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.

Robust state estimation for states evolving on compact manifolds is achieved by employing a point-mass filter. The proposed implementation emphasizes a sane treatment of the geometry of the problem, and advocates separation of the filtering algorithms from the implementation of particular manifolds.

This paper examines an index reduction method for linear time-invariant differential algebraic equations, with uncertainty in the equation coefficients. When the bottom block of a block upper triangular leading matrix contains no elements that can be distinguished from zero, the natural action to take is to replace all numbers in the block by exact zeros, and then proceed with index reduction by differentiation. Conditions are given under which zeroing of an uncertain small block gives a small deviation in the solution.  

Type 2 diabetes is characterized by insulin resistance of target organs, which is due to impaired insulin signal transduction. The skeleton of signaling mediators that provide for normal insulin action has been established. However, the detailed kinetics, and their mechanistic generation, remain incompletely understood. We measured time-courses in primary human adipocytes for the short-term phosphorylation dynamics of the insulin receptor (IR) and the IR substrate-1 in response to a step increase in insulin concentration. Both proteins exhibited a rapid transient overshoot in tyrosine phosphorylation, reaching maximum within 1 min, followed by an intermediate steady-state level after approximately 10 min. We used model-based hypothesis testing to evaluate three mechanistic explanations for this behavior: (A) phosphorylation and dephosphorylation of IR at the plasma membrane only, (B) the additional possibility for IR endocytosis, (C) the alternative additional possibility of feedback signals to IR from downstream intermediates. We concluded that (A) is not a satisfactory explanation, that (B) may serve as an explanation only if both internalization, dephosphorylation, and subsequent recycling are permitted, and that (C) is acceptable. These mechanistic insights cannot be obtained by mere inspection of the datasets, and they are rejections and thus stronger and more final conclusions than ordinary model predictions.

Methods for index reduction of general nonlinear differential-algebraic equations are generally difficult to implement due to the recurring use of functions defined only via the implicit function theorem. By adding structure to the equations, these implicit funcitons may become possible to implement. In particular, this is so for the quasilinear and linear time-invariant (LTI) structures, and it turns out that there exists an algorithm for the quasilinear form that is a generalization of the shuffle algorithm for the LTI form in the sense that, when applied to the LTI form, it reduces to the shuffle algorithm. For this reason, the more general algorithm is referred to as a quasilinear shuffle algorithm. One can then say that the LTI form is invariant under the quasilinear shuffle algorithm, and it is expected that the algorithm can be fruitfully tailored to take care of the structural information in any such invariant form. In this paper a class of forms ranging from quasilinear to LTI DAE is searched for forms that are invariant under the quasilinear shuffle algorithm, and it is suggested that this kind of survey be extended to a more complete mapping between index reduction algorithms and their invariant forms.

When massaging differential-algebraic equations during an index reduction process, one usually keeps track of structural zeros in order to be able to detect when non-differential equations appear. However, in a numerical setting structural zeros may turn out as small numbers, impossible to distinguish from zero when numerical precision is taken into account.Such coefficients are referred to as non-structural zeros. This note aims to provide a better understanding of the ad hoc procedure of replacing those and other small numbers by structural zeros, at critical stages in the index reduction process.

Den kvasilinjära formen av differential-algebraiska ekvationer är både en mycket allmängiltig generalisering av den linjära tidsinvarianta formen, och en form som visar sig lämpa sig väl för indexreduktionsmetoder som vi hoppas ska komma att bli både praktiskt tillämpbara och väl förstådda i framtiden.

Kuperingsalgoritmen (engelska: the shuffle algorithm) användes ursprungligen för att bestämma konsistenta initialvillkor för linjära tidsinvarianta differential-algebraiska ekvationer, men har även andra tillämpningar, till exempel det grundläggande problemet numerisk integration. I syfte att förstå hur kuperingsalgoritmen kan tillämpas på kvasilinjära differential-algebraiska ekvationer som inte låter sig analyseras utifrån mönstret av nollor, har problemet att förstå singulära perturbationer i differential-algebraiska ekvationer uppstått. Den här avhandlingen presenterar en indexreduktionsmetod där behovet framgår tydligt, och visar att algoritmen inte bara generaliserar kuperingsalgoritmen, utan även är ett specialfall av den mer allmänna strukturalgoritmen (engelska: the structure algorithm) för att invertera system av Li och Feng.

Ett kapitel av den här avhandlingen söker av en klass av ekvations-former efter former som är mindre generella än den kvasilinjära, men som en algoritm lik vår kan anpassas till. Det visar sig att indexreduktionen ofta förstör strukturella egenskaper hos ekvationerna, och att det därför är naturligt att arbeta med den mest allmänna kvasilinjära formen.

Avhandlingen innehåller också några tidiga resultat gällande hur perturbationerna kan hanteras. Huvudresultaten är inspirerade av den modellering i skilda tidskalor som görs i teorin om singulära perturbationer (engelska: singular perturbation theory). Medan teorin om singulära perturbationer betraktar inverkan av en försvinnande skalär i ekvationerna, betraktar analysen häri en okänd matris vars norm begränsas av en liten skalär. Resultaten är begränsade till linjära tidsinvarianta ekvationer av index inte högre än 1, men det är värt att notera att index 0-fallet självt innebär en intressant generalisering av teorin för singulära perturbationer för ordinära differentialekvationer.

The quasilinear form of differential-algebraic equations is at the same time both a very versatile generalization of the linear time-invariant form, and a form which turns out to suit methods for index reduction which we hope will be practically applicable and well understood in the future.

The shuffle algorithm was originally a method for computing consistent initial conditions for linear time-invariant differential algebraic equations, but has other applications as well, such as the fundamental task of numerical integration. In the prospect of understanding how the shuffle algorithm can be applied to quasilinear differential-algebraic equations that cannot be analyzed by zero-patterns, the question of understanding singular perturbation in differential-algebraic equations has arose. This thesis details an algorithm for index reduction where this need is evident, and shows that the algorithm not only generalizes the shuffle algorithm, but also specializes the more general structure algorithm for system inversion by Li and Feng.

One chapter of this thesis surveys a class of forms of equations, searching less general forms than the quasilinear, to which an algorithm like ours can be tailored. It is found that the index reduction process often destroys structural properties of the equations, and hence that it is natural to work with the quasilinear form in its full generality.

The thesis also contains some early results on how the perturbations can be handled. The main results are inspired by the separate timescale modeling found in singular perturbation theory. While the singular perturbation theory considers the influence of a vanishing scalar in the equations, the analysis herein considers an unknown matrix bounded in norm by a small scalar. Results are limited to linear time-invariant equations of index at most 1, but it is worth noting that the index 0 case in itself holds an interesting generalization of the singular perturbation theory for ordinary differential equations.