Calculating Positive Invariant Sets: A Quantifier Elimination Approach

Abstract

A Lyapunov-based approach for calculating positive invariant sets in an automatic manner is presented. This is done using real algebraic geometry techniques, which are summed up under the term quantifier elimination (QE). Using available tools, the approach presented yields an algorithmizable procedure whose conservatism only depends on the initial choice for the Lyapunov candidate function. The performance of the approach is illustrated on a variant of the Rössler system and on the Lorenz-Haken system.

Algebraic Stability Analysis of Particle Swarm Optimization Using Stochastic Lyapunov Functions and Quantifier Elimination

“…A very powerful method to achieve such a description is the so-called quantifier elimination (QE). Before we continue with the determination of the convergence set using this technique, let us briefly introduce some necessary notions and definitions, see also [49,50]. Equations ( 32) and ( 36) can be generalized using the prenex formula with Q i ∈ {∃, ∀} .…”

“…For obtaining a proof of stability for the system's motion, while also attenuating the conservatism of the Lyapunov approach to PSO stability, we propose in this paper an analysis consisting of two steps. First, we employ stochastic Lyapunov functions [23,29,38,52] and subsequently determine the convergence set by quantifier elimination [49,50]. A main feature of such an analysis is that it is an algebraic approach relying entirely on symbolic computations.…”

Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

“…The usage of quantifier elimination for estimation respective approximation of positively invariant sets has been reported in [9,10], where the Lorenz system, the Lorenz-Haken system and a version of the Rössler system have been investigated. Now, we will apply this approach to compute bounds on the Lorenz family (1).…”

Convergence analysis of particle swarm optimization using stochastic Lyapunov functions and quantifier elimination

“…To attenuate the conservatism of the Lyapunov approach to PSO stability, we propose in this paper an analysis consisting of two steps. First, we employ stochastic Lyapunov functions [17,21,27,39] and subsequently determine the convergence set by quantifier elimination [37,38]. It is shown that the convergence set we obtain by such a procedure gives a reevaluation and extension of previously know stability regions for PSO using a Lyapunov approach under stagnation assumptions.…”

“…A very powerful method to achieve such a description is the so-called quantifier elimination (QE). Before we continue with the determination of the convergence set using this technique, let us briefly introduce some necessary notions and definitions, see also [37,38]. Eqs.…”