CN114296449B - Water surface unmanned ship track rapid tracking control method based on fixed time H-infinity control - Google Patents
Water surface unmanned ship track rapid tracking control method based on fixed time H-infinity control Download PDFInfo
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Abstract
本发明公开了一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法,包括步骤:S1、建立包括运动学和动力学模型的水面无人艇控制系统;S2、基于运动学和动力学模型,搭建水面无人艇轨迹跟踪误差模型;S3、基于水面无人艇运动学和动力学设计固定时间扩张状态观测器;S4、建立固定时间H∞控制器;S5、基于水面无人艇轨迹跟踪误差模型、固定时间扩张状态观测器和固定时间H∞控制器设计辅助控制器建立动态系统;S6、基于动态系统设计固定时间H∞轨迹跟踪控制器,并进行稳定性分析。本发明不仅提高了水面无人艇控制系统的鲁棒性,而且系统的收敛时间不依赖系统初始值,在保证跟踪误差快速收敛的前提下,系统具有较高的稳定性。
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
技术领域technical field
本发明属于水面无人艇控制的技术领域,具体涉及一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法。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.
背景技术Background technique
随着军事侦察、环境监测、海上救援和潜艇通道检查在内的许多海洋活动的迅速兴起,水面无人艇的轨迹跟踪控制问题已成为现今研究的热点。其核心工作是设计控制律,探索控制方法,使船舶在稳定状态下,轨迹跟踪更加精确。然而,由于风、浪和海流对船舶的速度和操纵性有着重要的影响,如何实现快速且精确的轨迹跟踪是控制器面临的一大挑战。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.
在当前水面无人艇控制器中,船体模型参数的不确定性以及复杂多变的海洋环境扰动给系统的稳定性带来巨大影响。现今研究控制器中常用的PID(比例-积分-微分)控制不需要精确的系统模型且大部分情况下具有可接受的控制性能,但是传统的PID控制存在抗干扰能力不强,控制速度缓慢且参数整定复杂等问题,通常情况下参数的整定依靠多次的试凑以及丰富的调参经验支持,即使在当前的系统状态下将参数调整至较好的控制效果时,时变的外界环境干扰也会对其控制效果造成很大的影响。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.
目前,针对水面无人艇参数的不确定性以及海洋环境扰动问题,常见的控制方法通常是设计鲁棒控制器或者采用神经网络逼近系统的集总扰动,设计有效的轨迹跟踪控制律,使得水面无人艇能够实现从初始状态下跟踪设定的期望轨迹完成规定任务,并且在较短的时间内保证轨迹跟踪误差的全局一致渐进稳定,进而实现在指定区域的高精度快速部署作业要求。但是,由于神经网络的运算复杂以及在线计算速度有限导致一些不可避免的问题,如控制器的调整速度较慢、控制精度低、无法实现在有限的时间内收敛等。在实际工程中,由于复杂的海洋环境对速度传感器精度影响较大,因此控制器的状态信息也是不易获取的。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.
因此,针对实际海洋环境设计一种在固定时间内实现全局快速稳定的水面无人艇系统控制方法,使其在保证跟踪误差快速收敛的前提下,系统具有较高的稳定性,成为业内亟待解决的问题。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.
发明内容SUMMARY OF THE INVENTION
为解决上述技术问题中的至少之一,本发明提出一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法。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.
本发明的目的通过以下技术方案实现:The object of the present invention is achieved through the following technical solutions:
本发明提供了一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法,包括如下步骤: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、建立包括运动学和动力学模型的水面无人艇控制系统;S1. Establish a surface unmanned vehicle control system including kinematics and dynamic models;
S2、基于运动学和动力学模型,搭建水面无人艇轨迹跟踪误差模型;S2. Based on the kinematics and dynamics model, build a trajectory tracking error model of the surface unmanned boat;
S3、基于水面无人艇运动学和动力学设计固定时间扩张状态观测器;S3. Design a fixed time expansion state observer based on the kinematics and dynamics of the surface UAV;
S4、建立固定时间H∞控制器;S4, establish a fixed time H ∞ controller;
S5、基于水面无人艇轨迹跟踪误差模型、固定时间扩张状态观测器和固定时间H∞控制器设计辅助控制器建立动态系统;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、基于动态系统设计固定时间H∞轨迹跟踪控制器,并进行稳定性分析。S6. Design a fixed-time H ∞ trajectory tracking controller based on the dynamic system, and conduct stability analysis.
