# A Hybrid Degradation Modeling and Prognostic Method for the Multi-Modal System
## 2 Degradation Modeling and Prognostic Hybrid
### Long-Term Cumulative Degradation Assessment with Cdd-Hi
#### 2.1.2 Composite Health Indicator Construction  
To avoid the redundancy among different features, this paper develops a feature-level fusion method to combine cumulative dynamic physical difference features to get a composite health indicator based on the multi-objective optimization model. The health indicator \(H(t)\) is regarded as a linear combination of each cumulative dynamic physical difference feature \(f_{o}\), as Equation (1),  
\[H(t)=w_{1}f_{1}(t)+w_{2}f_{2}(t)+\cdots+w_{o}f_{o}(t), \tag{1}\]  
Figure 1: Structure of the proposed hybrid degradation model. The offline data are used to adjust parameters of prognostic model. The learned prognostic model is used to predict remaining useful life based on the online data. where \(o\) is the dimension of cumulative dynamic physical difference feature, and \(w\) is the weight coefficient of each cumulative dynamic difference feature, \(-1<w<1\). To represent the actual degradation trend of system, the interpretable health indicator is constructed based on three properties. Property 1 The degradation of the system should be monotonic in general without external maintenance. To establish the monotonic health indicator, the slack variable \(\varepsilon_{i,t}\) is introduced to measure the violation of monotonicity of health indicator. The \(\varepsilon_{i,t}\) of system \(i\) and action time \(t\) is \(\varepsilon_{i,t}=\max(H_{i,t}-H_{i,t+1},0)\). Assuming that the indicator is monotonically increasing, Equation (2) tries to minimize the total weighted sum of violations to ensure the monotonicity of the health indicator. \[ \begin{split}&\min_{w\
eq i,t}\sum\limits_{i=1}^{p}\sum\limits_{t =1}^{q_{i}-1}\varepsilon_{i,t}\\ & s.t.\ \ w^{\prime}M^{\prime}1=1,\ \ Mw\geq 0,\ \ \ \varepsilon_{i,t}\geq 0,\end{split}  \tag{2}\]  
where \(p\) is number of system samples, and \(q\) is the maximum observation epochs (action times) of each sample. \(M\) is a diagonal matrix to express the degradation trend information. For example, if the feature shows an increasing (decreasing) trend, the diagonal entry of \(M\) is 1 (\(-1\)). The different system \(i\) varies in the life cycle length \(q_{i}\). Property 2 The first hitting time of failure is regarded as the failure threshold \(\theta_{i}\) of each system. The systems with the same functions and components should have similar fault thresholds under the same environmental conditions. Figure 2: The example of cumulative dynamic physical difference feature construction. Calculate the proposed features of \(t=25\). Compared points are the last action times of all modes. \(F_{1}\) is the value of smooth data. \(F_{2,1},F_{2,2}\dots F_{2,6}\) is the cumulative time of each mode. \(F_{3,1},F_{3,2}\dots F_{3,6}\) are the difference times to compared points. \(F_{4,1},F_{4,2}\dots F_{4,6}\) are the difference data to compared points. The predictability is expressed by minimizing the variance of \(\theta_{i}\), as \(\sum_{i=1}^{p}{(\theta_{i}-\bar{\theta})^{2}}/(p-1)\). Assuming \(A\in R^{p\times o}\) is the matrix recording the feature of the last observation epoch in full life-cycle of each system, the variance can be translated as the quadratic term \(w^{\prime}A^{\prime}Baw\) based on Equation (3):  
\[ \begin{array}{l}\left((Aw)^{\prime}(Aw)-p((1^{\prime}Aw)/(p))^{2}\right)/(p-1 )\\ =(w^{\prime}A^{\prime}Aw-(w^{\prime}A^{\prime}11^{\prime}Aw)/p)/(p-1)\\ =w^{\prime}A^{\prime}((l-(11^{\prime})/p)/(p-1))Aw\\ =w^{\prime}A^{\prime}Baw,\end{array}  \tag{3}\]  
where \(B\) is a symmetric matrix; \(B=(l-O/p)/(p-1)\). \(l\) is the identity matrix, and \(O\) is the matrix which each entry equals 1. Above all, Property 2 could be formulated with an optimization problem, as Equation (4):  
\[ \begin{array}{l}\min_{w}w^{\prime}A^{\prime}Baw\\ s.t.w^{\prime}M^{\prime}1=1,Mw\geq 0.\end{array}  \tag{4}\]  
Property 3: For multi-modal systems, although different mode combinations would result in different short-term fluctuations of each sample, there are still similar degradation trends for those samples from the long-term perspective.