CN104049534A

CN104049534A - Self-adaption iterative learning control method for micro-gyroscope

Info

Publication number: CN104049534A
Application number: CN201410179089.2A
Authority: CN
Inventors: 陆晓春; 费峻涛
Original assignee: Hohai University HHU
Current assignee: Hohai University HHU
Priority date: 2014-04-29
Filing date: 2014-04-29
Publication date: 2014-09-17
Anticipated expiration: 2034-04-29
Also published as: CN104049534B

Abstract

The invention discloses an adaptive iterative learning control method of a micro gyroscope, comprising the following steps: 1) establishing a dimensionless dynamic model of the micro gyroscope; Trajectories, including position and speed signals; 3) The adaptive law module receives the reference trajectory and the output of the micro-gyroscope system, and uses the adaptive law to estimate the increment of the parameters; 4) The controller module receives the new parameter estimates, and the trajectory Tracking error and velocity tracking error jointly produce the control signal output of the adaptive iterative learning control method; 5) receive the output signal of the controller module, and output the position and velocity information of the vibrating part of the micro gyroscope; 6) repeat step 3 by using the iterative method )-step 5) to obtain the final position and velocity information of the vibrating part of the micro gyroscope. The invention can improve the tracking performance of the micro gyroscope system to the reference track.

Description

Adaptive Iterative Learning Control Method for Micro Gyroscope

技术领域 technical field

本发明涉及振动式微陀螺仪的控制系统及方法，特别是涉及振动式微陀螺仪的自适应迭代学习控制系统及方法。 The invention relates to a control system and method of a vibrating micro gyroscope, in particular to an adaptive iterative learning control system and method of a vibrating micro gyroscope. the

背景技术 Background technique

振动式微陀螺仪(MEMS Vibratory Gyroscopes，以下简称微陀螺仪)是利用微电子技术和微加工技术加工而成的用来感测角速度的传感器件。它通过一个由硅制成的振动的微机械部件来检测角速度，因此微陀螺仪非常容易小型化和批量生产，具有成本低和体积小等特点，因而被广泛应用在航空、航天、航海、陆地车辆的导航与定位、消费电子领域及油田勘探开发等军事、民用领域中。但是，由于生产制造过程中不可避免的加工误差以及环境温度的影响，会造成原件特性与设计之间的差异，导致微陀螺仪存在参数不确定性，难以建立精确的数学模型。再加上工作环境中的外界扰动作用不可忽略，使得微陀螺仪的轨迹追踪控制难以实现，且鲁棒性较低。 Vibrating micro gyroscopes (MEMS Vibratory Gyroscopes, hereinafter referred to as micro gyroscopes) are sensor devices processed by microelectronics and micromachining technologies to sense angular velocity. It detects angular velocity through a vibrating micromechanical component made of silicon, so the micro gyroscope is very easy to miniaturize and mass produce, and has the characteristics of low cost and small size, so it is widely used in aviation, aerospace, navigation, land Vehicle navigation and positioning, consumer electronics and oilfield exploration and development and other military and civilian fields. However, due to the inevitable processing errors in the manufacturing process and the influence of ambient temperature, there will be differences between the characteristics of the original and the design, resulting in the uncertainty of the parameters of the micro gyroscope, and it is difficult to establish an accurate mathematical model. In addition, the external disturbance in the working environment cannot be ignored, which makes the trajectory tracking control of the micro gyroscope difficult to realize, and the robustness is low. the

目前，国内外对于微陀螺仪的研究目前主要集中在结构设计及制造技术方面，以及上述的机械补偿技术和驱动电路研究，而对于微陀螺仪振动轨迹的跟踪控制方面的研究却很少，特别是利用现代智能控制方法实现轨迹跟踪方面的研究和成果十分缺乏。 At present, domestic and foreign research on micro-gyroscopes is mainly focused on structural design and manufacturing technology, as well as the above-mentioned mechanical compensation technology and drive circuit research, but there is little research on the tracking control of micro-gyroscope vibration trajectory, especially It is very lack of research and achievements in trajectory tracking using modern intelligent control methods. the

现有的应用到微陀螺仪的控制方法有自适应控制和滑模控制等方法，但这些设计方法较为复杂，计算量大，难以应用，且对外界扰动的鲁棒性很低，易使系统变得不稳定。 The existing control methods applied to micro-gyroscopes include adaptive control and sliding mode control, but these design methods are relatively complex, with a large amount of calculation, difficult to apply, and the robustness to external disturbances is very low, which is easy to make the system become unstable. the

由此可见，上述现有的陀螺仪在使用上，显然仍存在有不便与缺陷，而亟待加以进一步改进。为了解决现有的陀螺仪在使用上存在的问题，相关厂商莫不费尽心思来谋求解决之道，但长久以来一直未见适用的设计被发展完成。 This shows that the above-mentioned existing gyroscope obviously still has inconvenience and defects in use, and needs to be further improved urgently. In order to solve the problems existing in the use of the existing gyroscopes, relevant manufacturers have tried their best to find a solution, but no suitable design has been developed for a long time. the

发明内容 Contents of the invention

本发明的目的在于，克服现有的微陀螺仪控制方法存在的缺陷，特别是在存在模型参数不确定以及外界噪声干扰情况下，为提高微陀螺仪系统对参考轨迹的跟踪性能而提供一种微陀螺仪的自适应迭代学习控制系统及方法。 The purpose of the present invention is to overcome the defects in the existing micro gyroscope control method, especially in the case of uncertain model parameters and external noise interference, to provide a method for improving the tracking performance of the micro gyroscope system to the reference track. An adaptive iterative learning control system and method for a micro gyroscope. the

本发明的目的及解决其技术问题是采用以下技术方案来实现的，微陀螺仪的自适应迭代学习控制系统，包括： Purpose of the present invention and solving its technical problem are to adopt following technical scheme to realize, the self-adaptive iterative learning control system of micro-gyro, comprising:

参考轨迹模块(101)，用于输出微陀螺仪x和y轴振动的参考轨迹，包括位置、速度信号； Reference track module (101), for outputting the reference track of microgyroscope x and y-axis vibration, including position, speed signal;

自适应律模块(102)，用于接收参考轨迹及微陀螺仪系统的输出，利用自适应律估计出参数的增量； The adaptive law module (102) is used to receive the output of the reference trajectory and the micro-gyroscope system, and utilizes the adaptive law to estimate the increment of the parameter;

控制器模块(103)，用于接收新的参数估计，并和轨迹跟踪误差、速度跟踪误差共同作用产生自适应迭代学习控制方法的控制信号输出； Controller module (103), is used for receiving new parameter estimation, and the control signal output that produces adaptive iterative learning control method together with trajectory tracking error, speed tracking error;

微陀螺仪系统(104)，被控对象的数学模型，考虑了机械噪声的影响，接收控制器模块的输出信号，输出微陀螺仪振动部件的位置和速度信息； Micro-gyroscope system (104), the mathematical model of the controlled object, has considered the influence of mechanical noise, receives the output signal of the controller module, and outputs the position and velocity information of the micro-gyroscope vibrating part;

存储模块(105)，用于保存本次迭代时的参数估计信息，用于下一次迭代时的参数估计。 A storage module (105), configured to store parameter estimation information during this iteration, and to be used for parameter estimation during the next iteration. the

微陀螺仪的自适应迭代学习控制方法，其特征在于，包括以下步骤： The adaptive iterative learning control method of micro-gyroscope is characterized in that, comprises the following steps:

1)建立微陀螺仪的无量纲动力学模型； 1) Establish a dimensionless dynamic model of the micro gyroscope;

2)参考轨迹模块输出微陀螺仪x和y轴振动的参考轨迹，包括位置、速度信号； 2) The reference trajectory module outputs the reference trajectory of the x and y axis vibrations of the micro gyroscope, including position and speed signals;

