CN114296449B - Water surface unmanned ship track rapid tracking control method based on fixed time H-infinity control - Google Patents

Abstract

The invention discloses a fast tracking control method for the trajectory of a surface unmanned boat based on fixed time H _∞ control, comprising the steps of: S1, establishing a surface unmanned boat control system including kinematics and dynamic models; S2, based on kinematics and Dynamic model, build the trajectory tracking error model of the surface unmanned vehicle; S3, design a fixed-time expansion state observer based on the kinematics and dynamics of the surface unmanned vehicle; S4, establish a fixed-time H _∞ controller; S5, based on the surface unmanned vehicle The boat trajectory tracking error model, the fixed-time dilated state observer and the fixed-time H _∞ controller are designed to assist the controller to establish a dynamic system; S6, based on the dynamic system, the fixed-time H _∞ trajectory tracking controller is designed, and stability analysis is performed. The invention not only improves the robustness of the surface unmanned boat control system, but also the convergence time of the system does not depend on the initial value of the system, and on the premise of ensuring the rapid convergence of the tracking error, the system has high stability.

Description

Fast Tracking Control Method of Surface Unmanned Vehicle Based on Fixed Time H∞ Control

technical field

The invention belongs to the technical field of surface unmanned boat control, and in particular relates to a fast tracking control method for a surface unmanned boat trajectory based on fixed time H _∞ control.

Background technique

With the rapid rise of many marine activities including military reconnaissance, environmental monitoring, marine rescue and submarine channel inspection, the problem of trajectory tracking and control of surface unmanned boats has become a hot research topic. Its core work is to design control laws and explore control methods to make the ship's trajectory tracking more accurate in a stable state. However, since wind, waves and currents have important effects on the speed and maneuverability of the ship, how to achieve fast and accurate trajectory tracking is a major challenge for the controller.

In the current surface unmanned vehicle controller, the uncertainty of the hull model parameters and the complex and changeable marine environment disturbance have a huge impact on the stability of the system. The PID (proportional-integral-derivative) control commonly used in research controllers today does not require an accurate system model and has acceptable control performance in most cases, but the traditional PID control has poor anti-interference ability, slow control speed and The parameter setting is complicated and other problems. Usually, the parameter setting is supported by multiple trials and rich experience in parameter adjustment. Even when the parameters are adjusted to a better control effect under the current system state, the time-varying external environment interferes. It will also have a great impact on its control effect.

At present, for the uncertainty of the parameters of the surface UAV and the disturbance of the marine environment, the common control method is usually to design a robust controller or use a neural network to approximate the lumped disturbance of the system, and design an effective trajectory tracking control law to make the water surface The unmanned boat can track the desired trajectory from the initial state to complete the specified task, and ensure the global consistent and progressive stability of the trajectory tracking error in a short period of time, thereby achieving the requirements of high-precision and rapid deployment in the designated area. However, due to the complex operation of neural network and the limited online calculation speed, some inevitable problems are caused, such as the slow adjustment speed of the controller, the low control accuracy, and the inability to achieve convergence in a limited time. In practical engineering, because the complex marine environment has a great influence on the accuracy of the speed sensor, the state information of the controller is not easy to obtain.

Therefore, it is an urgent problem to be solved in the industry to design a control method for the surface unmanned vehicle system that can achieve global fast and stable in a fixed time according to the actual marine environment, so that the system has high stability under the premise of ensuring the rapid convergence of the tracking error. The problem.

SUMMARY OF THE INVENTION

In order to solve at least one of the above technical problems, the present invention proposes a fast tracking control method for the trajectory of the surface unmanned boat based on fixed time H _∞ control.

The object of the present invention is achieved through the following technical solutions:

The present invention provides a fast tracking control method for the trajectory of the surface unmanned boat based on fixed time H _∞ control, comprising the following steps:

S1. Establish a surface unmanned vehicle control system including kinematics and dynamic models;

S2. Based on the kinematics and dynamics model, build a trajectory tracking error model of the surface unmanned boat;

S3. Design a fixed time expansion state observer based on the kinematics and dynamics of the surface UAV;

S4, establish a fixed time H _∞ controller;

S5. The auxiliary controller is designed to establish a dynamic system based on the trajectory tracking error model of the surface unmanned vehicle, the fixed-time expansion state observer and the fixed-time H _∞ controller;

S6. Design a fixed-time H _∞ trajectory tracking controller based on the dynamic system, and conduct stability analysis.

As a further improvement, in the step S1, establishing a surface unmanned vehicle controller including kinematics and dynamic models includes the following steps:

S11. Establish the kinematics and dynamics model of the surface unmanned vehicle;

S12. Perform coordinate transformation on the kinematics and dynamics model of the surface unmanned vehicle;

S13, processing the parameter uncertainty and time-varying interference of the surface unmanned vehicle control system on the kinematics and dynamics model of the surface unmanned vehicle.

S14, obtain the desired trajectory of the kinematic and dynamic model of the surface unmanned vehicle.

