Implicit integral equations for modeling systems with a transport delay

In this chapter, we present a particular class of transport delay systems (e.g. systems involving transportation of material), in which the delay is defined through an implicit integral equation. To illustrate the practical interest of this class, experimental use of such models is presented for two different examples of physical systems, both from the field of automotive gasoline engines (specifically, exhaust gas recirculation and exhaust catalyst thermal dynamics). We also discuss related control challenges, together with some solutions. 1 Some motivations for investigating transport delay modeling Time-delay systems have been widely investigated in the past decade following the rise of telecommunications and network exchanges. Due to the practical relevance of such cases, this research effort has yielded a substantial number of monographs and studies devoted to time-varying delays [4, 31, 44]. In this field of research, the variability of the studied delay is usually unstructured. Another important class of delays consists in the ones arising from transportation of material. Prime examples of such physics-driven systems are mixing processes [35] for liquid or gaseous fluids, chemical reactors [17], automotive engine and exhaust line [21], heat collector plant [38], and blending in liquid or solid networks [10], to name a few. Despite this record, control oriented modeling and control design for transport delay systems is still an underdeveloped field. The varying delays are either represented by purely uncertain time-varying models or, in the worst case, by a constant Delphine Bresch-Pietri GIPSA-lab, Department of Automatic Control, 11 rue des Mathématiques, 38000 Grenoble, France e-mail: [email protected] Nicolas Petit MINES ParisTech, PSL Research University, CAS – Centre Automatique et Systèmes, 60 bd StMichel, 75006 Paris, France e-mail: [email protected]