Identification and the Information Matrix: How to Get Just Sufficiently Rich?

Abstract

Abstract-In prediction error identification, the information matrix plays a central role. Specifically, when the system is in the model set, the covariance matrix of the parameter estimates converges asymptotically, up to a scaling factor, to the inverse of the information matrix. The existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix, and the rate of convergence of the parameter estimate depends on its "size". The information matrix is also the key tool in the solution of optimal experiment design procedures, which have become a focus of recent attention. Introducing a geometric framework, we provide a complete analysis, for arbitrary model structures, of the minimum degree of richness required to guarantee the nonsingularity of the information matrix. We then particularize these results to all commonly used model structures, both in open loop and in closed loop. In a closed-loop setup, our results provide an unexpected and precisely quantifiable trade-off between controller degree and required degree of external excitation.

An Algorithm for Finding Process Identification Intervals from Normal Operating Data

“…However, for a reader only interested in the final algorithm and its application this section may be omitted. The section is mainly based on [12][13][14].…”

“…A number of such scenarios are discussed in [12]; nonlinear control, switching linear controllers or fixed linear controller with high enough order. The case of fixed linear controllers is further studied in [14]. Distinguish between two cases, either r(k) ≡ 0, i.e., the system is driven solely by the noise, or r(k) is changing.…”

“…Checking whether the inputs are sufficiently rich is however impractical since the definition of persistent excitation is based on an ensemble of potential observations, with theĒ operator given in Equation (14). Furthermore, many signals used in practice, during the operation of the plant, are of low SRn.…”

Accuracy improvement in Least-Squares estimation with harmonic regressor: New preconditioning and correction methods

“…For a certain step number the matrix becomes a full rank matrix and invertible. This matrix is a positive definite matrix in all subsequent steps since harmonic regressor is persistently exciting [1], [2], [4], [5]. For a sufficiently large i this matrix becomes an SDD (Strictly Diagonally Dominant) matrix [6].…”

A geometric characterisation of persistently exciting signals generated by continuous-time autonomous systems

“…Thereafter, persistently exciting signals have attracted considerable interest in the control engineering area prompted by the possibility of proving stability results for adaptive control algorithms (see, e.g., Morgan and Narendra [1977], Anderson and Johnson [1982], Boyd and Sastry [1983], Bitmead [1984], Goodwin and Teoh [1985], Boyd and Sastry [1986], Narendra and Annaswamy [1987]). This has generated a vast and diverse body of contributions, e.g., Ljung [1971], Yuan and Wonham [1977], Stoica [1981], Moore [1983], Mareels [1984], Mareels and Gevers [1988], Glad [1990, 1994], Willems et al [2005], Gevers et al [2009], Ortega and Fradkov [1993], Zhang et al [1996], Panteley et al [2001], Loría et al [2005], which have provided extensions and applications of the persistence of excitation condition.…”