As a further improvement, in the step S2, based on the kinematics and dynamics model, a trajectory tracking error model of the surface unmanned boat is built, including the following steps:

S21. Define the trajectory tracking error of the surface unmanned vehicle and the speed error of the surface unmanned vehicle in a fixed coordinate system, and obtain the tracking error dynamics of the surface unmanned vehicle;

S22, combining steps S13 and S14 to rewrite the tracking error dynamics of the surface unmanned boat.

As a further improvement, in the step S3, the fixed time dilation state observer is designed based on the surface unmanned vehicle controller and the surface unmanned vehicle trajectory tracking error model, including the following steps:

S31. Design a fixed time dilation state observer, and then define its observation error;

S32, further design a fixed time dilation state observer according to the defined observation error.

As a further improvement, in the step S4, establishing a fixed-time H _∞ controller includes the following steps:

S41. Define a closed-loop system including a causal dynamic compensator;

S42. Use the theorem to prove that the closed-loop system defined by technical analysis is stable in global fixed time, then the causal dynamic compensator is a global fixed time H _∞ controller.

As a further improvement, in the step S5, a dynamic system is established based on the trajectory tracking error model of the surface unmanned vehicle, the fixed time dilation state observer and the fixed time H _∞ controller design auxiliary controller, including the following steps:

S51, according to steps S22, S32 and S42, change the tracking error dynamics of the surface unmanned boat within a fixed time;

S52, introduce an auxiliary controller and define a tracking error variable;

S53, combine the tracking error dynamics of the surface unmanned vehicle with the defined tracking error variables to establish a dynamic system.

As a further improvement, in step S6, a fixed-time H _∞ trajectory tracking controller is designed based on the dynamic system, and stability analysis is performed, including the following steps:

S61. Define an auxiliary function for realizing the stability of the dynamic system in a fixed time;

S62, a performance vector is defined in the dynamic system;

S63. Design a fixed-time H _∞ trajectory tracking controller strategy based on a fixed-time-expanded state observer according to the auxiliary function and the performance vector;

S64 , based on the fixed-time H _∞ trajectory tracking controller strategy, using theorem proving technology to analyze that the surface unmanned vehicle can track the desired trajectory within a fixed time and does not depend on the initial state of the surface unmanned vehicle.

As a further improvement, in the step S11, the kinematics and dynamics model of the surface unmanned vehicle is established as follows:

代表水面无人艇在大地坐标系下的速度与角速度向量，

代表水面无人艇在船体坐标系下的加速度与角加速度向量，M为惯性矩阵，C(v)为科里奥利向心力矩阵，D(v)为流体阻尼矩阵，上角标T代表矩阵的转置，τ为水面无人艇的控制输入向量，w(t)为风浪以及海流海洋环境所造成的时变扰动。Among them, n represents the three-degree-of-freedom pose vector of the surface unmanned vehicle in the horizontal plane under the geodetic coordinate system, y represents the heading angle, R(ψ) represents the coordinate transformation matrix between the geodetic coordinate system and the hull coordinate system, R Represents a real number, v represents the velocity and angular velocity vector of the surface unmanned vehicle in the horizontal plane under the hull coordinate system,

represents the velocity and angular velocity vector of the surface unmanned vehicle in the geodetic coordinate system,

Represents the acceleration and angular acceleration vectors of the surface UAV in the hull coordinate system, M is the inertia matrix, C(v) is the Coriolis centripetal force matrix, D(v) is the fluid damping matrix, and the superscript T represents the matrix Transpose, τ is the control input vector of the surface UAV, and w(t) is the time-varying disturbance caused by wind, waves and ocean currents.

As a further improvement, in the step S21, the tracking error dynamics of the UAV obtained are as follows:

Among them, _ne is the trajectory tracking error of the surface unmanned vehicle, _ve is the velocity error of the surface unmanned vehicle in the fixed coordinate system, and v _d represents the expected value of the speed and angular velocity of the surface unmanned vehicle in the horizontal plane under the hull coordinate system. .

As a further improvement, in the step S31, the observation error of the fixed time dilation state observer is defined as follows:

为n的观测值，

为μ的观测值，

为χ的观测值，

均为估计误差值。Among them, μ=R(ψ)v, χ is the aggregate disturbance of the surface unmanned boat control system,

is the observed value of n,

is the observed value of μ,

is the observed value of χ,

Both are estimated error values.

