On the Solution to Matrix Equation $TA - FT = LC$ and Its Applications

Tsui, Chia-Chi

doi:10.1137/0614003

Cited by 46 publications

(47 citation statements)

References 20 publications

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“…In particular, it was Van Dooren who presented some fundamental concepts in this direction using block Hessenberg forms [8]. Later Tsui in [5] presented a parametrization using the aforementioned Sylvester equation.…”

Section: Introductionmentioning

confidence: 99%

A Computational Method for Reduced-Order Observers in Linear Systems *

Syrmos

Zagalak²

1992

IFAC Proceedings Volumes

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Section: Introductionmentioning

confidence: 99%

A Computational Method for Reduced-Order Observers in Linear Systems *

Syrmos

Zagalak²

1992

IFAC Proceedings Volumes

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“…In fact, this special feature of observer order design/adjustment freedom is a key breakthrough that enabled all significant developments of observer design, as described in the rest of this survey. Fortunately, such a closed-form solution of (5) was derived in the middle of 1980s [34,35] , and is described in the following. For simplicity of presentation, only distinct and real observer poles (eigenvalues of F ) λi(i = 1, 2, · · · ) are used.…”

Section: Once This Convergence Z Z Z(t) → Tx X X(t) Is Achieved It Imentioning

confidence: 99%

“…For simplicity of presentation, only distinct and real observer poles (eigenvalues of F ) λi(i = 1, 2, · · · ) are used. For general eigenvalue cases, this solution can easily be generalized [5,34,35] . Let F be in Jordan form Λ (diagonal in this eigenvalue case).…”

Section: Once This Convergence Z Z Z(t) → Tx X X(t) Is Achieved It Imentioning

confidence: 99%

“…where t t ti(i = 1, 2, · · · ) is the i-th row of T , the mdimensional row c c ci is the complete remaining freedom of t t ti, and the m × n dimensional basis vector matrix Di of t t ti can always be computed from the corresponding right n−m columns of (5), [5,34,35] .…”

Section: Once This Convergence Z Z Z(t) → Tx X X(t) Is Achieved It Imentioning

confidence: 99%

“…of (5) must also be completely decoupled because a decoupled design solution has many technical and computational advantages [5,34] . For example, a designer can freely design the observer order (or the number of rows of (F, T, L)) only if these rows are decoupled.…”

Section: Once This Convergence Z Z Z(t) → Tx X X(t) Is Achieved It Imentioning

confidence: 99%

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Observer design — A survey

Tsui

2015

Int. J. Autom. Comput.

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This paper surveys the results of observer design for linear time-invariant (L-T-I) deterministic irreducible open-loop systems (OLS), the most basic type of OLS. An observer estimates Kx x x(t) signal where K is a constant and x x x(t) is the state vector of the OLS. Thus, an observer can be used as a feedback controller that implements state feedback control (SFC) or Kx x x(t)-control, and observer design is therefore utterly important in all feedback control designs of state space theory. In this survey, the observer design results are divided into three categories and for three respective main purposes. The first category of observers estimate signal Kx x x(t) only with a given K, and this survey has four conclusions: 1) Function observer that estimates Kx x x(t) directly is more general than state observer that estimates x x x(t), and may be designed with order lower than that of state observer, and the additional design objective is to minimize observer order; 2) The function observer design problem has already been simplified to the solving of a single set of linear equations only while seeking the lowest possible number of rows of the solution matrix, and an apparently most effective and general algorithm of solving such a problem can guarantee unified upper and lower bounds of the observer order; 3) Because such a single set of linear equations is the simplest possible theoretical formulation of the design problem and such theoretical observer order bounds are the lowest possible, and because the general, simple, and explicit theoretical formula for the function observer order itself do not exist, the theoretical part of this design problem is solved; 4) Because the function observer order is generically near its upper bound, further improvement on the computational design algorithm so that the corresponding observer order can be further reduced, is generically not worthwhile. The second category of observers further realize the loop transfer function and robustness properties of the direct SFC, and the conclusion of this survey is also fourfold: 1) To fully realize the loop transfer function of a practically designed Kx x x(t)-control, the observer must be an output feedback controller (OFC) which has zero gain to OLS input; 2) If parameter K is separately designed before the observer design, as in the separation principle which has been followed by almost all people for over half of a century, then OFC that estimates Kx x x(t) does not exist for almost all OLS s; 3) As a result, a synthesized design principle that designs an OFC first and is valid for almost all OLS s, is proposed and fully developed, the corresponding K will be designed afterwards and will be constrained by the OFC order as well as the OFC parameters; 4) Although the Kx x x(t)-control is constrained in this new design principle and is therefore called the "generalized SFC" (as compared to the existing SFC in which K is unconstrained), it is still strong enough for most OLS s and this new design principle overcomes many fund...

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Eigenstructure assignment by the differential Sylvester equation for linear time-varying systems

Choi

Lee

Yoo

1999

KSME International Journal

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This work is concerned with the assignment of the desired eigenstructure for linear time -varying systems such as missiles, rockets, fighters, etc. Despite its well-known limitations, gain scheduling control continues to be a major focus of research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper, we i) introduce a differential algebraic eigenvalue theory for linear time-varying systems, and ii) propose an eigenstructure assignment scheme for linear time-varying systems via the differential Sylvester equation based upon newly developed notions. The whole design procedure of the proposed eigenstructure assignment scheme is very systematic. The scheme can be used to determine the stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for linear time-varying systems. The presented method is illustrated by a numerical example.

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On the Solution to Matrix Equation $TA - FT = LC$ and Its Applications

Cited by 46 publications

References 20 publications

A Computational Method for Reduced-Order Observers in Linear Systems *

A Computational Method for Reduced-Order Observers in Linear Systems *

Observer design — A survey

Eigenstructure assignment by the differential Sylvester equation for linear time-varying systems

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