3)自适应律模块接收参考轨迹及微陀螺仪系统的输出，利用自适应律估计出参数的增量； 3) The adaptive law module receives the reference trajectory and the output of the micro-gyroscope system, and uses the adaptive law to estimate the increment of the parameters;

4)控制器模块接收新的参数估计，并和轨迹跟踪误差、速度跟踪误差共同作用产生自适应迭代学习控制方法的控制信号输出； 4) The controller module receives the new parameter estimation, and works together with the trajectory tracking error and the speed tracking error to generate the control signal output of the adaptive iterative learning control method;

5)接收控制器模块的输出信号，输出微陀螺仪振动部件的位置和速度信息； 5) Receive the output signal of the controller module, and output the position and speed information of the vibrating part of the micro gyroscope;

6)利用迭代方法重复步骤3)-步骤5)，得到最终的微陀螺仪振动部件的位置和速度信息。 6) Step 3)-step 5) are repeated using an iterative method to obtain the final position and velocity information of the vibrating part of the micro gyroscope. the

在所述步骤1)中，微陀螺仪的无量纲动力学模型为： In described step 1), the dimensionless dynamic model of microgyroscope is:

在考虑外界干扰时，振动式微陀螺仪模型用式(6)表达： When considering external interference, the vibrating micro-gyroscope model is expressed by formula (6):

其中：d表示外界干扰； Among them: d represents external interference;

$q = [\begin{matrix} x \\ y \end{matrix}], D = [\begin{matrix} d_{xx} & d_{xy} \\ d_{xy} & d_{yy} \end{matrix}], K = [\begin{matrix} ω_{x}^{2} & ω_{xy} \\ ω_{xy} & ω_{y}^{2} \end{matrix}], u = [\begin{matrix} u_{x} \\ u_{y} \end{matrix}], Ω = [\begin{matrix} 0 & - Ω_{z} \\ Ω_{z} & 0 \end{matrix}];$ u_x、u_y表示沿x轴、y轴方向的驱动力；和d_xx、d_yy、d_xy分别表示x轴、y轴、x轴和y轴之间的弹簧系数和阻尼系数；m为质量；Ω_z表示沿z轴方向的角速度；ω₀为两轴的共振频率；分别是速度向量和加速度向量； $q = [\begin{matrix} x \\ the y \end{matrix}], D. = [\begin{matrix} d_{xx} & d_{xy} \\ d_{xy} & d_{yy} \end{matrix}], K = [\begin{matrix} ω_{x}^{2} & ω_{xy} \\ ω_{xy} & ω_{the y}^{2} \end{matrix}], u = [\begin{matrix} u_{x} \\ u_{the y} \end{matrix}], Ω = [\begin{matrix} 0 & - Ω_{z} \\ Ω_{z} & 0 \end{matrix}];$ u _x , u _y represent the driving force along the x-axis and y-axis; and d _xx , d _yy , d _xy represent the spring coefficient and damping coefficient of the x-axis, y-axis, and between the x-axis and the y-axis, respectively; m is the mass; Ω _z represents the angular velocity along the z-axis direction; ω ₀ is the two axes the resonant frequency; are velocity vector and acceleration vector respectively;

在迭代控制过程中，式(6)表示为： In the iterative control process, formula (6) is expressed as:

${\overset{\cdot &Center Dot; \cdot &Center Dot;}{q q}}_{k k} + + ((D D. + + 22 Ω Ω)) {\overset{\cdot &Center Dot;}{q q}}_{k k} + + K K {q q}_{k k} = = {u u}_{k k} + + {d d}_{k k} - - - - - - ((77))$

式中k为迭代次数，k为正整数，q_k、u_k、d_k分别为q、u、d的第k次的加速度信号向量、速度信号向量、位置信号向量、输入控制向量、干扰向量。 In the formula, k is the number of iterations, k is a positive integer, q _k , u _k , d _k are respectively Acceleration signal vector, velocity signal vector, position signal vector, input control vector, and disturbance vector of the kth order of q, u, and d.

在所述步骤3)中，根据微陀螺仪的无量纲动力学模型进行自适应参数估计，自适应参数估计算法为： In described step 3) in, carry out adaptive parameter estimation according to the dimensionless dynamic model of micro-gyroscope, adaptive parameter estimation algorithm is:

${\overset{^^}{θ θ}}_{k k} ((t t)) = = {\overset{&OverBar; &OverBar;}{θ θ}}_{k k} + + {\overset{^^}{θ θ}}_{k k - - 11} ((t t)) - - - - - - ((99))$

其中θ为由系统未知参数组成的向量，为θ的估计值，为θ估计值的初始值，为第k次迭代时θ的估计值的增量；矩阵Γ为正定对称矩阵，Γ＝diag(90,90,90,90,90,90,90,90)； where θ is a vector composed of unknown parameters of the system, is the estimated value of θ, is the initial value of the estimated value of θ, is the increment of the estimated value of θ at the kth iteration; the matrix Γ is a positive definite symmetric matrix, Γ=diag(90,90,90,90,90,90,90,90);

$θ = (\begin{matrix} d_{xx} \\ d_{xy} \\ d_{yy} \\ Ω_{z} \\ ω_{x}^{2} \\ ω_{xy} \end{matrix}),$ β为正实数； $θ = (\begin{matrix} d_{xx} \\ d_{xy} \\ d_{yy} \\ Ω_{z} \\ ω_{x}^{2} \\ ω_{xy} \end{matrix}),$ β is a positive real number;

${\overset{\cdot}{e}}_{k} (t) = {\overset{\cdot}{q}}_{d} (t) - {\overset{\cdot}{q}}_{k} (t);$ 为期望的速度信号。 ${\overset{&Center Dot;}{e}}_{k} (t) = {\overset{\cdot}{q}}_{d} (t) - {\overset{\cdot}{q}}_{k} (t);$ is the desired velocity signal.

在所述步骤4)中，根据微陀螺仪的无量纲动力学模型和自适应参数估计算法，进行自适应迭代学习控制，控制信号u_k(t)定义为： In said step 4), according to the dimensionless dynamic model of the micro-gyroscope and the adaptive parameter estimation algorithm, the adaptive iterative learning control is carried out, and the control signal u _k (t) is defined as:

其中矩阵K_P、K_D都为正定对称矩阵， $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ $K_{D} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ 则当满足迭代条件未知误差初始信号e_k(0)＝0和速度误差初始信号时，未知误差信号e_k(t)和速度误差初始信号有界，且时间t处于迭代周期[0,T]中。 where matrix Both K _P and K _D are positive definite symmetric matrices, $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ $K_{D.} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ Then when the iteration condition is satisfied, the unknown error initial signal e _k (0)=0 and the velocity error initial signal When , the unknown error signal e _k (t) and the velocity error initial signal bounded, and Time t is in iteration period [0,T].