The present invention provides a fast tracking control method for the trajectory of a surface unmanned boat based on fixed time H _∞ control, comprising the steps of: S1, establishing a surface unmanned boat control system including kinematics and dynamic models; S2, based on kinematics and Dynamic model, build the trajectory tracking error model of the surface unmanned vehicle; S3, design a fixed-time expansion state observer based on the kinematics and dynamics of the surface unmanned vehicle; S4, establish a fixed-time H _∞ controller; S5, based on the surface unmanned vehicle The boat trajectory tracking error model, the fixed-time dilated state observer and the fixed-time H _∞ controller are designed to assist the controller to establish a dynamic system; S6, based on the dynamic system, the fixed-time H _∞ trajectory tracking controller is designed, and stability analysis is performed. The present invention is further studied on the basis of H _∞ control, and the designed fixed time dilation state observer is used to estimate the state information and aggregate disturbance of the system, which is more in line with the practical application requirements of engineering, and establishes the entire closed loop of the fixed time H _∞ controller. The system achieves global stability in a fixed time, which greatly improves the convergence rate and stability of tracking control. It not only improves the robustness of the surface unmanned vehicle control system, but also the convergence time of the system does not depend on the initial value of the system. Under the premise of fast convergence of tracking error, the system has high stability.

Description of drawings

The present invention will be further described by using the accompanying drawings, but the embodiments in the accompanying drawings do not constitute any limitation to the present invention. For those of ordinary skill in the art, under the premise of no creative work, other Attached.

FIG. 1 is a schematic flow chart of the present invention.

FIG. 2 is a schematic diagram of the kinematics and dynamics model of the present invention.

FIG. 3 is a schematic diagram of the expected trajectory and the actual trajectory of the kinematics and dynamics model of the present invention.

4 is a schematic diagram of the tracking situation of the surge trajectory of the present invention; in the figure, x is the longitudinal position coordinate of the surface unmanned boat under the geodetic coordinate system, and x _d is the desired longitudinal position coordinate of the surface unmanned boat under the geodetic coordinate system.

5 is a schematic diagram of the sway track tracking situation of the present invention; in the figure, y is the lateral position coordinate of the surface unmanned boat under the geodetic coordinate system, and y _d is the desired lateral position coordinate of the surface unmanned boat under the geodetic coordinate system.

FIG. 6 is a schematic diagram of the tracking situation of the yaw trajectory of the present invention; in the figure, ψ is the heading angle, and ψ _d is the desired heading angle.

为水面无人艇在船体坐标系下纵向估计速度坐标。Fig. 7 is the schematic diagram of the velocity estimation situation in the surge direction of the present invention; in the figure, u is the actual longitudinal velocity coordinate of the surface unmanned boat in the hull coordinate system,

Estimate the longitudinal velocity coordinates for the surface UAV in the hull coordinate system.

为水面无人艇在船体坐标系下横荡估计速度坐标。8 is a schematic diagram of the speed estimation situation in the sway direction of the present invention; in the figure, v is the actual speed coordinate of the unmanned surface swaying in the hull coordinate system,

Estimate the velocity coordinates for the sway of the surface unmanned vehicle in the hull coordinate system.

为水面无人艇在船体坐标系下艏摇方向估计速度坐标。9 is a schematic diagram of the speed estimation in the yaw direction of the present invention; in the figure, r is the actual speed coordinate of the surface unmanned boat in the yaw direction of the hull coordinate system,

The velocity coordinates are estimated for the yaw direction of the surface unmanned vehicle in the hull coordinate system.

Fig. 10 is a schematic diagram of the control input of the surge direction according to the present invention; in the figure, τ _u is the longitudinal control input of the surface unmanned boat along the hull coordinate system.

Fig. 11 is a schematic diagram of the control input of the sway direction of the present invention; in the figure, τ _v is the control input of the surface unmanned boat along the lateral direction of the hull coordinate system.

12 is a schematic diagram of the control input of the yaw direction of the present invention; in the figure, τ _r is the control input of the yaw direction of the surface unmanned boat along the hull coordinate system.

Detailed ways

In order to make those skilled in the art better understand the technical solutions of the present invention, the present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments. and features of the embodiments may be combined with each other.

An embodiment of the present invention provides a fast tracking control method for a surface unmanned boat trajectory based on a fixed time H _∞ control, as shown in FIG. 1 , including the following steps:

S1. Establish a surface unmanned vehicle control system including kinematics and dynamic models, including the following steps:

S11, establish the kinematics and dynamics model of the surface unmanned boat, as shown in Figure 2, as follows:

是n的一阶导数，

代表水面无人艇在大地坐标系下的速度与角速度向量；

是v的一阶导数，

代表水面无人艇在船体坐标系下的加速度与角加速度向量；M为惯性矩阵；C(v)为科里奥利向心力矩阵；D(v)为流体阻尼矩阵；上角标T代表矩阵的转置；τ＝[τ_u，τ_v，τ_r]^T为水面无人艇系统的控制输入向量；w(t)为风浪以及海流海洋环境所造成的时变扰动；其中，转换矩阵R(ψ)满足以下性质：Among them, n=[x, y, ψ] ^T represents the three-degree-of-freedom pose vector of the surface unmanned vehicle in the horizontal plane under the geodetic coordinate system Xe-Oe-Ye, and x and y represent the geodetic coordinate of the surface unmanned vehicle, respectively. Longitudinal and lateral position coordinates in the system Xb-Ob-Yb, ψ represents the heading angle; R(ψ) represents the coordinate transformation matrix between the geodetic coordinate system and the hull coordinate system, R(ψ)∈R ^3×3 , R represents Real number; v=[u, v, r] ^T , v represents the velocity and angular velocity vector of the surface unmanned vehicle in the horizontal plane under the hull coordinate system,

