Carlo Novara - Data-Driven Modeling, Filtering and Control

One of the main challenges of modern engineering applications is how to effectively handle the increasingly large amount of data and information available in todays complex dynamical systems []. Data can not only be used for mathematical modeling but also for monitoring, filtering and control.

For instance, it is often estimated that more than 75% of the costs associated to an advanced control project goes into accurate mathematical modeling []. For some real plants, some work might be dedicated to devise a model of the physics underlying the process, but nowadays most of the efforts are usually devoted to data-driven modeling, that is, experiment design and implementation, model learning from data and data-driven validation.

Within this framework, system identification, namely, the science of deriving dynamical models from input/output measurements has played a key role throughout the last 50 years. The traditional objective of system identification has been that of finding the model which best fits the data, so that model-based filters and controllers could be designed standing on the assumption that the identified mathematical description coincides with the real system (the so-called certainty equivalence principle).

Application-oriented system identification. Sincequoting G. Boxall models are wrong, some are useful [].

In any case, the model (and the related uncertainty region) which best fits the data might be a bad choice for model-based design, in which the loss function accounted for during the identification phase (typically, a norm of the data-fitting error) does not usually coincide with the final application-oriented cost to minimize (e.g., in control it is a measure of the mismatch between the desired closed-loop behavior and the achieved one). Nonetheless, robust design is safe but may lead in some cases to very conservative performance, depending on the size of the uncertainty set. Also, deriving the required uncertainty region model may be a difficult task, especially in the case of nonlinear system. Some additional knowledge about the specific domain or the model application is then necessary to deal with real-world systems in an effective way.

Identification for control (I4C). Based on the above observations, in the 1990s, a few leading experts in system identification started to analyze the interplay between identification and control, see, e.g., the plenary [] at the 1991 IFAC Symposium on System Identification. The new field originated from such studies, called I4C, proposes a change of perspective, in which modeling is seen as a design problem. In other words, one could think of designing the identification process in order to find a model (and an uncertainty set) which does not necessarily fit the data at best but which obtains satisfactory closed-loop performance when the (robust) model-based controller is applied to the real system, byat the same timeminimizing the level of conservativeness.

An alternative yet appealing approach is to map the data directly onto the controller parameters. From now on, this approach will be referred to as direct data-driven control (D3C) []. D3C can be seen as a subfield of I4C, in which the identification process is designed so that the system to identify from data is directly the controller. An interesting feature of this approach is that the control design stage is automatically incorporated into the modeling part, thus simplifying the procedure.

The overall I4C rationale is depicted in (solid lines). Unlike standard identification and control, the modeling step is control-oriented, in which data are mapped onto the model which is best suited for the control design. In the same figure, the D3C alternative is illustrated using dashed lines.

Figure 1.1 The I4C design procedure based on experimental data (solid lines). First, data are mapped onto a model of the system, by taking the control objective into account. Then, the controller is designed based on the identified model and, possibly, an estimate of the uncertainty set. If a D3C approach is adapted, the data are directly mapped onto the control parameters (dashed lines)

Identification for filtering (I4F). Similar to what happens in I4C, also in the context of I4F, two main approaches can be distinguished: the two-step approach, where first a model of the system of interest is identified from data and then a filter/observer is designed from the identified model; the direct approach, where the filter/observer is directly designed from the available data. As discussed in [].

The direct approach can overcome these issues, allowing the design of optimal filters for linear and nonlinear systems []. Another advantage of the direct approach is that it is in general simpler.

In summary, in I4F, we can say that the direct approach may be more convenient than the two-step approach in many situations. A similar statement does not hold in I4C, where the direct approach may present some advantages with respect to the two-step approach, but, in some situations, the latter may be more convenient (see the discussion in Section 1.2).

The overall I4F rationale is depicted in .

Figure 1.2 The I4F design procedure based on experimental data (solid lines). First, data are mapped onto a model of the system, by taking the control objective into account. Then, the filter/observer is designed based on the identified model and, possibly, an estimate of the uncertainty set. If a direct approach is adapted, the data are directly mapped onto the filter/observer (dashed lines)