Comments on “Exact Output Regulation for Nonlinear Systems Described by Takagi–Sugeno Fuzzy Models”

Abstract

This note considers the work entitled "Exact Output Regulation for Nonlinear Systems Described by Takagi-Sugeno Fuzzy Models" published in IEEE Trans. on Fuzzy Systems [1], where the authors try to provide a fuzzy approach to the well known problem of exact output regulation [2] via a dynamic implementation of the nonlinear mapping between the state space and the steady-state zero error manifold. Two problems invalidate this methodology: 1) The results do not depend on the fuzzy structure as claimed by the authors; 2) The results have been obtained without properly taking into account the time-derivative of the aforementioned nonlinear mapping. Example B in [1] is revised and corrected to illustrate these issues.

A practical solution to implement nonlinear output regulation via dynamic mappings

“…Solving nonlinear partial differential equations such as ( 16) -( 17) to find the steady-state nonlinear mappings π(w) and γ(w) is a difficult task, especially for high-order systems; moreover, even if these mappings are found, they might be too involved for real-time implementation. The methodology in [24], corrected in [29], tackles this problem by relaxing the requirement of finding an explicit static mapping π(w); instead, implicit dynamic mappings are used. This novel methodology -depicted in Figure 1 -is hereby extended to the error-feedback case.…”

“…Feasibility of LMIs ( 7) -( 9) implies the existence of a local controller with gain K and a local observer with gain G whose linearization is the same as that of ( 25); thus, condition 2 for the solution of the NLEFORP is also satisfied by defining F and H in the same way as in section 2 (linear case). Defining nonlinear mappings x = π(w) = Π(w)w and u = γ(w) = Γ(w)w, condition 3 for solving the NLEFORP translates into rewriting ( 16) -( 17 29) -( 30) can be straightforwardly obtained [24,29].…”

“…Scheme implementation: The proposed scheme requires solving equations ( 29) -(30) (a) for each entry of Γ and Π whenever an explicit solution is possible, or (b) forΠ ij at those entries (i, j) of Π where no explicit solution can be found: these are dynamically implemented, i. e., they are obtained on-line via integration of the differential equations that describe them [29].…”

“…Problem statement: Solving a nonlinear output regulation problem requires dealing with nonlinear partial differential equations, whose analytical solution -if availablemight be impossible to obtain [21]; this has precluded real-time applications of this theory [27,33]. Several works aimed to tackle this problem via Takagi-Sugeno models, but the solutions thus found remained approximate [23,3]; some others converted some of the partial differential equations into ordinary ones, thus easing the former requirement of finding explicit analytical solutions [24,29]. The latter has been referred to as dynamic mappings; it is only available for state feedback, which hinders its application to realtime setups where very often the full state is not available.…”