## Algebraically Approximate and Noisy Realization of Discrete-Time Systems and Digital ImagesThis monograph deals with approximation and noise cancellation of dyn- ical systems which include linear and nonlinear input/output relationships. It also deal with approximation and noise cancellation of two dimensional arrays. It will be of special interest to researchers, engineers and graduate students who have specialized in ?ltering theory and system theory and d- ital images. This monograph is composed of two parts. Part I and Part II will deal with approximation and noise cancellation of dynamical systems or digital images respectively. From noiseless or noisy data, reduction will be made. A method which reduces model information or noise was proposed in the reference vol. 376 in LNCIS [Hasegawa, 2008]. Using this method will allow model description to be treated as noise reduction or model reduction without having to bother, for example, with solving many partial di?er- tial equations. This monograph will propose a new and easy method which produces the same results as the method treated in the reference. As proof of its advantageous e?ect, this monograph provides a new law in the sense of numerical experiments. The new and easy method is executed using the algebraic calculations without solving partial di?erential equations. For our purpose,manyactualexamplesofmodelinformationandnoisereductionwill also be provided. Using the analysis of state space approach, the model reduction problem may have become a major theme of technology after 1966 for emphasizing e?ciency in the ?elds of control, economy, numerical analysis, and others. |

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### Contents

3 | |

InputOutput Map and Additive Noises | 11 |

22 Analysis for Algebraically Approximate and Noisy Realization | 13 |

23 Measurement Data with Noise | 14 |

25 Algebraically Constrained Least Square Method | 17 |

251 Algebraic CLS Method and Analytic CLS Method | 19 |

Algebraically Approximate and Noisy Realization of Linear Systems | 21 |

32 Finite Dimensional Linear Systems | 23 |

65 Algebraically Approximate Realization of Pseudo Linear Systems | 120 |

66 Algebraically Noisy Realization of Pseudo Linear Systems | 131 |

67 Historical Notes and Concluding Remarks | 144 |

Algebraically Approximate and Noisy Realization of Affine Dynamical Systems | 146 |

71 Basic Facts about Affine Dynamical Systems | 148 |

72 Finite Dimensional Aﬃne Dynamical Systems | 150 |

73 Partial Realization Theory of Affine Dynamical Systems | 153 |

74 Algebraically Approximate Realization of Affine Dynamical Systems | 155 |

33 Partial Realization Theory of Linear Systems | 25 |

34 Algebraically Approximate Realization of Linear Systems | 26 |

35 Algebraically Noisy Realization of Linear Systems | 36 |

351 Comparative Table of the Algebraic CLS and AIC Method | 48 |

36 Historical Notes and Concluding Remarks | 49 |

Algebraically Approximate and Noisy Realization of Socalled Linear Systems | 53 |

41 Basic Facts about Socalled Linear Systems | 54 |

42 Finite Dimensional Socalled Linear Systems | 55 |

43 Partial Realization of Socalled Linear Systems | 58 |

44 RealTime Partial Realization of Almost Linear Systems | 60 |

45 Algebraically Approximate Realization of Socalled Linear Systems | 61 |

46 Algebraically Noisy Realization of Socalled Linear Systems | 72 |

47 Historical Notes and Concluding Remarks | 83 |

Algebraically Approximate and Noisy Realization of Almost Linear Systems | 86 |

51 Basic Facts of Almost Linear Systems | 88 |

52 Finite Dimensional Almost Linear Systems | 89 |

53 Algebraically Approximate Realization of Almost Linear Systems | 90 |

54 Algebraically Noisy Realization of Almost Linear Systems | 99 |

55 Historical Notes and Concluding Remarks | 108 |

Algebraically Approximate and Noisy Realization of Pseudo Linear Systems | 111 |

62 Finite Dimensional Pseudo Linear Systems | 113 |

63 Partial Realization of Pseudo Linear Systems | 116 |

64 RealTime Partial Realization of Pseudo Linear Systems | 118 |

75 Algebraically Noisy Realization of Affine Dynamical Systems | 170 |

76 Historical Notes and Concluding Remarks | 182 |

Algebraically Approximate and Noisy Realization of Linear Representation Systems | 185 |

81 Basic Facts about Linear Representation Systems | 186 |

82 Finite Dimensional Linear Representation Systems | 187 |

83 Partial Realization Theory of Linear Representation Systems | 190 |

84 Algebraically Approximate Realization of Linear Representation Systems | 191 |

85 Algebraically Noisy Realization of Linear Representation Systems | 201 |

86 Historical Notes and Concluding Remarks | 211 |

Algebraically Approximate and Noisy Realization of TwoDimensional Images | 214 |

91 Commutative Linear Representation Systems | 216 |

92 FiniteDimensional Commutative Linear Representation Systems | 218 |

93 Partial Realization Theory of TwoDimensional Images | 223 |

94 Measurement Data with Approximate and Noisy Error | 226 |

95 Analyses for Approximate and Noisy Data | 227 |

96 Nonlinear Integer Programming for Digital Images | 228 |

97 Algebraically Approximate Realization of TwoDimensional Images | 229 |

98 Algebraically Noisy Realization of TwoDimensional Images | 237 |

99 Historical Notes and Concluding Remarks | 246 |

249 | |

252 | |

### Other editions - View all

Algebraically Approximate and Noisy Realization of Discrete-Time Systems and ... Yasumichi Hasegawa No preview available - 2009 |

Algebraically Approximate and Noisy Realization of Discrete-Time Systems and ... Yasumichi Hasegawa No preview available - 2009 |

Algebraically Approximate and Noisy Realization of Discrete-Time Systems and ... Yasumichi Hasegawa No preview available - 2012 |

### Common terms and phrases

affine dynamical system aﬃnedynamical system algebraic CLS method algebraically approximate realization algebraically noisy realization algebraicCLS analytic CLS method analyticCLS approximate and noisy approximate realization problem Based on Proposition behavior canonical characteristic polynomial CLS error commutative linear representation covariance matrix eigenvalues covariance matrix square diﬀerence dimen dimensional linear dynamical system obtained eigenvectors equation finite dimensional Hankel matrix impulse responses I(0 independent vectors input response map Input/output matrix Let a matrix linear operator linear representation system linear space linear system obtained linearly independent matrix composed matrix norm matrix square root modelobtained modiﬁed impulse responses noisy realization algorithm noisy realization problem ofdimensions original input response original signal Proposition 2.14 pseudo linear system Q in Proposition reference Hasegawa root for sum root of eigenvalues sense ofthenumerical calculation signal by CLS sion ratio so-called linear system sum of cosine Theorem time-invariant values of square vector index Wecan show