# Unfalsified LTI models with bounded complexity in the behavioral framework

The notion of the most powerful unfalsified model (MPUM), introduced by Jan C. Willems in the late 1980s, led to a growing interest in exact model identification by researchers from the systems and control community. Willems defined the MPUM for a noise-free infinite time series as the least complex linear time-invariant (LTI) model explaining a given time series. The complexity of a model is defined by the pair (number of inputs, system order), where a model with a higher number of inputs is always considered a more complex model (lexicographic ordering). Willems proved that the MPUM always exists and is unique. Under certain assumptions, it can be shown that the MPUM coincides with the data generating system.

In real-world applications, however, time series are finite. Therefore, it is desirable to define the MPUM for a given finite time series. Specifically, we aim to develop systematically the MPUM for an observed finite time series and derive necessary and sufficient conditions for its existence and uniqueness. However, in the finite case, there are two crucial issues: i) the MPUM may not exist or be non-unique, and ii) due to the above definition of model complexity, the MPUM if exists always coincides with a finite dimensional autonomous system, as it has no inputs.

We tackle these complications by adopting an intuitive approach: we assume that the number of inputs is a priori known and we minimize the order in order to minimize the complexity of unfalsified model. In this setting, we have formulated necessary and sufficient conditions for the existence and uniqueness of the MPUM. Our initial findings suggest that, if we restrict the complexity of the model, we can identify the MPUM as a system with inputs. Furthermore, in the case of non-uniqueness of unfalsified models, it has been shown that every unfalsified model can be expressed as a sum of the MPUM and an autonomous model of bounded order. In other words, every unfalsified model must include the MPUM.

In real-life applications the data are usually noisy. The next step is to extend this study to noisy data set.

## References

- J. C. Willems, “From time series to linear system—Part II. Exact modelling,” Automatica, vol. 22, no. 6, pp. 675–694, 1986.
- J. C. Willems, P. Rapisarda, I. Markovsky, and B. De Moor, “A note on persistency of excitation,” Syst. Control Lett., vol. 54, no. 4, pp. 325–329, 2005.
- J. C. Willems, “Thoughts on system identification,” in Control of uncertain systems: modelling, approximation, and design. Springer, 2006, pp. 389–416.
- I. Markovsky, “On the behavior of autonomous Wiener systems,” Automatica, vol. 110, article no. 108601, 2019.