Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron

Abstract

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.2010 Mathematics Subject Classification. Primary: 34A30, 37M25, 65K05.

A novel unified approach to invariance conditions for a linear dynamical system

“…When the optimal solution η * of (40) exists, then by Theorem 3. 35 we can claim that the Lorenz cone is an invariant set for the continuous system. Various celebrated SDO solvers, e.g., SeDuMi, CVX, and SDPT3, can be used to solve SDO problem (40).…”

Invariance under discretization for positive systems

“…Positivity of the system means that the nonnegative cone is invariant with respect to the dynamics of the system. In [23,24], the authors considered systems for which instead of the nonnegative cone other sets, like polyhedrons or ellipsoids, were used. Both continuous-time and discrete-time systems were studied, as well as constant step discretization.…”

Invariance Conditions for Nonlinear Dynamical Systems

“…Invariant set for discrete system is studied in [14], and an application to model predictive control is provided. The steplength threshold for preserving invariance of a set when applying a discretization method to continuous systems is studied in [10,12].…”