CN102778889B

CN102778889B - The intermittent defect robust parsing method of spacecraft attitude control system

Info

Publication number: CN102778889B
Application number: CN201210242175.4A
Authority: CN
Inventors: 杨浩; 姜斌; 程月华; 张化光
Original assignee: Nanjing University of Aeronautics and Astronautics
Current assignee: Nanjing University of Aeronautics and Astronautics
Priority date: 2012-07-12
Filing date: 2012-07-12
Publication date: 2015-07-29
Anticipated expiration: 2032-07-12
Also published as: CN102778889A

Abstract

The invention discloses a fault-tolerant analysis method for intermittent faults of a spacecraft attitude control system. The steps are: firstly, establishing a mathematical model of the spacecraft attitude control system; secondly, establishing a mathematical model of intermittent controller faults; using the Lyapunov method to design the controller, and Describe the behavior of the attitude control system under normal conditions and fault conditions, and use the random switching system model to describe the entire operation process of the spacecraft attitude control system with intermittent faults; then transform the fault-tolerant analysis problem of the attitude control system into a The problem of stability analysis of switching systems in unstable modes provides a fault-tolerant judgment criterion to judge whether the current system is stable in real time. As long as the criterion is met, no fault-tolerant control measures are required. Under the action of balance, it can still maintain stability. The invention can avoid adopting fault-tolerant control and high control energy consumption, high computational complexity and risks generated in the implementation process of fault-tolerant control.

Description

Intermittent Fault Tolerance Analysis Method for Spacecraft Attitude Control System

技术领域 technical field

本发明属于航天器姿态容错控制领域，特别涉及一种针对航天器控制器的间歇性故障的容错分析方法。The invention belongs to the field of spacecraft attitude fault-tolerant control, in particular to a fault-tolerant analysis method for intermittent faults of spacecraft controllers.

背景技术 Background technique

航天器姿态控制问题，因其重要的工程及学术价值，已经引起了人们极大的兴趣。航天器姿控系统对安全性有着极高的要求，而姿控系统的控制器、执行器、传感器以及系统内部都有可能发生故障，这些故障会严重影响系统的性能，甚至使系统崩溃。因此姿控系统需要具有一定的容错能力，才能保证其运行的稳定性和可靠性。关于航天器容错控制的方法和技术已经有很多成果，例如文献(IEEETransactions on Control Systems Technology,2008,16(4),799-808),（Journal of Guidance,Control and Dynamics,2008,31(5),1456-1463），（IET Control Theory & Applications,2011,5(2),271-282）等。这些容错控制方案大多只考虑执行器和传感器的故障情况，容错的核心思想是当故障发生后，通过调节或重构控制器以补偿故障对系统产生的影响，从而使故障系统依然稳定运行。The problem of spacecraft attitude control has aroused great interest because of its important engineering and academic value. The spacecraft attitude control system has extremely high requirements for safety, and the controllers, actuators, sensors, and internal systems of the attitude control system may fail. These failures will seriously affect the performance of the system, and even cause the system to crash. Therefore, the attitude control system needs to have a certain fault tolerance to ensure the stability and reliability of its operation. There have been many results on the methods and technologies of spacecraft fault-tolerant control, such as literature (IEEETransactions on Control Systems Technology, 2008, 16(4), 799-808), (Journal of Guidance, Control and Dynamics, 2008, 31(5) , 1456-1463), (IET Control Theory & Applications, 2011, 5(2), 271-282), etc. Most of these fault-tolerant control schemes only consider the failure of actuators and sensors. The core idea of fault tolerance is to compensate for the impact of the fault on the system by adjusting or reconfiguring the controller after the fault occurs, so that the fault system still operates stably.

发生在航天器姿控系统的控制器的间歇性故障自身时而产生，时而消失。当它发生时，会使得系统工作发生异常；当它消失时，系统又回到正常的工作状态。间歇性故障是导致电路系统失效的主要原因,文献（IEEE Transactions onReliability,1997,46,(2),269-274）提到这类故障的发生概率是永久性故障的10至30倍。对于航天器姿控系统的控制电路单元，各个电气元件和电路板上的焊接点随着时间的推移会产生物理磨损而破裂，或者产生时断时续的连接现象。这种松散的连接是一类典型的间歇性故障，可以直接影响到控制器的输出，即转矩指令电压，进而影响电机的输出。因此，对姿控系统的控制器间歇性故障进行容错分析是十分必要的。Intermittent faults occurring in the controllers of the spacecraft attitude control system appear and disappear by themselves. When it occurs, it will make the system work abnormally; when it disappears, the system will return to the normal working state. Intermittent faults are the main cause of circuit system failures. The literature (IEEE Transactions on Reliability, 1997, 46, (2), 269-274) mentions that the probability of such faults is 10 to 30 times that of permanent faults. For the control circuit unit of the spacecraft attitude control system, the solder joints on the various electrical components and circuit boards will be physically worn and broken over time, or intermittent connections will occur. This kind of loose connection is a typical intermittent fault, which can directly affect the output of the controller, that is, the torque command voltage, and then affect the output of the motor. Therefore, it is very necessary to carry out fault-tolerant analysis on the intermittent failure of the controller of the attitude control system.

然而，针对航天器间歇性故障的容错分析和设计的成果鲜有报道。众所周知，根据故障情况调节控制器以实现容错控制目标需要一定的时间，并且消耗一定的控制能量，这种方法很难适用于控制器的间歇性故障，原因有三：However, the results of fault-tolerant analysis and design for spacecraft intermittent failures are rarely reported. As we all know, adjusting the controller according to the fault situation to achieve the fault-tolerant control goal takes a certain amount of time and consumes a certain amount of control energy. This method is difficult to apply to the intermittent fault of the controller for three reasons:

1）间歇性故障发生频率比较高，如果每次当这些故障发生时都去调节控制器，势必要耗费很多的控制能耗；1) The frequency of intermittent faults is relatively high. If you adjust the controller every time these faults occur, it will consume a lot of control energy consumption;

2）过于频繁地调节控制器，会产生严重的超调，令系统的性能下降，甚至不稳定；2) If the controller is adjusted too frequently, serious overshoot will occur, which will degrade the performance of the system or even become unstable;

3）当控制器自身发生故障时，很难再去调节控制器以实现容错目标。3) When the controller itself fails, it is difficult to adjust the controller to achieve the fault tolerance goal.