本发明所达到的有益效果：本发明针对参数未知和存在外界干扰的振动式微陀螺仪系统，在推导其数学模型的基础上，提出了自适应迭代学习控制方案来实现微陀螺仪系统的轨迹跟踪控制。该控制方案由传统的PD反馈控制、未知参数和跟踪误差组成的迭代项组成，控制律通过类Lyapunov稳定性理论说明了其稳定性，保证了控制系统的全局稳定性和跟踪误差的渐近收敛性。本发明为微陀螺仪应用范围的扩展提供了良好基础本发明的微陀螺仪控制方法，在存在模型参数不确定以及外界噪声干扰情况下，可以提高微陀螺仪系统对参考轨迹的跟踪性能。 The beneficial effect achieved by the present invention: the present invention aims at the vibrating micro-gyroscope system with unknown parameters and external interference, and on the basis of deriving its mathematical model, proposes an adaptive iterative learning control scheme to realize the trajectory tracking of the micro-gyroscope system control. The control scheme is composed of traditional PD feedback control, unknown parameters and iterative items consisting of tracking error. The stability of the control law is explained by the Lyapunov-like stability theory, which ensures the global stability of the control system and the asymptotic convergence of the tracking error. sex. The present invention provides a good foundation for the expansion of the application range of the micro gyroscope. The micro gyroscope control method of the present invention can improve the tracking performance of the micro gyroscope system to the reference track in the case of uncertain model parameters and external noise interference. the

附图说明 Description of drawings

图1为本发明的原理结构图； Fig. 1 is a schematic structural diagram of the present invention;

图2为基于本发明的微陀螺仪x轴的轨迹跟踪曲线； Fig. 2 is the trajectory tracking curve based on micro-gyroscope x-axis of the present invention;

图3为基于本发明的微陀螺仪y轴的轨迹跟踪曲线； Fig. 3 is the trajectory tracking curve based on the micro-gyroscope y-axis of the present invention;

图4为本发明中的微陀螺仪的x轴的速度跟踪曲线； Fig. 4 is the velocity tracking curve of the x-axis of micro-gyroscope among the present invention;

图5为本发明中的微陀螺仪的y轴的速度跟踪曲线。 Fig. 5 is the speed tracking curve of the y-axis of the micro gyroscope in the present invention. the

具体实施方式 Detailed ways

本发明的微陀螺仪的自适应迭代学习控制方法，包括以下步骤： The adaptive iterative learning control method of the micro gyroscope of the present invention comprises the following steps:

在所述步骤1)中，建立微陀螺仪的无量纲动力学模型，具体为： In said step 1), set up the dimensionless dynamic model of micro-gyroscope, specifically:

当微陀螺仪沿z轴方向旋转时，利用旋转系中的牛顿定律可以对微陀螺仪进行受力分析： When the micro gyroscope rotates along the z-axis direction, the force analysis of the micro gyroscope can be carried out by using Newton's law in the rotation system:

F_r＝F_phy+F_centri+F_Colis+F_Eular＝ma_r (1) F _r ＝F _phy +F _centri +F _Colis +F _Eular ＝ma _r (1)

其中F_r表示质量块在旋转系中受到的合力，F_phy表示质量块在惯性参考系下受到的合力，F_centri是离心力，F_Colis是科里奥利力，F_Euler是欧拉力，a_r是质量块相对旋转系的加速度,m为质量。 Among them, F _r represents the resultant force of the mass block in the rotating system, F _phy represents the resultant force of the mass block in the inertial reference system, F _centri is the centrifugal force, F _Colis is the Coriolis force, F _Euler is the Euler force, a _r is the acceleration of the mass block relative to the rotating system, and m is the mass.

假定微陀螺仪输入角速度Ω在足够长的时间内保持不变，即输入角速度Ω为常量，且对z轴陀螺仪而言，可以认为质量块被限制在x-y平面内运动，不能沿z轴运动，因此沿x轴和y轴方向的角速度Ω_x＝Ω_y＝0。 Assume that the input angular velocity Ω of the micro-gyroscope remains constant for a long enough time, that is, the input angular velocity Ω is constant, and for the z-axis gyroscope, it can be considered that the mass block is restricted to move in the xy plane and cannot move along the z-axis , so the angular velocity Ω _x =Ω _y =0 along the x-axis and y-axis directions.

式(1)中欧拉力可表示为其中r_r是质量块相对于旋转系的位移。由于输入角速度Ω为常量，F_Eular＝0，t为时间。 The Euler force in formula (1) can be expressed as where r _r is the displacement of the mass relative to the rotation system. Since the input angular velocity Ω is constant, F _Eular =0, and t is time.

离心力F_centri＝-mΩ×(Ω×r_r)，由于Ω_x＝Ω_y＝0，因此沿x轴方向的离心力可表示为F_centri-x＝-mΩ_z ²x，其中x表示质量块沿x轴方向的位移，沿x轴方向的离心力可表示为F_centri-y＝-mΩ_z ²y，其中y表示质量块沿y轴方向的位移。由于F_centri-x和F_centri-y的值非常小，通常不超过其他作用力的千分之一，因此可以忽略不计。 Centrifugal force F _centri =-mΩ×(Ω×r _r ), since Ω _x =Ω _y =0, the centrifugal force along the x-axis can be expressed as F _centri-x =-mΩ _z ² x, where x represents the The displacement in the x-axis direction and the centrifugal force along the x-axis direction can be expressed as F _centri-y = -mΩ _z ² y, where y represents the displacement of the mass block along the y-axis direction. Since the values of F _centri-x and F _centri-y are very small, usually no more than one-thousandth of the other forces, they can be ignored.

科里奥利力F_Colis＝-2mΩ×v_r，其中v_r表示质量块相对旋转系的速度。根据科里奥利力作用原理，沿x轴方向的科里奥利力可表示为式中表示沿y轴方向的运动速度，沿y轴方向的科里奥利力可表示为式中表示沿x轴方向的运动速度。Ω_z表示沿z轴方向的角速度。 Coriolis force F _Colis =-2mΩ×v _r , where v _r represents the velocity of the mass relative to the rotating system. According to the principle of Coriolis force, the Coriolis force along the x-axis direction can be expressed as In the formula represents the motion velocity along the y-axis direction, and the Coriolis force along the y-axis direction can be expressed as In the formula Indicates the movement speed along the x-axis direction. Ω _z represents the angular velocity along the z-axis direction.

质量块在惯性参考系下受到的合力F_phy主要由驱动力、弹簧弹力和阻尼力组成。分别用u_x、u_y表示沿x轴、y轴方向的驱动力。用k_xx、k_yy、k_xy和d_xx、d_yy、d_xy分别表示x轴、y轴、x轴和y轴之间的弹簧系数和阻尼系数，其中k_xy、d_xy主要是由于制造误差造成的结构不对称引起的两轴耦合。惯性参考系中沿x轴方向的合力为 $F_{phy - x} = u_{x} - k_{xx} x - k_{xy} y - d_{xx} \overset{\cdot}{x} - d_{xy} \overset{\cdot}{y},$ 沿y轴方向的合力为 $F_{phy - y} = u_{y} - k_{yy} y - k_{xy} x - d_{yy} \overset{\cdot}{y} - d_{xy} \overset{\cdot}{x} .$ The resultant force F _phy on the mass block in the inertial reference system is mainly composed of driving force, spring force and damping force. Use u _x , u _y to denote the driving forces along the x-axis and y-axis directions, respectively. Use k _xx , k _yy , k _xy and d _xx , d _yy , d _xy to denote the spring coefficient and damping coefficient between x-axis, y-axis, x-axis and y-axis respectively, where k _xy , d _xy are mainly due to manufacturing Two-axis coupling caused by structural asymmetry caused by errors. The resultant force along the x-axis in the inertial reference frame is $f_{phy - x} = u_{x} - k_{xxx} x - k_{xy} the y - d_{xxx} \overset{&Center Dot;}{x} - d_{xy} \overset{\cdot}{the y},$ The resultant force along the y-axis is $f_{phy - the y} = u_{the y} - k_{yy} the y - k_{xy} x - d_{yy} \overset{&Center Dot;}{the y} - d_{xy} \overset{\cdot}{x} .$

根据以上分析，式(1)可展开为： According to the above analysis, formula (1) can be expanded as:

$\begin{matrix} m m \overset{\cdot &Center Dot; \cdot &Center Dot;}{x x} + + {d d}_{xx xx} \overset{\cdot &Center Dot;}{x x} + + {d d}_{xy xy} \overset{\cdot &Center Dot;}{y the y} + + {k k}_{xx xx} x x + + {k k}_{xy xy} y the y = = {u u}_{x x} + + 22 m m {Ω Ω}_{z z} \overset{\cdot &Center Dot;}{y the y} \\ m m \overset{\cdot \cdot \cdot \cdot}{y the y} + + {d d}_{xy xy} \overset{\cdot &Center Dot;}{x x} + + {d d}_{yy yy} \overset{\cdot &Center Dot;}{y the y} + + {k k}_{xy xy} x x + + {k k}_{yy yy} y the y = = {u u}_{y the y} - - 22 m m {Ω Ω}_{z z} \overset{\cdot &Center Dot;}{x x} \end{matrix} - - - - - - ((22))$

