LPV approximation of distributed parameter systems in environmental modelling

Environmental systems often involve phenomena that are continuous functions not only of time, but also of other independent variables, such as space coordinates. Typical examples are transportation phenomena of mass or energy, such as heat transmission and/or exchange, humidity diffusion or concentration distributions. These systems are intrinsically distributed parameter systems whose description usually requires the introduction of partial differential equations (PDE). Therefore, their modelling can be quite complex, both for what concerns the model construction and its identification. Indeed, a typical approach for the simulation of such systems is the use of finite elements techniques. However, this kind of description usually involves a huge number of parameters and requires time-consuming computations while not being suited for identification. For this reason, such models are generally not suitable for control purposes. In many cases, however, the involved phenomena depend on the independent (space) variables in a smooth way, and for fixed values of the independent variables, input–output relations can be satisfactorily represented by linear time-invariant models. In such conditions, a possible alternative to PDE consists in representing the physical system with a Linear Parameter Varying (LPV) model whose parameters are functions of the independent variables. The advantage of this approach is the relatively simple model obtained, which is directly suitable for control purposes and can be easily identified from input–output data by means of classical techniques. Moreover, optimal identification schemes can be derived for such models, allowing the optimization of the number of measurements. This can be particularly useful in several environmental applications for which the cost of measurements represents a severe constraint. In this paper, the derivation of LPV models for the representation of distributed phenomena in environmental systems is discussed, and the issue of model uncertainty is addressed. In particular, it is shown that the derived models are linear in the parameters, and therefore classical methods for handling uncertainty are directly applicable. The proposed approach is illustrated by means of a simulated and a practical example concerning soil disinfestation by solarization.