作为进一步的改进,所述S1步骤中,建立包括运动学和动力学模型的水面无人艇控制器包括如下步骤:As a further improvement, in the step S1, establishing a surface unmanned vehicle controller including kinematics and dynamic models includes the following steps:
S11、建立水面无人艇的运动学和动力学模型;S11. Establish the kinematics and dynamics model of the surface unmanned vehicle;
S12、对水面无人艇的运动学和动力学模型进行坐标转换;S12. Perform coordinate transformation on the kinematics and dynamics model of the surface unmanned vehicle;
S13、对水面无人艇的运动学和动力学模型进行水面无人艇控制系统参数不确定性以及时变干扰处理。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、获得水面无人艇运动学和动力学模型的期望轨迹。S14, obtain the desired trajectory of the kinematic and dynamic model of the surface unmanned vehicle.
作为进一步的改进,所述S2步骤中,基于运动学和动力学模型,搭建水面无人艇轨迹跟踪误差模型,包括如下步骤: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、定义水面无人艇轨迹跟踪误差和水面无人艇在固定坐标系下的速度误差并获得水面无人艇跟踪误差动态;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、结合步骤S13和步骤S14对水面无人艇跟踪误差动态进行改写。S22, combining steps S13 and S14 to rewrite the tracking error dynamics of the surface unmanned boat.
作为进一步的改进,所述S3步骤中,基于水面无人艇控制器和水面无人艇轨迹跟踪误差模型设计固定时间扩张状态观测器,包括如下步骤: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、设计一个固定时间扩张状态观测器,后定义其观测误差;S31. Design a fixed time dilation state observer, and then define its observation error;
S32、根据定义的观测误差进一步设计固定时间扩张状态观测器。S32, further design a fixed time dilation state observer according to the defined observation error.
作为进一步的改进,所述S4步骤中,建立固定时间H∞控制器,包括如下步骤:As a further improvement, in the step S4, establishing a fixed-time H ∞ controller includes the following steps:
S41、定义一个包含因果动态补偿器的闭环系统;S41. Define a closed-loop system including a causal dynamic compensator;
S42、利用定理证明技术分析定义的闭环系统是全局固定时间稳定的,则该因果动态补偿器是一个全局固定时间H∞控制器。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.
作为进一步的改进,所述S5步骤中,基于水面无人艇轨迹跟踪误差模型、固定时间扩张状态观测器和固定时间H∞控制器设计辅助控制器建立动态系统,包括如下步骤: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、根据步骤S22、S32和S42,在固定时间内转变水面无人艇的跟踪误差动态;S51, according to steps S22, S32 and S42, change the tracking error dynamics of the surface unmanned boat within a fixed time;
S52、引入一个辅助控制器并定义跟踪误差变量;S52, introduce an auxiliary controller and define a tracking error variable;
S53、将水面无人艇的跟踪误差动态和定义的跟踪误差变量结合以建立动态系统。S53, combine the tracking error dynamics of the surface unmanned vehicle with the defined tracking error variables to establish a dynamic system.
作为进一步的改进,所述S6步骤中,基于动态系统设计固定时间H∞轨迹跟踪控制器,并进行稳定性分析,包括如下步骤: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、定义用于实现动态系统在固定时间内稳定的辅助函数;S61. Define an auxiliary function for realizing the stability of the dynamic system in a fixed time;
S62、在动态系统中定义了一个性能向量;S62, a performance vector is defined in the dynamic system;
S63、根据辅助函数和性能向量设计一个基于固定时间扩张状态观测器的固定时间H∞轨迹跟踪控制器策略;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、基于固定时间H∞轨迹跟踪控制器策略利用定理证明技术分析水面无人艇能够在固定的时间内跟踪期望轨迹且不依赖于水面无人艇的初始状态。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.
作为进一步的改进,所述S11步骤中,建立水面无人艇的运动学和动力学模型如下所示:As a further improvement, in the step S11, the kinematics and dynamics model of the surface unmanned vehicle is established as follows:
其中,n表示水面无人艇在大地坐标系下的水平面内的三自由度位姿向量,y代表航向角,R(ψ)表示代表大地坐标系与船体坐标系之间的坐标转换矩阵,R代表实数,v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向量,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.