因此，本发明人试图提出一种简便的控制器间歇性故障容错分析方法，本案由此产生。Therefore, the inventor attempted to propose a simple fault-tolerant analysis method for intermittent faults of the controller, and this case arose from it.

发明内容 Contents of the invention

本发明的目的，在于提供一种航天器姿态控制系统的间歇性故障容错分析方法，其针对控制器的间歇性故障，分析间歇性故障的发生和消失对系统稳定性的影响，得到一个容错判断准则，使得当间歇性故障满足一定条件时，不需要采取任何容错控制措施，带有控制器间歇性故障的航天器姿控系统依然稳定运行，从而避免采取容错控制，以及容错控制实施过程中产生的高控制能耗、高计算复杂度和风险。The object of the present invention is to provide a fault-tolerant analysis method for intermittent faults of a spacecraft attitude control system, which aims at the intermittent faults of the controller, analyzes the impact of the occurrence and disappearance of intermittent faults on the stability of the system, and obtains a fault-tolerant judgment The criterion makes it unnecessary to take any fault-tolerant control measures when the intermittent fault meets certain conditions, and the attitude control system of the spacecraft with the intermittent fault of the controller still operates stably, thereby avoiding the adoption of fault-tolerant control and the occurrence of fault-tolerant control during the implementation process. High control energy consumption, high computational complexity and risk.

为了达成上述目的，本发明的解决方案是：In order to achieve the above object, the solution of the present invention is:

一种航天器姿态控制系统的间歇性故障容错分析方法，包括如下步骤：A method for intermittent fault-tolerant analysis of a spacecraft attitude control system, comprising the steps of:

（1）建立航天器的姿态控制系统数学模型：(1) Establish the mathematical model of the attitude control system of the spacecraft:

$J J \overset{\cdot \cdot}{ω ω} = = - - {ω ω}^{\times \times} Jω Jω + + μ μ + + d d$

$\overset{\cdot &Center Dot;}{q q} = = \frac{11}{22} (({q q}_{44} ω ω + + {ω ω}^{\times \times} q q))$

${\overset{\cdot \cdot}{q q}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q$

其中，表示惯性角速度；q₄是标量，q₁,q₂,q₃,q₄表示为四元数；J=J^T表示正定惯性矩阵，μ表示控制器输出，即控制扭矩；d表示系统的不确定和扰动；in, Indicates the inertial angular velocity; q ₄ is a scalar, q ₁ , q ₂ , q ₃ , and q ₄ are represented as quaternions; J=J ^T represents a positive definite inertia matrix, μ represents the output of the controller, that is, the control torque; d represents the uncertainty and disturbance of the system;

选择姿控系统的平衡点为：ω=q=0，q₄=1，则有：Select the balance point of the attitude control system as: ω=q=0, q ₄ =1, then:

$\overset{\cdot \cdot}{q q} = = \frac{11}{22} (({\overset{~ ~}{q q}}_{44} + + 11)) ω ω + + \frac{11}{22} {ω ω}^{\times \times} q q$

${\overset{\cdot \cdot}{\overset{~ ~}{q q}}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q$

其中， ${\tilde{q}}_{4} \overset{Δ}{=} q_{4} - 1;$ in, ${\tilde{q}}_{4} \overset{Δ}{=} q_{4} - 1;$

（2）建立带有间歇性故障的控制器数学模型：(2) Establish a mathematical model of the controller with intermittent faults:

用μ_σ表示带有间歇性故障的控制器输出，其中下标σ(t)是一个随时间变化的切换函数，在{0,1}中取值，其中0表示控制器处在正常情况，1表示控制器处在间歇性故障情况，根据间歇性故障的发生特点，该切换函数用马尔科夫链来描述，即Denote the controller output with intermittent faults by μ _σ , where the subscript σ(t) is a time-varying switching function, taking values in {0,1}, where 0 indicates that the controller is in normal condition, 1 means that the controller is in an intermittent fault situation. According to the characteristics of intermittent faults, the switching function is described by a Markov chain, that is,

$P P {{σ σ ((t t + + Δ Δ)) = = j j | | σ σ ((t t)) = = i i}} = = \{\begin{matrix} {ρ ρ}_{ij ij} Δ Δ + + o o ((Δ Δ)),, & i i &NotEqual; &NotEqual; j j \\ 11 + + {ρ ρ}_{ii i} Δ Δ + + o o ((Δ Δ)) & i i = = j j \end{matrix}$

其中，0≤ρ_ij≤1表示从模式i到模式j(i≠j)的转移率，ρ_ii=-∑_j≠iρ_ij；△>0是无穷维转移时间间隔，ο(△)是高阶无穷小；Among them, 0≤ρ _ij ≤1 represents the transfer rate from mode i to mode j (i≠j), ρ _ii =-∑ _j≠i ρ _ij ; △>0 is the infinite-dimensional transfer time interval, ο(△) is high-order infinitesimal;

根据姿态控制系统模型，设计镇定状态控制器μ₀，该控制器在故障情况下变化为μ₁；According to the attitude control system model, design a steady state controller μ ₀ , which changes to μ ₁ under the fault condition;

（3）用带有不稳定模态的随机切换系统描述系统运行过程：(3) Use a random switching system with unstable modes to describe the system operation process:

姿态控制系统模型在μ_σ的作用下写为切换系统：The attitude control system model is written as a switching system under the action of μ _σ :

dx(t)=f_σ(t)(x(t))dtdx(t)=f _σ(t) (x(t))dt

其中状态f_σ由姿态控制系统模型获得；which state f _σ is obtained from the attitude control system model;

（4）建立容错判断准则：(4) Establish fault-tolerant judgment criteria:

