control//state estimation//state augmentation

State augmentation is the technique of promoting a slowly varying parameter or sensor error to a state variable of the estimator, so that the filter tracks it alongside position, velocity or temperature instead of treating it as a constant known in advance. It is how a drone's estimator follows the bias of its gyroscopes and accelerometers, how a navigation filter follows the barometer's drift with the weather, and how a controller learns the new mass of a drone that has just dropped its payload. A parameter becomes a state when it changes slowly and it matters.

State augmentation is the technique of promoting a slowly varying parameter or sensor error to a state variable of the estimator, so that the filter tracks it alongside position, velocity or temperature instead of treating it as a constant known in advance. It is how a drone's estimator follows the bias of its gyroscopes and accelerometers, how a navigation filter follows the barometer's drift with the weather, and how a controller learns the new mass of a drone that has just dropped its payload. A parameter becomes a state when it changes slowly and it matters.

The model only needs a law for how the new state evolves. A bias that wanders is usually written as a first-order Gauss-Markov process, bk=ϕ bk−1+ηkb_k=\phi,b_{k-1}+\eta_kbk​=ϕbk−1​+ηk​ with ϕ=e−Δt/τ\phi=e^{-\Delta t/\tau}ϕ=e−Δt/τ, or as a random walk when it has no preferred value; a constant parameter gets bk=bk−1b_k=b_{k-1}bk​=bk−1​ plus a little process noise, so the filter stays able to follow it. The dynamics matrix gains a block for it, and the sensor model gains a column where the bias adds to the reading. A gyroscope integrated with its bias, θk+1=θk+Δt (ωk−bk)\theta_{k+1}=\theta_k+\Delta t,(\omega_k-b_k)θk+1​=θk​+Δt(ωk​−bk​), is the textbook case: the angle and the bias become a two-state filter, and every correction from the accelerometer moves both, through the correlation the model creates between their errors (unmeasured state estimation).

What was noise with memory becomes structure the filter can use. A drifting error left out of the state makes the filter arrogant, counting the same slow error as many independent readings (colored noise); put in the state, it is estimated and subtracted.

The added state must be observable, or the filter only spreads the error between the two. A single sensor reading position plus a constant bias cannot separate them; a second sensor with a different error (the GPS beside the barometer) or a manoeuvre (an accelerometer bias in hover is indistinguishable from a tilt error) is what makes the bias visible.

Each added state costs a row and a column of the covariance, and the filter's arithmetic grows roughly as the cube of the state size. Twenty-odd states are routine on an autopilot; a hundred slow parameters belong in a separate estimator, such as recursive least squares when the measurement is linear in them.

Control uses the same trick. A state-feedback controller that cannot cancel a constant push gains an integrator by augmenting its state with the integral of the error (LQR).