Remarks on Suucient Information for Adaptive Nonlinear Regulation

The present work is an attempt to design universal regulatorss for nonlinear systems under some minimal informations. It presents a controller which is a gen-eralisation of one proposed previously for the linear case, and based on the knowledge of a nite dimen-sionnal family of controllers one of which is known to be a stabilizing one. 1 Problem statement Consider a control system _ x = f(x; u) (1) with state x 2 IR n and input u 2 IR m. We assume that the complete state x is available for control (i.e. is measured and can be taken as the input of a controller), but we consider that the map f : IR n IR m ?! IR n itself is not known explicitly. For the sake of simplicity, we suppose that it is smooth. The objective is to design a dynamic controller _ = a certain function of and x u = a certain (other) function of and x (2) where is the state of this controller, such that this controller regulatess x in the sense that the closed-loop system (its state is (x;)) has the following property : for any initial condition, (x(t); (t)) is bounded and x(t) goes to zero. Of course, this is absolutely impossible without any information at all on the function f : if it were possible , the obtained controller would achieve the desired objective for any f, and, for example, would also work in the case f(x; u) = x, i.e. render the solutions of _ x = x bounded. An interesting question is : what information on f is necessary to achieve the desired objective ? First of all, to avoid absurd situations as the one mentioned above (f(x; u) = x), it is necessary to be certain that f is such that the objective could be achieved if it were known, i.e. there exists a controller achieving the objective at least for this precise f. It is however very doubtfull that the following information on f: f is smooth and there exists a controller of the form (2) which regulates x is suucient, i.e. it seems there most probably exists no controller of the form (2) satisfying the objective for any possible f satisfying these two properties. Note that this non-existence is only a conjecture. A more reasonable framework is the case where not only is it known that a …