents the angular velocity, which indicates the number of radians the autonomous vehicle rotates per unit time. Let be the moment of inertia of a single autonomous vehicle; This refers to the uncertainties in the linear and angular velocity directions of the system model; r i h is the radius of the wheels of a single autonomous vehicle. i It is half the distance between the two drive wheels of a single autonomous vehicle. Figure 2 A diagram of a multi-driver system guided by a virtual leader is shown. The input torque is applied to the drive wheels of the autonomous vehicle; For external disturbances, satisfy in and A positive number indicates an upper bound on the external disturbance;

[0141] Control objectives include geometric control objectives and dynamic control objectives. The geometric control objective is:

[0142]

[0143] This means maintaining an ideal formation distance between followers and leaders, with distance error controlled within a very small and finite range;

[0144] The objective of dynamic control is:

[0145]

[0146] That is, controlling the leader's speed to bring it close to the set desired speed, with the speed error controlled within a very small and finite range;

[0147] Where, p i (t)=[x i (t),y i (t)] T Let p0(θ(t)) represent the position of the i-th autonomous vehicle, where p0(θ(t)) = [x0(θ(t)), y0(θ(t))]. TThe virtual leader represents the position of the virtual leader, which is not a real autonomous vehicle but serves to provide a reference path for the autonomous vehicle to follow. θ(t) represents the path variable, used to describe the positional changes of the virtual leader, satisfying the expression... in Let θ(t) represent the derivative of θ(t) with respect to time t, used to update θ(t); v0(t) represents the forward speed of the virtual leader; χ(t) represents the update law, used to enable the autonomous vehicle to better follow the reference path provided by the virtual leader; q i0 (t) represents the expected distance between the i-th autonomous vehicle and the virtual leader (the virtual leader is marked with subscript 0); Representing a very small positive number, the significance of the geometric control objective is that, as time increases, the difference between the positional distance between the vehicles in the autonomous vehicle platoon and the set desired positional distance can be controlled within a very small range; Representing a very small positive number, the meaning of the dynamic control objective is to control the difference between the speed of the autonomous vehicle and the desired speed within a very small range as time increases;

[0148] Adaptive neural networks handle uncertainties, specifically:

[0149] An adaptive neural network is used to approximate the uncertainty term. and Its expression is:

[0150]

[0151] Among them, c i (t)=[v i (t),w(t)] T ε is the input vector of the neural network; iv (t), ε iw (t) represents the estimation error of the neural network, and satisfies Two unknown, very small positive numbers are used to illustrate that the estimation error of the neural network is bounded; σ(·) is the basis function; The ideal weight matrix;

[0152] The form of σ(·) is as follows:

[0153]

[0154] Where, r m x is a positive number, set to 1, where m represents the number of neurons; m This is the input acting on each neuron;

[0155] For ideal weights The estimation is performed, and the corresponding estimation error is:

[0156]

[0157] in, To and The estimate;

[0158] Therefore, the estimation error of the uncertainty function using neural network estimation is:

[0159] in,

[0160] That is, the uncertainty term and The estimation error is:

[0161]

[0162] Obstacle avoidance is achieved based on artificial potential fields. This involves introducing a potential field function to calculate the corresponding repulsive force, treating obstacles as points with high repulsive force fields. When an unmanned vehicle (UAV) passes through the area affected by an obstacle, a repulsive force is generated within that area, causing the UAV to maintain a safe distance from the obstacle. This allows multiple UAVs in a platoon to bypass the obstacle and continue moving forward. Figure 3 A schematic diagram of the artificial potential structure with two obstacles is shown. Figure 3 The unmanned vehicle did not enter the maximum influence range of obstacle 2, so the actual repulsive force F generated was... rep2 The value is 0. In this invention, the attractive force is generated by the formation control itself, so no additional calculation is required; only the repulsive force exerted by each obstacle on the autonomous vehicle needs to be calculated. Obstacle avoidance based on artificial potential energy specifically includes:

[0163] Define the distance z between the autonomous vehicle and the obstacle. il (t) is:

[0164] z il (t)=p i (t)―o l

[0165] Where, p i (t)=[x i (t),y i (t)] T Let o be the position coordinates of the unmanned vehicle i. l Let l be the position vector of obstacle l (i.e., the position vector composed of horizontal and vertical coordinates); when the distance between the autonomous vehicle and the obstacle is less than or equal to... When the distance between the autonomous vehicle and the obstacle approaches a certain value, the repulsive field is triggered, generating a repulsive force that allows the vehicle to move around the obstacle. d l When this happens, the resulting repulsive force will approach infinity, forcibly preventing the driverless car from colliding with obstacles;

[0166] The power field function is:

[0167]

[0168] Among them, Ψ l For ||z il (t)|| is the symbolic representation of the power field function of the independent variable;

[0169] Calculate the repulsive force using the field function:

[0170]

[0171] When the autonomous vehicle enters the defined range of the force field function, the obstacles begin to exert a repulsive force on the vehicle. This represents the maximum distance at which the l-th obstacle can generate a repulsive force. d l This means that when the distance between the autonomous vehicle and the obstacle is less than this value, the repulsive force will approach infinity, forcing the autonomous vehicle to avoid the obstacle;

[0172] The design of a non-fragile sliding mode controller, including the design of kinematic and dynamic control laws, aims to achieve obstacle avoidance in autonomous vehicle formation. First, by setting the positional error between autonomous vehicles, a virtual kinematic control law is derived to stabilize the error system (it's called a virtual control law because there's no actual control input u in the kinematic part of the established system model, but a virtual control law is designed to correspond to the dynamic part). Then, the desired linear velocity and desired heading angle can be obtained from the designed virtual control law; these two parameters will be used in the subsequent design of the dynamic controller. Finally, to achieve coordination between the leader and followers, an update law is designed to update the virtual leader's path, thus providing a suitable reference path for the followers. Specifically:

[0173] Designing kinematic control laws includes:

[0174] The positional error between autonomous vehicles is defined as:

[0175]

[0176] Differentiating with respect to time t, we get:

[0177]

[0178] Where, p j (t)=[x j (t),y j (t)] T Let q be the position of the j-th driverless car; ij (t) represents the position offset of the ideal formation; a ij The elements in the adjacency matrix represent whether there is communication interaction between autonomous vehicles. ij =1 indicates that driverless car j has a channel to transmit information to driverless car i, when a ij = 0 indicates that there is no channel for autonomous vehicle j to transmit information to autonomous vehicle i. The adjacency matrix is ​​a matrix used in graph theory to represent the adjacency relationship between nodes; therefore, in this invention, element a from the matrix is ​​used. ij This indicates whether there is a channel for information transmission between driverless vehicles;

[0179]

[0180] Where N represents the number of autonomous vehicles following the virtual leader in the entire autonomous vehicle platoon; This represents the partial derivative of p0(θ(t)) with respect to θ(t); α ei (t) represents the kinematic tracking error; Let v represent the expected linear velocity of driverless car i. i (t) represents the actual linear velocity of driverless vehicle i, and col(·) represents a column vector; Let ψ represent the expected heading angle of driverless vehicle i. i (t) represents the actual heading angle of the unmanned vehicle;

[0181] The virtual control law is derived based on the position error:

[0182]

[0183] in, Indicates that for a si Find the inverse, K i4 ,i=1,2,3,…,N represents the gain of the kinematic controller to be designed, which is a two-dimensional positive definite matrix, and its value range can be determined through subsequent stability analysis.

