_2}(t - \tau ) + \sigma _2^2{{(t - \tau )}^2})} }}\left[ \begin{aligned} &\frac{{{\varphi _2}'(t - \tau )}}{{{\varphi _2}(t - \tau )}}(D - {X_\tau }) +
\left(1 - \frac{{{\varphi _2}'(t - \tau )}}{{{\varphi _2}(t - \tau )}}(t - \tau )\right) \\ & \frac{{{u_b}{\varphi _2}(t - \tau ) + (D - {X_\tau })\sigma
_b^2(t - \tau )}}{{{\varphi _2}(t - \tau ) + \sigma _b^2{{(t - \tau )}^2}}} \end{aligned} \right]\cdot\\ &\qquad\exp \left\{ { - \frac{{{{(D - {X_\tau } -
{u_b}(t - \tau ))}^2}}}{{2({\varphi _2}(t - \tau ) + \sigma _b^2{{(t - \tau )}^2})}}} \right\} \cdot {g_\tau }({X_\tau }|{{{u}}_a},{\sigma
_a}){\rm{d}}{X_\tau } \approx {{{A}}_1}{\rm{ - }}{{{B}}_1},{\rm{ }} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\qquad\qquad {{ t > }}\tau \end{aligned} \right.$$
推论 1. 若已知当前时刻${t_k}$的退化状态${x_k}$, 用${l_k}$表示设备剩余寿
命, ${f_L}({l_k})$表示设备 RUL 分布的 PDF, 在随机退化速率${\lambda _1}$和${\lambda
_2}$的影响下, 可获得首达意义下两阶段自适应 Wiener 过程模型 RUL 的 PDF, 其形式与首
达意义下得到的寿命分布 PDF, 即与式(10)、(11)类似, 具体可分为以下两种情况:
情况 1. 当前时刻${t_k}$位于变点前, 即${t_k} < \tau $, 此时随机设备退化失效又存
在两种情况: 1) 失效阈值位于变点前, 即${l_k} + {t_k} \leq \tau$; 2) 失效阈值位于变点后,
即${l_k} + {t_k} > \tau $, 此时 RUL 的 PDF, 如式(12)所示.
情况 2. 当前时刻${t_k}$位于变点后, 即${t_k} > \tau $, 此时退化设备 RUL 的 PDF
为
$$\begin{split} &{A_1} = \frac{{\varphi' _2(t - \tau )}}{{{\varphi _2}(t - \tau )}}\sqrt {\frac{1}{{2\pi (\sigma _{a1}^2 + \sigma _{b1}^2)}}} \exp
\left[ { - \frac{{{{({u_{a1}} - {u_{b1}})}^2}}}{{2(\sigma _{a1}^2 + \sigma _{b1}^2)}}} \right] \cdot \left\{\frac{{{u_{b1}}\sigma _{a1}^2 +
{u_{a1}}\sigma _{b1}^2}}{{\sigma _{a1}^2 + \sigma _{b1}^2}}\Phi \left( {\frac{{{u_{b1}}\sigma _{a1}^2 + {u_{a1}}\sigma _{b1}^2}}{{\sqrt
{\sigma _{a1}^2\sigma _{b1}^2(\sigma _{a1}^2 + \sigma _{b1}^2)} }}} \right)+ \right.\\ &\qquad\left. \sqrt {\frac{{\sigma _{a1}^2\sigma
_{b1}^2}}{{\sigma _{a1}^2 + \sigma _{b1}^2}}} \phi \left( {\frac{{{u_{b1}}\sigma _{a1}^2 + {u_{a1}}\sigma _{b1}^2}}{{\sqrt {\sigma
_{a1}^2\sigma _{b1}^2(\sigma _{a1}^2 + \sigma _{b1}^2)} }}} \right) \right\} \\ &{B_1} = \exp \left\{ {\frac{{2{u_a}D}}{{\varphi' _1(\tau )}} +
\frac{{2({D^2}\sigma _a^4\tau + {D^2}\sigma _a^2\varphi' _1(\tau ))}}{{(\varphi' _1(\tau ) + \sigma _a^2\tau )\varphi' _1{{(\tau )}^2}}}}
\right\}\frac{{\varphi' _2(t - \tau )}}{{{\varphi _2}(t - \tau )}}\sqrt {\frac{1}{{2\pi (\sigma _{a1}^2 + \sigma _{b1}^2)}}} \exp \left[ { -
\frac{{{{({u_{a1}} - {u_{c1}})}^2}}}{{2(\sigma _{a1}^2 + \sigma _{b1}^2)}}} \right] \cdot\\ &\qquad\left\{ {\frac{{{u_{c1}}\sigma _{a1}^2 +
{u_{a1}}\sigma _{b1}^2}}{{\sigma _{a1}^2 + \sigma _{b1}^2}}\Phi \left( {\frac{{{u_{c1}}\sigma _{a1}^2 + {u_{a1}}\sigma _{b1}^2}}{{\sqrt
{\sigma _{a1}^2\sigma _{b1}^2(\sigma _{a1}^2 + \sigma _{b1}^2)} }}} \right) + \sqrt {\frac{{\sigma _{a1}^2\sigma _{b1}^2}}{{\sigma _{a1}^2 +
\sigma _{b1}^2}}} \phi \left( {\frac{{{u_{c1}}\sigma _{a1}^2 + {u_{a1}}\sigma _{b1}^2}}{{\sqrt {\sigma _{a1}^2\sigma _{b1}^2(\sigma _{a1}^2
+ \sigma _{b1}^2)} }}} \right)} \right\} \\ &{u_{a1}} = {u_b}(t - \tau ),\;\;{\rm{ }}{u_{b1}} = D - {u_a}\tau ,\;\;{\rm{ }}{u_{c1}} = - D - {u_a}\tau -
\frac{{\sigma _a^2\tau }}{{\varphi _1^{'}(\tau )}} \\ & \sigma _{a1}^2 = {\varphi _2}(t - \tau ) + \sigma _b^2{(t - \tau )^2},\;\;{\rm{ }}\sigma
_{b1}^2 = {\varphi _1}(\tau ) + \sigma _a^2{\tau ^2} \end{split} $$
$$\begin{split} &{f_L}({l_k}) = \left\{ \begin{aligned} &\frac{1}{{\sqrt {2\pi ({\varphi _{\rm{1}}}({l_k}) + \sigma
_a^2{l_k}^2)} }}\left[ \frac{{{\varphi' _{\rm{1}}}({l_k})}}{{{\varphi _{\rm{1}}}({l_k})}}(D - {X_k}) + \left(1 - \frac{{{\varphi'