### 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Integrating Reliability Computation

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Integrating reliability information into the control

Component reliability and system reliability have been less closely examined in the literature on control theory. [GOK 05] proposed integrating the parameters to increase the life of actuators to reduce the maintenance costs. The method is based on the estimation of time before failure according to the past component use and the modification of the component’s functioning state if the estimated remaining lifetime is less than expected. [GOK 06] presented some algorithms for adaptive control. The first algorithm maintains the expected actuator lifetime by adjusting its performance level. The other algorithm offers a compromise between the actuator performance and the expected lifetime according to mission requirements.
[PER 10] proposed a solution based on model predictive control (MPC) strategy that is used to allocate the effort among the redundant actuators by fixing constraints on the actuator degradation. This degradation is computed by cumulating the control inputs. This constraint is integrated in the MPC strategy to protect against the dangerous degradation levels of some critical actuators. This method is not based on reliability computation but integrates the co-variables (control input) that have an impact on the component reliability. The principle is to focus on increasing the component reliability without considering the system reliability.

[GUE 07, GUE 11] focused on defining a structure that combines components to elaborate a system with higher reliability level after a component failure. It is based on a fault-tolerant control whose fundamental principle is to keep the performance levels closer to the performance level defined before the occurrence of failure. Fault tolerance is a control reconfiguration or a restructuring strategy integrating reliability analysis and component costs [GUE 04a, GUE 04b]. From the fault detection and isolation process, the reconfiguration task consists of determining the possible structures that ensure the initial system performance or accepted degraded performances by isolating the faulty components or switching to operating subsystems. For this purpose, an optimal structure is searched for from among all the possible structures [GUE 04a, GUE 05, GUE 06].
[KHE 11] proposed a fault-tolerant control strategy to warrant the system reliability. This new methodology requires adaptation of several reliability models or parameters to integrate them as constraints or conditioning criteria of the control law. The integration of the impact of reliability on the end of mission is a key point of this work.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Control integrating reliability modeled by DBN

The goal is to define a control strategy for over-actuated systems that allows us to optimally allocate the effort on actuators under the constraint of preserving the system reliability in the normal case or when component failure occurs. To optimize the actuator inputs, it is necessary to have sufficient free degrees in the control law. Clearly, this is the case in over-actuated systems. An over-actuated system is not necessarily a system with redundant components but a system where the control goals can be attained in a different manner.

From a general point of view, an over-actuated system can be considered as a linear system with $m$ actuators and described by the following discrete equation:
$$\left{\begin{array}{c} \tilde{x}(k+1)=A \tilde{x}(k)+B_{u} \tilde{u}(k) \ \tilde{y}(k+1)=C \tilde{x}(k) \end{array}\right.$$
with $A \in R^{n \times n}, B_{u} \in R^{n \times m}$ and $C \in R^{p \times n}$ being the state, control and output matrices respectively. $\tilde{x} \in R^{n}$ is the state vector of the system, $\tilde{u} \in R^{m}$ is the input control vector and $\tilde{y} \in R^{p}$ is the system output vector. The condition $\operatorname{rank}\left(B_{u}\right)=r<m$ characterizes over-actuated systems. Figure $5.1$ shows the control principle of an over-actuated system integrating reliability information. The reliability model is used to allocate the control efforts on the actuators.
Matrix $B_{u}$ can be factorized:
$$B_{u}=B_{v} \cdot B$$

with $B_{v} \in R^{n \times r}$ and $B \in R^{r \times m}$ all of rank $r$. The system is then modeled by:
\left{\begin{aligned} \tilde{x}(k+1) &=& A \tilde{x}(k)+B_{v} \tilde{v}(k) \ \tilde{v}(k) &=& & B \tilde{u}(k) \ \tilde{y}(k+1) &=& & C \tilde{x}(k) \end{aligned}\right.
with $\tilde{v}(k) \in R^{r}$ representing the whole controlling effort required for the system to function is also called the virtual input vector. Control allocation aims to define the real control inputs of the system $\tilde{u}(k)$ from the expected virtual control input, such as:
$$\begin{gathered} \tilde{v}^{d}(k)=B \tilde{u}(k) \ \tilde{u}{\min } \leq \tilde{u} \leq \tilde{u}{\max } \end{gathered}$$