is the first derivative of n,

Represents the velocity and angular velocity vector of the surface unmanned vehicle in the geodetic coordinate system;

is the first derivative of v,

Represents the acceleration and angular acceleration vectors of the surface unmanned vehicle in the hull coordinate system; M is the inertia matrix; C(v) is the Coriolis centripetal force matrix; D(v) is the fluid damping matrix; the superscript T represents the matrix Transpose; τ=[τ _u , τ _v , τ _r ] ^T is the control input vector of the surface unmanned aerial vehicle system; w(t) is the time-varying disturbance caused by the wind, waves and current ocean environment; among them, the transformation matrix R ( ψ) satisfies the following properties:

是R的一阶导数；

为反对称矩阵。where R ^-1 is the inverse of the R matrix,

is the first derivative of R;

is an antisymmetric matrix.

S12, define μ=R(ψ)v, after the coordinate transformation of the kinematics and dynamics model of the surface unmanned vehicle is as follows:

是μ的一阶导数；M^-1是M矩阵的逆矩阵；f(n,μ)为简化矩阵；f(n,μ)＝S(r)μ-R(ψ)M^-1(C(v)+D(v))v；考虑到水面无人艇系统船模的参数不确定性，f(n,μ)可以如下表达：where μ is an auxiliary variable,

is the first derivative of μ; M ^-1 is the inverse matrix of M matrix; f(n, μ) is a simplified matrix; f(n, μ)=S(r)μ-R(ψ)M ^-1 (C( v)+D(v))v; Considering the parameter uncertainty of the surface unmanned boat system model, f(n, μ) can be expressed as follows:

f(n,μ)＝f ₀ (n,μ)+△f (Equation 4)

Among them, f ₀ (n, μ) is the nominal value form of the uncertain term in the surface unmanned boat control system, and △f is the uncertainty part of the surface unmanned boat control system. The specific mathematical expression is as follows:

where C ₀ (v) represents the nominal value of the Coriolis centripetal force matrix, D ₀ (v) represents the nominal value of the fluid damping matrix, ΔC(v) represents the uncertain value of the Coriolis centripetal force matrix, ΔD(v) represents the uncertainty value of the fluid damping matrix.

S13. After the uncertainty of the parameters of the control system of the surface unmanned vehicle and the time-varying interference are processed for the kinematics and dynamics model of the unmanned surface vehicle, the kinematics and dynamics model of the unmanned surface vehicle are as follows:

Among them, χ=△f(n, μ)+w(t) is regarded as the aggregate disturbance received by the surface unmanned vehicle.

S14. The desired trajectory of the kinematic and dynamic model of the surface UAV is obtained as follows:

和

分别是n_d的一阶导数和二阶导数；v_d＝[u_d,v_d,r_d]^T表示水面无人艇在船体坐标系下水平面内的速度与角速度期望值，u_d为u的期望值，v_d为v的期望值，r_d为r的期望值；

是v_d是一阶导数。where n _d =[x _d , y _d , ψ _d ] ^T represents the expected value of the three-degree-of-freedom pose and attitude of the surface unmanned vehicle in the horizontal plane under the geodetic coordinate system, x _d is the expected value of x, y _d is the expected value of y, ψ _d is the expected value of ψ;

and

are the first derivative and second derivative of n _d respectively; v _d =[ _ud ,v _d ,r _d ] ^T represents the expected value of the speed and angular velocity of the surface unmanned vehicle in the horizontal plane under the hull coordinate system, and _ud is the value of u Expected value, v _d is the expected value of v, r _d is the expected value of r;

is v _d is the first derivative.

S2. Based on the kinematics and dynamics model, build a trajectory tracking error model of the surface unmanned boat, including the following steps:

S21. Define n _e =nn _d , _ve =vv _d , and obtain the tracking error dynamics of the surface unmanned boat as follows:

是n_e是一阶导数；

是v_e的一阶导数。In the formula, n _e represents the trajectory tracking error of the surface unmanned vehicle in the geodetic coordinate system; _ve represents the speed error of the surface unmanned vehicle in the hull coordinate system;

is n _e is the first derivative;

is the first derivative of _ve .

结合步骤S13的式6和步骤S14的式7对水面无人艇跟踪误差动态进行改写如下所示：S22, define x ₁ = _ne ,

Combined with the formula 6 of step S13 and the formula 7 of step S14, the dynamic rewriting of the tracking error of the surface unmanned boat is as follows:

Among them, x ₁ represents the trajectory tracking error of the surface UAV in the geodetic coordinate system; x ₂ is the first derivative of x ₁ .

S3. Design a fixed time dilation state observer based on the kinematics and dynamics of the surface UAV, including the following steps:

S31. Design a fixed time dilation state observer, and then define its observation error as follows:

为n的估计值；

为μ的估计值；

为χ的估计值；

均为估计误差值。in,

is the estimated value of n;

is the estimated value of μ;

is the estimated value of χ;

Both are estimated error values.

