His main interest is in feedback stabilization of controlled dynamical systems under various aspects-linear and nonlinear, dynanuc, output, with oarametric or dvnamic uncertaintv ,. I [8] K. Nam and A. Arapostathis, "A model reference adaptive control scheme for pure feedback nonlinear systems, "

This paper gives a systematic way to design time-variant feedback control laws for a class of controllable non-linear systems. This class contains a lot of systems which cannot be stabilized via a time-invariant feedback control law.The interest of this work lies in the design method since a general existence result is already available. The techniques employed here are basic: they mainly involve classical Lyapunov analysis.

To cite this version:Jean-Baptiste Pomet. A differential geometric setting for dynamic equivalence and dynamic linearization.

However, it is straightforward to verify that there exist constants u1 > 0 and u2 > 0 such that we can write 01(1111(.)11; + ll.r(.)ll; + ~~llE~(.)ll;, and moreover, c0[11.1-011' + do + E:=, TJ,) 5 n[11.r011' + E,"=, d,]. Hence, by combining these inequalities with (4.14), it follows that 111((.)112' + ll.r(.)[lg + E,"=, llEJ(.)llg 5 m/fl1[11.~011~ + E,"=, d,]. That is, condition iii) of Definition 2.2 is satisfied. Conditions ii) and iv) of Definition 2.2 follows directly from the stability of the matrix P. Thus, we can now conclude that the system (2.1), (2.3) is absolutely stabilizable via the control (4.4). 0 Remark: The above theorem gives a necessary and sufficient condition for absolute stabilizability in terms of the solution to a corresponding H m control problem. It is well known that such an H m control problem can be solved in terms of two algebraic Riccati equations; e.g., see [lo]. The following corollary is an immediate consequence of the above theorem. Corollary 4.1: If the uncertain system (2.1), (2.3) satisfies Assumptions 4.1 t 4. 6) and is absolutely stabilizable via nonlinear control, then it will be absolutely stabilizable via a linear controller of the form (4.4). E;=, llEa(-)ll;) 5 (1 / 2 C l)lI4.)lK + ;ll.4.)ll; + Ilso~.

Abstract. This paper considers control a ne systems in R 4 with two inputs, and gives necessary and su cient conditions for dynamic feedback linearization of these systems with the restriction that the \lin-earizing outputs" must be some functions of the original state and inputs only. This also gives conditions for non-a ne systems in R 3 .