作为进一步的改进,所述S21步骤中,获得水面无人艇跟踪误差动态如下所示:As a further improvement, in the step S21, the tracking error dynamics of the UAV obtained are as follows:
其中,其中,ne为水面无人艇轨迹跟踪误差,ve为水面无人艇在固定坐标系下的速度误差,vd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。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. .
作为进一步的改进,所述步骤S31中,定义固定时间扩张状态观测器的观测误差如下所示:As a further improvement, in the step S31, the observation error of the fixed time dilation state observer is defined as follows:
其中,μ=R(ψ)v,χ为水面无人艇控制系统受到的集总扰动,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.
本发明提供的一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法,包括步骤:S1、建立包括运动学和动力学模型的水面无人艇控制系统;S2、基于运动学和动力学模型,搭建水面无人艇轨迹跟踪误差模型;S3、基于水面无人艇运动学和动力学设计固定时间扩张状态观测器;S4、建立固定时间H∞控制器;S5、基于水面无人艇轨迹跟踪误差模型、固定时间扩张状态观测器和固定时间H∞控制器设计辅助控制器建立动态系统;S6、基于动态系统设计固定时间H∞轨迹跟踪控制器,并进行稳定性分析。本发明在H∞控制的基础上进一步研究,设计的固定时间扩张状态观测器用于估计系统的状态信息以及集总扰动,更符合工程的实际应用要求,建立的固定时间H∞控制器的整个闭环系统在固定时间内实现全局稳定,大大的提高了跟踪控制的收敛速率以及稳定性,不仅提高了水面无人艇控制系统的鲁棒性,而且系统的收敛时间不依赖于系统初始值,在保证跟踪误差快速收敛的前提下,系统具有较高的稳定性。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.
附图说明Description of drawings
利用附图对本发明作进一步说明,但附图中的实施例不构成对本发明的任何限制,对于本领域的普通技术人员,在不付出创造性劳动的前提下,还可以根据以下附图获得其它的附图。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.
图1为本发明的流程示意图。FIG. 1 is a schematic flow chart of the present invention.
图2为本发明的运动学和动力学模型示意图。FIG. 2 is a schematic diagram of the kinematics and dynamics model of the present invention.
图3为本发明的运动学和动力学模型的期望轨迹和实际轨迹示意图。FIG. 3 is a schematic diagram of the expected trajectory and the actual trajectory of the kinematics and dynamics model of the present invention.
图4为本发明的纵荡轨迹跟踪情况示意图;图中,x为水面无人艇在大地坐标系下纵向位置坐标,xd为水面无人艇在大地坐标系下的期望纵向位置坐标。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.
图5为本发明的横荡轨迹跟踪情况示意图;图中,y为水面无人艇在大地坐标系下横向位置坐标,yd为水面无人艇在大地坐标系下的期望横向位置坐标。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.
图6为本发明的艏摇轨迹跟踪情况示意图;图中,ψ为航向角,ψd为期望航向角。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.
图7为本发明的纵荡方向速度估计情况示意图;图中,u为水面无人艇在船体坐标系下纵向实际速度坐标,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.
图8为本发明的横荡方向速度估计情况示意图;图中,v为水面无人艇在船体坐标系下横荡实际速度坐标,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.
图9为本发明的艏摇方向速度估计情况示意图;图中,r为水面无人艇在船体坐标系下艏摇方向实际速度坐标,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.
图10为本发明的纵荡方向控制输入情况示意图;图中,τu为水面无人艇沿船体坐标系下纵向的控制输入。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.
图11为本发明的横荡方向控制输入情况示意图;图中,τv为水面无人艇沿船体坐标系下横向的控制输入。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.
图12为本发明的艏摇方向控制输入情况示意图;图中,τr为水面无人艇沿船体坐标系下艏摇方向的控制输入。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.
具体实施方式Detailed ways
为了使本领域的技术人员更好地理解本发明的技术方案,下面结合附图和具体实施例对本发明作进一步详细的描述,需要说明的是,在不冲突的情况下,本申请的实施例及实施例中的特征可以相互组合。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.