定义符号η₀和η₁分别为系统在正常情况和故障情况下的状态收敛率和发散率，定义△t₁为时间段[0,t]中系统正常情况下的工作总时间，△t₂为时间段[0,t]中系统故障情况下的工作总时间， $\overset{&OverBar;}{λ} \overset{Δ}{=} \max {| ρ_{ii} | | i &Element; M},$ $\tilde{λ} \overset{Δ}{=} \max {ρ_{ij} | i, j &Element; M};$ Define the symbols η ₀ and η ₁ to be the state convergence rate and divergence rate of the system under normal conditions and fault conditions respectively, define Δt ₁ as the total working time of the system under normal conditions in the time period [0,t], Δt ₂ is the total working time under the condition of system failure in the time period [0,t], $\overset{&OverBar;}{λ} \overset{Δ}{=} \max {| ρ_{i} | | i &Element; m},$ $\tilde{λ} \overset{Δ}{=} \max {ρ_{ij} | i, j &Element; m};$

容错判断准则设计如下：The fault-tolerant judgment criterion is designed as follows:

在t时刻，如果存在一个常数β>0使得At time t, if there exists a constant β>0 such that

${e e}^{- - {η η}_{00} Δ Δ {t t}_{11} + + {η η}_{11} Δ Δ {t t}_{22}} \leq \leq β β {e e}^{((\overset{~ ~}{λ λ} - - \overset{&OverBar; &OverBar;}{λ λ})) t t},,$ $&ForAll; &ForAll; t t &GreaterEqual; &Greater Equal; 00$

那么航天器姿态控制系统在间歇性故障的作用下是稳定的。Then the spacecraft attitude control system is stable under the action of intermittent faults.

上述步骤（1）中，不确定项为系统状态的未知函数d(x)，满足Lipschitz条件，即为已知的正数。In the above step (1), the uncertain item is the unknown function d(x) of the system state, which satisfies the Lipschitz condition, namely is a known positive number.

上述步骤（2）中，正常情况下的控制器设计如下：In the above step (2), the controller design under normal conditions is as follows:

${μ μ}_{00} = = {ω ω}^{\times \times} Jω Jω - - {k k}_{11} Jω Jω - - \frac{11}{22} Jq Q + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22} {q q}^{T T} q q - - {k k}_{33} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44}))$

其中，k₁>0，k₂>0，k₃>0，ε>0是一个任意小的常数；Among them, k ₁ >0, k ₂ >0, k ₃ >0, ε>0 is an arbitrarily small constant;

故障情况下，控制器的电路性能异常，导致放大增益发生变化，控制器μ₀变为In the case of a fault, the circuit performance of the controller is abnormal, resulting in a change in the amplification gain, and the controller μ ₀ becomes

${μ μ}_{11} = = {ω ω}^{\times \times} Jω Jω {k k}_{11}^{f f} Jω Jω - - \frac{11}{22} Jq Q + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22}^{f f} {q q}^{T T} q q - - {k k}_{33}^{f f} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44}))$

其中，均表示故障下的增益系数，这使得控制器的输出偏离正常值。in, Both represent the gain coefficient under the fault, This makes the output of the controller deviate from the normal value.

上述步骤（3）中，通过以下方法求取正常情况下的状态收敛率η₀和故障情况下的状态发散率η₁：In the above step (3), the state convergence rate η ₀ under normal conditions and the state divergence rate η ₁ under fault conditions are calculated by the following methods:

在正常情况下，定义一个李雅普诺夫条件函数定义： ${LV}_{p} (x) \overset{Δ}{=} \frac{&PartialD; V_{p} (x)}{&PartialD; x} f_{p} (x), p = 0,1$ In the normal case, define a Lyapunov conditional function definition: ${LV}_{p} (x) \overset{Δ}{=} \frac{&PartialD; V_{p} (x)}{&PartialD; x} f_{p} (x), p = 0,1$

通过选择适当的k₁、k₂、k₃，有By choosing appropriate k ₁ , k ₂ , k ₃ , we have

${LV LV}_{00} = = 22 {ω ω}^{T T} {J J}^{- - 11} ((- - {ω ω}^{\times \times} Jω Jω + + {μ μ}_{00})) + + {q q}^{T T} (({\overset{~ ~}{q q}}_{44} + + 11)) ω ω - - {\overset{~ ~}{q q}}_{44}^{T T} {ω ω}^{T T} q q + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00}$

$= = - - 22 {k k}_{11} {ω ω}^{T T} ω ω + + \frac{{ω ω}^{T T} ω ω}{{ω ω}^{T T} ω ω + + ϵ ϵ} ((- - {k k}_{22} {q q}^{T T} q q - - {k k}_{33} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00}$

$\leq \leq - - {η η}_{00} {V V}_{00}$

其中，η₀大于0，表明控制器可以镇定正常情况下的系统，V₀指数收敛；Among them, _η0 is greater than 0, indicating that the controller can stabilize the system under normal conditions, and _V0 converges exponentially;

在故障情况下，有In the event of a fault, there is

${LV LV}_{00} = = 22 {k k}_{11}^{f f} {ω ω}^{T T} ω ω + + \frac{{ω ω}^{T T} ω ω}{{ω ω}^{T T} ω ω + + ϵ ϵ} ((- - {k k}_{22}^{f f} {q q}^{T T} q q - - {k k}_{33}^{f f} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00} \leq \leq {η η}_{11} {V V}_{00}$

其中η₁>0，表明故障情况下系统不再稳定，函数V₀指数发散。Among them, η ₁ >0 indicates that the system is no longer stable under the fault condition, and the function V ₀ diverges exponentially.

上述步骤（4）中，容错判断准则适用于任意时刻，即在任意t时刻，如果存在一个常数β>0，使得In the above step (4), the fault-tolerant judgment criterion is applicable at any time, that is, at any time t, if there is a constant β>0 such that

那么航天器姿态控制系统在间歇性故障的作用下依然是稳定的。Then the spacecraft attitude control system is still stable under the action of intermittent faults.