分别为x轴和y轴的加速度信号。下面对对公式(2)描述的模型进行非量纲化处理。取无量纲时间t^*＝ω₀t，其中ω₀为两轴的共振频率，取值在1KHz左右，则： are the acceleration signals of the x-axis and y-axis respectively. The model described by formula (2) will be non-dimensionalized as follows. Take the dimensionless time t ^* = ω ₀ t, where ω ₀ is the resonance frequency of the two axes, and the value is around 1KHz, then:

$\overset{\cdot &Center Dot;}{x x} = = {ω ω}_{00} {\overset{\cdot \cdot}{x x}}^{* *},, \overset{\cdot \cdot \cdot \cdot}{x x} = = {ω ω}_{00}^{22} {\overset{\cdot \cdot \cdot &Center Dot;}{x x}}^{* *},, \overset{\cdot &Center Dot;}{y the y} = = {ω ω}_{00} {\overset{\cdot \cdot}{y the y}}^{* *},, \overset{\cdot &Center Dot; \cdot \cdot}{y the y} = = {ω ω}_{00}^{22} {\overset{\cdot \cdot \cdot \cdot}{y the y}}^{* *} - - - - - - ((33))$

讲上式带入(2)式，并在等式左右两边同除以质量m、ω₀ ²和参考长度q₀可以得到： Bring the above formula into the formula (2), and divide the left and right sides of the equation by the mass m, ω ₀ ² and the reference length q ₀ to get:

$\begin{matrix} \frac{{\overset{\cdot &Center Dot; \cdot &Center Dot;}{x x}}^{* *}}{{q q}_{00}} + + \frac{{d d}_{xx xx}}{m m {ω ω}_{00}} \frac{{\overset{\cdot \cdot}{x x}}^{* *}}{{q q}_{00}} + + \frac{{d d}_{xy xy}}{m m {ω ω}_{00}} \frac{{\overset{\cdot &Center Dot;}{y the y}}^{* *}}{{q q}_{00}} + + \frac{{k k}_{xx xx}}{m m {ω ω}_{00}^{22}} \frac{{x x}^{* *}}{{q q}_{00}} + + \frac{{k k}_{xy xy}}{m m {ω ω}_{00}^{22}} \frac{{y the y}^{* *}}{{q q}_{00}} = = \frac{{u u}_{x x}}{m m {ω ω}_{00}^{22} {q q}_{00}} + + 22 \frac{{Ω Ω}_{z z}}{{ω ω}_{00}} \frac{{\overset{\cdot &Center Dot;}{y the y}}^{* *}}{{q q}_{00}} \\ \frac{{\overset{\cdot &Center Dot; \cdot &Center Dot;}{y the y}}^{* *}}{{q q}_{00}} + + \frac{{d d}_{xy xy}}{m m {ω ω}_{00}} \frac{{\overset{\cdot \cdot}{x x}}^{* *}}{{q q}_{00}} + + \frac{{d d}_{yy yy}}{m m {ω ω}_{00}} \frac{{\overset{\cdot &Center Dot;}{y the y}}^{* *}}{{q q}_{00}} + + \frac{{k k}_{xy xy}}{m m {ω ω}_{00}^{22}} \frac{{x x}^{* *}}{{q q}_{00}} + + \frac{{k k}_{yy yy}}{m m {ω ω}_{00}^{22}} \frac{{y the y}^{* *}}{{q q}_{00}} = = \frac{{u u}_{y the y}}{m m {ω ω}_{00}^{22} {q q}_{00}} - - 22 \frac{{Ω Ω}_{z z}}{{ω ω}_{00}} \frac{{\overset{\cdot &Center Dot;}{x x}}^{* *}}{{q q}_{00}} \end{matrix} - - - - - - ((44))$

x^*、y^*、分别表示无量纲的x轴和y轴的位置信号、速度信号和加速度信号。 x ^* , y ^* , Represents the position signal, velocity signal and acceleration signal of the dimensionless x-axis and y-axis, respectively.

以向量形式重写上式可以得到： Rewriting the above formula in vector form can get:

$\overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} + + D D. \overset{\cdot &Center Dot;}{q q} + + Kq Q = = u u - - 22 Ω Ω \overset{\cdot &Center Dot;}{q q} - - - - - - ((55))$

式中， $q = [\begin{matrix} \frac{x^{*}}{q_{0}} \\ \frac{y^{*}}{q_{0}} \end{matrix}], D = [\begin{matrix} \frac{d_{xx}}{m ω_{0}} & \frac{d_{xy}}{m ω_{0}} \\ \frac{d_{xy}}{m ω_{0}} & \frac{d_{yy}}{m ω_{0}} \end{matrix}], K = [\begin{matrix} \frac{k_{xx}}{m {ω_{0}}^{2}} & \frac{k_{xy}}{m {ω_{0}}^{2}} \\ \frac{k_{xy}}{m {ω_{0}}^{2}} & \frac{k_{yy}}{m {ω_{0}}^{2}} \end{matrix}], u = [\begin{matrix} \frac{u_{x}}{m {ω_{0}}^{2} q_{0}} \\ \frac{u_{y}}{m {ω_{0}}^{2} q_{0}} \end{matrix}], Ω = [\begin{matrix} 0 & - \frac{Ω_{z}}{ω_{0}} \\ \frac{Ω_{z}}{ω_{0}} & 0 \end{matrix}],$ 分别是位置信号向量q的一阶导数和二阶导数，即速度信号向量和加速度信号向量。为书写简洁，令则 $q = [\begin{matrix} x \\ y \end{matrix}],$ $D = [\begin{matrix} d_{xx} & d_{xy} \\ d_{xy} & d_{yy} \end{matrix}], K = [\begin{matrix} ω_{x}^{2} & ω_{xy} \\ ω_{xy} & ω_{y}^{2} \end{matrix}], u = [\begin{matrix} u_{x} \\ u_{y} \end{matrix}], Ω = [\begin{matrix} 0 & - Ω_{z} \\ Ω_{z} & 0 \end{matrix}] .$ In the formula, $q = [\begin{matrix} \frac{x^{*}}{q_{0}} \\ \frac{{the y}^{*}}{q_{0}} \end{matrix}], D. = [\begin{matrix} \frac{d_{xx}}{m ω_{0}} & \frac{d_{xy}}{m ω_{0}} \\ \frac{d_{xy}}{m ω_{0}} & \frac{d_{yy}}{m ω_{0}} \end{matrix}], K = [\begin{matrix} \frac{k_{xxx}}{m {ω_{0}}^{2}} & \frac{k_{xy}}{m {ω_{0}}^{2}} \\ \frac{k_{xy}}{m {ω_{0}}^{2}} & \frac{k_{yy}}{m {ω_{0}}^{2}} \end{matrix}], u = [\begin{matrix} \frac{u_{x}}{m {ω_{0}}^{2} q_{0}} \\ \frac{u_{the y}}{m {ω_{0}}^{2} q_{0}} \end{matrix}], Ω = [\begin{matrix} 0 & - \frac{Ω_{z}}{ω_{0}} \\ \frac{Ω_{z}}{ω_{0}} & 0 \end{matrix}],$ are the first and second derivatives of the position signal vector q, namely, the velocity signal vector and the acceleration signal vector. For simplicity of writing, let but $q = [\begin{matrix} x \\ the y \end{matrix}],$ $D. = [\begin{matrix} d_{xx} & d_{xy} \\ d_{xy} & d_{yy} \end{matrix}], K = [\begin{matrix} ω_{x}^{2} & ω_{xy} \\ ω_{xy} & ω_{the y}^{2} \end{matrix}], u = [\begin{matrix} u_{x} \\ u_{the y} \end{matrix}], Ω = [\begin{matrix} 0 & - Ω_{z} \\ Ω_{z} & 0 \end{matrix}] .$

