Identification Experiments, Excitation & Data Conditioning

An identification algorithm can only reorganize information that reached the recorded signals. If the input never excites a mode, the sensor filters it out, the actuator saturates, or timestamps are shifted, the missing information cannot be recovered by choosing a cleverer estimator. The experiment and measurement chain are therefore part of the model.

Begin with a model-use contract #

Declare the decision the model will support before choosing a waveform:

• manipulated inputs $u$, measured outputs $y$, known references $r$, measured disturbances $d_m$, and important unmeasured disturbances $d_u$;
• operating point and allowed excursions in load, temperature, speed, configuration and controller mode;
• frequency band, time horizon, amplitudes and accuracy that matter to the use;
• required representation: simulation model, one-step predictor, frequency response, transfer function, state space or physical parameters;
• actuator position/rate/energy limits and plant safety interlocks;
• whether the final controller will run open loop, closed loop, continuously or at a declared sample period.

A model used to predict the next sample may benefit from an elaborate disturbance model. A controller-design model must instead preserve gain, phase, delay and uncertainty around the intended loop bandwidth. The two objectives can select different models from the same data.

Excitation creates information #

For a regression $y_k=\varphi_k^T\theta+e_k$, the data information matrix is proportional to

$$R_\varphi=\frac{1}{N}\sum_{k=1}^{N}\varphi_k\varphi_k^T.$$

If $R_\varphi$ is singular, some parameter direction does not change the recorded prediction and cannot be identified. A well-conditioned positive-definite matrix means the regressors span the requested parameter directions with usable separation. This finite-data condition is the operational meaning behind persistent excitation.

For dynamic systems, the same idea is spectral: the input must carry energy across the frequencies where plant dynamics are to be distinguished. More amplitude improves signal-to-noise ratio but can cross into saturation, hysteresis, friction regimes, gain scheduling or unsafe operation. More bandwidth reveals faster dynamics but also excites resonances and unmodeled modes. Experiment design negotiates these constraints; it does not maximize “randomness.”

Choosing an input #

The chosen signal should be independently generated when possible, reproducible, bounded, and recorded after the actuator command path. Store its exact seed or deterministic sequence. “White noise” is not a complete specification: state its distribution, filter, amplitude, update period, clipping and realization.

Amplitude, duration and repeated runs #

Use a pilot test to find the smallest amplitude that produces output above sensor and disturbance noise across the target band. Then check whether estimated dynamics change when amplitude or operating point changes. If they do, the plant is nonlinear or time-varying over the tested envelope; averaging the runs into one LTI model hides that fact.

Duration must cover both spectral resolution and slow settling. A record of length $T$ has Fourier-bin spacing $\Delta f=1/T$; resolving two nearby modes requires a long record even when the sample interval is small. Several independent periods or runs are more useful than one unrepeatable trace because they expose run-to-run variability and support honest validation.

Sampling and intersample behavior #

The analog signal must be low-pass filtered before sampling so energy above Nyquist cannot alias into the retained band. Digital filtering after acquisition cannot undo aliasing. Record the analog anti-alias filter, sensor bandwidth, sample period and any decimation filters as part of the experiment.

Choose $T_s$ from the fastest mode to be modeled, the phase accuracy required near control bandwidth, sensor noise and computational limits—not from Nyquist alone. A DC-normalized zero-order hold contributes approximately half a sample of effective delay at low frequency; its frequency response is

$$H_{ZOH}(s)=\frac{1-e^{-sT_s}}{sT_s} =e^{-sT_s/2}\frac{\sinh(sT_s/2)}{sT_s/2}.$$

If a continuous-time model is estimated from sampled data, declare whether the physical input was zero-order held, first-order interpolated or continuously varying. Different intersample assumptions can produce the same sample values and different continuous-time dynamics.

Hardware timestamps matter more than array indices. Measure or calibrate:

• command-to-actuator delay;
• sensor group delay and filtering;
• clock offset, drift and jitter between devices;
• multiplexed-channel skew;
• buffering, packetization and resampling delay.

A one-sample alignment error creates a factor $q^{-1}$ and phase $-\omega T_s$. The estimator will happily report that instrumentation delay as a plant delay.

