Differential Algebraic Relations Research Articles

Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability

Elimination of unknowns in a system of differential equations is often required when analyzing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from the literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available online as a Julia package.

Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation [Formula: see text] where the coefficients [Formula: see text] are rational functions in [Formula: see text] with coefficients in [Formula: see text]. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.

Application of the "stretching" function for solving electrical circuits with differential-algebraic relations

In the article the problem of an accurately and effectively solving complex electrical circuits was considered. Application of the Kirchhoff’s laws enables us to model such circuits by both differential and algebraic equations. The presented approach is based on the direct shooting method. To preserve the continuity of the state variables and consistency of the initial conditions, in the nonlinear optimization problem the "stretching" function was applied. The presented method can be used to carry out the circuits simulation in a computationally efficient manner. Streszczenie. W artykule poruszono problem dokladnego i skutecznego rozwiązywania zlozonych obwodow elektrycznych. Zastosowanie praw Kirchoffa umozliwia modelowanie takich obwodow zarowno przez rownania rozniczkowe jak i algebraiczne. Zaprezentowane podejście bazuje na bezpośredniej metodzie strzalow. Aby zapewnic ciąglośc zmiennych stanu oraz spojne warunki początkowe, w zadaniu programowanie nieliniowego zastosowano funkcje "rozciągającą". Zaprezentowana metoda moze byc wykorzystana w symulowaniu obwodow w sposob wydajny obliczeniowo. (Zastosowanie funkcji "rozciągających" do rozwiązywania obwodow elektrycznych z relacjami rozniczkowo-algebraicznymi)

Geometric Methods in Dynamical Systems Modelling: Electrical, Mechanical and Control Systems

The problems concerning dynamical systems modelling are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalized hypersurface) being the available space of the system. The constitutive space of the (differential) dynamical system implicitly encloses all the necessary information which is sufficient to select the solution space and the infinitesimal generator of the system which enables the design of the dynamic behavior of the system. The differential inclusions then appear in a natural way in the modelling of physical systems as implicitly written differential systems, where sets of differential algebraic relations describe the constitution of the system. The constructive definition of a differential dynamical system considered here has been obtained through generalization of the mathematical model of a non-linear electrical network extended next to lumped mechanical systems and control systems. The constructive approach to differential dynamical systems presented in this paper corresponds to the procedure commonly used when a mathematical model of a physical system is set up. The subbundles of the tangent bundles to manifolds being mathematical models of the configuration spaces of dynamical systems are the basic mathematical tools. A general iterative procedure for selecting the solution space of the dynamical system is proposed, which extends the formulation given in the references. Examples illustrating the considerations are given.

An Algorithm to Test Identifiability of Non-Linear Systems

This paper considers an approach for analyzing the identifiability of nonlinear controlled or uncontrolled dynamical systems. The method is based on the computation of the ideal containing the differential algebraic relations between the input and the output of the model, a such ideal is called input-output ideal.Our contribution consists in developing the corresponding algorithm in a symbolic computing language. This algorithm is based on differential algebra.

Geometric characterization of singular differential algebraic equations

Singular differential algebraic equations (differential inclusions) are considered from a geometric point of view. The differential algebraic equations are defined as implicitly written ordinary differential equations. They appear in a natural way in the modelling of physical systems, as sets of differential algebraic relations describing the constitution of the systems. The main problem considered is that of finding the solution space and the infinitesimal generator (the dynamic equation) for a given differential algebraic equation. This question is far from being trivial, due to the fact that the solutions of a general differential algebraic equation are restricted to a lower dimensional manifold embedded in the configuration space. In that case, the differential algebraic equation is said to be singular, or algebraic incomplete. It is proved that the solution space of the equation is the set union of all (maximal) invariant submanifolds of the configuration space. A general iterative procedure for sele...