Stability analysis and controller synthesis for hybrid dynamical systems

Abstract

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.

Scheduled Controller Design for Systems With Varying Sensor Configurations: A Frequency-Domain Approach

“…condition is verified by means of Fig. 9, which represents the polar plot of G(s) as in (13), showing that ρRe(G(jω)) > −1, for ρ > 1 is met for all ω ∈ R with sufficient margins to account for plant uncertainty. In this respect, note that the user can influence Re(G(jω)), and thus condition (II) in Theorem IV.1, by means of the design of C 0 (s) and C 1 (s).…”

“…9. Polar plot of G(jω) as in (13) in which the dashed line represents −1. condition is verified by means of Fig.…”

“…Compared to our data-based design and analysis conditions, in [12] parametric models are required whereas bilinear matrix inequalities (BMIs) need to be solved which not only result in a suboptimal control solution, it also does not meet the standards in industry. Also many designs for the control of switched systems, see, e.g., [2], [13], [14] require accurate parametric models and LMI-based designs that are not so easily embraced by control engineers in industry. Consequently, there is often a need for design tools for switched systems, such as the class of dynamical systems with position-dependent switching of sensor configurations studied in this paper, that bridge the gap between hybrid systems theory (with formal stability guarantees) and industrial control practice, commonly exploiting frequency-domain design tools and non-parametric models such as measured FRF descriptions of the motion system dynamics.…”

Piecewise affine systems modelling for optimizing hormone therapy of prostate cancer

“…Based on the model constructed above, the optimal scheduling is computed through model predictive control (MPC; see Bemporad & Morari 1999;Heemels et al 2010 for details). For simplicity, we assume that we can observe the internal state x.…”

Hybrid Modelling in Biology: a Classification Review

“…Reviews dedicated to some specific applications of hybrid models can also be found, for example in biological networks modelling [12], in cancer modelling [10,21,30,40], and more specifically tumour growth [24,29], tumour immunology [41], brain cancer [42] or angiogenesis [16]. Many special issues or reviews are also dedicated to the more specific field of hybrid systems [2,17,31,39] which can be considered as a subclass of hybrid models mainly (but not only) related to the field of automatic control.…”