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Analysis of Over and Underdetermined Nonlinear DifferentialAlgebraic Systems with Application to Nonlinear Control Problems
, 2000
"... We study over and underdetermined systems of nonlinear differentialalgebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems via a behaviour a ..."
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Cited by 21 (10 self)
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We study over and underdetermined systems of nonlinear differentialalgebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems via a behaviour approach. We give a general (local) existence and uniqueness theory and apply the results to nonlinear control problems. In particular, we study regularization by state or output feedback. The theoretical analysis also leads immediately to numerical methods for the simulation as well as the construction of regularizing controls.
Analysis and Numerical Solution of Control Problems in Descriptor Form
 Math. Control Signals Systems
, 1999
"... We study linear variable coefficient control problems in descriptor form. Based on a behaviour approach and the general theory for linear differential algebraic systems we give the theoretical analysis and describe numerically stable methods to determine the structural properties of the system like ..."
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Cited by 20 (13 self)
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We study linear variable coefficient control problems in descriptor form. Based on a behaviour approach and the general theory for linear differential algebraic systems we give the theoretical analysis and describe numerically stable methods to determine the structural properties of the system like solvability, regularity, model consistency and redundancy. We also discuss regularization via feedback. Keywords: descriptor systems, differentialalgebraic equations, sindex, regularization, feedback design AMS(MOS) subject classification: 93C50, 65L05, 34H05, 93B10, 93B11, 93B40 1 Introduction In this paper we study control problems of the form E(t) x = A(t)x +B(t)u + f(t); (1) y = C(t)x + g(t) (2) in a given interval [t 0 ; t f ], with initial condition x(t 0 ) = x 0 : (3) Here x is the state, u is the input, y is the output of the system. If we denote by C r ([t 0 ; t f ]; C n;` ) the set of rtimes continuously differentiable functions from the interval [t 0 ; t f ] to th...
Regularization of Linear Descriptor Systems with Variable Coefficients
 SIAM J. Matrix Anal. Applicat
, 1997
"... We study linear descriptor control systems with rectangular variable coefficient matrices. We introduce condensed forms for such systems under equivalence transformations and use these forms to detect, whether the system can be transformed to a uniquely solvable closed loop system via state or deriv ..."
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Cited by 14 (10 self)
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We study linear descriptor control systems with rectangular variable coefficient matrices. We introduce condensed forms for such systems under equivalence transformations and use these forms to detect, whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback. We show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system, plus some solution components that are constrained to be zero.
A New Look At Pencils of Matrix Valued Functions
, 1992
"... this paper we study matrix pencils ..."
NavierStokes equations as a differentialalgebraic system
, 1996
"... Nonsteady NavierStokes equations represent a differentialalgebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to ..."
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Cited by 10 (0 self)
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Nonsteady NavierStokes equations represent a differentialalgebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to take care of the manifolds contained in the differentialalgebraic equation (DAE). For several discretization schemes for the NavierStokes equations we investigate how the consideration of the manifolds is taken into account and propose a variant of solving these equations along the lines of the theoretically best index reduction.
Linear DifferentialAlgebraic Equations of HigherOrder and the Regularity or Singularity of Matrix Polynomials
 of Matrix Polynomials, PhD thesis, Institut für Mathematik, Technische Universität
, 2004
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RankRevealing "TopDown" ULV Factorizations
, 1997
"... Rankrevealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for nullspaces of a matrix. However, in the practical ULV (resp. URV) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via trian ..."
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Cited by 7 (0 self)
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Rankrevealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for nullspaces of a matrix. However, in the practical ULV (resp. URV) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via triangular solves. Triangular solves are required at initial factorization, refinement and updating. As a result, algorithms based on these factorizations may be expensive, especially on parallel computers where triangular solves are expensive. In this paper we propose an alternative approach. Our new rankrevealing ULV factorization, which we call "topdown" ULV factorization (TDULV factorization) is based on right null vectors of lower triangular matrices and therefore no triangular solves are required. Right null vectors are easy to estimate accurately using condition estimators such as incremental condition estimator (ICE). The TDULV factorization is shown to be equivalent to the URV fact...
Regular Solutions of Nonlinear DifferentialAlgebraic Equations and Their Numerical Determination
, 1998
"... . For a general class of nonlinear (possibly higher index) differentialalgebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, nu ..."
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Cited by 7 (5 self)
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. For a general class of nonlinear (possibly higher index) differentialalgebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included. Key words. Nonlinear differentialalgebraic equations, regular solutions, strangeness index, existence and uniqueness, consistent initial values, numerical methods. AMS subject classifications. 65L99, 34A09. 1. Introduction. If physical systems contain constraints such as links in mechanical systems or if they are governed by some conservation laws as Kirchhoff's laws for electric circuits, their modelling usually leads to nonlinear equations of the form F (t; x; x) = 0; (1) socalled differentialalgebraic equations (DAEs), wi...
Index reduction for differentialalgebraic equations by minimal extension
, 2001
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Generalized inverses of differentialalgebraic operators
 SIAM J. Matr. Anal. Appl
, 1996
"... In the theoretical treatment of linear differentialalgebraic equations one must deal with inconsistent initial conditions, inconsistent inhomogeneities, and undetermined solution components. Often their occurrence is excluded by assumptions to allow a theory along the lines of differential equation ..."
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Cited by 5 (4 self)
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In the theoretical treatment of linear differentialalgebraic equations one must deal with inconsistent initial conditions, inconsistent inhomogeneities, and undetermined solution components. Often their occurrence is excluded by assumptions to allow a theory along the lines of differential equations. The present paper aims at a theory that generalizes the wellknown least squares solution of linear algebraic equations to linear differentialalgebraic equations and that fixes a unique solution even when the initial conditions or the inhomogeneities are inconsistent or when undetermined solution components are present. For that a higher index differentialalgebraic equation satisfying some mild assumptions is replaced by a socalled strangenessfree differentialalgebraic equation with the same solution set. The new equation is transformed into an operator equation and finally generalized inverses are developed for the underlying differentialalgebraic operator. Key words. Differentialalgebraic equations, standard form, MoorePenrose pseudoinverse, generalized inverse, least squares regularization. AMS(MOS) subject classifications. 34A09, 47E05, 15A09, 58E25. 1