S32, according to the defined observation error and formula 6, the fixed time dilation state observer is further designed as follows:

是

的一阶导数；

是

的一阶导数；

是

的一阶导数；

是χ的一阶导数，χ_d表示正实数；sgn代表符号函数，选择合适的状态观测器增益系数k_i(i＝1,2,3)和l_i(i＝1,2,3)，使得P₁ 矩阵和P₂ 矩阵为Hurwitz(赫尔维茨)矩阵，则n、μ和χ可在固定时间T₀内分别被

和

估计。in,

Yes

the first derivative of ;

Yes

the first derivative of ;

Yes

the first derivative of ;

is the first derivative of χ, χ _d represents a positive real number; sgn represents the sign function, select the appropriate state observer gain coefficients ki ( _i =1, 2, 3) and li ( _i =1, 2, 3), Let the P ₁ matrix and the P ₂ matrix be Hurwitz (Hurwitz) matrices, then n, μ and χ can be divided into fixed time T ₀

and

estimate.

Convergence time is satisfied

正常数

Q₁,Q₂,Ω₁,Ω₂都是非奇异、对称且正定的矩阵。此外，上诉参数矩阵分别满足

Among them, r ₁ represents a simplified matrix and satisfies r ₁ =λ _min (Q ₁ )/λ _max (Ω ₁ ); r ₂ represents a simplified matrix and satisfies r ₂ =λ _min (Q ₂ )/λ _max (Ω ₂ );

normal number

Q ₁ , Q ₂ , Ω ₁ , Ω ₂ are all nonsingular, symmetric and positive definite matrices. Furthermore, the appeal parameter matrix satisfies the

S4, establishing a fixed-time H _∞ controller, including the following steps:

S41. Definition 1: Consider a closed-loop system including a causal dynamic compensator as follows:

Among them, x is the vector reflecting the state information of the system, f(x) and g(x) both represent the nonlinear term about x in the closed-loop system, Ξ(x) is the controller input vector of the system, and w(x) is the unknown disturbance, Z is a performance vector, and w is external disturbance.

A causal dynamic compensator looks like this:

Ξ=φ(x, t) (Equation 15)

The causal dynamic compensator Ξ is a global fixed-time _H∞ controller if the following two conditions are met.

(1) When w=0, the closed-loop system in Equation 14 and Equation 15 is globally fixed-time stable;

(2) Given a real number γ>0, for all t ₁ >t ₀ and all nonlinear perturbations, if the output z produced by w and the initial state x(t ₀ )=0 satisfy the following inequality involving definite integrals:

where z(t) represents the performance function vector varying with time t; w(t) represents the disturbance term with respect to time t; then the L2 gain of the closed _- loop system in Equations 14 and 15 is less than or equal to γ.

S42. Use the theorem to prove that the closed-loop system defined by technical analysis is stable in global fixed time, then the causal dynamic compensator is a global fixed time H _∞ controller.

中定义了一个函数V(x)，对于在原点附近的V(x)，存在实数c₁＞0，c₂＞0和0＜a₁＜1，a₂＞1使得：Theorem 1: Consider Equation 9, assuming that in a neighborhood

A function V(x) is defined in , for V(x) near the origin, there exist real numbers c ₁ >0, c ₂ >0 and 0 < _{a 1} <1, a ₂ >1 such that:

(1) V(x) is a positive definite function in U ₀ ;

(2)

Among them, c ₁ , c ₂ are convergence speed control coefficients; a ₁ , a ₂ are fixed-time convergence control parameters.

Then, the origin of the closed-loop system Eq. 14 is locally fixed _- time stable, and the L2 gain of the system is less than or equal to γ. If U ₀ =R ⁿ and V(x) is radially unbounded (ie V(x)→+∞ when ||x||→+∞), the origin of the closed-loop system Eq. 14 is globally fixed-time stable of.

Proof: From Definition 1, two conditions must be satisfied.

(1) When w=0, the condition (2) in Theorem 1 can be written as:

According to Lemma 1, this closed-loop system consisting of Eqs. 14 and 15 is fixed-time stable.

(2) When w≠0, V(x)>0, then the following inequality holds:

Condition (2) in Definition 1 is satisfied, so the L ₂ gain of Equation 15 of the closed-loop system is less than or equal to γ, and the proof of Theorem 1 is completed.

S5. The auxiliary controller is designed to establish a dynamic system based on the trajectory tracking error model of the surface unmanned vehicle, the fixed-time expansion state observer and the fixed-time H _∞ controller, including the following steps:

可知，在固定时间T₀内转变水面无人艇的跟踪误差动态如下：S51. According to Equation 9 of Step S22, Equation 13 of S32, Lemma 1 of S42, and

It can be seen that the tracking error dynamics of transforming the surface unmanned vehicle within a fixed time T ₀ are as follows:

是

的一阶导数，

是水面无人艇在大地坐标系下轨迹跟踪的误差估计值；

是

的一阶导数。in,

Yes

The first derivative of ,

is the error estimate of the trajectory tracking of the surface unmanned vehicle in the geodetic coordinate system;

Yes

the first derivative of .