本发明实施例提供一种基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法,如图1所示,包括如下步骤: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、建立包括运动学和动力学模型的水面无人艇控制系统,包括如下步骤:S1. Establish a surface unmanned vehicle control system including kinematics and dynamic models, including the following steps:
S11、建立水面无人艇的运动学和动力学模型,结合图2所示,如下所示:S11, establish the kinematics and dynamics model of the surface unmanned boat, as shown in Figure 2, as follows:
其中,n=[x,y,ψ]T表示水面无人艇在大地坐标系Xe-Oe-Ye下的水平面内的三自由度位姿向量,x和y分别代表水面无人艇在大地坐标系Xb-Ob-Yb下纵向和横向位置坐标,ψ代表航向角;R(ψ)表示代表大地坐标系与船体坐标系之间的坐标转换矩阵,R(ψ)∈R3×3,R代表实数;v=[u,v,r]T,v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向量,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=[τu,τv,τr]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:
其中,R-1是R矩阵的逆矩阵,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、定义μ=R(ψ)v,对水面无人艇的运动学和动力学模型进行坐标转换后如下:S12, define μ=R(ψ)v, after the coordinate transformation of the kinematics and dynamics model of the surface unmanned vehicle is as follows:
其中,μ是辅助变量,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)=f0(n,μ)+△f (式4)f(n,μ)=f 0 (n,μ)+△f (Equation 4)
其中,f0(n,μ)为水面无人艇控制系统中含有不确定项的标称值形式,△f为水面无人艇控制系统的不确定性部分,具体的数学表达形式如下: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:
其中,C0(v)代表科里奥利向心力矩阵的标称值,D0(v)代表流体阻尼矩阵的标称值,△C(v)代表科里奥利向心力矩阵的不确定值,△D(v)代表流体阻尼矩阵的不确定值。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、对水面无人艇的运动学和动力学模型进行水面无人艇控制系统参数不确定性以及时变干扰处理后,水面无人艇的运动学和动力学模型如下所示: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:
其中,χ=△f(n,μ)+w(t)视为水面无人艇受到的集总扰动。Among them, χ=△f(n, μ)+w(t) is regarded as the aggregate disturbance received by the surface unmanned vehicle.
S14、获得水面无人艇运动学和动力学模型的期望轨迹如下所示:S14. The desired trajectory of the kinematic and dynamic model of the surface UAV is obtained as follows:
其中nd=[xd,yd,ψd]T表示水面无人艇在大地坐标系下水平面内的三自由度位姿期望值,xd为x的期望值,yd为y的期望值,ψd为ψ的期望值;和分别是nd的一阶导数和二阶导数;vd=[ud,vd,rd]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,ud为u的期望值,vd为v的期望值,rd为r的期望值;是vd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、基于运动学和动力学模型,搭建水面无人艇轨迹跟踪误差模型,包括如下步骤:S2. Based on the kinematics and dynamics model, build a trajectory tracking error model of the surface unmanned boat, including the following steps:
S21、定义ne=n-nd,ve=v-vd,获得水面无人艇跟踪误差动态如下所示:S21. Define n e =nn d , ve =vv d , and obtain the tracking error dynamics of the surface unmanned boat as follows:
式中,ne表示水面无人艇在大地坐标系下的轨迹跟踪误差;ve表示水面无人艇在船体坐标系下的速度误差;是ne是一阶导数;是ve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 .
S22、定义x1=ne,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:
其中,x1表示水面无人艇在大地坐标系下的轨迹跟踪误差;x2是x1的一阶导数。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、基于水面无人艇运动学和动力学设计固定时间扩张状态观测器,包括如下步骤:S3. Design a fixed time dilation state observer based on the kinematics and dynamics of the surface UAV, including the following steps:
S31、设计一个固定时间扩张状态观测器,后定义其观测误差如下所示:S31. Design a fixed time dilation state observer, and then define its observation error as follows:
其中,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、根据定义的观测误差以及式6进一步设计固定时间扩张状态观测器如下所示:S32, according to the defined observation error and formula 6, the fixed time dilation state observer is further designed as follows:
其中,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,χd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ki(i=1,2,3)和li(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.
收敛时间满足Convergence time is satisfied
其中,r1表示简化矩阵,并且满足r1=λmin(Q1)/λmax(Ω1);r2表示简化矩阵,并且满足r2=λmin(Q2)/λmax(Ω2);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、建立固定时间H∞控制器,包括如下步骤:S4, establishing a fixed-time H ∞ controller, including the following steps:
S41、定义1:考虑一个包含因果动态补偿器的闭环系统如下所示:S41. Definition 1: Consider a closed-loop system including a causal dynamic compensator as follows:
其中,x为反映了系统状态信息的向量,f(x)和g(x)均表示闭环系统中关于x非线性项,Ξ(x)是系统的控制器输入向量,w(x)为未知扰动,Z是一个性能向量,w为外部干扰。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.
