The Experts below are selected from a list of 9228600 Experts worldwide ranked by ideXlab platform
Froilan M Dopico - One of the best experts on this subject based on the ideXlab platform.
-
on minimal bases and indices of rational matrices and their Linearizations
Linear Algebra and its Applications, 2021Co-Authors: A Amparan, Froilan M Dopico, S Marcaida, Ion ZaballaAbstract:Abstract A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong Linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneering results obtained by Verghese, Van Dooren and Kailath in 1979-1980, which were the first proving results of this type. It is shown that the definitions of Linearizations and strong Linearizations do not guarantee any relationship between the minimal bases and indices of the Linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases Linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of Linearizations available in recent literature for square rational matrices.
-
block full rank Linearizations of rational matrices
arXiv: Numerical Analysis, 2020Co-Authors: Froilan M Dopico, S Marcaida, Maria C Quintana, Paul Van DoorenAbstract:Block full rank pencils introduced in [Dopico et al., Local Linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information about zeros that are not poles of rational matrices. In this paper we extend the structure of those block full rank pencils to construct Linearizations of rational matrices that allow us to recover locally not only information about zeros but also about poles, whenever certain minimality conditions are satisfied. In addition, the notion of degree of a rational matrix will be used to determine the grade of the new block full rank Linearizations as Linearizations at infinity. This new family of Linearizations is important as it generalizes and includes the structures appearing in most of the Linearizations for rational matrices constructed in the literature. In particular, this theory will be applied to study the structure and the properties of the Linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, submitted].
-
on minimal bases and indices of rational matrices and their Linearizations
arXiv: Numerical Analysis, 2019Co-Authors: A Amparan, Froilan M Dopico, S Marcaida, Ion ZaballaAbstract:A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong Linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneer results obtained by Verghese, Van Dooren and Kailath in 1979-80, which were the first proving results of this type under different nonequivalent conditions. It is shown that the definitions of Linearizations and strong Linearizations do not guarantee any relationship between the minimal bases and indices of the Linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases Linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of Linearizations available in recent literature for square rational matrices.
-
strong Linearizations of rational matrices with polynomial part expressed in an orthogonal basis
Linear Algebra and its Applications, 2019Co-Authors: Froilan M Dopico, S Marcaida, Maria C QuintanaAbstract:Abstract We construct a new family of strong Linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al. (2018) [4] , and the new Linearizations of polynomial matrices introduced by Fasbender and Saltenberger (2017) [15] . In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its Linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong Linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al. (2018) [4] , allows us to use essentially all the strong Linearizations of polynomial matrices developed in the last fifteen years to construct strong Linearizations of any rational matrix by expressing such a matrix in terms of its polynomial and strictly proper parts.
-
a block symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error
Calcolo, 2018Co-Authors: Maria Bueno, Froilan M Dopico, Susana Furtado, L MedinaAbstract:The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric Linearizations have proven to be very useful for constructing structured Linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space \(\mathbb {DL}(P)\) of block-symmetric Linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.
Steven D. Mackey - One of the best experts on this subject based on the ideXlab platform.
-
Linearizations of singular matrix polynomials and the recovery of minimal indices
2015Co-Authors: O De Teran, Frolian Dopico, Steven D. Mackey, Mims Eprint, Fernando De TeránAbstract:Abstract. A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil – a process known as linearization. Two vector spaces of pencils L1(P) and L2(P) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are Linearizations for P (λ) when P is regular. The goal of this work is to show that most of the pencils in L1(P) and L2(P) are still Linearizations when P (λ) is a singular square matrix polynomial, and that these Linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) from the minimal indices and bases of Linearizations in L1(P) and L2(P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P) = L1(P) ∩ L2(P) will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other Linearizations of singular polynomials defined in terms of more general polynomial bases
-
spectral equivalence of matrix polynomials and the index sum theorem
Linear Algebra and its Applications, 2014Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:Abstract The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using Linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that Linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of Linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and l-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. The properties of l-ifications are studied in depth; in this process a fundamental result on matrix polynomials, the “Index Sum Theorem”, is recovered and extended to arbitrary fields. Although this result is known in the systems theory literature for real matrix polynomials, it has remained unnoticed by many researchers. It establishes that the sum of the (finite and infinite) partial multiplicities, together with the (left and right) minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial. The “Index Sum Theorem” turns out to be a key tool for obtaining a number of significant results: on the possible sizes and degrees of l-ifications and companion forms, on the minimal index preservation properties of companion forms of arbitrary degree, as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree. This paper presents many new results, blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields. Therefore we have written the paper in an expository and self-contained style that makes it accessible to a wide variety of readers.
-
fiedler companion Linearizations for rectangular matrix polynomials
Linear Algebra and its Applications, 2012Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The development of new classes of Linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding Linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on Linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of Linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler Linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler Linearizations are one of the most important classes of Linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new Linearizations that has been generalized to rectangular polynomials.
