Control and simulation of hybrid systems

Microbial metabolism and growth under conditions of starvation modelled as the sliding mode of a differential inclusion

“…By contrast, if the nutrient concentration lies below a critical threshold, the crossing condition ( 25) is not met and the system is prevented from re-entering S + . To establish that a sliding mode ensues, we turn to the attracting sliding mode conditions (6). These take on the following form for X ∈ Σ:…”

“…Filippov and Utkin pioneered the mathematical treatment of PWS systems [2,3], which have subsequently proven to be invaluable in engineering and biological applications. In particular, PWS systems arise in control theory [3,4] and the study of complementary and hybrid systems [5,6], as well as in several other applications [7][8][9]. In biology, genetic regulatory networks have profitably been treated as PWS systems [10][11][12][13][14].…”

Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach

“…The former is not hard to do in practical cases (such as those arising when the discontinuity naturally arises because of the presence of a sign-function), and it has the advantage of greatly simplifying the theory, but typically leads to very stiff differential equations to solve and a hard problem to tackle numerically (see [27,28,29]). The latter technique (sigmoid blending) was introduced in [2,3], and also re-derived more recently in [12] from the point of view of complementarity systems (see also [19,30,33]). In the end, this technique gives a vector field on the intersection, call it f B , by forming a bilinear interpolant amongst the four vector fields and then imposing the orthogonality conditions:…”

“…Systems with discontinuous right-hand sides appear pervasively in applications of various nature (see, e.g. [5,8,18,19,23,24,30,33]). For a sample of references in the context of control, see e.g.…”

Modeling, simulation, sensitivity analysis, and optimization of hybrid systems

“…However, we note that the phenomenon of having an infinite number of transitions occurring in a finite amount of time has also been studied extensively. Sometimes called "chattering" behavior, this phenomenon of having infinitely fast mode switches is also called sliding in variable structure systems and relay control systems [Utkin 1992;Malmborg and Bernhardsson 1997]. The term Zeno was used to characterize a physical model of a bouncing ball in which the ball covers a finite distance in a finite time but with an infinite number of transitions [Galán and Barton 1998].…”