采用上述方案后，本发明充分利用间歇性故障的时而出现和时而消失的特点，只要满足本发明提出的容错判断准则，姿控系统在正常状态和故障状态的平衡作用下依然可以保持稳定，在这种情况下，不需要采取任何容错控制措施，系统在正常状态和故障状态的平衡作用下依然可以保持稳定，即不需要像传统的容错控制方法那样，对控制器进行重构。该方法避免了容错控制实施过程中产生的高控制能耗、高计算复杂度和风险，对于航天器这种要求高可靠性并且需要高能耗的复杂系统有着重要的实际意义。After adopting the above scheme, the present invention makes full use of the characteristics of intermittent faults appearing and disappearing from time to time. As long as the fault-tolerant judgment criterion proposed by the present invention is satisfied, the attitude control system can still maintain stability under the balance between normal state and fault state. In this case, there is no need to take any fault-tolerant control measures, and the system can still maintain stability under the balance of normal state and fault state, that is, the controller does not need to be reconfigured like traditional fault-tolerant control methods. This method avoids high control energy consumption, high computational complexity and risks in the implementation of fault-tolerant control, and has important practical significance for spacecraft, a complex system that requires high reliability and high energy consumption.

附图说明 Description of drawings

图1是本发明中姿态控制系统的结构框图；Fig. 1 is the structural block diagram of attitude control system among the present invention;

图2是本发明基于切换系统的容错方法示意图；Fig. 2 is a schematic diagram of the fault tolerance method based on the switching system of the present invention;

图3是第一个切换函数的轨迹图；Fig. 3 is a locus diagram of the first switching function;

图4是姿态控制系统对应图3的状态轨迹图；Fig. 4 is a state trajectory diagram corresponding to Fig. 3 of the attitude control system;

图5是第二个切换函数的轨迹图；Fig. 5 is a locus diagram of the second switching function;

图6是姿态控制系统对应图5的状态轨迹图。Fig. 6 is a state trajectory diagram of the attitude control system corresponding to Fig. 5 .

具体实施方式 Detailed ways

以下将结合附图，对本发明的技术方案及有益效果进行详细说明。The technical solutions and beneficial effects of the present invention will be described in detail below in conjunction with the accompanying drawings.

本发明提供一种航天器姿态控制系统的间歇性故障容错分析方法，包括如下步骤：The invention provides a method for intermittent fault-tolerant analysis of a spacecraft attitude control system, comprising the following steps:

（1）建立航天器的姿态控制系统数学模型，即为图1中的航天器机体模块、执行器模块和传感器模块；(1) Establish the mathematical model of the attitude control system of the spacecraft, which is the spacecraft body module, actuator module and sensor module in Figure 1;

$J J \overset{\cdot &Center Dot;}{ω ω} = = - - {ω ω}^{\times \times} Jω Jω + + μ μ + + d d - - - - - - ((11))$

$\overset{\cdot \cdot}{q q} = = \frac{11}{22} (({q q}_{44} ω ω + + {ω ω}^{\times \times} q q)) - - - - - - ((22))$

${\overset{\cdot \cdot}{q q}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q - - - - - - ((33))$

其中，表示惯性角速度；q₄是标量，q₁,q₂,q₃,q₄表示为四元数；J=J^T表示正定惯性矩阵，μ表示控制器输出，即控制扭矩。不确定项为系统状态的未知函数d(x)，d(x)满足Lipschitz条件，即为已知的正数。叉积形式为：in, Indicates the inertial angular velocity; q ₄ is a scalar, q ₁ , q ₂ , q ₃ , and q ₄ are represented as quaternions; J=J ^T represents a positive definite inertia matrix, and μ represents the controller output, that is, the control torque. The uncertain item is the unknown function d(x) of the system state, d(x) satisfies the Lipschitz condition, namely is a known positive number. The cross product form is:

${ω ω}^{\times \times} = = [\begin{matrix} 00 & - - {ω ω}_{33} & {ω ω}_{22} \\ {ω ω}_{33} & 00 & - - {ω ω}_{11} \\ - - {ω ω}_{22} & {ω ω}_{11} & 00 \end{matrix}]$

选择姿控系统的平衡点为：ω=q=0，q₄=1，则(2)-(3)式可改写为：The equilibrium point of the attitude control system is selected as: ω=q=0, q ₄ =1, then (2)-(3) can be rewritten as:

$\overset{\cdot \cdot}{q q} = = \frac{11}{22} (({\overset{~ ~}{q q}}_{44} + + 11)) ω ω + + \frac{11}{22} {ω ω}^{\times \times} q q - - - - - - ((44))$

${\overset{\cdot \cdot}{\overset{~ ~}{q q}}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q - - - - - - ((55))$

其中， ${\tilde{q}}_{4} \overset{Δ}{=} q_{4} - 1 .$ in, ${\tilde{q}}_{4} \overset{Δ}{=} q_{4} - 1 .$

（2）建立间歇性控制器故障数学模型，确定马尔科夫链的转移率ρ₁₀（即间歇性故障消失的概率）和ρ₀₁（间隙性故障发生的概率）；(2) Establish a mathematical model of intermittent controller faults, and determine the transfer rate ρ ₁₀ (probability of intermittent faults disappearing) and ρ ₀₁ (probability of intermittent faults occurring) of the Markov chain;

用μ_σ表示采用状态反馈设计的控制器输出，即控制扭矩。其中下标σ(t)是一个随时间变化的切换函数，在{0,1}中取值，其中0表示控制器处在正常情况，1表示控制器处在间歇性故障情况，这意味着控制器的输出值可能因故障情况而跳变。根据间歇性故障的发生特点，该切换函数用马尔科夫链来描述，即Use μ _σ to represent the output of the controller designed with state feedback, that is, the control torque. where the subscript σ(t) is a time-varying switching function, taking values in {0,1}, where 0 indicates that the controller is in a normal condition, and 1 indicates that the controller is in an intermittent failure condition, which means The output value of the controller may jump due to fault conditions. According to the occurrence characteristics of intermittent faults, the switching function is described by Markov chain, namely

其中，0≤ρ_ij≤1表示从模式i到模式j(i≠j)的转移率，ρ_ii=-∑_j≠iρ_ij。△>0是无穷过渡的时间间隔，ο(△)是高阶无穷小。Wherein, 0≤ρ _ij ≤1 represents the transfer rate from mode i to mode j (i≠j), ρ _ii =-∑ _j≠i ρ _ij . △>0 is the time interval of infinite transition, ο(△) is the higher order infinitesimal.