-
palindromic companion forms for matrix polynomials of odd degree
Journal of Computational and Applied Mathematics, 2011Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The standard way to solve polynomial eigenvalue problems P ( λ ) x = 0 is to convert the matrix polynomial P ( λ ) into a matrix pencil that preserves its spectral information - a process known as linearization. When P ( λ ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P ( λ ) have certain symmetries that can be lost when using the classical first and second Frobenius companion Linearizations for numerical computations, since these Linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding Linearizations that retain whatever structure the original P ( λ ) might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic Linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of Linearizations for odd degree polynomials P ( λ ) which are palindromic whenever P ( λ ) is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the Linearizations in the new family.
-
structured Linearizations for palindromic matrix polynomials of odd degree
2010Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The standard way to solve polynomial eigenvalue problems $P(\la)x=0$ is to convert the matrix polynomial $P(\la)$ into a matrix pencil that preserves its spectral information-- a process known as linearization. When $P(\la)$ is palindromic, the eigenvalues, elementary divisors, and minimal indices of $P(\la)$ have certain symmetries that can be lost when using the classical first and second companion Linearizations for numerical computations, since these Linearizations do not preserve the palindromic structure. Recently new families of Linearizations have been introduced with the goal of finding Linearizations that retain whatever structure that the original $P(\la)$ might possess, with particular attention paid to the preservation of palindromic structure. However, no general construction of palindromic Linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of Linearizations for odd degree polynomials $P(\la)$ which are palindromic whenever $P(\la)$ is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the Linearizations in the new family.
Fernando De Teran - One of the best experts on this subject based on the ideXlab platform.
-
spectral equivalence of matrix polynomials and the index sum theorem
Linear Algebra and its Applications, 2014Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:Abstract The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using Linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that Linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of Linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and l-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. The properties of l-ifications are studied in depth; in this process a fundamental result on matrix polynomials, the “Index Sum Theorem”, is recovered and extended to arbitrary fields. Although this result is known in the systems theory literature for real matrix polynomials, it has remained unnoticed by many researchers. It establishes that the sum of the (finite and infinite) partial multiplicities, together with the (left and right) minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial. The “Index Sum Theorem” turns out to be a key tool for obtaining a number of significant results: on the possible sizes and degrees of l-ifications and companion forms, on the minimal index preservation properties of companion forms of arbitrary degree, as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree. This paper presents many new results, blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields. Therefore we have written the paper in an expository and self-contained style that makes it accessible to a wide variety of readers.
-
fiedler companion Linearizations for rectangular matrix polynomials
Linear Algebra and its Applications, 2012Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The development of new classes of Linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding Linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on Linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of Linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler Linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler Linearizations are one of the most important classes of Linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new Linearizations that has been generalized to rectangular polynomials.
-
palindromic companion forms for matrix polynomials of odd degree
Journal of Computational and Applied Mathematics, 2011Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The standard way to solve polynomial eigenvalue problems P ( λ ) x = 0 is to convert the matrix polynomial P ( λ ) into a matrix pencil that preserves its spectral information - a process known as linearization. When P ( λ ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P ( λ ) have certain symmetries that can be lost when using the classical first and second Frobenius companion Linearizations for numerical computations, since these Linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding Linearizations that retain whatever structure the original P ( λ ) might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic Linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of Linearizations for odd degree polynomials P ( λ ) which are palindromic whenever P ( λ ) is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the Linearizations in the new family.
-
recovery of eigenvectors and minimal bases of matrix polynomials from generalized fiedler Linearizations
SIAM Journal on Matrix Analysis and Applications, 2011Co-Authors: Maria Bueno, Fernando De Teran, Froilan M DopicoAbstract:A standard way to solve polynomial eigenvalue problems P(λ)x=0 is to convert the matrix polynomial P(λ) into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This process is known as linearization and is not unique, since there are infinitely many Linearizations with widely varying properties associated with P(λ). This freedom has motivated the recent development and analysis of new classes of Linearizations that generalize the classical first and second Frobenius companion forms, with the goals of finding Linearizations that retain whatever structures that P(λ) might possess and/or of improving numerical properties, as conditioning or backward errors, with respect the companion forms. In this context, an important new class of Linearizations is what we name generalized Fiedler Linearizations, introduced in 2004 by Antoniou and Vologiannidis as an extension of certain Linearizations introduced previously by Fiedler for scalar polynomials. On the other hand, the mere ...