在考虑外界干扰时，振动式微陀螺仪模型可用下式表达： When considering external interference, the vibrating micro-gyroscope model can be expressed by the following formula:

$\overset{\cdot &Center Dot; \cdot \cdot}{q q} + + ((D D. + + 22 Ω Ω)) \overset{\cdot &Center Dot;}{q q} + + K K q q = = u u + + d d - - - - - - ((66))$

式中d表示外界干扰； In the formula, d represents external interference;

在迭代学习控制中，系统动力学方程表示为： In iterative learning control, the system dynamics equation is expressed as:

${\overset{\cdot &Center Dot; \cdot &Center Dot;}{q q}}_{k k} + + ((D D. + + 22 Ω Ω)) {\overset{\cdot \cdot}{q q}}_{k k} + + K K {q q}_{k k} = = {u u}_{k k} + + {d d}_{k k} - - - - - - ((77))$

前述的步骤3)，根据微陀螺仪的无量纲动力学模型进行自适应参数估计的方法，具体步骤为： Aforesaid step 3), carry out the method for self-adaptive parameter estimation according to the dimensionless dynamic model of microgyroscope, concrete steps are:

由分析可知，系统模型还满足以下特性： It can be seen from the analysis that the system model also satisfies the following characteristics:

(1)由于q_k和为可检测的质量块位移和速度，所以为已知矩阵，ξ未知向量，由系统结构参数组成，其具体形式如下式所示： (1) Since q _k and is the detectable mass displacement and velocity, so is a known matrix, and ξ is an unknown vector, which is composed of system structure parameters, and its specific form is shown in the following formula:

${(\begin{matrix} {\overset{\cdot &Center Dot;}{q q}}_{11} & {\overset{\cdot \cdot}{q q}}_{22} & 00 & - - 22 {\overset{\cdot &Center Dot;}{q q}}_{22} & {q q}_{11} & {q q}_{22} & 00 \\ 00 & {\overset{\cdot &Center Dot;}{q q}}_{11} & {\overset{\cdot &Center Dot;}{q q}}_{22} & - - 22 {\overset{\cdot &Center Dot;}{q q}}_{11} & 00 & {q q}_{11} & {q q}_{22} \end{matrix})}_{k k} = = Ψ Ψ (({q q}_{k k},, {\overset{\cdot &Center Dot;}{q q}}_{k k}))$

$(\begin{matrix} {d d}_{xx xx} \\ {d d}_{xy xy} \\ {d d}_{yy yy} \\ {Ω Ω}_{z z} \\ {ω ω}_{x x}^{22} \\ {ω ω}_{xy xy} \\ {ω ω}_{y the y}^{22} \end{matrix}) = = ξ ξ$

$K K {q q}_{k k} ((t t)) + + ((D D. + + 22 Ω Ω)) {\overset{\cdot \cdot}{q q}}_{k k} ((t t)) = = {(\begin{matrix} {\overset{\cdot \cdot}{q q}}_{11} & {\overset{\cdot &Center Dot;}{q q}}_{22} & 00 & - - 22 {\overset{\cdot \cdot}{q q}}_{22} & {q q}_{11} & {q q}_{22} & 00 \\ 00 & \overset{\cdot \cdot}{{q q}_{11}} & {\overset{\cdot \cdot}{q q}}_{22} & - - 22 {\overset{\cdot \cdot}{q q}}_{11} & 00 & {q q}_{11} & {q q}_{22} \end{matrix})}_{k k} (\begin{matrix} {d d}_{xx xx} \\ {d d}_{xy xy} \\ {d d}_{yy yy} \\ {Ω Ω}_{z z} \\ {ω ω}_{x x}^{22} \\ {ω ω}_{xy xy} \\ {ω ω}_{y the y}^{22} \end{matrix}) = = Ψ Ψ (({q q}_{k k},, {\overset{\cdot \cdot}{q q}}_{k k})) ξ ξ$

(2)β为正实数，为期望加速信号向量； (2) β is a positive real number, is the desired acceleration signal vector;

(3)θ(t)＝[ξ^T(t)β]^T，θ由系统结构参数和β组成，即 $θ = (\begin{matrix} d_{xx} \\ d_{xy} \\ d_{yy} \\ Ω_{z} \\ ω_{x}^{2} \\ ω_{xy} \\ ω_{x}^{2} \\ β \end{matrix}) .$ (3)θ(t)＝[ξ ^T (t)β] ^T , θ is composed of system structure parameters and β, namely $θ = (\begin{matrix} d_{xxx} \\ d_{xy} \\ d_{yy} \\ Ω_{z} \\ ω_{x}^{2} \\ ω_{xy} \\ ω_{x}^{2} \\ β \end{matrix}) .$

(4)sgn()为符号函数，R为实数，位置误差信号向量e_k(t)＝q_d(t)-q_k(t)，速度误差信号向量具体形式如下式所示： (4) sgn() is a symbolic function, R is a real number, position error signal vector e _k (t) = q _d (t)-q _k (t), speed error signal vector The specific form is as follows:

根据以上系统特性，现提出自适应参数估计算法： According to the above system characteristics, an adaptive parameter estimation algorithm is proposed:

${\overset{&OverBar; &OverBar;}{θ θ}}_{k k} ((t t)) = = {\overset{&OverBar; &OverBar;}{θ θ}}_{k k} + + {\overset{^^}{θ θ}}_{k k - - 11} ((t t)) - - - - - - ((99))$

其中为θ的估计值，为第k次迭代时θ的估计值的增量。矩阵 Γ为正定对称矩阵，Γ＝diag(90,90,90,90,90,90,90,90)。 in is the estimated value of θ, is the increment of the estimated value of θ at the kth iteration. matrix Γ is a positive definite symmetric matrix, Γ=diag(90,90,90,90,90,90,90,90).

前述的步骤4)，根据微陀螺仪的无量纲动力学模型和自适应参数估计算法，自适应迭代学习控制方法控制信号u_k(t)定义为： In the aforementioned step 4), according to the dimensionless dynamic model of the micro-gyroscope and the adaptive parameter estimation algorithm, the control signal u _k (t) of the adaptive iterative learning control method is defined as:

其中矩阵K_P、K_D都为正定对称矩阵， $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ $K_{D} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ 则当满足迭代条件未知误差初始信号e_k(0)＝0和速度误差初始信号时，未知误差信号e_k(t)和速度误差初始信号有界，且 $\lim_{k &RightArrow; 0} e_{k} (t) = \lim_{k &RightArrow; 0} {\overset{\cdot}{e}}_{k} (t) = 0,$ t处于迭代周期[0,T]中。 where matrix Both K _P and K _D are positive definite symmetric matrices, $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ $K_{D.} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ Then when the iteration condition is satisfied, the unknown error initial signal e _k (0)=0 and the velocity error initial signal When , the unknown error signal e _k (t) and the velocity error initial signal bounded, and $\lim_{k &Right Arrow; 0} e_{k} (t) = \lim_{k &Right Arrow; 0} {\overset{\cdot}{e}}_{k} (t) = 0,$ t is in iteration period [0,T].