[0184] Based on the virtual control law, the desired linear velocity and desired heading angle are obtained:

[0185]

[0186] Where, ||α i (t)|| represents the virtual control law α i (t) Find the norm; k is an integer, k = 1, 2, 3, ...; With y i (t) and x i (t) has the same meaning, representing the ordinate and abscissa of the position of the unmanned vehicle i, respectively;

[0187] The update law is obtained:

[0188]

[0189] in, λ1>0 and λ2>0 are the derivatives of χ(t) with respect to time t, used to update χ(t); λ1>0 and λ2>0 are the adjustment parameters of the update law, and the specific values ​​are adjusted according to the actual effect: the values ​​of λ1 and λ2 need to satisfy the matrix inequality D<0 in the stability analysis. After solving this matrix inequality, the approximate range can be determined. Then, based on the stability of the system, by checking the position and speed error waveforms and the formation effect diagram of multiple unmanned vehicles, the values ​​that make the error close to 0 and the formation trajectory smooth and non-oscillating can be adjusted.

[0190] After the kinematic control laws are designed, the dynamic control laws can be designed. The design of the dynamic control laws includes:

[0191] The sliding mode function is:

[0192] S iv (t)=K i1 v ie (t)

[0193] S iw (t)=K i2 w ie (t)

[0194] in, This represents the difference between the actual linear velocity and the desired linear velocity of driverless vehicle i. K represents the difference between the actual angular velocity and the desired angular velocity of driverless vehicle i. i1 and K i2 For controller gain;

[0195] The controller gain perturbation is as follows:

[0196]

[0197]

[0198] Among them, controller gain perturbation refers to a random parameter fluctuation, ΔK, that occurs in the originally designed controller gain. i1 (t), ΔK i2 (t) represents possible controller gain perturbations; Let be a random variable, following a Bernoulli distribution: ΔK i1 (t), ΔK i2 (t) is:

[0199] ΔK i1 (t)=μ i1 m i (t)μ i2

[0200] ΔK i2 (t)=μ i1 m i (t)μ i2

[0201] Where, m i (t) is a time-varying constant, satisfying (m i (t)) 2 <1;μ i1 =0.6, μ i2 =0.5, the time-varying constant can be set to m i (t) = 0.5sin(t);

[0202] Designing non-fragile sliding mode controllers also includes:

[0203]

[0204] The two expressions above constitute the final controller design, corresponding to the kinematic model in... and These will serve as the input torque to control the autonomous vehicle, guiding it along the desired path. This controller addresses the impact of controller gain perturbations and mitigates the effects of uncertainties on the system through neural network estimation of uncertainties in system modeling. The introduction of sliding mode control effectively eliminates the influence of external disturbances on the system. and Let be the derivatives of the desired linear velocity and the desired angular velocity with respect to time t, respectively; sgn(S iv (t)) and sgn(S iw (t) represents the values ​​of S respectively. iv (t) and S iw (t) takes the sign function sgn(·); where, and If it is a positive number, it satisfies and External disturbances and The upper bound when external disturbances are considered. and When set to 0.5sin(t) and 0.6sin(t) respectively, It can be set to 0.51. It can be set to 0.61; k i1 and k i2 For the controller gain, the range of values ​​is obtained through Lyapunov reachability and stability analysis. The specific value is adjusted according to the specific situation: within the range of values, the magnitude is adjusted by viewing the waveforms of position error and speed error, as well as the overall formation effect diagram of multiple unmanned vehicles, so that the error is as close to 0 as possible and the formation trajectory does not oscillate. and Used to handle external interference and The adverse effects on the system; and Used to handle uncertainties in the system; and Used to handle controller gain perturbations generated on the controller. and

[0205] The desired angular velocity is designed as follows:

[0206]

[0207] in K represents the difference between the actual heading angle and the desired heading angle of driverless vehicle i; i3 The gain of the controller to be designed is obtained through Lyapunov reachability and stability analysis to determine its range. Within this range, the gain is adjusted by observing the waveforms of position error and velocity error, as well as the overall formation effect of multiple unmanned vehicles, so that the error is as close to 0 as possible and the formation trajectory does not oscillate.

