State Estimation in P2V

State Estimation in P2V

State Estimation

• Consider $x_k$ is digital state vectors, $y_k$ is noisy measurement (observation) vector, $u_k$ is measured input vector, $\omega_k$ vector of process noise variable, $v_k$ is vector of measurement noise variable, $f(\cdot)$ is state transition function and $g(\cdot)$ measurement function.
  • State transition: $xk=f(x{k-1},u_{k-1})+\omega_k$
  • Measurement: $y_k= g(x_k)+v_k$
• The state transition equation models the time evolution of the states as a dynamic system $f(\cdot)$, perturbed by noise $\omegak$. The state transition equation can be rewritten as the transition probability density $p(x_k|x{k-1})$.
• The measurement equation can be written as $p(y_k|x_k)$. Conditional density of measurement $y_k$ given the state $x_k$
• It can be represented as a hidden Markov model as below
  
• The objective of the state estimation is to estimate (in recursive way) the hidden state ($xk$) from the measurements ($y$). To estimate it, we compute the marginal conditional distribution of $x_k$ given all available measurements $y{1:k}$
• The marginal conditional of the digital state can be denoted as $p(xk|y{1:k})$ and computed using Bayes theorem as $$ p({x}{k}|{y}{1:k})=\frac{p({y}{k}|{x}{k})p({x}{k}|{y}{1:k-1})}{\int p({y}{k}|{x}{k})p({x}{k}|{y}{1:k-1})d{x}{k}}, \propto p(y_k|x_k)p(x_k|y{1:k}) $$

where $y_{1:k}$ are observation from $t_1$ to $t_k$, $\propto$ is proportional to, and $p(\cdot|\cdot)$ is a conditional probability density function.

See Recursive Bayesian Estimation.

State Estimation Problems

• Filtering
  • estimate present (current) states
  • applications: fault diagnosis
• Prediction
  • estimate future states
  • Prediction methods are concerned with estimating future states using historical data.
  • Prediction is typically used for understanding future system state.
  • Prediction in digital twin is typically used for planning and control
  • applications: health forecasting, RUL predictions
• Smoothing:
  • estimate past states
  • Smoothing methods operate on previously collected data and generally reduce the size of the data in the process.
  • Smoothing was used to understand historical system state.

Bayesian Filters

• Four recursive Bayesian filter as below
  
• Applications of particle filters for state estimation in the digital twin context have been mostly limited to a small number of state dimensions (typically < 5)

State Estimation vs Parameter Estimation

• Parameter estimation: estimating digital state variables that change very slowly or do not change with time.
• State estimation: estimating digital state variables that change rapidly with time

References