定义跟踪误差变量

如下：S52. Introduce an auxiliary controller

Define tracking error variable

as follows:

S53, combine the tracking error dynamic equation 19 of the surface unmanned vehicle with the defined tracking error variable equation 20 to establish a dynamic system as follows:

S6. Design a fixed-time H _∞ trajectory tracking controller based on the dynamic system, and conduct stability analysis.

S61. Define the auxiliary function used to realize the stability of the dynamic system formula 21 in a fixed time as follows:

表示一个自变量为轨迹跟踪误差

的辅助函数；

表示一个自变量为轨迹跟踪误差

的辅助函数；α、β表示固定时间收敛控制参数并且满足0<a<1，β>1；p_i(i＝1,2,3,4)＝diag{p_i1,p_i2,...,p_in}均为正定对角矩阵，均表示辅助函数控制增益系数；符号函数sign满足关系：sig^a(·)＝sign(·)|·|^a,sig^β(·)＝sign(·)|·|^β。in,

Represents an independent variable as trajectory tracking error

helper function;

Represents an independent variable as trajectory tracking error

_{Auxiliary functions of} ,p _in } are all positive definite diagonal matrices, all representing the auxiliary function control gain coefficient; the sign function sign satisfies the relationship: sig ^a (·)=sign(·)|·| ^a ,sig ^β (·)=sign(·) |·| ^β .

S62, a performance vector is defined in the dynamic system formula 21 as follows:

的加权系数。Among them, λ ₁ >0, λ ₂ >0, representing x ₁ and

weighting factor.

S63. Design a fixed-time H _∞ trajectory tracking controller strategy based on a fixed-time-expanded state observer according to the auxiliary function and the performance vector as follows:

是辅助函数

的一阶导数；

是辅助函数

的一阶导数；通过选择适当的参数，动态系统式21在有限的时间内达到稳定状态，并且闭环系统的L₂增益小于或者等于γ。in,

is the helper function

the first derivative of ;

is the helper function

The first derivative of ; by choosing appropriate parameters, the dynamic system Eq. 21 reaches a steady state in a finite time, and the L2 gain of the closed _- loop system is less than or equal to γ.

S64. Based on the fixed-time H _∞ trajectory tracking controller strategy, the theorem proving technique is used to analyze that the surface unmanned vehicle can track the desired trajectory within a fixed time and does not depend on the initial state of the surface unmanned vehicle, as follows:

Theorem 2: When the surface unmanned vehicle tracks the desired trajectory, the system uncertainty, marine environment disturbance and unpredictable state are considered, and the designed fixed-time H _∞ trajectory tracking controller strategy Equation 24 can ensure that the surface unmanned The boat tracks the desired trajectory in a fixed time, and the error convergence time may not depend on the initial state of the surface unmanned boat. The upper bound T _s of the convergence time is as follows:

T _s =T ₀ +T ₁ (Equation 25)

Proof: The proof of reaching the stable stage in a fixed time is as follows:

Substitute the fixed-time H _∞ trajectory tracking controller strategy equation 24 into the dynamic system equation 21, we get:

Design a Lyapunov function (Lyapunov function, namely V function):

Combining Equation 22, Equation 24 and Equation 26, and taking the derivative of V, we get:

Define function H:

且p^*＝min{p_0i}，那么可以得到：Substitute Equation 22, Equation 26 and Equation 28 into Equation 29, and select a positive real number p ^* that satisfies

And p ^* = min{p _0i }, then we can get:

根据引理1可得：Assuming σ=(1+α)/2, 1/2<σ<1, δ=(1+β)/2, p _1min =min{p _1i },p _2min =min{p _2i },p _3min = min{p _3i }p _4min =min{p _4i },

According to Lemma 1, we can get:

Combining Equation 29 and Equation 31, we can get:

均表示收敛速度控制系数；σ、δ均表示固定时间收敛控制参数。in,

Both represent the convergence rate control coefficient; σ and δ both represent the fixed-time convergence control parameters.

From Theorem 1, we know that the dynamic system Eq. 21 is globally stable in fixed time, and from Lemma 2, the upper bound of the convergence time can be calculated:

和

在固定时间T₁内收敛到0，结合固定时间扩张状态观测器的收敛时间T₀，依据分离原理，整个闭环系统是在固定时间内达到全局稳定且不依赖于系统的初始状态，定理2证明完毕。According to the fixed-time convergent system theory, the error can be known

and

Convergence to 0 within a fixed time T ₁ , combined with the convergence time T ₀ of the fixed time dilation state observer, according to the separation principle, the entire closed-loop system is globally stable within a fixed time and does not depend on the initial state of the system. Theorem 2 proves that complete.

