Global properties of the triangular systems in the singular case

Abstract

We introduce a new class of the triangular (multi-input and multi-output) control systems, of O.D.E., which are not feedback linearizable, and investigate its global behavior. The triangular form introduced is a generalization of the classes of triangular systems, considered before. For our class, we solve the problem of global robust controllability. Combining our main result with that of [F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag, A.I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control 42 (1997) 1394-1407], we obtain a corollary on the global discontinuous sampled stabilization (an example showing that global smooth stabilization can be irrelevant to the singular case is considered). To prove our main result, we apply a certain "back-stepping" algorithm and combine the technique proposed in [V.I. Korobov, S.S. Pavlichkov, W.H. Schmidt, Global robust controllability of the triangular integro-differential Volterra systems, J. Math. Anal. Appl. 309 (2005) 743-760] with solving a specific problem of global "practical stabilization" by means of a discontinuous, time-varying feedback law.

Global uniform input-to-state stabilization of large-scale interconnections of MIMO generalized triangular form switched systems

“…, v n ] T in R m = R m 1 +···+m n to the system (7) defined by (18)- (20). For each t ∈ R we denote the transformation that is inverse to (17) …”

“…Later, in order to obviate this restriction, some authors [1,2,17,21,27,28,34,35] focused on more general "singular case", when the above-mentioned "input-output maps" are not necessarily invertible and the system is neither flat nor feedback linearizable. On this way efficient methods of local stabilization based on homogeneity properties were developed not only for triangular forms [2,4] but also for some cases of non-triangular forms [3].…”

“…On this way efficient methods of local stabilization based on homogeneity properties were developed not only for triangular forms [2,4] but also for some cases of non-triangular forms [3]. This motivated to introduce and to explore the concept of the so-called "generalized triangular form" (GTF) [17,27], which is defined by the only assumption that f i (x 1 , . .…”

“…The following three reasons justify these efforts: first, there are physical examples of the triangular systems in the singular case [17,29], second the backstepping approach appears to be applicable to the triangular form systems with singular inputoutput links [17,21,27,34,35], and third some coordinate-free criteria which extend the Respondek-Jakubczyk result were obtained in some important cases of triangular systems with uncontrollable linearization [5,28]. This promises a way to extend most results of the above-mentioned classical theory but such a theory has not been completed yet.…”

Stepwise synthesis of constrained controls for single input nonlinear systems of special form

“…The first such class of systems called "the class of triangular systems" was introduced in the paper [16], where the feedback linearization was given. In the paper [17] global properties of the triangular systems in the singular case is considered. In the present paper we introduce the new class of nonlinear systems called "the class of staircase systems" which are mapped on the systems (1.5) and give the corresponding changes of variables (Section 7).…”