当迭代次数趋于无穷时，微陀螺仪的轨迹跟踪误差和速度跟踪误差趋近于零，实际应用中，只需要5次迭代就可以实现在参数未知和存在外界干扰的情况下实现轨迹跟踪误差和速度跟踪误差趋近于零，具有实际应用价值。 When the number of iterations tends to infinity, the trajectory tracking error and velocity tracking error of the micro gyroscope approach zero. In practical applications, only 5 iterations are needed to realize the trajectory tracking error in the case of unknown parameters and external interference. And the speed tracking error is close to zero, which has practical application value. the

为更进一步阐述本发明为达成预定发明目的所采取的技术手段及功效，以下结合附图及较佳实施例，对依据本发明提出的微陀螺仪的自适应迭代学习控制系统及方法进行详细说明如后。 In order to further explain the technical means and effects adopted by the present invention to achieve the intended purpose of the invention, the self-adaptive iterative learning control system and method of the micro gyroscope according to the present invention will be described in detail below in conjunction with the accompanying drawings and preferred embodiments As later. the

为了说明本发明提出的微陀螺仪的自适应迭代学习控制方法对微陀螺仪控制系统的稳定性和有效性，现通过类Lyapunov理论加以说明。 In order to illustrate the stability and effectiveness of the micro-gyroscope self-adaptive iterative learning control method proposed in the present invention to the micro-gyroscope control system, it will now be described through the Lyapunov-like theory. the

Lyapunov函数设计为： The Lyapunov function is designed as:

${W W}_{k k} (({e e}_{k k} ((t t)),, {\overset{\cdot &Center Dot;}{e e}}_{k k} ((t t)),, {\overset{~ ~}{θ θ}}_{k k} ((t t)))) = = {V V}_{k k} (({e e}_{k k} ((t t)),, {\overset{\cdot \cdot}{e e}}_{k k} ((t t)))) + + \frac{11}{22} {&Integral; &Integral;}_{00}^{t t} {\overset{~ ~}{θ θ}}_{k k}^{T T} ((t t)) {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k} ((t t)) dτ dτ - - - - - - ((44))$

其中 $V_{k} (e_{k} (t), {\overset{\cdot}{e}}_{k} (t)) = \frac{1}{2} (e_{k}^{T} K_{p} e_{k} + {\overset{\cdot}{e}}_{k}^{T} {\overset{\cdot}{e}}_{k}), {\tilde{θ}}_{k} (t) = θ (t) - {\hat{θ}}_{k} (t), {\hat{θ}}_{k} (t) = {[\begin{matrix} {\hat{ξ}}_{k}^{T} (t) & {\hat{β}}_{k} \end{matrix}, (t)]}^{T}$ 为θ(t)的估计值。 in $V_{k} (e_{k} (t), {\overset{&Center Dot;}{e}}_{k} (t)) = \frac{1}{2} (e_{k}^{T} K_{p} e_{k} + {\overset{&Center Dot;}{e}}_{k}^{T} {\overset{\cdot}{e}}_{k}), {\tilde{θ}}_{k} (t) = θ (t) - {\hat{θ}}_{k} (t), {\hat{θ}}_{k} (t) = {[\begin{matrix} {\hat{ξ}}_{k}^{T} (t) & {\hat{β}}_{k} \end{matrix}, (t)]}^{T}$ is the estimated value of θ(t).

(1)W_k为非增序列 (1) W _k is a non-increasing sequence

$Δ Δ {W W}_{k k} = = {W W}_{k k} - - {W W}_{k k - - 11} = = {V V}_{k k} - - {V V}_{k k - - 11} + + \frac{11}{22} {&Integral; &Integral;}_{00}^{t t} (({\overset{~ ~}{θ θ}}_{k k}^{T T} ((t t)) {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k} ((t t)) - - {\overset{~ ~}{θ θ}}_{k k - - 11}^{T T} ((t t)) {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k - - 11} ((t t)))) dτ dτ - - - - - - ((55))$

取 ${\overset{&OverBar;}{θ}}_{k} = {\hat{θ}}_{k} (t) - {\hat{θ}}_{k - 1} (t),$ 则 ${\overset{&OverBar;}{θ}}_{k} = θ - {\tilde{θ}}_{k} (t) - θ + {\tilde{θ}}_{k - 1} (t) = - {\tilde{θ}}_{k} (t) + {\tilde{θ}}_{k - 1} (t),$ 且 Pick ${\overset{&OverBar;}{θ}}_{k} = {\hat{θ}}_{k} (t) - {\hat{θ}}_{k - 1} (t),$ but ${\overset{&OverBar;}{θ}}_{k} = θ - {\tilde{θ}}_{k} (t) - θ + {\tilde{θ}}_{k - 1} (t) = - {\tilde{θ}}_{k} (t) + {\tilde{θ}}_{k - 1} (t),$ and

${\overset{~ ~}{θ θ}}_{k k}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k} - - {\overset{~ ~}{θ θ}}_{k k - - 11}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k - - 11} = = - - {\overset{&OverBar; &OverBar;}{θ θ}}_{k k}^{T T} {Γ Γ}^{- - 11} {\overset{&OverBar; &OverBar;}{θ θ}}_{k k} - - 22 {\overset{&OverBar; &OverBar;}{θ θ}}_{k k}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k}$

将该式带入式(11)得到： Put this formula into formula (11) to get:

$Δ Δ {W W}_{k k} = = {W W}_{k k} - - {W W}_{k k - - 11} = = {V V}_{k k} - - {V V}_{k k - - 11} - - \frac{11}{22} {&Integral; &Integral;}_{00}^{t t} (({\overset{&OverBar; &OverBar;}{θ θ}}_{k k}^{T T} {Γ Γ}^{- - 11} {\overset{&OverBar; &OverBar;}{θ θ}}_{k k} + + 22 {\overset{&OverBar; &OverBar;}{θ θ}}_{k k}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{k k})) dτ dτ - - - - - - ((66))$

由于 ${&Integral;}_{0}^{t} {\overset{\cdot}{V}}_{k} (t) dτ = V_{k} (t) - V_{k} (0),$ 即 $V_{k} (t) = V_{k} (0) + {&Integral;}_{0}^{t} {\overset{\cdot}{V}}_{k} (t) dτ .$ 又由于 because ${&Integral;}_{0}^{t} {\overset{\cdot}{V}}_{k} (t) dτ = V_{k} (t) - V_{k} (0),$ Right now $V_{k} (t) = V_{k} (0) + {&Integral;}_{0}^{t} {\overset{\cdot}{V}}_{k} (t) dτ .$ And because of

则 but

式(13)中，将该式带入式(14)可得到： In formula (13), Put this formula into formula (14) to get:

由迭代条件e_k(0)＝0， ${\overset{\cdot}{e}}_{k} (0) = 0,$ 所以 $V_{k} ({\overset{\cdot}{e}}_{k} (0), (0)) = 0,$ 则 By the iteration condition e _k (0)=0, ${\overset{\cdot}{e}}_{k} (0) = 0,$ so $V_{k} ({\overset{&Center Dot;}{e}}_{k} (0), (0)) = 0,$ but

由自适应律可知根据该式，可有如下结果： From the adaptive law we know According to this formula, the following results can be obtained:

将式(16)～(18)带入式(13)得 Put formula (16)～(18) into formula (13) to get

因此W_k为非增序列。 Therefore W _k is a non-increasing sequence.