Further, Lemma 1 in the proof step is as follows:

和

那么对于

以下不等式被满足：if it exists

and

then for

The following inequalities are satisfied:

为幂次项，x_i为正实数。In the formula,

is a power term, and x _i is a positive real number.

Further, Lemma 2 in the appeal proof step is as follows:

Consider the following system:

是y的一阶导数，y(0)为系统状态y在0时刻初始值，y₀为系统状态在0时刻初始值；b₁、b₂均为收敛速度控制系数并且满足b₁>0，b₂>0；q₁、q₂为固定时间收敛控制参数并且满足0<q₁<1，q₂>1；系统的平衡点是固定时间稳定的，且其收敛时间的上界T_max可不依赖初始状态计算得到，如下：Among them, y is the system output value,

is the first derivative of y, y(0) is the initial value of the system state y at time 0, y ₀ is the initial value of the system state at time 0; b ₁ and b ₂ are both convergence speed control coefficients and satisfy b ₁ >0, b ₂ >0; q ₁ and q ₂ are fixed-time convergence control parameters and satisfy 0<q ₁ <1, q ₂ >1; the equilibrium point of the system is stable at fixed time, and the upper bound T _max of its convergence time may not be It is calculated depending on the initial state, as follows:

Among them, _Tmax is the upper bound of the fixed time convergence.

The present invention is further studied on the basis of _{H ∞} control, and the designed fixed time dilation state observer is used to estimate the state information and lumped disturbance of the control system, which is more in line with the practical application requirements of engineering. The closed-loop system achieves global stability in a fixed time, which greatly improves the convergence rate and stability of the tracking control, not only improves the robustness of the surface UAV control system, but also the convergence time of the system does not depend on the initial value of the system. Under the premise of ensuring the fast convergence of tracking error, the system has high stability.

In order to verify the effectiveness of the embodiments of the present invention, simulation experiments were carried out, as follows:

Combined with the Cybership II (Cybership II) ship model, simulation experiments are carried out to verify the effectiveness of the fast tracking control method for the trajectory of the surface UAV based on the fixed-time H _∞ control. The Cybership II ship model parameters are shown in Table 1.

Table 1

The parameter settings in the surface unmanned boat control system are as follows in Table 2:

Table 2

The time-varying perturbation is as follows:

The reference track is as follows:

The simulation results are shown in Figure 3-Figure 12, showing that the designed trajectory tracking controller enables the surface UAV to accurately track the desired trajectory in about 5s, and has good robustness. Obviously, the proposed control scheme can guarantee that the convergence time is less than the maximum value and can be calculated independently of the substituted initial state.

Many specific details are set forth in the above description to facilitate a full understanding of the present invention. However, the present invention can also be implemented in other ways different from those described herein, so it should not be construed as limiting the protection scope of the present invention.

In a word, although the present invention lists the above-mentioned preferred embodiments, it should be noted that although those skilled in the art can make various changes and modifications, unless such changes and modifications deviate from the scope of the present invention, they should be included in the present invention. within the scope of protection of the invention.

Claims (8)

1. Based on fixed time H _∞ The fast track control method for the unmanned surface vehicle track is characterized by comprising the following steps:

s1, establishing a water surface unmanned ship control system comprising a kinematics and dynamics model;

s2, building a water surface unmanned ship trajectory tracking error model based on a kinematics and dynamics model;

s3, designing a fixed time extended state observer based on the kinematics and dynamics of the unmanned surface vehicle;

s4, establishing fixed time H _∞ A controller;

s5, tracking error model based on unmanned surface vehicle trajectory, fixed time extended state observer and fixed time H _∞ The controller design assists the controller to establish a dynamic system, and comprises the following steps:

s51, changing the tracking error dynamics of the unmanned surface vehicle within a fixed time:

wherein,

is that

The first derivative of (a) is,

is an error estimation value of track tracking of the unmanned surface vehicle under a geodetic coordinate system,

is that

The first derivative of (a) is,

is that

The first derivative of (a), R (psi) is a coordinate transformation matrix between a geodetic coordinate system and a ship body coordinate system, M is an inertia matrix, tau is a control input vector of a water surface unmanned ship control system, f ₀ (n, mu) is in a nominal value form containing uncertain items in the water surface unmanned ship control system,

is an estimated value of chi, the chi is the lumped disturbance of the unmanned surface vehicle control system,

is n _d Second derivative of, n _d The expected value of the three-degree-of-freedom pose of the unmanned surface vehicle in the horizontal plane under the geodetic coordinate system is obtained;

s52, introducing an auxiliary controller

Defining tracking error variables

The following were used:

s53, combining the steps S51 and S52 to establish a dynamic system as follows:

wherein:

representing an argument as a tracking error

The auxiliary function of (2);

s6, designing fixed time H based on dynamic system _∞ A trajectory tracking controller and performing stability analysis, comprising the steps of:

s61, defining the auxiliary function for realizing the stability of the dynamic system of the step S53 in a fixed time as follows:

wherein,

representing an argument as a tracking error

α, β represent fixed time convergence control parameters and satisfy 0<a<1，β>1；p _i (i＝1,2,3,4)＝diag{p _i1 ,p _i2 ,...,p _in All positive definite diagonal matrixes represent auxiliary function control gain coefficients, and the sign function sign satisfies the relation: sig ^a (·)＝sign(·)|·| ^a ,sig ^β (·)＝sign(·)|·| ^β ；