[0208] The obstacle avoidance items are:

[0209]

[0210] Among them, U R Let β be the sum of the repulsive forces generated by k obstacles. l >0 represents the obstacle avoidance coefficient, which is adjusted according to the actual obstacle avoidance effect; l represents the l-th obstacle, and in the above non-fragile sliding mode controller, this term controls the autonomous vehicle to avoid obstacles; and

[0211] Accessibility and stability analyses are performed to demonstrate that the tracking error system of the multi-unmanned vehicle platooning system enters the slip diaphragm domain and stabilizes within a finite time, and inequality conditions for the gain of the controller to be designed are obtained, specifically:

[0212] Accessibility analysis includes:

[0213] Choosing Lyapunov functions Taking its infinitesimal operator yields:

[0214]

[0215] in

[0216] tr(·) represents the trace of an expression within parentheses; This represents the expectation of the content within the parentheses; Δt represents the time interval; min{·} represents the minimum value of the content within the parentheses; through the above derivation process, the linear velocity error v is obtained. ie (t) will converge to the sliding region. Where τ1∈(0,1) is an adjustable parameter; and the controller gain K is obtained simultaneously. i1 The range of values ​​for needs to satisfy the inequality

[0217] Choosing Lyapunov functions Taking its infinitesimal operator yields:

[0218]

[0219] in

[0220] The above derivation yields the following:

[0221] angular velocity error w ie (t) will converge to the sliding region. Where τ2∈(0,1) is an adjustable parameter; and the controller gain K is obtained simultaneously. i2 The range of values ​​for needs to satisfy the inequality

[0222] Stability analysis includes:

[0223] Choosing Lyapunov functions

[0224]

[0225] Taking its infinitesimal operator yields:

[0226]

[0227]

[0228] in

[0229] Based on the above derivation, we can conclude that:

[0230] The dynamic subsystem is stable and the dynamic error will be resolved in a finite time. The convergence is achieved, where t0 is an arbitrary initial time; simultaneously, the controller gain K is derived. i1 The range of values ​​needs to satisfy Controller gain K i3 The value range needs to satisfy 2K i3 -1>0; Controller gain K i2 The range of values ​​needs to satisfy

[0231] Through Lyapunov stability analysis, the adaptive law design for the neural network weights that ensures system stability is determined as follows:

[0232]

[0233] Among them, γ1 and γ2 are adaptive law parameters, γ1>0 and γ2>0. Both values ​​need to be adjusted according to the actual situation, and there is no specific theoretical derivation method. and They represent as well as The derivative with respect to time t is used to estimate the neural network weights using the above formula. and Update;

[0234] Choosing Lyapunov functions Taking its derivative with respect to time t, we get:

[0235]

[0236] make

[0237]

[0238] in

[0239]

[0240] K i4 The values ​​of i = 1, 2, 3, ..., N represent the kinematic controller gain to be designed, which is a two-dimensional positive definite matrix. This gain is obtained by solving the matrix inequality D < 0. Adjusting the two values: After solving the matrix inequality D<0 to obtain the range of values, adjust the magnitude within this range by observing the waveforms of position error and velocity error, as well as the overall formation effect diagram of multiple unmanned vehicles, so that the error is as close to 0 as possible and the formation trajectory does not oscillate; a 10This indicates whether the virtual leader has a channel to transmit information to vehicle number 1, and so on. N0 This indicates whether a virtual leader is transmitting information to vehicle N; for The upper bound of the norm, It is a positive number, and the specific value depends on the actual situation: it is set by the amplitude of the reference trajectory. For example, when the reference trajectory of the virtual leader is set to 40sin(0.5t), Set it to a number slightly larger than 20; I represents the identity matrix, P 11 , P 21 , P N1 All are positive definite matrices, solved together using the matrix inequality D < 0; G3 = col{z(t), χ(t)}, where col{·} represents a column vector; z(t) = col{z1(t), ..., z N (t)};when hour, This indicates that the kinematic subsystem is stable and the kinematic error converges; where ||·|| represents the norm of the expression within the symbol; λ max (·) indicates taking the largest eigenvalue of the expression within the parentheses; P1=diag{P 11 ,P 21 ,…,P N1}, diag{·} denotes a diagonal matrix; a s =diag{a s1 ,a s2 ,…,a sN};α e (t)=col{α e1 (t),α e2 (t),…,α eN (t)};

