Extended observer form: simple existence conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed. 1. Introduction Conditions for transformability of the discrete-time nonlin-ear state equations by state coordinate transformation into the observer (input–output injection) form are known to be very restrictive, see Lee and Nam (1991). This has motivated to generalise the observer form in such a manner that one may still construct the observers with linear error dy-A in the observer form is allowed to depend on control variable u. In Besançon and Bornard (1995), only state transformation was allowed whereas Califano et al. (2003) and Lee and Hong (2011) allow the output transformation also. The paper by Califano et al. (2009) extends the results to multi-input multi-output case. The paper by Lin and Wei (2009) relies on the time-scaling technique, which means that the right-hand side of the standard observer form is multiplied by a single-variable function, depending on the output variable , and the paper only allows the state transformation. A method called dynamic observer error linearisation was provided in Zhang, Feng, and Xu (2010), where in order to transform the system into the generalised observer form, it is suggested to augment the system by means of the so-called dynamic auxiliary system of the specific linear form. …