S62, defining a performance vector in the dynamic system of step S53 as follows:

wherein λ is ₁ >0,λ ₂ >0, each represents

And

the weighting coefficient of (2);

s63, designing a fixed time H infinity trajectory tracking controller strategy based on the fixed time extended state observer according to the auxiliary function of the step S61 and the performance vector of the step S62 as follows:

wherein the superscript T represents the transpose of the matrix,

is a secondary function

The first derivative of (a) is,

is a secondary function

Is given as a real number;

s64, analyzing that the unmanned surface vehicle can track the expected track in a fixed time and is independent of the initial state of the unmanned surface vehicle by utilizing a theorem proving technology based on a fixed time H infinity track tracking controller strategy.

2. Fixed time base H according to claim 1 _∞ The method for controlling the rapid tracking of the trajectory of the unmanned surface vehicle is characterized in that in the step S1, the step of establishing the unmanned surface vehicle controller comprising a kinematics and dynamics model comprises the following steps:

s11, establishing a kinematics and dynamics model of the unmanned surface vehicle;

s12, carrying out coordinate transformation on the kinematics and dynamics model of the unmanned surface vehicle;

s13, carrying out parameter uncertainty and time-varying interference processing on the kinematics and dynamics model of the unmanned surface vehicle;

and S14, obtaining the expected track of the kinematics and dynamic model of the unmanned surface vehicle.

3. Fixed time base H according to claim 2 _∞ The controlled water surface unmanned ship track rapid tracking control method is characterized in that in the step S2, a water surface unmanned ship track tracking error model is built based on a kinematics and dynamics model, and the method comprises the following steps:

s21, defining track tracking errors of the unmanned surface vehicle and speed errors of the unmanned surface vehicle in a fixed coordinate system and obtaining the tracking error dynamics of the unmanned surface vehicle;

and S22, combining the step S13 and the step S14 to rewrite the tracking error dynamic of the unmanned surface boat.

4. Fixed time base H according to claim 3 _∞ The controlled water surface unmanned ship track rapid tracking control method is characterized in that in the step S3, a fixed time extended state observer is designed based on a water surface unmanned ship controller and a water surface unmanned ship track tracking error model, and the method comprises the following steps:

s31, designing a fixed time extended state observer and defining the observation error of the observer;

and S32, further designing the fixed-time extended state observer according to the defined observation error.

5. Fixed time base H according to claim 4 _∞ The method for controlling the rapid tracking of the trajectory of the unmanned surface vehicle is characterized in that in the step S4, fixed time H is established _∞ A controller comprising the steps of:

s41, defining a closed loop system containing a causal dynamic compensator;

s42, the closed loop system analytically defined by the theorem proving technology is stable in global fixed time, and the causal dynamic compensator is a global fixed time H _∞ And a controller.

6. Fixed time base H according to claim 2 _∞ The method for controlling the rapid tracking of the trajectory of the unmanned surface vehicle is characterized in that in the step S11, a kinematics and dynamics model of the unmanned surface vehicle is established as follows:

wherein n represents the water of the unmanned surface vehicle in the geodetic coordinate systemThree-degree-of-freedom pose vector in a plane, psi represents a course angle, R represents a real number, v represents the velocity and angular velocity vector of the unmanned surface vessel in the horizontal plane under a vessel body coordinate system,

representing the velocity and angular velocity vector of the unmanned surface vehicle under a geodetic coordinate system,

representing the acceleration and angular acceleration vectors of the unmanned surface vehicle in a hull coordinate system, wherein C (v) is a Coriolis centripetal force matrix, D (v) is a fluid damping matrix, and w (t) is time-varying disturbance caused by storms and ocean current marine environments.

7. Fixed time base H according to claim 3 _∞ The controlled water surface unmanned ship track rapid tracking control method is characterized in that in the step S21, the following dynamic states of the tracking error of the water surface unmanned ship are obtained:

wherein n is _e Tracking error of unmanned surface vehicle trajectory, v _e Is the speed error v of the unmanned surface vehicle under a fixed coordinate system _d And the expected values of the speed and the angular speed of the unmanned surface vehicle in the horizontal plane under the ship body coordinate system are shown.

8. Fixed time base H according to claim 4 _∞ The method for controlling the rapid tracking of the trajectory of the unmanned surface vehicle is characterized in that in the step S31, the observation error of the fixed-time extended state observer is designed and defined as follows:

wherein, mu ═ R (psi) nu,

is the observed value of n and is,

is the observed value of mu, and the measured value of mu,

the observed value of x is the value of x,

are both estimated error values.

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