一个因果动态补偿器如下所示:A causal dynamic compensator looks like this:
Ξ=φ(x,t) (式15)Ξ=φ(x, t) (Equation 15)
如果满足以下两个条件,那么这个因果动态补偿器Ξ是一个全局固定时间H∞控制器。The causal dynamic compensator Ξ is a global fixed-time H∞ controller if the following two conditions are met.
(1)当w=0时,式14和式15中的闭环系统是全局固定时间稳定的;(1) When w=0, the closed-loop system in Equation 14 and Equation 15 is globally fixed-time stable;
(2)给定实数γ>0,对于所有t1>t0和所有的非线性扰动,如果由w产生的输出z和初始状态x(t0)=0满足以下包含定积分的不等式:(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:
其中,z(t)表示随时间t变化的性能函数向量;w(t)表示关于时间t的扰动项;那么式14和式15中闭环系统的L2增益小于或者等于γ。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、利用定理证明技术分析定义的闭环系统是全局固定时间稳定的,则该因果动态补偿器是一个全局固定时间H∞控制器。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.
定理1:考虑式9,假设在一个邻域中定义了一个函数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)在U0内是一个正定的函数;(1) V(x) is a positive definite function in U 0 ;
(2) (2)
其中,c1、c2为收敛速度控制系数;a1、a2为固定时间收敛控制参数。Among them, c 1 , c 2 are convergence speed control coefficients; a 1 , a 2 are fixed-time convergence control parameters.
那么,闭环系统式14的原点是局部固定时间稳定的,且系统的L2增益小于或者等于γ。如果U0=Rn并且V(x)是径向无界的(即当||x||→+∞时,V(x)→+∞),闭环系统系统式14的原点是全局固定时间稳定的。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.
证明:从定义1可知,两个条件必须满足。Proof: From Definition 1, two conditions must be satisfied.
(1)当w=0,定理1中的条件(2)可以写成:(1) When w=0, the condition (2) in Theorem 1 can be written as:
根据引理1,这由式14和式15组成闭环系统是固定时间稳定的。According to Lemma 1, this closed-loop system consisting of Eqs. 14 and 15 is fixed-time stable.
(2)当w≠0,V(x)>0,那么以下不等式成立:(2) When w≠0, V(x)>0, then the following inequality holds:
满足定义1中的条件(2),因此闭环系统式15的L2增益小于或者等于γ,定理1证明完毕。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、基于水面无人艇轨迹跟踪误差模型、固定时间扩张状态观测器和固定时间H∞控制器设计辅助控制器建立动态系统,包括如下步骤: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:
S51、根据步骤S22的式9、S32的式13、S42的引理1和可知,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:
其中,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 .
S52、引入一个辅助控制器定义跟踪误差变量如下:S52. Introduce an auxiliary controller Define tracking error variable as follows:
S53、将水面无人艇的跟踪误差动态式19和定义的跟踪误差变量式20结合以建立动态系统如下: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、基于动态系统设计固定时间H∞轨迹跟踪控制器,并进行稳定性分析。S6. Design a fixed-time H ∞ trajectory tracking controller based on the dynamic system, and conduct stability analysis.
S61、定义用于实现动态系统式21在固定时间内稳定的辅助函数如下:S61. Define the auxiliary function used to realize the stability of the dynamic system formula 21 in a fixed time as follows:
其中,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;pi(i=1,2,3,4)=diag{pi1,pi2,...,pin}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:siga(·)=sign(·)|·|a,sigβ(·)=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、在动态系统式21中定义了一个性能向量如下:S62, a performance vector is defined in the dynamic system formula 21 as follows:
其中,λ1>0,λ2>0,分别代表x1和的加权系数。Among them, λ 1 >0, λ 2 >0, representing x 1 and weighting factor.
