Globale Oplossing voor Problemen in Systeemidentificatie en Modelreductie

Systems theory provides a mathematical framework for describing dynamical phenomena through the abstract notion of a ``system''. Such a system is typically described by differential (continuous-time) or difference (discrete-time) equations that govern the evolution of internal states, and their relationship to inputs and outputs, over time. The abstract notion of a system makes it applicable across a wide range of disciplines: systems may represent physical entities, such as industrial processes, biological networks, or mechanical structures, but equally abstract constructs in economics or sociology. The obtained mathematical models form the basis for analysis, simulation, prediction, and control, the latter of which refers to the manipulation of the system's inputs to achieve certain objectives (e.g., stabilization, trajectory tracking or disturbance rejection). This dissertation addresses two closely related topics: system identification, the task of inferring dynamical models from observed data, and model reduction, the approximation of a given high-order model by a model of lower complexity. Both problems naturally lead to non-convex optimization formulations, even when restricted to linear model classes. Consequently, the state of the art is dominated by iterative approaches that compute (at best) locally optimal solutions. While such methods are highly effective in engineering practice, they leave important questions unresolved: global optimality cannot be guaranteed, so one is never certain whether even better solutions exist, the outcome often depends strongly on the chosen initialization, and reproducibility is not always guaranteed. From a mathematical standpoint, these problems can therefore not yet be considered ``solved.'' In this dissertation, we develop a novel deterministic methodology that leads to globally optimal solutions. We specifically focus on the (weighted) least squares realization problem for identifying autonomous linear time-invariant models and the single-input single-output H2-norm model reduction problem. The key insight is that, for both problems, the first-order optimality conditions can be expressed in polynomial form, with their real common roots fully characterizing the set of (local) minimizers. By exploiting the partially linear structure, we reformulate the polynomial system as a multiparameter eigenvalue problem (MEP), which generalizes the standard eigenvalue problem by allowing more than one spectral parameter and removing the restriction of linearity in its parameter(s). The solutions of these MEPs, the so-called eigentuples, contain the globally minimizing solutions of the original problem. The proposed framework is powerful in that it provides a unified methodology for two problems that are closely related yet fundamentally distinct. The resulting formulations also represent a substantial advancement over the state of the art in globally optimal approaches, both in terms of improved computational efficiency as theoretical simplicity. Finally, the contributions are non-trivial: in particular, the role of Walsh's Theorem as an important building block in the development of globally optimal solution methods had not previously been recognized. The pursuit of global optimality, however, comes at a considerable computational cost. The efficiency of the proposed methodologies is governed by the difficulty of computing all common roots of the system of polynomial equations, or equivalently, the eigentuples of the resulting MEPs, the number of which grows rapidly with the problem size. For this reason, the proposed framework is best viewed as a theoretical benchmark, rather than a practical tool for large-scale applications. To better understand these computational limitations, we present an algebro-geometric analysis of the solution sets of the obtained MEPs. This comprehensive analysis, which forms our second main contribution, leads to novel insights into the solution set to the critical equations. Most notably, we establish reducedness of the solutions of the MEPs (i.e., each solution occurs with multiplicity one) and, through the Thom--Porteous formula, we derive exact solution counts.