正常情况下的控制器设计如下：The controller design under normal circumstances is as follows:

${μ μ}_{00} = = {ω ω}^{\times \times} Jω Jω - - {k k}_{11} Jω Jω - - \frac{11}{22} Jq Q + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22} {q q}^{T T} q q - - {k k}_{33} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) - - - - - - ((66))$

其中，k₁>0，k₂>0，k₃>0，ε>0是一个任意小的常数。Wherein, k ₁ >0, k ₂ >0, k ₃ >0, ε>0 is an arbitrarily small constant.

所考虑的控制器间歇性故障是由于控制电路芯片中的各个焊接点由于时间的推移产生的物理磨损而破裂或者松动造成的。这种松散的连接是一个典型间歇性故障，可以使控制电路的性能异常，导致放大增益发生变化。故障情况下，控制器μ₀（6）变为The considered intermittent failure of the controller is caused by cracking or loosening of individual solder joints in the control circuit chip due to physical wear over time. This loose connection is a typical intermittent fault that can cause abnormal performance of the control circuit, causing changes in the amplification gain. In the fault case, the controller μ ₀ (6) becomes

${μ μ}_{11} = = {ω ω}^{\times \times} Jω Jω {k k}_{11}^{f f} Jω Jω - - \frac{11}{22} Jq Q + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22}^{f f} {q q}^{T T} q q - - {k k}_{33}^{f f} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) - - - - - - ((77))$

其中，均表示故障下的增益系数，这使得控制器的输出偏离正常值。in, Both denote the gain coefficients under faults, which make the output of the controller deviate from the normal value.

（3）基于Lyapunov函数（李雅普诺夫函数）方法设计正则控制器μ₀，即为图1中的控制器模块，并计算出Lyapunov函数V₀的收敛率η₀，将其传给容错分析模块；(3) Design the canonical controller μ ₀ based on the Lyapunov function (Lyapunov function) method, which is the controller module in Figure 1, and calculate the convergence rate η ₀ of the Lyapunov function V ₀ , and pass it to the fault-tolerant analysis module ;

姿态控制系统模型（1）（4）（5）在μ_σ的作用下可写为切换系统：The attitude control system model (1) (4) (5) can be written as a switching system under the action of μ _σ :

dx(t)=f_σ(t)(x(t))dtdx(t)=f _σ(t) (x(t))dt

其中，状态f_σ可以由姿态控制系统模型获得。Among them, the state f _σ can be obtained from the attitude control system model.

$\leq \leq - - {η η}_{00} {V V}_{00}$

其中，η₀大于0，由上式可以看出控制器式（6）的作用下，李雅普诺夫条件函数V₀以指数形式收敛，收敛率为η₀。Among them, η ₀ is greater than 0. It can be seen from the above formula that under the action of the controller formula (6), the Lyapunov conditional function V ₀ converges exponentially, and the convergence rate is η ₀ .

在故障情况下，有In the event of a fault, there is

其中η₁>0，由上式可以看出发生故障后，控制器式（7）不再能够镇定系统，函数V₀以指数形式发散，收敛率为η₁。Where η ₁ >0, it can be seen from the above formula that after a fault occurs, the controller formula (7) can no longer stabilize the system, the function V ₀ diverges in an exponential form, and the convergence rate is η ₁ .

因此切换系统dx(t)=f_σ(t)(x(t))dt的模态0稳定，模态1不稳定。Therefore, mode 0 of the switched system dx(t)=f _σ(t) (x(t))dt is stable and mode 1 is unstable.

运用这种随机切换系统有两个优点：There are two advantages to using this random switching system:

1）可以方便地分析姿控系统整个过程的行为，而不是单独研究故障前和故障后的情况。这对于分析在间歇性故障下的系统行为是很重要的；1) It is convenient to analyze the behavior of the attitude control system in the whole process, instead of studying the pre-fault and post-fault situations separately. This is important for analyzing system behavior under intermittent failures;

2）姿控系统的容错问题可以转化为带有不稳定模态的切换系统的稳定性问题，可以使用一些对切换系统很有用的分析工具。2) The fault tolerance problem of the attitude control system can be transformed into the stability problem of the switched system with unstable modes, and some useful analysis tools for the switched system can be used.

定义△t₁为时间段[0,t]中系统正常情况（模式0运行）下的工作总时间，△t₂为时间段[0,t]中系统故障情况（模式1）下的工作总时间，定义Define △t ₁ as the total working time of the system under normal conditions (mode 0 operation) in the time period [0, t], and △t ₂ as the total working time under the system failure condition (mode 1) in the time period [0, t]. time, definition

$\overset{&OverBar; &OverBar;}{λ λ} \overset{Δ Δ}{= =} max max {{| | {ρ ρ}_{ii i} | | | | i i &Element; &Element; M m}},,$ $\overset{~ ~}{λ λ} \overset{Δ Δ}{= =} max max {{{ρ ρ}_{ij ij} | | i i,, j j &Element; &Element; M m}};;$

${e e}^{- - {η η}_{00} Δ Δ {t t}_{11} + + {η η}_{11} Δ Δ {t t}_{22}} \leq \leq β β {e e}^{((\overset{~ ~}{λ λ} - - \overset{&OverBar; &OverBar;}{λ λ})) t t},,$ $&ForAll; &ForAll; t t &GreaterEqual; &Greater Equal; 00 - - - - - - ((88))$

简便起见，下面用共同的η代替η₀和η₁。用记号N_σ(t)表示系统在[0,t]时间段内的切换次数。用t_k代表系统的第k次切换时刻。For simplicity, η ₀ and η ₁ are replaced by the common η below. Use the symbol N _σ(t) to represent the switching times of the system in [0,t] time period. Use t _k to represent the kth switching time of the system.

