State Estimation for Non-linear Fully-implicit, Index-1 Differential Algebraic Equation Systems
01 Jul 2020-pp 743-748
TL;DR: A linear transformation of the mass matrix is proposed which enables us to find candidate algebraic states for the fully–implicit index-1 DAE system and is relatively simpler than constructing the transformation matrices in the Weierstrass–Kronecker canonical form.
Abstract: The Kalman filter and its variants have been developed for state estimation in semi-explicit, index-1 DAE systems in current literature. In this work, we develop a method for state estimation in non-linear fully-implicit, index-1 differential algebraic equation (DAEs) systems. In order to extend the Kalman filtering techniques for fully-implicit index-1 DAE systems, in the correction step we convert the fully-implicit DAE into a system of ordinary differential equations (ODEs). This is achieved by the index reduction of DAE using the method of successive differentiation of algebraic equations. This is a challenging problem as the fully-implicit DAE system does not contain explicit algebraic states unlike in the semi-explicit case. In this work, we propose a linear transformation of the mass matrix which enables us to find candidate algebraic states for the system. This transformation on the mass matrix is relatively simpler than constructing the transformation matrices in the Weierstrass-Kronecker canonical form. We illustrate our proposed method with two examples, a linear and a non-linear fully-implicit index-1 DAE system.
Citations
More filters
10 Jun 2005
TL;DR: This work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher, and the classical ERK methods.
Abstract: During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.
Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
4
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher
U n and Jonhson [94]; and the classical ERK methods. The order p of
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.
665 citations
References
More filters
01 Apr 1976
1,524 citations
"State Estimation for Non-linear Ful..." refers background in this paper
...Kalman filter (KF) is an optimal estimator for linear dynamic systems with process and sensor uncertainties [5], [6]....
[...]
08 May 2002
TL;DR: In this article, a generalisation of the unscented transformation (UT) which allows sigma points to be scaled to an arbitrary dimension is described. But the scaling issues are illustrated by considering conversions from polar to Cartesian coordinates with large angular uncertainties.
Abstract: This paper describes a generalisation of the unscented transformation (UT) which allows sigma points to be scaled to an arbitrary dimension. The UT is a method for predicting means and covariances in nonlinear systems. A set of samples are deterministically chosen which match the mean and covariance of a (not necessarily Gaussian-distributed) probability distribution. These samples can be scaled by an arbitrary constant. The method guarantees that the mean and covariance second order accuracy in mean and covariance, giving the same performance as a second order truncated filter but without the need to calculate any Jacobians or Hessians. The impacts of scaling issues are illustrated by considering conversions from polar to Cartesian coordinates with large angular uncertainties.
1,122 citations
10 Jun 2005
TL;DR: This work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher, and the classical ERK methods.
Abstract: During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.
Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
4
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher
U n and Jonhson [94]; and the classical ERK methods. The order p of
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.
665 citations
TL;DR: A number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form ${\bf F(t, t, y, y') = {\bf 0}$.
Abstract: This paper outlines a number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form ${\bf F}(t,{\bf y},{\bf y}') = {\bf 0}$. Problems which can be written in this general form include standard ODE systems as well as problems which are substantially different from standard ODE'S. Some of the differential/algebraic systems can be solved using numerical methods which are commonly used for solving stiff systems of ordinary differential equations. Other problems can be solved using codes based on the stiff methods, but only after extensive modifications to the error estimates and other strategies in the code. A further class of problems cannot be solved at all with such codes, because changing the stepsize causes large errors in the solution. We describe in detail the causes of these difficulties and indicate solutions in some cases.
497 citations