[0241] The controller gain K can be obtained through reachability analysis and stability analysis. i1 The range of values ​​for needs to satisfy the inequality Controller gain K i2 The range of values ​​for needs to satisfy the inequality Controller gain K i3 The value range needs to satisfy 2K i3 -1>0; Controller gain K i4 The controller gain K is obtained by solving the matrix inequality D<0. The specific value can be adjusted according to actual needs. i4 Multiple solutions are obtained by solving the matrix inequality D<0. Then, based on the position error, velocity error and formation effect diagram of different solutions, the solution that makes the error as close to 0 as possible and prevents formation oscillation is selected.

[0242] In this embodiment, the autonomous vehicle platoon includes four follower vehicles and one leader vehicle, and the communication topology diagram is as follows. Figure 1 As shown, the direction of information transmission between autonomous vehicles can be determined by the arrows in the communication topology. Here, 0 represents the virtual leader, transmitting information to autonomous vehicles 1 and 2; autonomous vehicle 1 transmits information to autonomous vehicle 4; and autonomous vehicle 2 transmits information to autonomous vehicle 3. From this, the elements in the adjacency matrix can be derived: a 10 =1,a 20 =1,a 41 =1,a 32 =1. The following are some parameter settings for the autonomous vehicle itself: h i =0.186m,r i =0.042m. Assume the uncertainty term present in the system modeling is... External interference is The desired distance between autonomous vehicles is set to q. 10 (t) = [3,3] T ,q 20 (t) = [3, -3] T ,q 32 (t) = [―6,0] T ,q 41 (t) = [―6,0] T The virtual leader's speed is set to v0(t) = 1 m / s, x0(θ(t)) = 0.12θ(t), y0(θ(t)) = 40sin(0.0075θ(t)); the neural network's adaptive parameters are set to γ1 = γ2 = 5; the obstacle avoidance coefficient is set to β. l =5, d l =6; ΔK i1 =ΔK i2 =0.6×0.5sin(6t)×0.5, the probability of the random variable is set to Therefore, through reachability stability analysis and experimental simulation, suitable controller parameters and update law parameters can be designed as λ1=20, λ2=12. K 11 =K 12 =15,K 13 =1,K 24 =diag{20,20},K 21 =K 22 =15,K23 =10,K 34 =diag{75,75},K 31 =K 32 =15,K 33 =10,K 44 =diag{60,60},K 41 =K 42 =15,K 43 =10, Figure 4 The diagram illustrates the obstacle avoidance of the convoy, specifically showing the obstacle avoidance effect when a convoy consisting of a virtual leader and four AGVs (AGV1, AGV2, AGV3, and AGV4) encounters an obstacle. Figure 4 As can be seen, apart from the virtual leader serving as a reference path (the virtual leader is not an actual car, but only serves to provide a reference path), the paths of the other four cars changed when they approached obstacles 1 and 2 (indicated by the black cross "×"), forming an arc effect and successfully avoiding the obstacles. When they passed obstacle 3, the distance from the obstacle exceeded the effective range, so the convoy continued to move forward along the expected path. Figure 5 The diagram shows the horizontal coordinate error of the autonomous vehicle formation, which represents the difference in the horizontal coordinate between the actual position of the autonomous vehicles and the desired formation position. Figure 6 The diagram shows the vertical coordinate error of the unmanned vehicle formation, which is the difference between the vertical coordinate of the actual position of the unmanned vehicle and the desired formation position. It can be seen that the errors of both the horizontal and vertical coordinates eventually stabilized at a position close to 0, indicating good control effect and achieving the goal of unmanned vehicle formation.

