Singular systems: A new approach in the time domain

Zhou, Zheng; Shayman, Mark A.; Tarn, Tzyh-Jong

doi:10.1109/cdc.1986.267438

Cited by 9 publications

(9 citation statements)

References 16 publications

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“…In particular, (E, A, B) is controllable if and only if the corresponding regular systems (A 1 , B 1 ) ∈ L r,m (F) and (A 2 , B 2 ) ∈ L n−r,m (F) are completely controllable [7,20]. We also have the following controllability criterion.…”

Section: Generalized Linear Systems and Group Actionsmentioning

confidence: 96%

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Generalized linear dynamical systems and the classification problem

Osetinskii¹

2008

Comput Math Model

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We consider the classification of generalized linear controllable systems over the field F = C or F = R under transformations defined by the action of the group GLn(F)×GLn(F). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O'Halloran, and others on the geometric structure of the set of orbits Cn,m(F) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of Cn,m(F) and evaluate its dimension. We show that Cn,m(F) is a natural compactification of the set of orbits of ordinary linear controllable systems Σn,m(F), and the boundary Cn,m(F) − Σn,m(F) consists of the orbits of singular generalized systems.

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Section: Generalized Linear Systems and Group Actionsmentioning

confidence: 96%

“…Various controllability criteria for generalized systems are given in [1,3,7,20]. In particular, (E, A, B) is controllable if and only if the corresponding regular systems (A 1 , B 1 ) ∈ L r,m (F) and (A 2 , B 2 ) ∈ L n−r,m (F) are completely controllable [7,20].…”

Section: Generalized Linear Systems and Group Actionsmentioning

confidence: 99%

Generalized linear dynamical systems and the classification problem

Osetinskii¹

2008

Comput Math Model

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“…In Lewis [7], it was shown that these systems could be decomposed into forwards and backwards propagating subsystems, so that their solution involves recursions in both time directions. However, in spite of these useful observations, it is fair to say that most of the literature on descriptor systems has focused mainly on issues of structure [8]- [10], and their implication for the control of descriptor systems [11]- [14]. This is primarily due to the fact that in continuous-time, descriptor systems display an impulsive behavior, which until recently has been the focus of most of the attention.…”

Section: Introductionmentioning

confidence: 95%

Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems

Nikoukhah

Levy

Willsky

1989

IEEE Trans. Automat. Contr.

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In this paper, we introduce the concept of internal stability for two-point boundary-value descriptor systems (TPBVDSs). Since TPBVDSs are defined only over a finite interval, the concept of stability is not easy to formulate for these systems. The definition which is used here consists in requiring that as the length of the interval of definition increases, the effect of boundary conditions on states located close to the center of the interval should go to zero. Stochastic TPBVDSs are studied, and the property of stochastic stationarity is characterized in terms of a generalized Lyapunov equation satisfied by the variance of the boundary vector. A second generalized Lyapunov equation satisfied by the state variance of a stochastically stationary TPBVDS is also introduced, and the existence and uniqueness of positive definite solutions to this equation is then used to characterize the property of internal stability.

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“…At the end of this section it is proved that every rational B(U; Y)-valued function (including those that have a pole at the origin) with finite McMillan degree n has a minimal s/s realization whose state space X has dimension n. This realization is minimal in the usual sense, and also spectrally minimal in the sense that the set Ω(Σ; U, Y) coincides with the points of holomorphy of the given rational function. Our minimal s/s realizations of rational matrix functions are quite different from the standard descriptor realizations discussed in, e.g., [28] and [29] and the references listed there. Among others, the dimension of the state space of a descriptor realization is bigger than the dimension of the state space of our s/s realization (i.e., bigger than the McMillan degree of the function) whenever the given function has a pole at the origin.…”

Section: Introductionmentioning

confidence: 98%

State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

Arov

Staffans

2007

Complex anal.oper. theory

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In the present article we continue to develop the linear discrete time-invariant state/signal systems theory that we have introduced and studied in three earlier articles (Parts I-III). The trajectories of a state/signal system Σ with a Hilbert state space X and a Hilbert or Kreȋn signal space W consists of a pair of sequences (x(·), w(·)) which after an admissible input/output decomposition W = Y U of the signal space can be obtained from the set of trajectories (x(·), u(·), y(·)) of a standard input/state/output system by taking w(·) = y(·) + u(·). In Part I we studied the families of all admissible decompositions of W for a given state/signal system Σ and the corresponding input/state/output representations and their transfer functions.Here we extend that theory to the the non-admissible case, and obtain generalized input/output transfer functions, as well as right and left affine transfer functions. As opposed to a standard transfer function, a generalized transfer function not need to be holomorphic at the origin. This makes it possible to realize a much larger class of transfer functions than what is possible by using the standard input/state/output theory. For example, every rational matrixvalued function (including those that have a pole at the origin) can be realized as the generalized transfer function of a state/signal system whose state space has a finite dimension equal to the McMillan degree of the given function. Likewise, every meromorphic J-contractive matrix-valued function can be realized as the generalized transfer function of a simple conservative state/signal system, or of a minimal passive state/signal system. Similar operator-valued results are also presented. As is well-known, a rational matrix-valued function with a pole at the origin can also be realized as the transfer function of a descriptor system, but this descriptor realization has the drawback that the dimension of its state space is bigger than the dimension of the state space of our s/s realization (i.e., bigger than the McMillan degree of the function). Our Damir Z. Arov thanksÅbo Akademi for its hospitality and the Academy of Finland and the Magnus Ehrnrooth Foundation for their financial support during his visits toÅbo in 2003-2006.458 D. Z. Arov and O. J. Staffans Comp.an.op.th.s/s realizations also differ significantly from other known realizations which depend on the choice of an auxiliary point in the complex plane where the given function is holomorphic. (2000). Primary 47A48, 93A05; Secondary 94C05. Mathematics Subject Classification

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Singular systems: A new approach in the time domain

Cited by 9 publications

References 16 publications

Generalized linear dynamical systems and the classification problem

Generalized linear dynamical systems and the classification problem

Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems

State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

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