Principles of MHE

Moving horizon estimation (MHE) is a multivariable estimation algorithm that uses:

• an internal dynamic model of the process
• a history of past measurements and
• an optimization cost function J over the estimation horizon,

to calculate the optimum states and parameters.

Moving horizon estimation scheme ^[4]

The optimization estimation function is given by:

${\displaystyle J=\sum _{i=1}^{N}w_{y}(x_{i}-y_{i})^{2}+\sum _{i=1}^{N}w_{\hat {x}}(x_{i}-{\hat {x}}_{i})^{2}+\sum _{i=1}^{N}w_{p_{i}}{\Delta p_{i}}^{2}}$

without violating state or parameter constraints (low/high limits)

With:

${\displaystyle x_{i}}$ = i -th model predicted variable (e.g. predicted temperature)

${\displaystyle y_{i}}$ = i -th measured variable (e.g. measured temperature)

${\displaystyle p_{i}}$ = i -th estimated parameter (e.g. heat transfer coefficient)

${\displaystyle w_{y}}$ = weighting coefficient reflecting the relative importance of measured values ${\displaystyle y_{i}}$

${\displaystyle w_{{\hat {x}}_{i}}}$ = weighting coefficient reflecting the relative importance of prior model predictions ${\displaystyle {\hat {x}}_{i}}$

${\displaystyle w_{p_{i}}}$ = weighting coefficient penalizing relative big changes in ${\displaystyle p_{i}}$

Moving horizon estimation uses a sliding time window. At each sampling time the window moves one step forward. It estimates the states in the window by analyzing the measured output sequence and uses the last estimated state out of the window, as the prior knowledge.