## 3 CPS Resilience Quantitative Assessment Framework
### Integrated CPS Modeling Strategy and Comparative Advantages
#### 3.4.1 Normalization and Parameterization  
This section introduces a coupling strategy that connects these two domains by aligning cyber states with phases in the performance evolution process. The key idea is to treat the evolution of physical performance as a piecewise dynamic process, where each segment corresponds to a specific cyber state or class of cyber states. When the system transitions between states in the cyber layer--such as from normal operation to degraded or failed modes--the physical performance curve correspondingly switches to a new segment governed by different parameters that reflect the severity of impact and the system's recovery capacity. To simplify the coupling logic and enable interpretability, we adopt a fixed canonical sequence of cyber states to guide the composition of the piecewise performance curve. This sequence--Normal \(\rightarrow\) Silent Failure \(\rightarrow\) Observable Failure \(\rightarrow\) Degraded \(\rightarrow\) Recovered--represents a typical trajectory of attack propagation and system response. While the actual CTMC may allow arbitrary transitions, using a fixed sequence enables clear mapping and comparative analysis across systems. This coupling strategy provides a structured foundation for the subsequent mathematical formulation, normalization, and simulation of the integrated cyber-physical resilience dynamics. To enable quantitative integration between the physical performance model and the cyber-layer CTMC model, a consistent parameterization and normalization process is essential. The physical performance function \(F(t)\) describes the evolution of system functionality over time. To generalize comparisons and simplify interpretation, we normalize the performance curve by dividing it by the nominal performance \(F_{N}\). The normalized performance function \(\widetilde{F}(t)=F(t)/F_{N}\) thus always lies in the range \([0,1]\), where 1 corresponds to full functionality and 0 indicates complete failure. While the CTMC model defines transitions between discrete cyber states, the performance model evolves continuously. To integrate the Markov-based security dynamics with the performance evolution model, the total modeling duration \(T\) is partitioned into time segments, each corresponding to a distinct Markov state. The duration of each segment, denoted as \(\Delta t_{i}\), is determined by a joint weighting mechanism that considers both the steady-state probability \(p_{i}\) and the mean sojourn time \(\Delta\tilde{t}_{i}\) of state \(i\). This reflects not only how frequently the system enters a state, but also how long it tends to remain there once entered. Formally, the segment duration \(\Delta t_{i}\) is calculated as follows:  
\[\Delta t_{i}=\frac{p_{i}\cdot\Delta\tilde{t}_{i}}{\sum_{j}p_{j}\cdot\Delta \tilde{t}_{j}}\cdot T \tag{13}\]  
where \(\Delta\tilde{t}_{i}=\frac{1}{\sum_{j\
eq i}q_{ij}}\) and \(q_{ij}\) is the transition rate from state \(i\) to state \(j\) in the Markov process. The total duration \(T\) represents the entire simulation or evaluation window for performance evolution. This value can be determined based on practical constraints such as mission time, operational period, or analysis horizon, depending on the specific CPS context. In scenarios where no absolute time reference is required, \(T\) can be normalized to 1, simplifying the analysis while preserving relative temporal dynamics across states. For each interval \(t_{i}\), a differential performance function is defined using a simplified model with two parameters: \(A_{i}\), representing the intensity of adverse effects, and \(R_{i}\), capturing the system's recovery capability. These parameters are selected or estimated according to the characteristics of the corresponding cyber state:  
* Available states: \(A_{i}\ll R_{i}\), representing effective control and recovery. * Degraded states: These states reflect abnormal but recognizable behavior.