假设在时刻t切换了j次，即t≥t_j。由得到：Assume j times of switching at time t, that is, t≥t _j . Depend on get:

$E E. [[{V V}_{00} ((x x ((t t))))]] \leq \leq E E. [[{e e}^{η η ((t t - - {t t}_{j j}))} {V V}_{00} ((x x (({t t}_{j j}))))]]$

$\leq \leq E E. [[{e e}^{η η ((t t - - {t t}_{j j - - 11}))} {V V}_{00} ((x x (({t t}_{j j - - 11}))))]]$

$\begin{matrix} . . \\ . . \\ . . \end{matrix}$

$\leq \leq E E. [[{e e}^{ηt ηt} {V V}_{00} ((x x ((00))))]]$

$\leq \leq {Σ Σ}_{j j = = 00}^{\infty \infty} P P (({N N}_{σ σ ((t t))} = = j j)) {e e}^{ηt ηt} {V V}_{00} ((x x ((00))))$

$\leq \leq {Σ Σ}_{j j = = 00}^{\infty \infty} \frac{{e e}^{- - \overset{~ ~}{λ λ} t t} {((\overset{&OverBar; &OverBar;}{λ λ} t t))}^{j j}}{j j!!} β β {e e}^{((\overset{~ ~}{λ λ} - - \overset{&OverBar; &OverBar;}{λ λ})) t t} {V V}_{00} ((x x ((00))))$

$\leq \leq β β {V V}_{00} ((x x ((00))))$

由上式可以看出，在整个系统运行过程中，如果容错条件（8）满足，则在间歇性故障的影响下，状态x(t)的数学期望仍然始终是有界的，且随着x(0)趋于原点而趋向于原点。From the above formula, it can be seen that during the operation of the whole system, if the fault-tolerant condition (8) is satisfied, under the influence of intermittent faults, the mathematical expectation of the state x(t) is always bounded, and as x (0) tends to the origin and tends to the origin.

容错判断准则（8）充分利用了间歇性故障的时而出现和时而消失的特点。该准则的核心在于运用不稳定模态和稳定模态之间的平衡去镇定系统。如果系统在正常情况（故障消失）下的工作时间足够长，而在故障出现的情况下工作时间足够短，那么系统仍然能保证稳定，系统性能和β的选取有很大关系，该值通常根据姿控系统的需求选择。The fault-tolerant judgment criterion (8) makes full use of the characteristics of intermittent faults appearing and disappearing from time to time. The core of this criterion is to use the balance between unstable and stable modes to stabilize the system. If the working time of the system is long enough under normal conditions (fault disappears), and the working time is short enough under the condition of faults, then the system can still guarantee stability, and the system performance has a great relationship with the selection of β, which is usually based on The demand selection of the attitude control system.

设计间歇性故障的检测与诊断机制，即为故障诊断模块，依据是Design the detection and diagnosis mechanism of intermittent faults, which is the fault diagnosis module, based on

LV₀≤-η₀V₀ （9）LV ₀ ≤ -η ₀ V ₀ (9)

如果在t时刻，上述不等式（9）不满足，则认为有故障在t时刻发生。如果某些故障发生了，不等式（9）仍然满足，则认为该故障没有破坏系统的稳定性，无需检测。If at time t, the above inequality (9) is not satisfied, it is considered that a fault occurs at time t. If some fault occurs, inequality (9) is still satisfied, it is considered that the fault does not destroy the stability of the system, no need to detect.

在系统运行的过程中，始终运行故障诊断模块，该模块实时采集系统的输入和状态信息，当检测出间歇性故障发生后，计算出故障情况下函数的发散率η₁，将故障发生时间和值η₁传给容错分析模块。如果在某时刻故障消失了，故障诊断模块也将故障消失的时刻传给容错分析模块；During the operation of the system, the fault diagnosis module is always running. This module collects the input and status information of the system in real time. When intermittent faults are detected, it calculates the divergence rate η ₁ of the function under fault conditions, and calculates the fault occurrence time and The value η ₁ is passed to the fault-tolerant analysis module. If the fault disappears at a certain moment, the fault diagnosis module will also pass the time when the fault disappears to the fault-tolerant analysis module;

（5）容错分析模块收到故障发生时间，发散率η₁以及故障消失时间后，根据容错判断准则（8），判断系统当前的稳定性，如果满足则系统继续运行，不采取任何容错控制或者保护措施，因为在过去的时间里，系统在正常情况（故障消失）下的工作时间足够长，而在故障出现的情况下工作时间足够短，因此系统仍然能保证稳定，如图2所示。如果判断出当前系统已不稳定，则报警，此时必须采取容错控制措施。(5) After receiving the fault occurrence time, divergence rate η ₁ and fault disappearance time, the fault-tolerant analysis module judges the current stability of the system according to the fault-tolerant judgment criterion (8). If it is satisfied, the system continues to run without any fault-tolerant control or Protection measures, because in the past time, the system works long enough under normal conditions (fault disappears) and short enough under the condition of faults, so the system can still guarantee stability, as shown in Figure 2. If it is judged that the current system is unstable, an alarm will be issued, and fault-tolerant control measures must be taken at this time.

下面以仿真验证本发明方法的有效性。Next, the validity of the method of the present invention is verified by simulation.

