Seonglae ChoA model where the state update rule (dynamics) that determines how the state changes over time is in a linear form.
Dynamics refers to the time update rule, and linear means that this rule is linear with respect to the state and input.
- Dynamics (state transition):
x_{t+1} = A x_t + B u_t + w_t
- Observation (measurement):
y_t = C x_t + v_t
Here, "linear" means it is expressed only as matrix multiplication + addition with respect to
x_t or u_t.In other words, it has the form
f(x) = Ax, so superposition (additivity and homogeneity) holds.In contrast, for the nonlinear case:
- If the system is linear with Gaussian noise, the Kalman filter provides an exact solution.
- If the system is nonlinear, approximation or sampling-based methods such as EKF, UKF, or particle filters are commonly used.
The dynamics change to arbitrary nonlinear functions
f and g. Examples include robot position and orientation, aircraft attitude, diffusion and epidemic models, neural dynamics, and more.