## A note on the stochastic realization problem (1976)

### Cached

### Download Links

Venue: | Hemisphere Publishing Corporation |

Citations: | 108 - 24 self |

### BibTeX

@INPROCEEDINGS{Lindquist76anote,

author = {Anders Lindquist and Giorgio Picci},

title = {A note on the stochastic realization problem},

booktitle = {Hemisphere Publishing Corporation},

year = {1976},

pages = {1--5}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations)require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families Of external realizations defined on the same probability space and totally ordered with respect to state covariances. 1. Introduction. One

### Citations

671 | The Theory of Matrices - Gantmacher - 1974 |

635 |
Stochastic processes
- Doob
- 1990
(Show Context)
Citation Context ...s x is the forward, and not the backward, realization problem. Proof of Theorem 4.1. For each fixed tR the process {so(z); -=>-t}, where so(z) X(t;-z), is a uniformly integrable wide sense martingale =-=[10]-=-, and therefore (t; T) tends to a limit x,(t) in mean square as T -oo. Moreover, (4.20) :(t, T)= .{x(t)lntr,o(dy)}-, J.{x(t)l/T<_tntr,o(dy)} in mean square [10], and hence (4.9) holds (a.s. for each t... |

471 | prediction: A tutorial review - Makhoul - 1975 |

463 | Algebraic coding theory - Berlekamp - 1984 |

429 | Singular Points of Complex Hypersurfaces - Milnor - 1968 |

408 | An iteration method for the solution of the eigenvalue problem of linear differential and integral operators - Lanczos - 1950 |

342 | All optimal Hankel-norm approximations of linear multivariable systems, and their L∞-error bound - Glover - 1984 |

201 | The Classical Moment Problem - Akhiezer - 1965 |

195 | The Theory of Matrices in Numerical Analysis - Householder - 1964 |

162 | Functions of One Complex Variable - Conway - 1978 |

124 |
Finite Dimensional Linear Systems
- Brockett
- 1970
(Show Context)
Citation Context ...ion , find all matrices W(s) of real rational functions with all its poles in Re (s)< 0 and satisfying (1.10). Such a function will be called a stable spectral factor. Let 3{.} denote McMillan degree =-=[8]-=-. Then 6{ W} -> 1/23{}; if there is equality we shall say that W is minimal. We have seen that the transfer function (1.8) of any wide sense stochastic realization of y is a stable spectral factor of ... |

97 | Introduction to the Theory of Random Processes - Gikhman, Skorohod - 1969 |

89 | Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen - Carathéodory - 1911 |

87 |
Network analysis and synthesis
- Anderson, Vongpanitlerd
- 1973
(Show Context)
Citation Context ... P P, + [M.(N)] -1 belongs to + if and only ifN >- O. Likewise, P P* [M*(N)] -1 _ belongs to and only ifN >-_ O. Finally, P* P, [M.(0)] -1 [M*(0)] -1. Various versions of this theorem can be found in =-=[7]-=- and 11]. It provides us with a procedure to determine all elements in + CI _: First compute P. and P*. Then varying N over the nonnegative cone will generate the other elements in / CI _. The corresp... |

82 | State Space Modeling of Time Series - Aoki - 1987 |

82 | Hyperstability of Control Systems - Popov - 1973 |

81 | Subspace algorithms for the stochastic identification problem,” Automatica 29 - Overschee, Moor - 1993 |

80 | Markovian representation of stochastic processes by canonical variables - Akaike - 1975 |

71 | Spectrum analysis-a modern perspective - Kay, Marple, et al. - 1981 |

68 | Effective construction of linear state-variable models from input/output functions. Re- gelungstechnik 14(1966 - Ho, Kalman |

67 | Linear Multivariable Systems - Wolovich - 1974 |

53 | Realization of power spectra from partial covariance sequences - Georgiou - 1987 |

53 | The Numerical Treatment of a Single Nonlinear Equation - Householder - 1970 |

48 | Matrix interpretations and applications of the continued fraction algorithm - Gragg - 1974 |

45 | The Pade table and its relation to certain algorithms of numerical analysis - Gragg - 1972 |

42 | Partial Realization of Covariance Sequences - Georgiou - 1983 |

38 | On the Fctorization of Rational Matrices - Youla - 1961 |

30 | Realization of covariance sequences - Kalman - 1981 |

27 | Spectral estimation: An overdetermined rational model equation approach - Cadzow - 1982 |

27 | Dooren, Speech modelling and the trigonometric moment problem - Delsarte, Kamp, et al. - 1982 |

24 | A complete parametrization of all positive rational extensions of a covariance sequence - Byrnes, Lindquist, et al. - 1995 |

24 | On partial realizations, transfer functions and canonical forms - Kalman - 1979 |

23 | Positive partial realization of covariance sequences", Modelling, Identi and Robust - Kimura - 1985 |

22 | On the partial realization problem, Linear Algebra AppZ - Gragg, Lindquist - 1983 |

20 | De fractionibus continuis dissertatio, Comm. Acad. Sci. Petropol. 9(1744) 98-137; also in Opera Omnia - Euler - 1925 |

19 | Sur la réduction en fraction continue d’une série procédant suivant les puissances descendantes d’une variable - Stieltjes |

17 | The stability and instability of partial realizations - Byrnes, Lindquist - 1982 |

17 | On power series which are bounded in the interior of the unit circle - Schur - 1918 |

16 | A new algorithm for optimal filtering of discrete-time stationary processes - Lindquist - 1974 |

15 | On the nonlinear dynamics of fast filtering algorithms - Byrnes, Lindquist, et al. |

14 | On the geometry of the Kimura-Georgiou parameterization of modeling filter - Byrnes, Lindquist - 1989 |

14 | Certain continued fractions associated with the Pad~ table - Magnus - 1960 |

14 | A minimal realization algorithm for matrix sequences - DICKINSON, MORF, et al. - 1974 |

12 | The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences - Demeure, Mullis - 1989 |

12 | On minimal partial realizations of a linear input/output map - Kalman - 1971 |

12 | Recursive identification of linear systems - Rissanen - 1971 |

12 | Digital Signal Processing and Control and Estimation Theory - Points of Tangency - Willsky - 1976 |

12 |
Stochastic Processes in Information and Dynamical Systems
- Wong
- 1971
(Show Context)
Citation Context ...orce Office of Scientific Research under Grant AFOSR-78-3519. 365366 ANDERS LINDQUIST AND GIORGIO PICCI does not depend on t, and it satisfies the Lyapunov equation (1.5) AP + PA’ + BB’ O. (See e.g. =-=[35]-=-.) The output process z has stationary increments. Each w /’k has a unique spectral representation (1.6) w(t) e ito 1 d(to) [12; p. 205], where dff is an orthogonal stochastic measure such that E{d(to... |

11 |
N-port synthesis via reactance extraction-Part I
- Youla, Tissi
- 1966
(Show Context)
Citation Context ...n, n m and rn n respectively. Hence F is a stability matrix, (F, G) is controllable and (H, F) is observable [8]. There are computational procedures for determining (F, G, H, R) from [8], [13], [31], =-=[38]-=-, so in the sequel we shall assume that such a quadruplet is given. It can be shown [5] that all wide sense minimal stochastic realizations are given by (1.16) [A,B, C, D] [TFT-’, T(B,,B2)S, HT-’, (R ... |

11 | Controllability and observability in multivariable control systems - Gilbert - 1963 |