On the computation of local invariant sets for nonlinear systems

Abstract

The importance of invariant sets in control is due to the fact that they define a region of the space where stability, and potentially asymptotic convergence, are assured. For this reason many control design strategies are related to the computation of an invariant set. This is particularly the case for receding horizon based strategies as model predictive control. This paper presents a method for computing a convex invariant set for nonlinear systems. Using properties of D.C. functions, which are functions that can be expressed as difference of two convex functions, and the fact that any continuous nonlinear function can be expressed as D.C. functions, or, at least, well approximated by them, we propose an algorithm for computing a polyhedral invariant set in which no global optimization problem has to be solved. The proposed strategy is guaranteed to provide a non-empty local invariant set provided the nonlinear system is locally stable.

Computation of the Largest Constraint Admissible Set for Linear Continuous-Time Systems with State and Input Constraints

“…The basic idea is to study the convergence rate of the unconstrained system to the origin. A similar approach was used in Fiacchini et al (2007) to compute λ-contractive polytopic sets for linear and nonlinear discrete-time systems. Basically, the proof of the following lemma represents a specialized variant of the proof of the well-known Lyapunov criterion for asymptotic stability (see, e.g., (Vidyasagar, 2002, pp.…”

Invariant sets computation for convex difference inclusions systems

“…A possibility to obtain the initial guess is to compute a contractive set for a system which is a local approximation, possibly linear, of the CDI one. Given a contractive set Ω for a linear approximation of the CDI (or nonlinear) system, there exists β > 0 such that Ω L = βΩ is contractive for the CDI one, under certain differentiability assumptions (see [17] for an analogous result). Standard algorithms can be employed, see for instance [3,5] and [7], to obtain Ω. Alternatively, an LDI system, local extension of the CDI one, can be computed.…”

“…Methods to obtain invariant polytopes are proposed for saturated systems, [14] and for Lur'e systems, [15]. The computation of invariant polytopes for general nonlinear systems is discussed in [16], using interval arithmetic, and in [17,18], employing properties of DC functions. The work [19] proposes approximations of the minimal invariant set for quantized systems.…”

Computing Robust Forward Invariant Sets of Multidimensional Non-linear Systems via Geometric Deformation of Polytopes

“…In contrast, the body of literature on nonlinear systems is relatively limited, where most of the work focuses on discrete-time systems or is restricted to certain classes of nonlinear systems. Examples include [39], where the authors study robust invariance in discrete-time nonlinear systems by lifting the feedback operation to the space of sets, and [40], [41], where the authors compute control invariant sets for such systems using differences of convex functions. For continuoustime nonlinear systems, the focus has primarily been on polynomial dynamics.…”