(2)W₀(t)的有界性 (2) Boundedness of W ₀ (t)

由式(11)可知W₀(t)的一阶导数为 From formula (11), it can be seen that the first derivative of W ₀ (t) is

${\overset{\cdot &Center Dot;}{W W}}_{00} (({\overset{\cdot \cdot}{e e}}_{00} ((t t)),, {e e}_{00} ((t t)),, {\overset{~ ~}{θ θ}}_{00} ((t t)))) = = {\overset{\cdot &Center Dot;}{V V}}_{00} (({e e}_{00} ((t t)),, {\overset{\cdot &Center Dot;}{e e}}_{00} ((t t)))) + + \frac{11}{22} {\overset{~ ~}{θ θ}}_{00}^{T T} ((t t)) {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{00} ((t t)) - - - - - - ((1313))$

当k＝0时将式(16)两边求导得带入式(19)得 When k=0, deriving both sides of formula (16) to get Bring into formula (19) to get

由于 ${\hat{θ}}_{- 1} (t) = 0,$ 所以则再代入式(20)得 because ${\hat{θ}}_{- 1} (t) = 0,$ so but Then substitute into formula (20) to get

${\overset{\cdot &Center Dot;}{W W}}_{00} \leq \leq - - {\overset{\cdot &Center Dot;}{e e}}_{00}^{T T} {K K}_{D D.} {\overset{\cdot &Center Dot;}{e e}}_{00} + + (({\overset{^^}{θ θ}}_{00}^{T T} + + \frac{11}{22} {\overset{~ ~}{θ θ}}_{00}^{T T})) {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{00} = = - - {\overset{\cdot &Center Dot;}{e e}}_{00}^{T T} {K K}_{D D.} {\overset{\cdot &Center Dot;}{e e}}_{00} - - \frac{11}{22} {\overset{~ ~}{θ θ}}_{00}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{00} + + {θ θ}_{00}^{T T} {Γ Γ}^{- - 11} {\overset{~ ~}{θ θ}}_{00} - - - - - - ((1515))$

再由Young’s inequality(a²+b²≥2ab)，则其中K₁>0。因为K_D为正定阵，其两个特征值λ_d2≥λ_d1≥0，所以同理有 $- \frac{1}{2} {\tilde{θ}}_{0}^{T} Γ^{- 1} {\tilde{θ}}_{0} \leq - \frac{1}{2} λ_{1} {| | {\tilde{θ}}_{0} | |}^{2}, K_{1} {| | Γ^{- 1} {\tilde{θ}}_{0} | |}^{2} \leq K_{1} λ_{2}^{2} {| | {\tilde{θ}}_{0} | |}^{2},$ 其中λ₁、λ₂分别为正定矩阵Γ^-1的最小、最大特征值。 Then by Young's inequality (a ² +b ² ≥ 2ab), then where K ₁ >0. Because K _D is a positive definite matrix, its two eigenvalues λ _d2 ≥ λ _d1 ≥ 0, so In the same way $- \frac{1}{2} {\tilde{θ}}_{0}^{T} Γ^{- 1} {\tilde{θ}}_{0} \leq - \frac{1}{2} λ_{1} {| | {\tilde{θ}}_{0} | |}^{2}, K_{1} {| | Γ^{- 1} {\tilde{θ}}_{0} | |}^{2} \leq K_{1} λ_{2}^{2} {| | {\tilde{θ}}_{0} | |}^{2},$ Among them, λ ₁ and λ ₂ are the minimum and maximum eigenvalues of positive definite matrix Γ ^-1 respectively.

$\begin{matrix} {\overset{\cdot &Center Dot;}{W W}}_{00} \leq \leq - - {λ λ}_{d d 11} {| | | | {\overset{\cdot &Center Dot;}{e e}}_{00} | | | |}^{22} - - \frac{11}{22} {λ λ}_{11} {| | | | {\overset{~ ~}{θ θ}}_{00} | | | |}^{22} + + {K K}_{11} {λ λ}_{22}^{22} {| | | | {\overset{~ ~}{θ θ}}_{00} | | | |}^{22} + + \frac{11}{{44 K K}_{11}} {| | | | θ θ | | | |}^{22} \\ = = - - {λ λ}_{d d 11} {| | | | {\overset{\cdot &Center Dot;}{e e}}_{00} | | | |}^{22} - - ((\frac{11}{22} {λ λ}_{11} - - {K K}_{11} {λ λ}_{22}^{22})) {| | | | {\overset{~ ~}{θ θ}}_{00} | | | |}^{22} + + \frac{11}{{44 K K}_{11}} {| | | | θ θ | | | |}^{22} \end{matrix} - - - - - - ((1616))$

取则式(23)进一步可以得到即W₀(t)在t∈[0,T]有界。 Pick but Formula (23) can be further obtained That is, W ₀ (t) is bounded at t∈[0,T].

(3)W_k(t)的有界性 (3) Boundedness of W _k (t)

因为W_k(t)为Lypunov能量函数、且在迭代轴上是非增序列，所以对于k≥1有0≤W_k(t)≤W₀(t)。又因为W₀(t)在t∈[0,T]有界，所以W_k(t)在t∈[0,T]有界。 Since W _k (t) is a Lypunov energy function and is a non-increasing sequence on the iteration axis, 0≤W _k (t)≤W ₀ (t) for k≥1. And because W ₀ (t) is bounded at t∈[0,T], W _k (t) is bounded at t∈[0,T].

根据以上三部分的说明，可以推导出有过程如下。 According to the description of the above three parts, it can be deduced that have The process is as follows.

W_k(t)可写为将式(19)和式(11)带入可得 W _k (t) can be written as Substituting formula (19) and formula (11) into

${W W}_{k k} \leq \leq {W W}_{00} - - {Σ Σ}_{j j = = 11}^{k k} {V V}_{j j - - 11} \leq \leq {W W}_{00} - - \frac{11}{22} {Σ Σ}_{j j = = 11}^{k k} (({\overset{\cdot &Center Dot;}{e e}}_{j j - - 11}^{T T} {\overset{\cdot &Center Dot;}{e e}}_{j j - - 11} + + {e e}_{j j - - 11}^{T T} {K K}_{P P} {e e}_{j j - - 11}))$

则 but

${Σ Σ}_{j j = = 11}^{k k} (({\overset{\cdot \cdot}{e e}}_{j j - - 11}^{T T} {\overset{\cdot \cdot}{e e}}_{j j - - 11} + + {e e}_{j j - - 11}^{T T} {K K}_{P P} {e e}_{j j - - 11})) \leq \leq 22 (({W W}_{00} - - {W W}_{k k})) \leq \leq {22 W W}_{00} - - - - - - ((1717))$

式(24)蕴含了当迭代次数趋于无穷时系统能完全跟踪上参考轨迹。 Equation (24) implies When the number of iterations tends to infinity, the system can completely track the upper reference trajectory.

最后，进行计算机仿真 Finally, computer simulations

本实施例中，利用数学软件Matlab/Simulink进行计算机仿真实验，选取微陀螺仪的参数为： In the present embodiment, utilize mathematical software Matlab/Simulink to carry out computer simulation experiment, select the parameter of micro-gyroscope as:

m＝1.8×10^-7kg，k_xx＝63.955N/m，k_yy＝95.92N/m，k_xy＝12.779N/m，d_xx＝1.8×10^-6N·s/m，d_yy＝1.8×10^-6N·s/m，d_xy＝3.6×10^-7N·s/m m=1.8×10 ^-7 kg, k _xx =63.955 N/m, k _yy =95.92 N/m, k _xy =12.779 N/m, d _xx =1.8×10 ^-6 N·s/m, d _yy = 1.8×10 ^-6 N·s/m, d _xy =3.6×10 ^-7 N·s/m

未知的输入角速度假定为Ω_z＝100rad/s。参考长度选取为q₀＝1μm，参考频率ω₀＝1000Hz，非量纲化后，各微陀螺仪参数如下： The unknown input angular velocity is assumed to be Ω _z =100 rad/s. The reference length is selected as q ₀ =1μm, and the reference frequency ω ₀ =1000Hz. After non-dimensionalization, the parameters of each micro gyroscope are as follows:

