A comprehensive review of digital twin — part 1: modeling and twinning enabling technologies

Abstract

As an emerging technology in the era of Industry 4.0, digital twin is gaining unprecedented attention because of its promise to further optimize process design, quality control, health monitoring, decision and policy making, and more, by comprehensively modeling the physical world as a group of interconnected digital models. In a two-part series of papers, we examine the fundamental role of different modeling techniques, twinning enabling technologies, and uncertainty quantification and optimization methods commonly used in digital twins. This first paper presents a thorough literature review of digital twin trends across many disciplines currently pursuing this area of research. Then, digital twin modeling and twinning enabling technologies are further analyzed by classifying them into two main categories: physical-to-virtual, and virtual-to-physical, based on the direction in which data flows. Finally, this paper provides perspectives on the trajectory of digital twin technology over the next decade, and introduces a few emerging areas of research which will likely be of great use in future digital twin research. In part two of this review, the role of uncertainty quantification and optimization are discussed, a battery digital twin is demonstrated, and more perspectives on the future of digital twin are shared. Code and preprocessed data for generating all the results and figures presented in the battery digital twin case study in part 2 of this review are available on Github.

Appendices

Appendix A: a generic particle filter algorithm

Particle filtering is a widely used Bayesian filtering technique for state estimation in generic state-space models and DBNs. It is a key enabler in many digital twin applications where probabilistic model updating with state estimation plays an essential role. Let us briefly look at how particle filtering works. Particle filters represent a family of algorithms that recursively execute Bayesian filtering through sequential Monte Carlo simulation (Arulampalam et al. 2002). In a particle filter, the marginal posterior of a state at the current time step k is approximated using a large set of weighted particles. This approximation takes the following form

$$\begin{aligned} \begin{aligned} p({{\mathbf{x}}_k}\vert {{\mathbf{y}}_{1:k}}) \approx \sum \limits _{j = 1}^{{N_p}} {w_k^j\delta \left( {{\mathbf{x}}_k} - {\mathbf{x}}_k^j\right) } , \end{aligned} \end{aligned}$$

(13)

where ${\mathbf{x}}_k^j$ is the jth particle of the state x at the kth time step, $w_k^j$ is its associated weight, NP is the total number of particles, and $\delta$ is the Dirac delta function. Algorithm 1 depicted in Fig. 28 gives the pseudo-code of a generic particle filter (Hu et al. 2018; Cappé et al. 2007). Three key steps of the particle filtering procedure are described as follows:

• State transition: Each particle of the state transitions forward in time by one step (from the previous, $(k - 1)$th step to the current, kth step according to the state transition equation in Eq. (13) (Line 5). These particles are equally weighed ($1/N_P$), and they form the prior of ${{\mathbf{x}}_k}$ before the current measurement is used.

• Weight evaluation and normalization: The weight of each particle is updated using the likelihood of the current measurement given the state values of the particle, i.e., ${\mathbf{w}}_k^i \propto {\mathbf{w}}_{k - 1}^ip({{\mathbf{y}}_k}\vert {\mathbf{x}}_k^i)$ (Line 6). The likelihood is calculated by comparing the measurement distribution predicted by the measurement equation ${\mathbf{g}}$ in Eq. (13) to the actual measurement ${{\mathbf{y}}_k}$. After that, the updated weights are normalized, and the normalized weights sum up to 1 (Lines 8–12).

• Resampling: Particles with negligible weights are replaced by new particles that are copies of particles with higher weights (Line 13). This step mitigates the issue of particle degeneracy, where the weights become overly concentrated on a very small subset of particles (in an extreme case, only all by one particle have close-to-zero weights).

Fig. 28

Pseudocode of a generic particle filter algorithm

Appendix B: decomposition of likelihood and Bayesian inference in a DBN

Using the DBN given in Fig. 16 as an example, the posterior distribution $p({x_{1,k}},{x_{2,k}},{x_{3,k}} ,{x_{4,k}} \vert {y_{1,k}})$ of state variables ${x_{1,k}}$, ${x_{2,k}}$, ${x_{3,k}}$, and ${x_{4,k}}$ for given observation ${y_{1,k}}$ at time $t_k$ is given by

$$\begin{aligned} \begin{aligned}&p({x_{1,k}},{x_{2,k}},{x_{3,k}},{x_{4,k}} \vert {y_{1,k}}) \\&\quad \propto p({y_{1,k}} \vert {x_{3,k}},{x_{4,k}})p({x_{3,k}} \vert {x_{2,k}},{x_{1,k}}) \\&\qquad \times p({x_{4,k}} \vert {x_{2,k}},{x_{1,k}})p'({x_{1,k}},{x_{2,k}},{x_{3,k}},{x_{4,k}}), \end{aligned} \end{aligned}$$

(14)

where $p({y_{1,k}} \vert {x_{3,k}},{x_{4,k}})$, $p({x_{3,k}} \vert {x_{2,k}},{x_{1,k}})$, and $p({x_{4,k}} \vert {x_{2,k}},{x_{1,k}})$ are the conditional probability tables or conditional probability distributions describing the probabilistic causality relationship between the parent nodes (e.g., $x_{2,k}$ and $x_{1,k}$) and a child node (e.g., $x_{3,k}$), $p'({x_{1,k}},{x_{2,k}},{x_{3,k}})$ is the prior distribution of state variables ${x_{1,k}},{x_{2,k}}$, ${x_{3,k}}$, and ${x_{4,k}}$ at $t_k$, which is obtained at each time step by recursively performing Bayesian inference and uncertainty propagation based on observations and state transition probabilities given in Eq. (6).

Theoretically, all the Bayesian inference methods discussed in Sect. 4.2.2 can be employed to update the posterior of state variables based on the formulation given in Eq. (14). In the context of digital twins, particle filters such as the one presented in Appendix A are usually used in conjunction with DBNs to update digital states for two main reasons: (1) particle filters have fewer assumptions on the nonlinearity of the state transition and measurement functions and noise distributions than Kalman filters (see Table 4); and (2) using surrogate models to approximate conditional probability distributions [see Eq. (5)] allows for an efficient evaluation of likelihood functions, such that a large number of particles can be used for the inference. Using Eq. (14) as an example, in a particle filter implementation, we first generate equally weighted particles of state variables ${x_{1,k}},{x_{2,k}}$, ${x_{3,k}}$, and ${x_{4,k}}$ as their prior at $t_k$ according to the transition probabilities of state variables defined by the transition BN (i.e., Step 5 in Fig. 28). After that, the likelihood of each particle is computed using the decomposed likelihood function $p({y_{1,k}} \vert {x_{3,k}},{x_{4,k}})p({x_{3,k}} \vert {x_{2,k}},{x_{1,k}}) \times p({x_{4,k}} \vert {x_{2,k}},{x_{1,k}})$ given in Eq. (14). This corresponds to the computation of particle weights in Step 6 of Fig. 28. The likelihood values of the particles are then normalized and the particles are subsequently re-sampled based on the normalized weights to obtain the posterior samples of the state variables at $t_k$. This process is implemented repeatedly over time to dynamically update the state variables.