# Differential-algebraic equations and matrix-valued singular perturbation

Abstract: With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero.The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions.Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations.While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly.One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution.One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

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