Forward and Backward Shifts of Vector Fields: Towards the Dual Algebraic Framework

Weak reachability and controllability of discrete-time nonlinear systems: generic approach and singular points

“…However, unlike the continuous-time case, D π ∞ cannot be directly used for finding the singular points from which the system is not weakly reachable in k steps. The present paper builds upon the results of (Mullari, Kotta, Bartosiewicz, Pawluszewicz and Moog, 2017), in particular on D π ∞ . This vector space is the starting point to construct the matrix which helps to find the reachability singular points.…”

“…The generic accessibility property of non-linear discrete-time system can be characterized using the concept of autonomous variables, see e.g. (Aranda-Bricaire, Kotta and Moog, 1996), (Mullari, Kotta, Bartosiewicz, Pawluszewicz and Moog, 2017). The existence of an autonomous variables can be easily checked with the help of the vector space of 1-forms, denoted by H ∞ , and introduced in (Aranda-Bricaire, Kotta and Moog, 1996).…”

“…In the continuous-time case alternative accessibility condition (Conte, Moog, Perdon, 2007) is formulated in terms of strong accessibility distribution, that allows to find the reachability singular points. In the discrete-time case the space of vector fields D π ∞ (Mullari, Kotta, Bartosiewicz, Pawluszewicz and Moog, 2017) is the discrete-time analogue of the strong accessibility distribution. Computing of D π ∞ requires to define the backward shift operators acting on the vector fields in a manner that is consistent with the geometric meaning of shifting the vector field along the (discrete) trajectory of the control system (Rieger, Schlacher, 2011).…”

On the finiteness of accessibility test for nonlinear discrete-time systems

“…where x 1 and x 2 denote the rotational angle and the angular velocity of the coil, respectively. It is easy to see that system (14) is generically accessible (in 2 steps), and I M2 = x 1 (x 1 + T x 2 ) , shows that from the points of V(I M2 ) the system (14) is not accessible in 2 steps. Using Algorithm 2, we obtain…”

“…accessibility from almost any point). Various finite criteria exist for generic accessibility [3], [14], that require no more than n steps, where n is the state dimension. However, for pointwise accessibility, in general, it is not clear a priori how many steps are needed to distinguish between the accessibility singular points (from which the system is not accessible in any finite number of steps) and the points from which the system Mohammad Amin Sarafrazi: Rezvan complex, Motahari Sq., Motahari Blvd., 71868 is forward accessible in a finite number of steps.…”