Linearization by Output Injection under Approximate Sampling

The Observer Error Linearization Problem via Dynamic Compensation

“…The proof, detailed in the Appendix, is based on the computation of an appropriate change of coordinates such that the dynamic compensator solving the problem is split into a subsystem necessary to solve the problem, and an additional part which can be neglected. In these coordinates the first subsystem has the structure (27).…”

“…Proof: If the algorithm ends at Step 5, then a compensator has been computed together with the appropriate change of coordinates which transforms the extended system in the canonical observer form up to input and output injection, thus proving the sufficiency part. As for the necessity the algorithm is based on the necessary conditions enounced in Theorem 2 with the compensator already written in the form (27). At Step 3, the compensator is updated by adding a dynamics of the form (29), where η 1i (ỹ) and η 2i (ỹ) are computed in order to satisfy the conditions on the coefficients Γj j−i and Θi−1 respectively given in Theorem 2, as a consequence the algorithm cannot end at step exit if a compensator exists.…”

“…In the early 80's, the transformation of a nonlinear system into a linear canonical observer form was obtained and geometric conditions were derived in [22]. This so-called static solution has been widely investigated both in the continuous and discrete time context (see for example the early works of [6], [23], [21], [33], right to [13], [12], [29], [5], [14], [24], [31], [30], [32] where more general transformations were considered for continuous time systems, and to [26], [11], [37] for continuous systems with delay; [25], [4], [20], [15], [19], [9], [27], [10] deal with discrete-time and sample-data systems). Whenever those conditions are not fulfilled, the problem may be still solvable by using a dynamic compensator.…”