New approach to robust observer design

Tsui, Chia-Chi

doi:10.1080/00207178808906052

Cited by 61 publications

(19 citation statements)

References 14 publications

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“…with A, E, B being given coefficient matrices, F a parameter matrix which may be unknown a priori, and V and W being matrix variables to be determined, has applications in many control systems design problems, such as, pole/eigenstructure assignment ([1]- [3]), observer designs ( [1], [4]- [6]), robust fault detection ( [7]- [9]), etc., and has been extensively examined ( [1], [10]- [15]). For (1) with F being in Jordan form, complete parametric solutions have been proposed in [1], [13] and [14].…”

Section: Introductionmentioning

confidence: 99%

Solution to a type of nonhomogeneous generalized Sylvester equations

Duan

2013

2013 25th Chinese Control and Decision Conference (CCDC)

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In this paper, a new type of nonhomogeneous generalized Sylvester equations (GSEs) are proposed. A complete general parametric solution in a neat explicit closed form is established using the F -coprimeness condition. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The matrix R, together with the matrix F, may be both set undetermined and used as a part of the degree of freedom beyond the completely free parameter matrix Z. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many control systems analysis and design problems involving dynamical systems.

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Section: Introductionmentioning

confidence: 99%

Solution to a type of nonhomogeneous generalized Sylvester equations

Duan

2013

2013 25th Chinese Control and Decision Conference (CCDC)

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“…There is another approximate solution to the same problem, which is aimed directly at minimizing T B by selecting the n − m observer poles clustered around the existing stable transmission zeros while designing a function observer [30,48] . However, this pattern of observer poles will result in a near singular matrix C which in turn will result in a large observer output gain K while trying to satisfy (6).…”

Section: It Is Also Obvious That the Feedback Of U U U(t) Into The Obmentioning

confidence: 99%

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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“…This case of a unique pair of semisimple eigenvalues was also considered in [19] in -Again, we would like to emphasize that to construct the Laurent series (22), we need to solve a series of new reduced Sylvester equations but with coefficient matrices of much smaller dimensions. This fact is especially useful in the case of higher order singularities.…”

Section: =1mentioning

confidence: 98%

“…Similarly, using again (9) and biorthogonality condition Again we would like to emphasize that the linear systems (21) and (23) have the same coefficient matrix on the left hand side, which permits an efficient calculation of the regular terms in the Laurent series (22).…”

Section: =1mentioning

confidence: 98%

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Analytic perturbation of Sylvester and Lyapunov matrix equations

Avrachenkov¹,

Lasserre²

Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)

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We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. Here we provide a necessary and sufficient condition for the existence of a Laurent series with a first order pole. An efficient recursive procedure for the calculation of the Laurent series' coefficients is given. Finally, we show that in the particular, but practically important case of semisimple eigenvalues, the recursive procedure can be written in a compact matrix form.First, all the computations needed t o obtain the coefficients of the Laurent series for X ( z ) are numerically stable, because we are using a reduction process which works on spaces of small dimension. Second, the Laurent series expansion of X ( z ) permits one t o compute a good approximation of the exact solution and provides interesting information on the behavior of the approximate solution for small values of z . For other applications involving parametric Sylvester matrix equations (1) see for example [19, 251 and the references therein. Therefore we will study (1) where A ( z ) = x F = o z k A ( k ) E PX", B ( z ) = C r = o~k B ( k ) E P x n , C ( z ) = ~~' o z k C ( k ) E C m x n . These series are assumed t o be convergent in some disc in the complex plane, around the origin z = 0. In particular, when z = 0, the equation

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New approach to robust observer design

Cited by 61 publications

References 14 publications

Solution to a type of nonhomogeneous generalized Sylvester equations

Solution to a type of nonhomogeneous generalized Sylvester equations

Observer design — A survey

Analytic perturbation of Sylvester and Lyapunov matrix equations

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