S63、根据辅助函数和性能向量设计一个基于固定时间扩张状态观测器的固定时间H∞轨迹跟踪控制器策略如下: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:
其中,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、基于固定时间H∞轨迹跟踪控制器策略利用定理证明技术分析水面无人艇能够在固定的时间内跟踪期望轨迹且不依赖于水面无人艇的初始状态,具体如下: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:
定理2:在水面无人艇跟踪期望轨迹时,考虑了系统不确定性、海洋环境扰动以及状态不可测等因素,所设计的基于固定时间H∞轨迹跟踪控制器策略式24能够保证水面无人艇在固定时间内跟踪期望轨迹,并且误差收敛时间可不依赖于水面无人艇的初始状态,收敛时间的上界Ts如下: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:
Ts=T0+T1 (式25)T s =T 0 +T 1 (Equation 25)
证明:固定时间内到达稳定阶段的证明如下:Proof: The proof of reaching the stable stage in a fixed time is as follows:
将固定时间H∞轨迹跟踪控制器策略式24代入动态系统式21,得到:Substitute the fixed-time H ∞ trajectory tracking controller strategy equation 24 into the dynamic system equation 21, we get:
设计一个Lyapunov函数(李雅普诺夫函数,即V函数):Design a Lyapunov function (Lyapunov function, namely V function):
结合式22、式24和式26,对V求导,可得:Combining Equation 22, Equation 24 and Equation 26, and taking the derivative of V, we get:
定义函数H:Define function H:
将式22、式26和式28代入式29,选择一个正实数p*,满足且p*=min{p0i},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:
假设σ=(1+α)/2,1/2<σ<1,δ=(1+β)/2,p1min=min{p1i},p2min=min{p2i},p3min=min{p3i}p4min=min{p4i}, 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:
结合式29和式31,可以得到:Combining Equation 29 and Equation 31, we can get:
其中,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.
由定理1可知动态系统式21是固定时间全局稳定的,由引理2,可计算得到收敛时间的上界: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:
根据固定时间收敛系统理论可知误差和在固定时间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.
进一步地,所述证明步骤中的引理1具体如下:Further, Lemma 1 in the proof step is as follows:
如果存在和那么对于以下不等式被满足:if it exists and then for The following inequalities are satisfied:
式中,为幂次项,xi为正实数。In the formula, is a power term, and x i is a positive real number.
进一步地,上诉证明步骤中的引理2具体如下:Further, Lemma 2 in the appeal proof step is as follows:
考虑以下系统:Consider the following system:
其中,y为系统输出值,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Tmax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:
其中,Tmax为固定时间收敛的上界。Among them, Tmax is the upper bound of the fixed time convergence.
本发明在H∞控制的基础上进一步研究,设计的固定时间扩张状态观测器用于估计控制系统的状态信息以及集总扰动,更符合工程的实际应用要求,建立的固定时间H∞控制器的整个闭环系统在固定时间内实现全局稳定,大大的提高了跟踪控制的收敛速率以及稳定性,不仅提高了水面无人艇控制系统的鲁棒性,而且系统的收敛时间不依赖于系统初始值,在保证跟踪误差快速收敛的前提下,系统具有较高的稳定性。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.
为了验证本发明实施方式的有效性,进行了仿真实验,具体如下:In order to verify the effectiveness of the embodiments of the present invention, simulation experiments were carried out, as follows:
结合Cybership II(赛博二号)船舶模型进行仿真实验来验证所设计的基于固定时间H∞控制的水面无人艇轨迹快速跟踪控制方法的有效性,Cybership II船模参数如下表1。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.
表1Table 1
水面无人艇控制系统中的参数设置如下表2:The parameter settings in the surface unmanned boat control system are as follows in Table 2:
表2Table 2
时变扰动如下:The time-varying perturbation is as follows:
参考轨迹如下:The reference track is as follows:
仿真结果如图3-图12所示,展示了设计的轨迹跟踪控制器使得水面无人艇能够在5s左右精确跟踪上期望的轨迹,并具备较好的鲁棒性。显然,所提出的控制方案可以保证收敛时间小于最大值,并且可以独立于代入的初始状态进行计算。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.
上面的描述中阐述了很多具体细节以便于充分理解本发明,但是,本发明还可以采用其他不同于在此描述的其他方式来实施,因此,不能理解为对本发明保护范围的限制。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.
总之,本发明虽然列举了上述优选实施方式,但是应该说明,虽然本领域的技术人员可以进行各种变化和改型,除非这样的变化和改型偏离了本发明范围,否则都应该包括在本发明的保护范围内。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.
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