On the construction of static stabilizers and static output trackers for dynamically linearizable systems, related results and applications

It is shown that the property of dynamic linearizability, to be understood as linearizability by means of the dynamic extension algorithm, implies the existence of static, possibly time varying, control laws yielding asymptotic output tracking with arbitrary speed of convergence and asymptotic stabilization with a computable bound on the region of attraction. Similar results hold for systems which are only input/output linearizable by means of dynamic state feedback, provided that the inverse dynamics possess certain stability properties. Applications of these results to the problem of regional stabilization of a VTOL aircraft is considered, together with the tracking problem for a class of flexible joints robots. Moreover, a novel parameterization for flexible joint robots is also proposed.

I. Introduction

Since the seminal contribution of R. Brockett [4], the problem of feedback linearization of nonlinear systems has attracted a wide attention in the nonlinear geometric control community, see [17], pp. 205–206] and [11], pp. 460–461 for a brief yet complete account of the main contributions. To date the problem can be regarded as completely solved, i.e. necessary and sufficient conditions for the solvability of the (local or global) feedback linearization problem have been derived. At the same time, the problem of Input/Output linearization has been addressed and solved. An important observation resulting from the general theory is that, for multi-input multi-output systems, the use of dynamic state feedback may result beneficial in the solution of linearization problems, i.e. systems which are not linearizable by means of static feedback transformations may be linearizable provided dynamic feedback transformations are allowed. A notion which is strongly related to that of feedback linearization is the notion of flatness, introduced by M. Fliess and co-workers in [8]. In fact, as noted in [16], Corollary 2, a flat system is linearizable by means of a (possibly dynamic) feedback transformation. The converse statement is also true: if a system is feedback linearizable then the linearizing outputs are flat outputs [16]. Finally, the notion of prime systems [14], [1] has been introduced to characterize those systems which are linearizable by means of a feedback transformation together with an output transformation. It is worth noting that the main interest in obtaining, after a feedback transformation, a linear system stands in the possibility to apply, to such transformed system, classical linear design tools. These in turn may be used to guarantee that the output tracks certain pre-assigned reference signals (while the state remains bounded) or to achieve local asymptotic stability of a certain equilibrium of the closed-loop system while allowing the computation of a bound on the region of attraction.