The Structure of Small-Time Reachable Sets in Low Dimensions

1989 ◽  
Vol 27 (1) ◽  
pp. 120-147 ◽  
Author(s):  
Arthur J. Krener ◽  
Heinz Schättler
Keyword(s):  
Small Time ◽  
Reachable Sets ◽  
Low Dimensions

scholarly journals THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

2020 ◽  
Vol 6 (1) ◽  
pp. 71
Author(s):  
Mikhail I. Gusev
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear Control ◽  
Sufficient Conditions ◽  
Linearization Method ◽  
Small Time ◽  
Reachable Set ◽  
Reachable Sets ◽  
Time Interval ◽  
Initial State ◽  
Linearized System

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets.  The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero.  The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension.  The results of numerical simulation illustrate the theoretical conclusions of the paper.

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