control//system identification//persistent excitation
Persistent excitation is the condition on a system's input and regressor signals that guarantees the data carry information about every parameter of a model, at every stretch of time, so that the parameters can be estimated and an adaptive law can converge; it is why a drone hovering perfectly still cannot learn its own mass. In hover with a constant command the data contain one number, the ratio of thrust to weight: the thrust efficiency can be estimated, but not the mass separately from the thrust coefficient, nor a friction term that appears only in motion.
Persistent excitation is the condition on a system's input and regressor signals that guarantees the data carry information about every parameter of a model, at every stretch of time, so that the parameters can be estimated and an adaptive law can converge; it is why a drone hovering perfectly still cannot learn its own mass. In hover with a constant command the data contain one number, the ratio of thrust to weight: the thrust efficiency can be estimated, but not the mass separately from the thrust coefficient, nor a friction term that appears only in motion.
Formally, the regressor vector φj\varphi_jφj (the measured signals that multiply the parameters) is persistently exciting if there are a window of MMM samples and an α>0\alpha>0α>0 such that, at all times,
∑j=kk+MφjφjT⪰αI.\sum_{j=k}^{k+M}\varphi_j\varphi_j^{\mathsf T}\succeq\alpha I.j=k∑k+MφjφjT⪰αI.
The sum is the information each window brings, and ⪰αI\succeq\alpha I⪰αI means all its eigenvalues exceed α\alphaα: in any window the data push in every direction of parameter space. It is observability applied to the parameters, read through the eigenvalues of the right matrix. A useful rule: a sum of nnn sinusoids of distinct frequencies excites up to 2n2n2n parameters.
No learning without excitation, and tracking well does not mean having estimated well.
An adaptive controller can drive its tracking error to zero along a trajectory that never showed the difference between parameter values; it then holds wrong parameters with perfect tracking, until the next manoeuvre finds them wrong (MRAC).
Its absence breaks estimators in a characteristic way. A recursive estimator with forgetting keeps discarding old information while no new information arrives, so its covariance grows and the next excitation makes the parameters leap (recursive least squares, robust adaptation).
The conflict is fundamental: a good controller keeps the system still, and a still system teaches nothing (dual control). Practice injects small PRBS signals in commissioning or maintenance windows, uses natural manoeuvres, and freezes adaptation when the data are uninformative.
In a batch fit the same lack shows as a near-singular information matrix and wild, opposite-signed coefficients; identifiability is then fixed by the test (a wider operating range, a richer input), never by a cleverer solver (experiment design).
More data of the same kind do not help. A thousand hours of hover contain no more about the mass than one minute; what the data lack is variety (system identification).