Control of distributed-parameter systems using normal forms

In control engineering and systems theory, the temporal behavior of technical systems is analyzed and methods are developed to manipulate this behavior in a desired way. This is expediently done on the basis of mathematical models of the underlying (physical) processes. The models are usually given in the form of differential equations, which can be used to describe the temporal evolution of the system. Within such models, it is often sufficient to consider only the evolution of a finite number of quantities, such as the position and speed of a vehicle or the mean temperature of a workpiece. In control engineering, such systems are called lumped parameter systems (LPS). They are described by ordinary differential equations (ODEs). In other applications, it is necessary not only to consider the time dependence of the system variables, but also to explicitly include their spatial dependence in the model, for example when the temperature distribution in a workpiece or the traffic density along a road is of interest. Such systems are called distributed parameter systems (DPS). They are the subject of this research project. To describe DPS, one needs partial differential equations (PDEs), which are much more demanding than ODEs with respect to the underlying mathematics. Therefore, it is often attempted to approximate DPS by LPS, e.g. by considering the spatially dependent quantities only at certain points. However, essential properties of the systems may be lost in the process. Moreover, this approach leads to huge models with a large number of system variables, which are often only manageable by numerical methods. The approach pursued in the actual research project retains the original description of the DPS by PDEs. The focus is on so-called normal or canonical forms, which result from equivalent transformations of the system equations and usually allow a particularly simple analysis of the system properties and solution of design tasks. Based on these normal forms, controllers are designed to appropriately influence the system behavior. Furthermore, observers are designed to reconstruct the unmeasured system variables from the measured quantities. For LPS, both the normal forms and the associated analysis and design methods belong to the standard repertoire of control engineering; for DPS, they have been intensively developed for some time. In particular, the backstepping method and the flatness-based design are used in the research project. The goal is to understand these methods even better than before and thus to open them up for new classes of DPS, not least by exploiting their similarities and differences. Thereby, a focus lies on the elaboration of systematic, easily applicable design algorithms.

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