Battery knowledge · Transport and modelling

State-space model

A model written as first-order equations for a set of internal states — such as state of charge and the voltages across RC pairs — plus an output equation for the measured terminal voltage. Kalman filters and other observers need this form, so equivalent circuit and reduced physics models are cast this way for a BMS. ^[1]

Also called: state-space representation
Relationship: ẋ = A·x + B·u; y = C·x + D·u

Example

Read a one-state thermal model

Let x be temperature rise and u heating power. With ẋ = −0.01x + 0.002u, the time constant is 100 s and a steady 10 W input gives x = 2 K. The output y = x directly reports temperature rise.

Explore the graph

A one-state linear model separates the stored state from its observed output.

First-order state-space step response

Unit input uOutput y = 1 − exp(−t/τ) (dashed)

At τ = 10 s, output reaches 63.2% after 10 s and 95.0% after 30 s.

dx/dt = −x/τ + u/τ; y = x; x(0) = 0

Illustrative model. Scalar continuous-time, stable linear model with no saturation, delay or measurement noise.

Use in Lemonergy

Specify phase, direction and boundaries.

This is a reference concept. No executable Lemonergy block has been verified for this term.

References & further reading

Technical references

Feedback Systems: An Introduction for Scientists and Engineers, second edition
Karl J. Åström, Richard M. Murray · Princeton University Press; author-maintained FBSwiki · 2021
Technical reference · State-space modeling, feedback and PID control. These mathematical foundations do not establish a particular software implementation.