Realization Theory of Discrete-Time Dynamical Systems

by Matsuo, Tsuyoshi; Hasegawa, Yasumichi

• ISBN13: 9783540406754
• ISBN10: 3540406751
• Format: Paperback
• Copyright: 2004-02-01
• Publisher: Springer Nature

Summary

This monograph extends Realization Theory to the discrete-time domain. It includes new results and constructs a new and very wide inclusion relation for various non-linear dynamical systems. After establishing some features of discrete-time dynamical systems it presents results concerning systems which are proposed by the authors for the first time. They introduce General Dynamical Systems, Linear Representation Systems, Affine Dynamical Systems, Pseudo Linear Systems, Almost Linear Systems and So-called Linear Systems for discrete-time and demonstrate the relationship between them and the other dynamical systems. This book is intended for graduate students and researchers who study control theory.

Table of Contents

Introduction	1	(9)
Input/Output Maps	9	(6)
Concatenation Monoid: Set of Experiments	9	(1)
Input Response Map (Input/Output Map with Causality)	10	(4)
Historical Notes and Concluding Remarks	14	(1)
General Dynamical Systems	15	(23)
Realization Theory of General Dynamical Systems	15	(3)
Finite General Dynamical Systems	18	(5)
Control Systems and Multi-experiments	23	(3)
Historical Notes and Concluding Remarks	26	(2)
Appendix	28	(10)
U-Actions	28	(2)
Pointed U-Actions	30	(2)
U-Actions with a Readout Map	32	(1)
General Dynamical Systems	33	(5)
Linear Representation Systems	38	(41)
Realization Theory of Linear Representation Systems	38	(3)
Finite Dimensional Linear Representation Systems	41	(7)
Partial Realization Theory of Linear Representation Systems	48	(2)
Historical Notes and Concluding Remarks	50	(2)
Appendix	52	(27)
Realization Theorem	52	(1)
Linear State Structure: Linear U-Actions	52	(5)
Pointed Linear U-Actions	57	(2)
Linear U-Actions with a Readout Map	59	(1)
Linear Representation Systems	60	(1)
Sophisticated Realization Theorem	61	(3)
Finite Dimensionality	64	(1)
Pointed Finite Dimensional Linear U-Actions	64	(3)
Finite Dimensional Linear U-Actions with a Readout Map	67	(2)
Finite Dimensional Linear Representation Systems	69	(1)
Existence Criterion for Linear Representation Systems	70	(1)
Realization Procedure for Linear Representation Systems	71	(1)
Partial Realization	71	(1)
Pointed Linear U-Actions	71	(1)
Linear U-Actions with a Readout Map	72	(1)
Partial Realization Problem	73	(6)
Affine Dynamical Systems	79	(34)
Realization Theorem of Affine Dynamical Systems	79	(5)
Finite Dimensional Affine Dynamical Systems	84	(7)
Historical Notes and Concluding Remarks	91	(2)
Appendix	93	(20)
Realization Theorem	93	(1)
Fundamentals for Affine Dynamical Systems	93	(1)
Affine State Structure: Linear U-Actions	94	(4)
Affine State Structure with a Readout Map	98	(3)
Affine Dynamical Systems	101	(2)
Sophisticated Realization Theorem	103	(2)
Finite Dimensionality	105	(1)
Pointed Finite Dimensional Linear U-Actions	105	(3)
Finite Dimensional Linear U-Actions with a Readout Map	108	(2)
Finite Dimensional Affine Dynamical Systems	110	(1)
Existence Criterion for Affine Dynamical Systems	111	(1)
Realization Procedure for Affine Dynamical Systems	112	(1)
Pseudo Linear Systems	113	(43)
Input Response Maps with Time-Invariance	113	(1)
Pseudo Linear Systems	114	(3)
Finite Dimensional Pseudo Linear Systems	117	(5)
Partial Realization of Pseudo Linear Systems	122	(3)
Real-Time Partial realization of Pseudo Linear Systems	125	(1)
Historical Notes and Concluding Remarks	126	(2)
Appendix	128	(28)
Realization Theory	128	(1)
Deriving Pseudo Linear Systems from Affine Dynamical Systems	128	(1)
Linear State Structure: Free Motions	129	(3)
Free Motions with an Affine Map	132	(3)
Free Motions with a Readout Map	135	(1)
Pseudo Linear Systems	136	(1)
Sophisticated Pseudo Linear Systems	137	(3)
Finite Dimensionality	140	(1)
Finite Dimensional Free Motions with an Affine Map	140	(3)
Finite Dimensional Free Motions with a Readout Map	143	(2)
Finite Dimensional Pseudo Linear Systems	145	(1)
Existence Criterion for Pseudo Linear Systems	146	(1)
Realization Procedure for Pseudo Linear Systems	146	(1)
Partial Realization	147	(1)
Free Motions with an Affine Map	147	(1)
Free Motion with a Readout Map	148	(1)
Partial Realization Problem	149	(5)
Real-Time Partial Realization Problem	154	(2)
Almost Linear Systems	156	(39)
Time-Invariant, Affine Input Response Map (Input Response Maps with Time-Invariance and Affinity)	156	(1)
Almost Linear Systems	157	(3)
Finite Dimensional Almost Linear Systems	160	(5)
Partial Realization Theory of Almost Linear Systems	165	(2)
Real-Time Partial Realization Theory of Almost Linear Systems	167	(2)
Historical Notes and Concluding Remarks	169	(1)
Appendix	170	(25)
Realization Theory	171	(1)
Deriving Almost Linear Systems from Pseudo Linear Systems	171	(1)
Free Motions with an Affine Map	172	(4)
Free Motions with a Readout Map	176	(2)
Almost Linear Systems	178	(1)
Sophisticated Almost Linear Systems	179	(2)
Finite Dimensionality	181	(1)
Finite Dimensional Free Motions with Affine Map	181	(3)
Finite Dimensional Free Motions with a Readout Map	184	(2)
Finite Dimensional Almost Linear Systems	186	(1)
Existence Criterion for Almost Linear Systems	187	(1)
Realization Procedure for Almost Linear Systems	187	(1)
Partial Realization	188	(1)
Free Motions with an Affine Map	188	(1)
Free Motions with a Readout Map	189	(1)
Partial Realization Problem	190	(4)
Real-Time Partial Realization of Almost Linear Systems	194	(1)
So-Called Linear Systems (Linear Systems with a Non-zero Initial State)	195	(24)
Input Output Relations for So-Called Linear Systems	196	(1)
So-Called Linear Systems and Almost Linear Systems	197	(6)
Partial Realization Theory of So-Called Linear Systems	203	(4)
Real-Time Partial Realization Theory of So-Called Linear Systems	207	(1)
Historical Notes and Concluding Remarks	208	(2)
Appendix	210	(9)
So-Called Linear Systems and Almost Linear Systems	210	(1)
Partial Realization	211	(6)
Real-Time Partial Realization	217	(2)
References	219	(8)
Index	227