Record the actual causal channels #

Record actuator feedback or the final applied command, not only the desired reference. Dead zones, rate limits, saturation, local drive loops and selector logic can make them different. Likewise record relevant mode bits, interlocks, gains, unit conversions and controller output limits.

For a closed-loop test,

$$u_k=C(q)(r_k-y_k)+u_{test,k},$$

so $u$ is correlated with output disturbance through feedback. Regressing $y$ on $u$ as if the input were independent can attribute disturbance-driven control action to the plant. Inject an external excitation with adequate independent content, record $r,u,y$, preserve the controller description, and use an estimator whose closed-loop assumptions match the experiment.

Closed loop is sometimes the only safe way to excite an unstable or tightly constrained plant. It is not inferior data by definition; it is a different causal experiment that requires explicit feedback-aware identification.

Conditioning without inventing dynamics #

Keep the raw immutable channels and produce conditioning as a reproducible transformation with parameters. Typical operations are:

1. calibrate units, signs and sensor scales;
2. align clocks and document delays;
3. mark saturation, dropouts, mode changes and operator interventions;
4. remove a justified constant operating-point offset;
5. detrend only when drift is demonstrably external to the dynamics of interest;
6. anti-alias, then decimate with the complete filter recorded;
7. select stationary intervals and preserve experiment boundaries.

Detrending can remove a real integrator or slow pole. Differencing can turn a level model into an increment model and amplify high-frequency noise. Filtering both $u$ and $y$ changes the estimation frequency weighting; filtering only one channel changes the apparent plant. Every transformation is a modeling decision.

Missing samples should not be silently filled by linear interpolation across a transient. Either use a method that represents missing observations, restrict the interval, or justify the interpolation from the signal bandwidth. Preserve a mask so validation does not score invented samples as measurements.

Estimation and validation are different experiments #

Randomly shuffling adjacent samples leaks almost the same state trajectory into both sets. Instead, split by complete runs, periods, operating points or contiguous time blocks. A useful campaign contains:

• estimation runs with rich excitation;
• validation runs with a different realization of the excitation;
• use-case runs such as references, disturbances and operating transitions;
• amplitude/operating-point sweeps that test the claimed LTI envelope.

Apply the same causal preprocessing pipeline to estimation and validation data, while fitting any preprocessing parameters only where appropriate. A model selected after repeatedly inspecting one validation run has indirectly trained on that run; retain a final untouched test campaign for high-stakes use.

A worked experiment plan #

Suppose a thermal-flow plant appears to contain a fast actuator/sensor mode near $0.2$ s, a dominant process mode near $5$ s, and operation must remain within $\pm8\%$ of nominal flow.

1. Declare a target identification band of roughly $0.03$–$15\ \mathrm{rad/s}$, wide enough to cover both modes but below an observed mechanical resonance.
2. Use a bounded multisine whose line amplitudes are reduced near frequencies where preliminary plant gain is high; optimize phase to reduce crest factor.
3. Sample applied actuator feedback and output at $T_s=0.02$ s with a documented analog anti-alias filter. This is not justified by Nyquist alone: it also limits sample/hold and alignment phase error near the upper target band.
4. Run several complete periods after reaching the operating point, then repeat with different multisine phases at nominal and two neighboring loads.
5. Reserve whole runs for validation; do not split individual samples.
6. Reject or separately model intervals with saturation, mode switching or sensor dropout.

The numbers are an experiment specification, not universal defaults. If the pilot spectrum shows low output signal-to-noise ratio or nonlinear distortion, redesign the amplitude distribution or narrow the claimed model band rather than hiding the failure in a high-order fit.

Experiment record #

Ship the data with:

• channel names, units, signs, calibration and physical locations;
• raw and applied inputs, outputs, references, measured disturbances and mode flags;
• timestamps, clock source, sample period, jitter and known delays;
• excitation generator, seed/phase, amplitudes, spectrum and safety limits;
• operating conditions and initial-state procedure;
• sensor, actuator, anti-alias and hold dynamics;
• immutable raw files plus versioned conditioning code;
• estimation/validation/test partition and exclusion reasons.

That record is not clerical metadata. It is the evidence needed to decide which poles, delays and uncertainty belong to the plant and which were manufactured by the experiment.