control//system identification//experiment design
Experiment design, in system identification, is the choice of the input signal, its amplitude, the sampling rate and the duration of a test so that the recorded data contain the dynamics a model must capture; it is the step that decides whether identification can succeed before any fitting starts. To learn a model, the system must move at the frequencies of interest. Months of a plant at steady state, terabytes of it, contain almost no information about its dynamics, because the controller kept everything still (persistent excitation).
Experiment design, in system identification, is the choice of the input signal, its amplitude, the sampling rate and the duration of a test so that the recorded data contain the dynamics a model must capture; it is the step that decides whether identification can succeed before any fitting starts. To learn a model, the system must move at the frequencies of interest. Months of a plant at steady state, terabytes of it, contain almost no information about its dynamics, because the controller kept everything still (persistent excitation).
The standard signals each have a use. A step is the simplest and reveals gain, time constant and dead time at a glance, enough for a FOPDT model. A PRBS, a pseudorandom binary sequence, switches the input between two levels at pseudorandom instants; it spreads energy over a wide band with small amplitude, so it can be injected during commissioning or a maintenance window without upsetting production. A frequency sweep (chirp) walks the input through frequencies one after another and maps the frequency response directly, the usual choice on a drone's attitude loop or a servo.
Unidentifiable data are fixed by changing the test.
On a drone motor bench, fitting thrust as T=k2ω2+k1ωT=k_2\omega^2+k_1\omegaT=k2ω2+k1ω from tests run only between 6,000 and 7,000 rpm fails: over that range ω2\omega^2ω2 and ω\omegaω are nearly proportional, the smallest singular value collapses (condition number) and the coefficients come out huge and of opposite sign. Sweeping the full speed range separates them.
Amplitude has a window: above the noise, so the response stands out, and below what breaks linearity or safety (a valve near its end stop, a drone near its tilt limit). Small and long beats large and short when the plant must keep producing.
Duration and sampling follow the dynamics: sample several times faster than the fastest dynamics of interest, and record for several of the slowest time constants, more if the noise is high.
An unstable plant is excited in closed loop, with the signal added to the setpoint, and the analysis must account for the feedback (system identification).
Record raw, with timestamps and metadata: which sensor, which calibration, which temperature. A misaligned clock between input and output adds a delay the model will dutifully learn.
Excitation is a cost (off-spec product, a less comfortable flight), which is the practical face of the dual control dilemma.