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Integrating reliability

The weighing matrix $W(k)$ is considered the key to integrating the actuators’ reliability into the control input allocation problem of overactuated systems. The control problem can be solved in several steps, as shown in Figure 5.2. To maximize system reliability, the weighing matrix $W(k)$ is set from the actuator contributions $\alpha_{i}^{k}$ to the system operation: This contribution depends on the structure function $\varphi\left(e^{k}\right)$, where $e^{k}=\left(e_{1}^{k}, e_{2}^{k}, \ldots, e_{m}^{k}\right)$ allows the calculation of system reliability according to the actuator $e_{i}^{k}$. The actuators are taken into account in the control strategy proportional to their contribution to the system

operation. The system state $S^{k}$ is defined from the structure function $\varphi\left(e^{k}\right)$ :
$$P\left(S^{k}=0\right)=P\left(\varphi\left(e^{k}\right)=0\right)$$
From the control point of view, the over-actuated system is hypothesised to be in a working state even if some actuators are in a failed state, i.e. $e_{j}^{k}=1$. To satisfy the system goals, the actuators to be used depend on their availability and the structure function $\varphi$.

The unavailable actuators $e_{j}^{k}=1$ are isolated by the Maintenance function, as shown in Figure 5.2, from the Diagnostic function. An available actuator $e_{i}^{k}$ is used by the control law if at least one operating scenario exists, i.e. $\varphi\left(e^{k}\right)=0$, using $e_{i}^{k}$. The probability of using an actuator and satisfying the system objectives is defined by the following conditional probability:
$$P\left(\alpha_{i}^{k}=0 \mid \varphi\left(e^{k}\right)=0\right)$$

To integrate the actuators’ reliability in the control strategy, the weighing matrix $W(k)$ is estimated online according to the actuators’ state given by the Diagnostic function. Consequently, if an actuator $e_{j}^{k}$ is unavailable, i.e. $e_{j}^{k}=1$, the system can work in the degraded mode

because it is over-actuated and the scalar $w_{i}(k)$ of each actuator is defined by the following probability:
$$w_{i}(k)=P\left(\alpha_{i}^{k}=0 \mid \varphi\left(e^{k}\right)=0, e_{j}^{k}=1\right)$$
The weight $w_{i}(k)$ corresponds to the contribution probability of the actuator $e_{i}^{k}$ when the system is working, given the unavailability of some failed actuators. This probability assessment is not only based on the actuator’s state of health but also considers the system structure and the availability of other actuators. By using usual reliability modeling, assessing this probability is complex or impossible, given the structure function $\varphi$. However, this assessment can easily be realized by the inference mechanism of DBN.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Integrating reliability information into the control

[PER 10] 提出了一种基于模型预测控制 (MPC) 策略的解决方案，该策略用于通过固定执行器退化的约束来在冗余执行器之间分配工作量。这种退化是通过累积控制输入来计算的。该约束被集成在 MPC 策略中，以防止某些关键执行器的危险退化水平。该方法不是基于可靠性计算，而是集成了对组件可靠性有影响的协变量（控制输入）。其原则是专注于提高组件可靠性而不考虑系统可靠性。

[GUE 07, GUE 11] 专注于定义一种结构，该结构将组件组合在一起，以在组件发生故障后制定具有更高可靠性级别的系统。它基于容错控制，其基本原则是保持性能水平接近故障发生前定义的性能水平。容错是一种控制重构或整合可靠性分析和组件成本的重构策略[GUE 04a，GUE 04b]。从故障检测和隔离过程中，重新配置任务包括确定可能的结构，这些结构通过隔离故障组件或切换到运行子系统来确保初始系统性能或可接受的降级性能。为此，从所有可能的结构[GUE 04a，GUE 05，
[KHE 11] 提出了一种容错控制策略来保证系统的可靠性。这种新方法需要调整几个可靠性模型或参数，以将它们整合为控制律的约束或条件标准。可靠性对任务结束的影响的整合是这项工作的一个重点。

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Control integrating reliability modeled by DBN

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$$\left{X~(ķ+1)=一种X~(ķ)+乙在在~(ķ) 在~(ķ)=乙在~(ķ) 是~(ķ+1)=CX~(ķ)\对。 在一世吨H在~(ķ)∈Rrr和pr和s和n吨一世nG吨H和在H这l和C这n吨r这ll一世nG和FF这r吨r和q在一世r和dF这r吨H和s是s吨和米吨这F在nC吨一世这n一世s一种ls这C一种ll和d吨H和在一世r吨在一种l一世np在吨在和C吨这r.C这n吨r这l一种ll这C一种吨一世这n一种一世米s吨这d和F一世n和吨H和r和一种lC这n吨r这l一世np在吨s这F吨H和s是s吨和米在~(ķ)Fr这米吨H和和Xp和C吨和d在一世r吨在一种lC这n吨r这l一世np在吨,s在CH一种s: 在~d(ķ)=乙在~(ķ) 在~分钟≤在~≤在~最大限度$$

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