在仿真时，选择惯性矩阵：When simulating, choose the inertia matrix:

$J J = = [\begin{matrix} 12001200 & 100100 & - - 200200 \\ 100100 & 22002200 & 300300 \\ - - 200200 & 300300 & 31003100 \end{matrix}] kg kg \cdot \cdot {m m}^{22}$

假定不确定项为d=0.1Wω，其中W为标准的布朗运动，代表噪声干扰。初始参数选为Assume that the uncertain item is d=0.1Wω, where W is the standard Brownian motion and represents noise interference. The initial parameter is chosen as

(ω₁ ω₂ ω₃)=(0.2 -0.1 0.1)(rad/s)(ω ₁ ω ₂ ω ₃ )=(0.2 -0.1 0.1)(rad/s)

(q₁ q₂ q₃ q₄)=(0.5 0.5 0.5 -0.5)(q ₁ q ₂ q ₃ q ₄ )=(0.5 0.5 0.5 -0.5)

假定间歇性故障满足-ρ₀₀=ρ₀₁=0.5且-ρ₁₁=ρ₁₀=0.8，控制器的参数设计k₁=k₂=k₃=10，ε=0.001，发生间歇性故障时，参数变为令β=2。Assuming that intermittent faults satisfy -ρ ₀₀ =ρ ₀₁ =0.5 and -ρ ₁₁ =ρ ₁₀ =0.8, the parameter design of the controller k ₁ =k ₂ =k ₃ =10, ε=0.001, when intermittent faults occur, the parameters becomes Let β=2.

由于故障的发生和消失都是随机的，因此做两组仿真。图3给出了在时间段0秒到20秒内的第一个切换函数的曲线，表明了故障发生和消失的各个时刻。经检验，在图3所示的故障情况下，容错判断准则在0秒到20秒内的任意时刻都是满足的，图4给出了对应的系统状态轨迹，可以看出虽然有间歇性故障发生，但系统状态（即姿态角速度和四元数）在保持原有控制器而没有采取任何容错控制的情况下依然是有界的，这表明了本发明所提供的容错判据的有效性。类似的，图5给出了在时间段0秒到20秒内的第二个切换函数的曲线，该曲线和图3所示的曲线不同，但经检验，在图5所示的故障情况下，容错判断准则在0秒到20秒内的任意时刻也都是满足的，图6给出了对应的系统状态轨迹，可以看出系统状态在没有采取任何容错控制的情况下依然是有界的，这同样表明了本发明所提供的容错判据的有效性。Since the occurrence and disappearance of faults are random, two sets of simulations are done. Figure 3 gives the curves of the first switching function in the time period 0 s to 20 s, indicating the various moments when the fault occurs and disappears. After inspection, in the case of the fault shown in Figure 3, the fault-tolerant judgment criterion is satisfied at any time from 0 seconds to 20 seconds. Figure 4 shows the corresponding system state trajectory. It can be seen that although there are intermittent faults occurs, but the system state (that is, attitude angular velocity and quaternion) is still bounded when the original controller is maintained without any fault-tolerant control, which shows the validity of the fault-tolerant criterion provided by the present invention. Similarly, Fig. 5 shows the curve of the second switching function in the time period 0 seconds to 20 seconds, which is different from the curve shown in Fig. 3, but after inspection, in the fault case shown in Fig. 5 , the fault-tolerant judgment criterion is also satisfied at any time between 0 seconds and 20 seconds. Figure 6 shows the corresponding system state trajectory. It can be seen that the system state is still bounded without any fault-tolerant control , which also shows the validity of the fault-tolerant criterion provided by the present invention.

以上实施例仅为说明本发明的技术思想，不能以此限定本发明的保护范围，凡是按照本发明提出的技术思想，在技术方案基础上所做的任何改动，均落入本发明保护范围之内。The above embodiments are only to illustrate the technical ideas of the present invention, and can not limit the protection scope of the present invention with this. All technical ideas proposed in accordance with the present invention, any changes made on the basis of technical solutions, all fall within the protection scope of the present invention. Inside.

Claims

1. a kind of intermittent failure fault-tolerant analysis method of spacecraft attitude control system, it is characterized in that comprising the steps:

(1) Establish the mathematical model of the attitude control system of the spacecraft:

J J \overset{. .}{ω ω} = = - - {ω ω}^{\times \times} Jω Jω + + μ μ + + d d

\overset{. .}{q q} = = \frac{11}{22} (({q q}_{44} ω ω + + {ω ω}^{\times \times} q q))

{\overset{. .}{q q}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q

in, Indicates the inertial angular velocity; q ₄ is a scalar, q ₁ , q ₂ , q ₃ , and q ₄ are represented as quaternions; J=J ^T represents a positive definite inertia matrix, μ represents the output of the controller, that is, the control torque; d represents the uncertainty and disturbance of the system;

The cross product form of ω is:

{ω ω}^{\times \times} = = [\begin{matrix} 00 & {- - ω ω}_{33} & {ω ω}_{22} \\ {ω ω}_{33} & 00 & - - {ω ω}_{11} \\ {- - ω ω}_{22} & {ω ω}_{11} & 00 \end{matrix}]

Select the balance point of the attitude control system as: ω=q=0, q ₄ =1, then:

\overset{. .}{q q} = = \frac{11}{22} (({\overset{~ ~}{q q}}_{44} + + 11)) ω ω + + \frac{11}{22} {ω ω}^{\times \times} q q

{\overset{\overset{. .}{~ ~}}{q q}}_{44} = = - - \frac{11}{22} {ω ω}^{T T} q q

in,

{\tilde{q}}_{4} \overset{Δ}{=} q_{4} - 1;

(2) Establish a mathematical model of the controller with intermittent faults:

Denote the controller output with intermittent faults by μ _σ , where the subscript σ(t) is a time-varying switching function, taking values in {0,1}, where 0 indicates that the controller is in normal condition, 1 means that the controller is in an intermittent fault situation. According to the characteristics of intermittent faults, the switching function is described by a Markov chain, that is,

P P {{σ σ ((t t + + Δ Δ)) = = j j | | σ σ ((t t)) = = i i}} = = \{\begin{matrix} {ρ ρ}_{ij ij} Δ Δ + + o o ((Δ Δ)),, & i i &NotEqual; &NotEqual; j j \\ 11 + + {ρ ρ}_{ii i} Δ Δ + + o o ((Δ Δ)) & i i = = j j \end{matrix}

