mathematics//dynamical systems//state-space model//measurement model
A measurement model is the equation of a state-space model that says what each sensor should read for a given state, \(y=h(x)+v\), and it is what lets a filter compare a prediction with a reading: the prediction lives in the state's units, the reading in the sensor's, and the measurement model translates one into the other. \(h\) is the ideal sensor and \(v\) the measurement noise, everything the sensor adds that no model predicts. In the linear case \(h\) is a matrix, \(\cc{z}=Hx+v\), with what the sensor brings in copper and the prediction before the reading in blue.
A measurement model is the equation of a state-space model that says what each sensor should read for a given state, y=h(x)+vy=h(x)+vy=h(x)+v, and it is what lets a filter compare a prediction with a reading: the prediction lives in the state's units, the reading in the sensor's, and the measurement model translates one into the other. hhh is the ideal sensor and vvv the measurement noise, everything the sensor adds that no model predicts. In the linear case hhh is a matrix, z=Hx+v\cc{z}=Hx+vz=Hx+v, with what the sensor brings in copper and the prediction before the reading in blue.
HHH selects, combines and converts. A GPS gives only the position of a car whose state is position and velocity; a thermocouple gives millivolts where the state holds degrees. Hx^−H\cb{\hat x^-}Hx^− answers what the sensor should read if the prediction were true, and the innovation z−Hx^−\cc{z}-H\cb{\hat x^-}z−Hx^− then compares like with like.
Sensor What it measures HHH HxHxHx
GPS position only [1 0][1;;0][10] ppp
Wheel encoder velocity only [0 1][0;;1][01] vvv
GPS and speed radar both [1001]\begin{bmatrix}1&0\0&1\end{bmatrix}[1001] [p, v]T[p,;v]^{\mathsf T}[p,v]T
Thermocouple millivolts, not degrees about 0.041 mV/°C 0.041 T0.041,T0.041T
A real sensor needs more than a matrix. Scale-factor error, a bias that drifts with temperature, latency and the rounding of the converter all belong in hhh (sensor error model). A bias that matters is appended to the state and estimated, which changes HHH to read p+bp+bp+b (sensor bias, state augmentation).
Many sensors are nonlinear. A range beacon measures (px−bx)2+(py−by)2\sqrt{(p_x-b_x)^2+(p_y-b_y)^2}(px−bx)2+(py−by)2, a camera projects a 3D point onto a pixel, a barometer turns pressure into altitude through an exponential; an extended Kalman filter linearizes hhh at each step and uses its Jacobian in place of HHH.
A reading describes the world at the moment it was taken. A GNSS fix processed for 100 ms describes x(t−0.1)x(t-0.1)x(t−0.1), and fusing it as if it were current injects a metre of error at 10 m/s, so the timestamp is part of the measurement model.
What hhh leaves out cannot be estimated from that sensor: whether the state can be reconstructed from the readings is observability, decided by hhh and the dynamics together. The dynamics half of the model is the state-space model.