Convert LP to a Standard Form

\[\min{}c^\top{}x,\text{subject to }Ax=b\land{}x\geq0\]

For Inequality

• \(x+y\geq{}a\rightarrow{}x+y-z=a,z\geq0\)
• \(x+y\leq{}a\rightarrow{}x+y+s=a,s\geq0\)
• Unrestricted \(\rightarrow{}x=y-z,y\geq0,z\geq0\)

Optimization problem in standard form

\[\text{minimize }f_0(x)\]

\[\text{subject to }f_i(x)\leq0\quad{}\land{}\quad{}Ax=b\]

可靠性逻辑框图

系统可靠性

• 在元件故障数据和系统结构已知的情况下，预测系统的可靠性
• 硬件可靠性/人员操作可靠性/软件可靠性

Basic properties and examples

Definition

\(f:R^n\rightarrow{}R\) is convex if dom\(f\) is a convex set and \[f(\theta{}x+(1-\theta)y)\leq\theta{}f(x)+(1-\theta)f(y)\] for all \(x,y\in\) dom\(f\), \(0\leq\theta\leq1\)

基本参数

可靠度与不可靠度

• \(n(t)\):在\(0\sim{}t\)时刻的工作时间内产品的累计失效数
• \(N_0\):\(t=0\)时在规定条件下进行工作的产品数

\[Reliability(t)=R(t)=\frac{N_0-n(t)}{N_0}\] \[Fallibility(t)=F(t)=\frac{n(t)}{N_0}\]

引言——成本总和的最小化

无限阶段问题的成本总和

定义非时变离散时间动态系统： \[x_{k+1}=f(x_k,u_k,w_k)\] \[x_k\in{}S,u_k\in{}U(x_k)\subset{}C,w_k\sim{}P(\cdot{}|x_k,u_k)\]

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