Control of nonlinear time-varying systems

Abstract

Consider the time-varying nonlinear system of m. . , being g" vector fields on Rn+l.We give necessary and sufficient conditions for this system to be transformable to a time-invariant controllable linear system. In order to control the nonlinear system, we map to the linear system, choose a desired control there, and return to the nonlinear system by the inverse of the transformation.

Linearization and input-output decoupling for general nonlinear systems

“…(1. 2) As such the paper is a continuation of [7], where the necessary and sufficient conditions for (local) controlled invariance for affine systems were generalized to general nonlinear systems, thereby solving for instance the disturbance decoupling problem for these systems (see [8]). First we recall the framework used in these references.…”

“…Linearization ItisclearthatA,cA,cA,c .a-.Themaintheorem is now as follows [2,4,14]. (B) Rank afj/axj-' = pi for j 2 2 and rank aft/au =p,.…”

Flatness of two-input control-affine systems linearizable via a two-fold prolongation

“…The problem of static feedback linearization was solved by Brockett [1] (for a smaller class of transformations) and then by Jakubczyk and Respondek [12] and, independently, by Hunt and Su [9], who gave geometric necessary and sufficient conditions. The following theorem recalls their result and, furthermore, gives an equivalent way of describing static feedback linearizable systems from the point of view of differential weight.…”

State and feedback linearizations of single-input control systems

“…Problem 1 was completely solved by Krener [16] and Problem 2 partially by Brockett [4] for m = 1 and β constant. A generalization was obtained independently by Hunt and Su [11], Jakubczyk and Respondek [13], who gave necessary and sufficient geometric conditions in terms of Lie brackets of vector fields defining the system. Indeed, attach to Σ the sequence of nested distributions …”

“…The problem of feedback classification for linear systems Λ is to find linear state coordinates w = T x and linear feedback u = Kx + Lv that map Λ into a simpler linear systemΛ. It is a classical result of the linear control theory (see, e.g., [2], [14]) that any linear controllable system is feedback equivalent to the following Brunovský canonical form (single-input case): For a complete description and geometric interpretation of the Brunovský controllability indices we refer to the literature [2], [11], [12] , [13], [14], [25] and references therein.…”