[0243] While some embodiments of the present invention have been described in this application, those skilled in the art will understand that these embodiments are merely illustrative. Numerous variations, alternatives, and improvements will arise in those skilled in the art under the teachings of this invention without departing from its scope. The appended claims are intended to define the scope of the invention and thereby cover methods and structures within the scope of the claims themselves and their equivalents.

Claims

1. A hierarchical control-based obstacle avoidance control method for unmanned vehicle platooning, characterized in that, include: Construct the hierarchical control system model for the i-th unmanned vehicle, including establishing kinematic and dynamic equations and defining the control objectives; Adaptive neural networks handle uncertainties; Obstacle avoidance is achieved based on artificial field potential, including introducing a potential field function to obtain the corresponding repulsive force; as well as Design non-fragile sliding mode controllers, including designing kinematic and dynamic control laws, to achieve obstacle avoidance in autonomous vehicle formations.

2. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, Constructing the i-th hierarchical control system model for an autonomous vehicle includes: Establish the kinematic equations: Among them, x i (t), y i (t) represents the x and y coordinates of the position of the i-th unmanned vehicle in the Earth coordinate system as a function of time t, where the unmanned vehicle acts as a follower. Let x be the x-coordinate of the i-th autonomous vehicle in the Earth coordinate system. i (t) and ordinate y i (t) is the derivative of ψ with respect to time t; i (t) is the yaw angle, ψ i (t)=atan2(y i (t),x i (t)), Represents ψ i (t) is the derivative of v with respect to time t. i (t) represents the linear velocity; Establish the dynamic equations: Where, ω i (t) is the angular velocity. Let be the moment of inertia of a single autonomous vehicle; This refers to the uncertainties in the linear and angular velocity directions of the system model; r i h is the radius of the wheels of a single autonomous vehicle. i It is half the distance between the two drive wheels of a single autonomous vehicle; The input torque is applied to the drive wheels of the autonomous vehicle; For external disturbances, satisfy in and A positive number indicates the upper bound of the external disturbance.

3. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, The control objectives include geometric control objectives and dynamic control objectives, wherein the geometric control objectives are: lim t→∞ ||p i (t)―p0(θ(t))―q i0 (t)||<l1 The objective of the dynamic control is: Where, p i (t)=[x i (t),y i (t)] T ,p0(θ(t))=[x0(θ(t)),y0(θ(t))] T The virtual leader's position is represented by θ(t); the path variable satisfies the expression... in Let θ(t) be the derivative of θ(t) with respect to time t, v0(t) be the forward velocity of the virtual leader, χ(t) be the update law, and q be the derivative of θ(t) with respect to time t. i0 (t) represents the expected distance between the i-th autonomous vehicle and the virtual leader; l1 represents a very small positive number; l2 represents a very small positive number.

4. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, Adaptive neural networks handle uncertainties including: An adaptive neural network is used to approximate the uncertainty term. and Its expression is: Among them, c i (t)=[v i (t),w(t)] T ε is the input vector of the neural network; iv (t), ε iw (t) represents the estimation error of the neural network, and satisfies Two unknown, very small positive numbers are used to illustrate that the estimation error of the neural network is bounded; σ(·) is the basis function; The ideal weight matrix; The form of σ(·) is as follows: Where, r m x is a positive number, set to 1, where m represents the number of neurons; m This is the input acting on each neuron; For ideal weights The estimation is performed, and the corresponding estimation error is: in, To and The estimate; Therefore, the estimation error of the uncertainty function using neural network estimation is: That is, the uncertainty term and The estimation error is:

5. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, Obstacle avoidance based on artificial field potential includes: Define the distance z between the autonomous vehicle and the obstacle. il (t) is: z il (t)=p i (t)―o l Where, p i (t)=[x i (t),y i (t)] T Let o be the position coordinates of the unmanned vehicle i. l Let l be the position vector of obstacle l; The force field function is: Among them, Ψ l For ||z il (t)|| is the symbolic representation of the power field function of the independent variable; The repulsive force is calculated using the force field function: When the autonomous vehicle enters the set range of the force field function, the obstacle begins to generate the repulsive force on the autonomous vehicle.

6. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, Designing kinematic control laws includes: The positional error between autonomous vehicles is defined as: Differentiating with respect to time t, we get: Where, p j (t)=[x j (t),y j (t)] T Let q be the position of the j-th driverless car; ij (t) represents the position offset of the ideal formation; a ij The elements in the adjacency matrix represent whether there is communication interaction between autonomous vehicles. ij =1 indicates that driverless car j has a channel to transmit information to driverless car i, when a ij =0 indicates that driverless car j does not have a channel to transmit information to driverless car i; Where N represents the number of autonomous vehicles following the virtual leader in the entire autonomous vehicle platoon; This represents the partial derivative of p0(θ(t)) with respect to θ(t); α ei (t) represents the kinematic tracking error; Let v represent the expected linear velocity of driverless car i. i (t) represents the actual linear velocity of driverless vehicle i, and col(·) represents a column vector; Let ψ represent the expected heading angle of driverless vehicle i. i (t) represents the actual heading angle of the unmanned vehicle; Based on the aforementioned position error, a virtual control law is derived: in, Indicates that for a si Find the inverse, K i4 ,i=1,2,3,…,N represents the gain of the kinematic controller to be designed, which is a two-dimensional positive definite matrix.

7. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 6, characterized in that, Based on the virtual control law, the desired linear velocity and desired heading angle are obtained: Where, ||α i (t)|| represents the virtual control law α i (t) Find the norm; k is an integer, k = 1, 2, 3, ...; With y i (t) and x i (t) has the same meaning, representing the ordinate and abscissa of the position of the unmanned vehicle i, respectively; The update law is obtained: in, λ1>0, λ2>0 are the derivative of χ(t) with respect to time t, used to update χ(t); λ1>0, λ2>0 are the adjustment parameters of the update law.

8. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, The design of dynamic control laws includes: The sliding mode function is: S iv (t)=K i1 v ie (t) S iw (t)=K i2 w ie (t) in, This represents the difference between the actual linear velocity and the desired linear velocity of driverless vehicle i. K represents the difference between the actual angular velocity and the desired angular velocity of driverless vehicle i. i1 and K i2 For controller gain; The controller gain perturbation is as follows: Where ΔK i1 (t), ΔK i2 (t) represents possible controller gain perturbations; Let be a random variable, following a Bernoulli distribution. ΔK i1 (t), ΔK i2 (t) is: ΔK i1 (t)=μ i1 m i (t)μ i2 ΔK i2 (t)=μ i1 m i (t)μ i2 Where, m i (t) is a time-varying constant, satisfying (m i (t)) 2 <1;μ i1 =0.6, μ i2 =0.

5.

9. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, Designing non-fragile sliding mode controllers also includes: in and Let be the derivatives of the desired linear velocity and the desired angular velocity with respect to time t, respectively; sgn(S iv (t)) and sgn(S iw (t) represents the values ​​of S respectively. iv (t) and S iw (t) takes the sign function sgn(·); where, and If it is a positive number, it satisfies and External disturbances and The upper bound of K; i1 and K i2 For controller gain; The desired angular velocity is designed as follows: in K represents the difference between the actual heading angle and the desired heading angle of driverless vehicle i; i3 The gain of the controller to be designed; The obstacle avoidance items are: Among them, U R Let β be the sum of the repulsive forces generated by k obstacles. l >0 represents the obstacle avoidance coefficient, which is adjusted according to the actual obstacle avoidance effect; l represents the l-th obstacle, and the role of this item in the above non-fragile sliding mode controller is to control the unmanned vehicle to avoid obstacles.

10. The unmanned vehicle platooning obstacle avoidance control method based on hierarchical control according to claim 1, characterized in that, It also includes reachability analysis and stability analysis to prove that the tracking error system of the multi-unmanned vehicle platooning system enters the slip diaphragm domain and stabilizes within a finite time, and obtains the inequality conditions for the gain of the controller to be designed.

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