Asymmetric delayed relay feedback identification based on the n -shifting approach

Abstract

The paper presents an improvement of the n-shifting technique to identify the frequency response of an industrial process using a fully asymmetric and delaying relay. The n-shifting approach allows the calculation of n + 1 points of G(s) by an asymmetric relay experiment. This set of n points is composed of G(0), G(jω osc ), ... , G(jnω osc ), being ω osc the oscillation frequency, and where G(jω osc ) is in most cases located in the third quadrant of the Nyquist map. By delaying the relay output and repeating a similar experiment it can be generated n additional points of G(s) where the first point is G(jω' osc ) with 0 < ω' osc < ω osc . In this way, it is possible to depict the full output spectrum of G(s) from zero to very high frequencies by a short relay experiment. An example of identification and tuning of a PID controller with data from the n-shifting are presented to show the validity of the approach.

Multiple frequency response points identification through single asymmetric relay feedback experiment

“…Another approach, called the 2-shifting method was described in [9], which allows obtaining 2 points of the frequency response. This method has been further developed resulting in the n-shifting method which has been used to identify n points instead [27,28]. Certainly, there exist other approaches to the identification problem, for example the one presented in [38], which proposes to replace the traditional relay by a parasitic relay or a cascade relay.…”

“…Some of the aforementioned works present different degrees of variation with regard to the relay structure, among these works, the variation presented in [28], which is a fully asymmetric and delaying relay, named FAD relay for short, can provide flexibility without compromising the difficulty of the setup. The FAD relay contains biases in its detection and in its output, and some user-defined delay to its output, which as it was shown in [28] allows decreasing the fundamental oscillation frequency, being, thus, possible to identify regions of the frequency response below the ultimate frequency. The information provided by this region is of interest, for example, for tuning PID controllers.…”

“…Since applying linear operations between rows and columns does not alter the value of the determinant, we have subtracted to every row, except from the first one, the first one, see expression (28), and then added to the first column the other columns, see expression (29), resulting in a triangular matrix, therefore the determinant is the product of the elements in the main diagonal:…”