-
structured Linearizations for palindromic matrix polynomials of odd degree
2010Co-Authors: Fernando De Teran, Froilan M Dopico, Steven D. MackeyAbstract:The standard way to solve polynomial eigenvalue problems $P(\la)x=0$ is to convert the matrix polynomial $P(\la)$ into a matrix pencil that preserves its spectral information-- a process known as linearization. When $P(\la)$ is palindromic, the eigenvalues, elementary divisors, and minimal indices of $P(\la)$ have certain symmetries that can be lost when using the classical first and second companion Linearizations for numerical computations, since these Linearizations do not preserve the palindromic structure. Recently new families of Linearizations have been introduced with the goal of finding Linearizations that retain whatever structure that the original $P(\la)$ might possess, with particular attention paid to the preservation of palindromic structure. However, no general construction of palindromic Linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of Linearizations for odd degree polynomials $P(\la)$ which are palindromic whenever $P(\la)$ is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the Linearizations in the new family.
Maria C Quintana - One of the best experts on this subject based on the ideXlab platform.
-
block full rank Linearizations of rational matrices
arXiv: Numerical Analysis, 2020Co-Authors: Froilan M Dopico, S Marcaida, Maria C Quintana, Paul Van DoorenAbstract:Block full rank pencils introduced in [Dopico et al., Local Linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information about zeros that are not poles of rational matrices. In this paper we extend the structure of those block full rank pencils to construct Linearizations of rational matrices that allow us to recover locally not only information about zeros but also about poles, whenever certain minimality conditions are satisfied. In addition, the notion of degree of a rational matrix will be used to determine the grade of the new block full rank Linearizations as Linearizations at infinity. This new family of Linearizations is important as it generalizes and includes the structures appearing in most of the Linearizations for rational matrices constructed in the literature. In particular, this theory will be applied to study the structure and the properties of the Linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, submitted].
-
Linearizations of rational matrices from general representations
arXiv: Numerical Analysis, 2020Co-Authors: Javier Perez, Maria C QuintanaAbstract:We construct a new family of Linearizations of rational matrices $R(\lambda)$ written in the general form $R(\lambda)= D(\lambda)+C(\lambda)A(\lambda)^{-1}B(\lambda)$, where $D(\lambda)$, $C(\lambda)$, $B(\lambda)$ and $A(\lambda)$ are polynomial matrices. Such representation always exists and are not unique. The new Linearizations are constructed from Linearizations of the polynomial matrices $D(\lambda)$ and $A(\lambda)$, where each of them can be represented in terms of any polynomial basis. In addition, we show how to recover eigenvectors, when $R(\lambda)$ is regular, and minimal bases and minimal indices, when $R(\lambda)$ is singular, from those of their Linearizations in this family.
-
strong Linearizations of rational matrices with polynomial part expressed in an orthogonal basis
Linear Algebra and its Applications, 2019Co-Authors: Froilan M Dopico, S Marcaida, Maria C QuintanaAbstract:Abstract We construct a new family of strong Linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al. (2018) [4] , and the new Linearizations of polynomial matrices introduced by Fasbender and Saltenberger (2017) [15] . In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its Linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong Linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al. (2018) [4] , allows us to use essentially all the strong Linearizations of polynomial matrices developed in the last fifteen years to construct strong Linearizations of any rational matrix by expressing such a matrix in terms of its polynomial and strictly proper parts.
-
strong Linearizations of rational matrices with polynomial part expressed in an orthogonal basis
arXiv: Numerical Analysis, 2018Co-Authors: Froilan M Dopico, S Marcaida, Maria C QuintanaAbstract:We construct a new family of strong Linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al., MIMS EPrint 2016.51, and the new Linearizations of polynomial matrices introduced by Fa{\ss}bender and Saltenberger, Linear Algebra Appl., 525 (2017). In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its Linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong Linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to use essentially all the strong Linearizations of polynomial matrices developed in the last fifteen years to construct strong Linearizations of any rational matrix by expressing such matrix in terms of its polynomial and strictly proper parts.
Jan Heiland - One of the best experts on this subject based on the ideXlab platform.
-
exponential stability and stabilization of extended Linearizations via continuous updates of riccati based feedback
International Journal of Robust and Nonlinear Control, 2018Co-Authors: Peter Benner, Jan HeilandAbstract:Summary Many recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady-state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well-developed state for which the linearization-based local approaches will not work. Considering extended Linearizations or state-dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati-based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.
-
exponential stability and stabilization of extended Linearizations via continuous updates of riccati based feedback
arXiv: Dynamical Systems, 2016Co-Authors: Peter Benner, Jan HeilandAbstract:Many recent works on stabilization of nonlinear systems target the case of locally stabilizing an unstable steady state solutions against small perturbation. In this work we explicitly address the goal of driving a system into a nonattractive steady state starting from a well developed state for which the linearization based local approaches will not work. Considering extended Linearizations or state-dependent coefficient representations of nonlinear systems, we develop sufficient conditions for stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. Based on these analytical results we propose an update scheme that is designed to maintain the stabilization property of Riccati based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in two examples.