Among them, 0≤ρ _ij ≤1 means the transfer rate from mode i to mode j, and i≠j, ρ _ii =-∑ _j≠i ρ _ij ; Δ>0 is the infinite-dimensional transfer time interval, ο(Δ) is high-order infinitesimal;

According to the attitude control system model, design a steady state controller μ ₀ , which changes to μ ₁ under the fault condition;

(3) Using a random switching system with unstable modes to describe the operation process of the system:

The attitude control system model is written as a switching system under the action of μ _σ :

dx(t)=f _σ(t) (x(t))dt

which state f _σ is obtained from the attitude control system model;

Among them, the state convergence rate η ₀ under normal conditions and the state divergence rate η ₁ under fault conditions are obtained by the following methods:

In the normal case, define a Lyapunov conditional function definition:

{LV LV}_{00} ((x x)) \overset{Δ Δ}{= =} \frac{{&PartialD; &PartialD; V V}_{00} ((x x))}{&PartialD; &PartialD; x x} {f f}_{00} ((x x)),,

By choosing appropriate k ₁ , k ₂ , k ₃ , we have

\begin{matrix} {LV LV}_{00} = = {22 ω ω}^{T T} {J J}^{- - 11} ((- - {ω ω}^{\times \times} Jω Jω + + {μ μ}_{00})) + + {q q}^{T T} (({\overset{~ ~}{q q}}_{44} + + 11)) ω ω - - {\overset{~ ~}{q q}}_{44}^{T T} {ω ω}^{T T} q q + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00} \\ = = - - {22 k k}_{11} {ω ω}^{T T} ω ω + + \frac{{ω ω}^{T T} ω ω}{{ω ω}^{T T} ω ω + + ϵ ϵ} (({- - k k}_{22} {q q}^{T T} q q - - {k k}_{33} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00} \\ \leq \leq {- - η η}_{00} {V V}_{00} \end{matrix}

Among them, k ₁ >0, k ₂ >0, k ₃ >0, ε>0 is an arbitrarily small constant, η ₀ is greater than 0, indicating that the controller can stabilize the system under normal conditions, V ₀ converges exponentially, is a known positive number;

In the event of a fault, there is

{LV LV}_{00} = = {22 k k}_{11}^{f f} {ω ω}^{T T} ω ω + + \frac{{ω ω}^{T T} ω ω}{{ω ω}^{T T} ω ω + + ϵ ϵ} (({- - k k}_{22}^{f f} {q q}^{T T} q q - - {k k}_{33}^{f f} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44})) + + \overset{&OverBar; &OverBar;}{d d} {V V}_{00} \leq \leq {η η}_{11} {V V}_{00}

Where η ₁ >0, indicating that the system is no longer stable under the fault condition, and the function V ₀ diverges exponentially,

k_{2}^{f} > 0,

k_{3}^{f} > 0,

Both represent the gain coefficient under the fault;

(4) Establish fault-tolerant judgment criteria:

Define the symbols η ₀ and η ₁ to be the state convergence rate and divergence rate of the system under normal conditions and fault conditions respectively, define Δt ₁ to be the total working time of the system under normal conditions in the time period [0,t], and Δt ₂ to be the time The total working time in the case of system failure in segment [0,t],

\overset{&OverBar;}{λ} \overset{Δ}{=} \max {| ρ_{i} | | i &Element; m |},

\overset{&OverBar;}{λ} \overset{Δ}{=} \max {| ρ_{i} | | i, j &Element; m |};

The fault-tolerant judgment criterion is designed as follows:

At time t, if there exists a constant β>0 such that

{e e}^{- - {η η}_{00} {Δt Δt}_{11} + + {η η}_{11} {Δt Δt}_{22}} \leq \leq {βe βe}^{((\overset{~ ~}{λ λ} - - \overset{&OverBar; &OverBar;}{λ λ})) t t},,

&ForAll; &ForAll; t t &GreaterEqual; &Greater Equal; 00

Then the spacecraft attitude control system is stable under the action of intermittent faults.

2. the intermittent failure fault-tolerant analysis method of spacecraft attitude control system as claimed in claim 1 is characterized in that: in described step (1), uncertain term is the unknown function d (x) of system state, satisfies Lipschitz condition, namely is a known positive number.

3. the intermittent failure fault-tolerant analysis method of spacecraft attitude control system as claimed in claim 1, is characterized in that: in described step (2), the controller design under normal conditions is as follows:

{μ μ}_{00} = = {ω ω}^{\times \times} Jω Jω - - {k k}_{11} Jω Jω - - \frac{11}{22} Jq Q + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22} {q q}^{T T} q q - - {k k}_{33} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44}))

Among them, k ₁ >0, k ₂ >0, k ₃ >0, ε>0 is an arbitrarily small constant;

In the case of a fault, the circuit performance of the controller is abnormal, resulting in a change in the amplification gain, and the controller μ ₀ becomes

{μ μ}_{11} = = {ω ω}^{\times \times} Jω Jω {k k}_{11}^{f f} Jω Jω - - \frac{11}{22} Jp Jp + + \frac{Jω Jω}{22 (({ω ω}^{T T} ω ω + + ϵ ϵ))} ((- - {k k}_{22}^{f f} {q q}^{T T} q q - - {k k}_{33}^{f f} {\overset{~ ~}{q q}}_{44}^{T T} {\overset{~ ~}{q q}}_{44}))

in, Both represent the gain coefficient under the fault, This makes the output of the controller deviate from the normal value.

4. the intermittent failure fault-tolerant analysis method of spacecraft attitude control system as claimed in claim 1, is characterized in that: in described step (4), fault-tolerant judgment criterion is applicable to any moment, promptly at any t moment, if exists A constant β>0 such that

{e e}^{- - {η η}_{00} {Δt Δt}_{11} + + {η η}_{11} {Δt Δt}_{22}} \leq \leq {βe βe}^{((\overset{~ ~}{λ λ} - - \overset{&OverBar; &OverBar;}{λ λ})) t t},,

&ForAll; &ForAll; t t &GreaterEqual; &Greater Equal; 00

Then the spacecraft attitude control system is still stable under the action of intermittent faults.