ω_x ²＝355.3，ω_y ²＝532.9，ω_xy＝70.99，d_xx＝0.01， ω _x ² =355.3, ω _y ² =532.9, ω _xy =70.99, d _xx =0.01,

d_yy＝0.01，d_xy＝0.002，Ω_z＝0.1 d _yy =0.01, d _xy =0.002, Ω _z =0.1

参考轨迹选取 $q_{d} = [\begin{matrix} x_{d} \\ y_{d} \end{matrix}] = [\begin{matrix} \sin (6.17 t) \\ 1.2 \sin (5.11 t) \end{matrix}], d_{k} = [\begin{matrix} rand (1) \sin (t) \\ rand (1) \sin (t) \end{matrix}]$ 被控对象的初始状态为q_k(0)＝q_d(0)， ${\overset{\cdot}{q}}_{k} (0) = {\overset{\cdot}{q}}_{d} (0) .$ 控制器参数设为 $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}], K_{D} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ 自适应律参数Γ＝diag(90,90,90,90,90,90,90,90)。 Reference track selection $q_{d} = [\begin{matrix} x_{d} \\ {the y}_{d} \end{matrix}] = [\begin{matrix} \sin (6.17 t) \\ 1.2 \sin (5.11 t) \end{matrix}], d_{k} = [\begin{matrix} rand (1) \sin (t) \\ rand (1) \sin (t) \end{matrix}]$ The initial state of the controlled object is q _k (0) = q _d (0), ${\overset{&Center Dot;}{q}}_{k} (0) = {\overset{&Center Dot;}{q}}_{d} (0) .$ The controller parameters are set to $K_{P} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}], K_{D.} = [\begin{matrix} 70 & 0 \\ 0 & 70 \end{matrix}],$ Adaptive law parameter Γ=diag(90,90,90,90,90,90,90,90).

仿真结果如图2～5所示。 The simulation results are shown in Fig. 2-5. the

图2显示了x轴和y轴方向第5次迭代周期中的位置跟踪图。图中实线代表x轴和y轴方向的参考轨迹，虚线代表x轴和y轴方向的实际轨迹。从图中可以很明显看出，当第5次迭代时，使用自适应迭代学习控制方案的位置曲线几乎完全跟踪上参考曲线。 Figure 2 shows the position tracking plot in the 5th iteration cycle in the x-axis and y-axis directions. The solid line in the figure represents the reference trajectory in the direction of the x-axis and the y-axis, and the dotted line represents the actual trajectory in the direction of the x-axis and the y-axis. It is obvious from the figure that when the 5th iteration, the position curve using the adaptive iterative learning control scheme almost completely tracks the upper reference curve. the

图3显示了5次迭代过程中每次迭代位置跟踪误差的最大绝对值变化情况，图中位置误差最大绝对值定义为e1＝max(|q1_d(t)-q1_k(t)|)，e2＝max(|q2_d(t)-q2_k(t)|)，t∈[0,T]。图中可以很明显看出x轴和y轴位置误差最大绝对值初值小，且下降很快，在第5次迭代时几乎接近于零。 Figure 3 shows the change of the maximum absolute value of the position tracking error in each iteration during the 5 iterations. The maximum absolute value of the position error in the figure is defined as e1=max(|q1 _d (t)-q1 _k (t)|), e2=max(| _q2d (t) _-q2k (t)|), t∈[0,T]. It can be clearly seen from the figure that the initial maximum absolute value of the position error of the x-axis and y-axis is small, and it drops rapidly, and it is almost close to zero in the fifth iteration.

图4和图5分别反映速度跟踪情况和速度跟踪误差最大绝对值变化情况。从图中可以很直观地观察到使用自适应迭代学习控制方案的速度跟踪性能较好。 Figure 4 and Figure 5 respectively reflect the speed tracking situation and the change of the maximum absolute value of the speed tracking error. It can be intuitively observed from the figure that the velocity tracking performance using the adaptive iterative learning control scheme is better. the

从以上仿真图可以看出，本发明提出的控制方法对微陀螺仪的轨迹跟踪有着很好的控制效果，大大提高了微陀螺仪系统的追踪性能和鲁棒性，对微陀螺仪两轴振动轨迹的高精度控制提供了理论依据和仿真基础，算法简单，易于实现，具有较好的实用价值。 As can be seen from the above simulation diagram, the control method proposed by the present invention has a good control effect on the trajectory tracking of the micro gyroscope, greatly improves the tracking performance and robustness of the micro gyroscope system, and has a good effect on the two-axis vibration of the micro gyroscope. The high-precision control of the trajectory provides a theoretical basis and a simulation basis. The algorithm is simple, easy to implement, and has good practical value. the

本发明说明书中未作详细描述的内容属于本领域专业技术人员公知的技术知识。 The content that is not described in detail in the specification of the present invention belongs to the technical knowledge known to those skilled in the art. the

以上所述，仅是本发明的较佳实施例而已，然而并非用以限定本发明，任何熟悉本专业的技术人员，在不脱离本发明技术方案范围内，当可利用上述揭示的技术内容作出些许更动或修饰为等同变化的等效实施例，但凡是未脱离本发明技术方案的内容，均仍属于本方明技术方案的保护范围。 The above is only a preferred embodiment of the present invention, but it is not intended to limit the present invention. Any skilled person who is familiar with the profession can make use of the technical content disclosed above without departing from the scope of the technical solution of the present invention. Some changes or modifications are equivalent embodiments with equivalent changes, but any content that does not deviate from the technical solution of the present invention still belongs to the protection scope of the technical solution of the present invention. the

Claims

1. the adaptive iterative learning control method of gyroscope, is characterized in that, comprises the following steps:

1) set up the dimensionless kinetic model of gyroscope;

2) reference locus of reference locus module output gyroscope x and y shaft vibration, comprises position, rate signal;

3) adaptive law module receives the output of reference locus and gyroscope system, utilizes adaptive law to estimate the increment of parameter;

4) controller module receives new parameter estimation, and exports with the control signal of track following error, speed tracking error acting in conjunction generation adaptive iterative learning control method;

5) output signal of reception controller module, position and the velocity information of output gyroscope vibrating mass;

6) utilize alternative manner repeating step 3)-step 5), obtain position and the velocity information of final gyroscope vibrating mass.

2. the adaptive iterative learning control method of gyroscope according to claim 1, is characterized in that, in described step 1) in, the dimensionless kinetic model of gyroscope is:

In the time considering external interference, oscillating micro gyroscope instrument is formula (6) expression for model:

Wherein: d represents external interference;

u _x, u _yrepresent along x axle, the axial driving force of y; ω _xy, and d _xx, d _yy, d _xyrepresent respectively spring constant and ratio of damping between x axle, y axle, x axle and y axle; M is quality; Ω _zrepresent along the axial angular velocity of z; ω ₀for the resonant frequency of diaxon; respectively velocity vector and vector acceleration;

In iteration control process, formula (6) is expressed as:

In formula, k is iterations, and k is positive integer, q _k, u _k, d _kbe respectively the acceleration signal vector of the k time of q, u, d, rate signal vector, position signalling vector, input control vector, interference vector.

3. the adaptive iterative learning control method of gyroscope according to claim 2, is characterized in that, in described step 3) in, carrying out adaptive parameter estimation according to the dimensionless kinetic model of gyroscope, adaptive parameter estimation algorithm is:

Wherein θ is the vector being made up of system unknown parameter, for the estimated value of θ, for the initial value of θ estimated value, the increment of the estimated value of θ while being the k time iteration; Matrix Γ ∈ R ^{8 × 8}, Γ is positive definite symmetric matrices, Γ=diag (90,90,90,90,90,90,90,90);

β is arithmetic number;

for the rate signal of expecting.

4. the adaptive iterative learning control method of gyroscope according to claim 3, it is characterized in that, in described step 4) in, according to the dimensionless kinetic model of gyroscope and adaptive parameter estimation algorithm, carry out adaptive iterative learning control, control signal u _k(t) be defined as:

Wherein matrix K _p∈ R ^{2 × 2}, K _d∈ R ^{2 × 2}, K _p, K _dbe all positive definite symmetric matrices, ought meet iterated conditional unknown errors initialize signal e _k(0)=0 and velocity error initialize signal time, unknown errors signal e _kand velocity error initialize signal (t) bounded, and time t is in iteration cycle [0, T].